Long Term Behavior of 2D and 3D Non-autonomous Random Convective Brinkman–Forchheimer Equations Driven by Colored Noise

Abstract

The long time behavior of Wong–Zakai approximations of 2D as well as 3D non-autonomous stochastic convective Brinkman–Forchheimer (CBF) equations with non-linear diffusion terms on some bounded and unbounded domains is discussed in this work. To establish the existence of pullback random attractors, the concept of asymptotic compactness (AC) is used. In bounded domains, AC is proved via compact Sobolev embeddings. In unbounded domains, due to the lack of compact embeddings, the ideas of energy equations and uniform tail-estimates are exploited to prove AC. In the literature, CBF equations are also known as Navier–Stokes equations (NSE) with damping, and it is interesting to see that the modification in NSE by linear and nonlinear damping provides better results than that available for NSE in 2D and 3D. The presence of linear damping term helps to establish the results in the whole space $\mathbb {R}^d$. The nonlinear damping term supports to obtain the results in 3D and to cover a large class of nonlinear diffusion terms also. In addition, we prove the existence of a unique pullback random attractor for stochastic CBF equations driven by additive white noise. Finally, for additive as well as multiplicative white noise cases, we establish the convergence of solutions and upper semicontinuity of pullback random attractors for Wong–Zakai approximations of stochastic CBF equations towards the pullback random attractors for stochastic CBF equations when the correlation time of colored noise converges to zero.

1 Introduction

1.1 Literature Survey and Motivations

In this article, we are interested in the long time behavior of Wong–Zakai approximations of stochastic convective Brinkman–Forchheimer (CBF) equations with non-autonomous forcing term (deterministic term) and nonlinear diffusion term. The analysis of CBF equations gained its attention for the past few years as it is considered as a regularized model for the well-known Navier–Stokes equations (NSE), see [1, 26, 30, 43, 44], etc. This mathematical model describes the motion of incompressible fluid flows in a saturated porous medium (cf. [43]). The CBF equations apply to flows when the velocities are sufficiently high and porosities are not too small, that is, when the Darcy law for a porous medium does not hold. Due to the lack of Darcy’s law, it is sometimes referred as non-Darcy model also (cf. [43]). Given $\mathfrak {s}\in \mathbb {R}$, we consider the non-autonomous stochastic CBF equations in $\mathcal {O}\subseteq \mathbb {R}^d\ (d=2,3)$ as

$$\begin{aligned} \left\{ \begin{aligned}\frac{\partial \textbf{u}}{\partial t}-\mu \Delta \varvec{u}+(\varvec{u}\cdot \nabla )\varvec{u}+\alpha \varvec{u}+\beta |\varvec{u}|^{r-1}\varvec{u}+\nabla p&= \varvec{f}+S(t,x,\varvec{u})\circ \frac{\text {d}\text {W}}{\text {d}t},{} & {} \text{ in } \mathcal {O}\times (\mathfrak {s},\infty ), \\ \nabla \cdot \varvec{u}&=0,{} & {} \text {in } \ \mathcal {O}\times (\mathfrak {s},\infty ), \\ \varvec{u}|_{t={\mathfrak {s}}}&= \varvec{u}_{{\mathfrak {s}}},{} & {} x\in {\mathcal {O}} { \text{ and } }{\mathfrak {s}}\in {\mathbb {R}}, \end{aligned} \right. \nonumber \\ \end{aligned}$$

(1.1)

where $\varvec{u}(x,t): {\mathcal {O}}\times (\mathfrak {s},\infty )\rightarrow \mathbb {R}^d$ denotes the velocity field, $p(x,t): \mathcal {O}\times (\mathfrak {s},\infty )\rightarrow \mathbb {R}$ represents the pressure field, $\varvec{f}(x,t): \mathcal {O}\times (\mathfrak {s},\infty )\rightarrow \mathbb {R}^d$ is an external body force, $S(t,x,\varvec{u})$ is a nonlinear diffusion term (cf. Sects. 3–5), the symbol $\circ$ means that the stochastic integral should be understood in the sense of Stratonovich integral and $\textrm{W}=\textrm{W}(t,\omega )$ is an one-dimensional two-sided Wiener process defined on a probability space $(\Omega ,\mathscr {F},\mathbb {P})$. Here $\Omega$ is given by $\Omega =\{\omega \in \textrm{C}(\mathbb {R};\mathbb {R}):\omega (0)=0\},$ $\mathscr {F}$ is the Borel sigma-algebra induced by the compact-open topology of $\Omega$, and $\mathbb {P}$ is the two-sided Gaussian measure on $(\Omega ,\mathscr {F})$. Consequently, $\textrm{W}$ has a form $\textrm{W}(t,\omega )=\omega (t)$. On $(\Omega ,\mathscr {F},\mathbb {P})$, consider Wiener shift operator $\{\vartheta _{t}\}_{t\in \mathbb {R}}$ defined by

$$\begin{aligned} \vartheta _{t}\omega (\cdot )=\omega (\cdot +t) -\omega (t), \ \ \ t\in \mathbb {R}\ \text { and }\ \omega \in \Omega . \end{aligned}$$

(1.2)

It is well-known from [2] that the Gaussian measure $\mathbb {P}$ is ergodic and invariant for $\vartheta _{t}$. Hence, $(\Omega ,\mathscr {F},\mathbb {P},\{\vartheta _{t}\}_{t\in \mathbb {R}})$ is a metric dynamical system (cf. [2]). Moreover, $\varvec{u}(\cdot ,\cdot )$ satisfies: $\varvec{u}={\textbf {0}}\ \text { on }\ \partial \mathcal {O}\times (\mathfrak {s},\infty )\ \text { or }\ |\varvec{u}(x,\mathfrak {s})|\rightarrow 0\ \ \text { as }\ \ |x|\rightarrow \infty ,$ when $\mathcal {O}\subset \mathbb {R}^d$ with smooth boundary $\partial \mathcal {O}$ or $\mathcal {O}=\mathbb {R}^d$, respectively. The positive constants $\mu ,\alpha , \beta >0$ are the Brinkman (effective viscosity), Darcy (permeability of porous medium) and Forchheimer (proportional to the porosity of the material) coefficients, respectively. The absorption exponent $r\in [1,\infty )$, $r=3$ is called the critical exponent and the model (1.1) with $r>3$ is referred as CBF equations with fast growing nonlinearites ([30]). The system (1.1) is also known as damped NSE (cf. [27]) because it becomes classical NSE for $\alpha =\beta =0$. Moreover, it is proved in [26] (cf. Proposition 1.1, [26]) that CBF equations ((1.1) with $S(t,x,\varvec{u})={\textbf {0}}$) have the same scaling as NSE only when $r=3$ and $\alpha =0$ but no scale invariance property for other values of $\alpha$ and r, therefore it is sometimes called NSE modified by an absorption term ([1]) or tamed NSE ([50]).

A good number of works are available in the literature on the solvability results for the system (1.1) and related models (cf. [7, 39,40,41, 45, 51], etc.). In particular, 3D NSE with a Brinkman–Forchheimer type term subject to an anisotropic viscosity driven by multiplicative noise and stochastic tamed 3D NSE on the whole space is considered in [7, 51], respectively. An improvement of the work [51] for a slightly simplified system is done in [11]. On bounded domains, the existence of martingale solutions and strong solutions (in the probabilistic sense) for stochastic CBF equations are established in [39, 45], respectively. The existence of a unique pathwise strong solution to 3D stochastic NSE is a well-known open problem, and the same is open for 3D stochastic CBF equations with $r\in [1,3)$ (cf. [45]) also. In the sequel, if we write $r\in [1,3),$ then it indicates that the two-dimensional case is considered.

In [9, 16], authors introduced the concept of random attractors for stochastic partial differential equations (SPDEs) having compact random dynamical systems (RDS) and applied it to some fluid flow models including 2D stochastic NSE. Later, this concept is used to establish the existence of random attractors for several fluid flow models (cf. [20, 24, 34, 72] etc. and references therein). In unbounded domains, due to the absence of compact Sobolev embeddings, RDS are no more compact. In the deterministic case, the difficulty discussed above was resolved by different methods, see [4, 46, 52, 59], etc. for autonomous case and [12, 13, 37], etc. for non-autonomous case. For SPDEs, the methods available in the deterministic case have also been generalized by several authors (cf. [5, 6, 10, 60], etc.). The concept of an asymptotically compact cocycle was introduced in [12] and the authors proved the existence of attractors for 2D non-autonomous NSE. Later, several authors used this method to prove the existence of random attractors on unbounded domains, see for example [5, 6, 32, 60], etc. Two of the methods used to prove asymptotic compactness without Sobolev embeddings are as follows: the method of energy equations (cf. [23, 32, 61], etc.) and the method of uniform tail-estimates (cf. [59, 63, 67], etc.).

For the existence of a unique random attractor (dynamics of almost all sample paths), it is required to define RDS or random cocycle through the solution operator of the Eq. (1.1). To the best our knowledge, it is still an open problem to define RDS for (1.1), when $S(t,x,\varvec{u})$ is a general nonlinear diffusion term. In this direction, when $S(t,x,\varvec{u})$ is a nonlinear diffusion term, the concept of weak pullback mean random attractors was introduced in [64] and this theory was applied to several physical models (cf. [22, 35, 65] etc.). On the other hand, using random transformations, SPDEs can generate RDS for either $S(t,x,\varvec{u})=\varvec{u}$ or $S(t,x,\varvec{u})$ is independent of $\varvec{u}$. Indeed, the existence of a unique random attractor for (1.1) is available for such $S(t,x,\varvec{u})$ only (see [33, 34] for bounded domains and [32, 37] for unbounded domains).

1.2 Wong–Zakai Approximations of Stochastic CBF Eq. (1.1)

In [70, 71], Wong and Zakai introduced the concept of approximating stochastic differential equations (SDEs) by random differential equations, where they have approximated one-dimensional Brownian motion. Later, this work was extended to SDEs in higher dimensions, see for example [29, 31, 54] etc., and references therein. In particular, the Wong–Zakai approximations results for stochastic CBF equations on bounded domains have been studied in [36]. Due to the lack of differentiability of sample paths of Wiener process $\textrm{W}(\cdot )$, the authors in [58, 69] first introduced the Ornstein–Uhlenbeck process (or colored noise) to approximate $\textrm{W}(\cdot )$. The Wong–Zakai approximations (with the help of colored noise) can be used to analyze the pathwise random dynamics of (1.1) with nonlinear diffusion term. The approach through approximations obeys the well-known framework given by Wong and Zakai [70, 71]. Despite of analyzing the corresponding random versions of (1.1) acquired by random transformations, we will rather analyze a system obtained by the Wong–Zakai approximations of (1.1) with nonlinear diffusion term. With reference to the nonlinear diffusion term, the idea of Wong–Zakai approximations for pullback random attractors is used for a wide class of physically admissible models, see [24, 25, 42, 53, 72] etc., for bounded domains, [23, 48, 66, 67] etc., for unbounded domains and [68, 73] etc., for discrete systems.

Let us now consider the following random partial differential equations as Wong–Zakai approximations of (1.1) for $\mathfrak {s}\in \mathbb {R}$ and $\delta \in \mathbb {R}\backslash \{0\}$:

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned} \frac{\partial \varvec{u}}{\partial t}-\mu \Delta \varvec{u}+(\varvec{u}\cdot \nabla )\varvec{u}+\alpha \varvec{u}+\beta |\varvec{u}|^{r-1}\varvec{u}+\nabla p&=\varvec{f}+S(t,x,\varvec{u})\mathcal {Z}_{\delta }(\vartheta _{t}\omega ),{} & {} \text {in } \mathcal {O}\times (\mathfrak {s},\infty ), \\ \nabla \cdot \varvec{u}&=0,{} & {} \text {in } \ \mathcal {O}\times (\mathfrak {s},\infty ), \\ \varvec{u}|_{t=\mathfrak {s}}&= \varvec{u}_{\mathfrak {s}},{} & {} x\in \mathcal {O} \text{ and } \mathfrak {s}\in \mathbb {R}, \end{aligned} \right. \end{aligned}\nonumber \\ \end{aligned}$$

(1.3)

where $\mathcal {Z}_{\delta }(\vartheta _{t}\omega )$ is defined in Lemma 2.12 below. Also, $\varvec{u}(x,t)$ satisfies $\varvec{u}={\textbf {0}}$ on $\partial \mathcal {O}\times (\mathfrak {s},\infty )$ or $\varvec{u}(x,\mathfrak {s})\rightarrow 0$ as $|x|\rightarrow \infty ,$ when $\mathcal {O}\subset \mathbb {R}^d$ with smooth boundary $\partial \mathcal {O}$ or $\mathcal {O}=\mathbb {R}^d$, respectively. Firstly, we establish the existence of pullback random attractors for the system (1.3) on bounded domains, the whole space $\mathbb {R}^d$ and the unbounded Poincaré domain under the appropriate assumptions on nonlinear diffusion and non-autonomous forcing terms. Next, we prove the existence of pullback random attractors for the system (1.1) on the whole space when $S(t,x,\varvec{u})$ is independent of $\varvec{u}$ (additive noise case). From the work [37], we infer that there exists a unique pullback random attractor for the system (1.1) on the whole space when $S(t,x,\varvec{u})=\varvec{u}$ (multiplicative noise case). Finally, for additive and multiplicative white noise cases we prove the upper semicontinuity of pullback random attractors for the system (1.3) towards the pullback random attractor of the system (1.1) when $\delta \rightarrow 0$. In appendix, we discuss the existence of pullback random attractors for 2D random NSE on unbounded Poincaré domains (the system (1.3) with $d=2$ and $\alpha =\beta =0$, where $\mathcal {O}$ is unbounded Poincaré domain).

1.3 Difficulties and Approaches

In bounded domains, as we are using compact Sobolev embeddings to get our required results, we assume that the nonlinear diffusion term $S(t,x,\varvec{u})$ satisfies the following assumption (for the functional settings, refer to Sect. 2).

Assumption 1.1

We assume that the nonlinear diffusion term

$$\begin{aligned} S(t,x,\varvec{u})=e^{\sigma t}\left[ \kappa \varvec{u}+\mathcal {S}(\varvec{u})+\varvec{h}(x)\right] , \end{aligned}$$

where $(t,x,\varvec{u})\in \mathbb {R}\times \mathbb {R}^d\times \mathbb {V}$, $\sigma \ge 0, \ \kappa \ge 0$ and $\varvec{h}\in \mathbb {H}$. Also, $\mathcal {S}:\mathbb {V}\rightarrow \mathbb {H}$ is a continuous function satisfying

$$\begin{aligned}&\Vert \mathcal {S}(\varvec{u})-\mathcal {S}(\varvec{v})\Vert _{\mathbb {H}} \le s_1\Vert \varvec{u}-\varvec{v}\Vert _{\mathbb {V}}, \qquad \qquad \quad \,\,\,\, \text { for all } \ \varvec{u},\varvec{v}\in \mathbb {V}, \end{aligned}$$

(1.4)

$$\begin{aligned}&|\left( \mathcal {S}(\varvec{u})-\mathcal {S}(\varvec{v}),\varvec{w}\right) |\le s_2\Vert \varvec{u}-\varvec{v}\Vert _{\mathbb {H}}\Vert \varvec{w}\Vert _{\mathbb {V}}, \quad \text { for all } \ \varvec{u},\varvec{v},\varvec{w}\in \mathbb {V}, \end{aligned}$$

(1.5)

$$\begin{aligned}&|\left( \mathcal {S}(\varvec{u}),\varvec{u}\right) | \le s_3 + s_4\Vert \varvec{u}\Vert _{\mathbb {V}}^{1+s_5}, \qquad \qquad \quad \quad \,\,\, \text { for all }\ \varvec{u}\in \mathbb {V}, \end{aligned}$$

(1.6)

where $s_1,\ldots ,s_4$ are non-negative constants and $s_5\in [0,1)$.

Example 1.2

Let us discuss an example for such a nonlinear diffusion term which satisfies the above conditions. Let $\mathcal {S}:\mathbb {V}\rightarrow \mathbb {H}$ be a nonlinear operator defined by $\mathcal {S}(\varvec{u})=\sin \varvec{u}+\textrm{B}({\textbf {g}}_1,\varvec{u})$ for all $\varvec{u}\in \mathbb {V}$, where ${\textbf {g}}_1\in \textrm{D}(\textrm{A})$ is a fixed element and operator $\textrm{B}$ is the bilinear operator defined in Sect. 2.4. It is easy to verify that $\mathcal {S}$ satisfies (1.4)–(1.6) (cf. [24]).

Due to the absence of compact Sobolev embeddings in unbounded domains, we prove asymptotic compactness of RDS via method of energy equations (by [4]) or method of uniform tail-estimates (by [59]), which require a stronger assumption than Assumption 1.1 on $S(t,x,\varvec{u})$. We observe that, in order to prove the results on some unbounded domains ($\mathbb {R}^d$ or unbounded Poincaré domains), the assumptions on $S(t,x,\varvec{u})$ which are required to use the method of energy equations and the method of uniform tail-estimates, are different and cover different classes of functions. Therefore we prove the results of this work using both the above discussed methods. In order to apply the method of energy equations, we need the following assumption on $S(t,x,\varvec{u})$.

Assumption 1.3

We assume that the nonlinear diffusion term

$$\begin{aligned} S(t,x,\varvec{u})=e^{\sigma t}\left[ \kappa \varvec{u}+\mathcal {S}(\varvec{u})+\varvec{h}(x)\right] , \end{aligned}$$

where $(t,x,\varvec{u})\in \mathbb {R}\times \mathbb {R}^d\times \mathbb {V}$, $\sigma \ge 0, \kappa \ge 0$ and $\varvec{h}\in \mathbb {H}$. Also, $\mathcal {S}:\mathbb {V}\rightarrow \mathbb {H}$ is a continuous function satisfying (1.4)–(1.5) and

$$\begin{aligned} \left( \mathcal {S}(\varvec{u}),\varvec{u}\right) =0, \ \text { for all }\ \varvec{u}\in \mathbb {V}. \end{aligned}$$

(1.7)

Note that the condition (1.6) is much weaker than (1.7) and for $s_3=s_4=0$ in (1.6), one can obtain (1.7).

Example 1.4

Let us discuss some examples for such nonlinear diffusion terms, which satisfy the above Assumption. Let $\mathcal {S}:\mathbb {V}\rightarrow \mathbb {H}$ be a nonlinear operator defined by $\mathcal {S}(\varvec{u})=\textrm{B}({\textbf {g}}_2,\varvec{u})$ for all $\varvec{u}\in \mathbb {V}$, where ${\textbf {g}}_2$ is a fixed element of $\textrm{D}(\textrm{A})$ and $\mathcal {S}$ satisfies (1.4)–(1.5) and (1.7), see [8, 23]. Hence $S(t,x,\varvec{u})=e^{\sigma t}\left[ \kappa {u}+\textrm{B}({\textbf {g}}_2,{u})+\varvec{h}(x)\right]$ satisfies Assumption 1.3 but not Assumption 1.5.

Note that $\mathcal {S}(\varvec{u})=\textrm{B}({\textbf {g}}_2,\varvec{u})$ is a noise coefficient which is linear in $\varvec{u}$. In this case, for $S(t,x,\varvec{u})=e^{\sigma t}\big [\kappa \varvec{u}+\textrm{B}({\textbf {g}}_2,\varvec{u})+\varvec{h}(x)\big ]$, the existence of pullback random attractors for the stochastic equations (1.1) is unknown. But as we will prove in this paper, the random equations (1.3) have a unique tempered pullback random attractor in $\mathbb {H}$.

It is worth mentioning here that this kind of noise has a physical background as well (see [8]). For example, in the classical derivation of NSE, the crucial point is the computation of the material derivative $\frac{\textrm{D}\varvec{u}}{\textrm{D}t}$, which gives NSE of the form

$$\begin{aligned} \frac{\textrm{D}\varvec{u}}{\textrm{D}t}=\nu \Delta \varvec{u}-\nabla p, \end{aligned}$$

where $\nu >0$ is the kinematic viscosity. The material derivative results from considering the acceleration of the fluid particle whose position will be denoted by X(t), that is,

$$\begin{aligned} \frac{\textrm{d}X(t)}{\textrm{d}t}=\varvec{u}(t,X(t)), \end{aligned}$$

and

$$\begin{aligned} \frac{\textrm{d}^2 X(t)}{\textrm{d}t^2}=\frac{\textrm{d}\varvec{u}(t,X(t))}{\textrm{d}t}=\frac{\partial \varvec{u}}{\partial t}+ (\varvec{u}\cdot \nabla )\varvec{u}=: \frac{\textrm{D}\varvec{u}}{\textrm{D}t}. \end{aligned}$$

Since, in the context of random perturbations, X(t) should be treated as a stochastic process satisfying the following SDE:

$$\begin{aligned} \textrm{d}X(t)=\varvec{u}(t,X(t))\textrm{d}t+ \varvec{b}(X(t))\circ {\textrm{dW}}(t), \end{aligned}$$

where $\varvec{b}(\cdot )$ is a given vector field which is independent of the white noise. Therefore, the stochastic material derivative is computed as follows:

Observe that the noise coefficient is nothing but $(\varvec{b}\cdot \nabla )\varvec{u}$ which satisfies Assumption 1.3.

In order to apply the method of uniform tail-estimates while working on the whole space $\mathbb {R}^3$, we need the following assumption on $S(\cdot ,\cdot ,\cdot )$.

Assumption 1.5

Let $S:\mathbb {R}\times \mathbb {R}^3\times \mathbb {R}^3\rightarrow \mathbb {R}^3$ be a continuous function such that for all $t\in \mathbb {R}$ and $x,y\in \mathbb {R}^3$,

$$\begin{aligned} |S(t,x,y)|&\le {|\mathcal {S}_1(t,x)||y|^{q-1}+|\mathcal {S}_2(t,x)|}, \end{aligned}$$

(1.8)

with $1\le q<\min \{6,r+1\}$, $\mathcal {S}_1\in \textrm{L}^{\infty }_{loc }(\mathbb {R};\mathbb {L}^{\frac{r+1}{r+1-q}}(\mathbb {R}^3)\cap \mathbb {L}^{\frac{6(r+1)}{3r+5-2q}}(\mathbb {R}^3)\cap \mathbb {L}^{\frac{6}{6-q}}(\mathbb {R}^3))$ and $\mathcal {S}_2\in \textrm{L}^{\infty }_{loc }(\mathbb {R};\mathbb {L}^{2}(\mathbb {R}^3)\cap \dot{\mathbb {H}}^{-1}(\mathbb {R}^3))$. Furthermore, we assume that S(t, x, y) is locally Lipschitz continuous with respect to the third variable.

Example 1.6

Let us discuss some examples for such nonlinear diffusion terms, which satisfy the above Assumption.

• 1. Take $S(t,x,\varvec{u})=\mathcal {S}_1(t,x)|\varvec{u}|$, where $\mathcal {S}_1\in \mathbb {L}^{\frac{r+1}{r-1}}(\mathbb {R}^3)\cap \mathbb {L}^{\frac{6(r+1)}{3r+1}}(\mathbb {R}^3)\cap \mathbb {L}^{\frac{3}{2}}(\mathbb {R}^3)$ for $r>1$.

• 2. Take $S(t,x,\varvec{u})=\mathcal {S}_1(t,x)\sin (\varvec{u})$ which satisfies Assumption 1.5 for $q=1$. Then we have

  $$\begin{aligned} |S(t,x,y)|\le |\mathcal {S}_1(t,x)|. \end{aligned}$$

  (1.9)

  As the left hand side of (1.9) is independent of y, we can consider $\mathcal {S}_1\in \textrm{L}^{\infty }_{loc }(\mathbb {R};\mathbb {L}^{2}(\mathbb {R}^3)\cap \dot{\mathbb {H}}^{-1}(\mathbb {R}^3))$ for $r\ge 1$.

Hence $S(t,x,\varvec{u})$ satisfies Assumption 1.5 but not Assumption 1.3.

Let $\mathcal {O}\subset \mathbb {R}^d$ $(d=2,3)$ be a domain satisfying the following assumption:

Assumption 1.7

Let $\mathcal {O}$ be an open and connected subset of $\mathbb {R}^d$ $(d=2,3)$, the boundary of which is uniformly of class $\textrm{C}^3$ (cf. [28]). For the domain $\mathcal {O}$, we also assume that, there exists a positive constant $\lambda$ such that the following Poincaré inequality is satisfied:

$$\begin{aligned} \lambda \int _{\mathcal {O}} |\psi (x)|^2 \textrm{d}x \le \int _{\mathcal {O}} |\nabla \psi (x)|^2 \textrm{d}x, \ \text { for all } \ \psi \in \mathbb {H}^{1}_0 (\mathcal {O}). \end{aligned}$$

(1.10)

Remark 1.8

A domain in which Poincaré inequality is satisfied, we call it as a Poincaré domain (cf. [57, p. 306] and [49, p. 117]). It can be easily seen that if $\mathcal {O}$ is bounded in some direction, then the Poincaré inequality holds. For example, in two-dimensions, if $x=(x_1,x_2)\in \mathbb {R}^2$, then one can take $\mathcal {O}$ is included in a region of the form $0<x_1<L$.

In order to apply the method of uniform tail-estimates while working on a Poincaré domain $\mathcal {O}$ satisfying Assumption 1.7, we consider the following assumption on $S(\cdot ,\cdot ,\cdot )$:

Assumption 1.9

Let $\mathcal {O}\subset \mathbb {R}^d$ satisfies Assumption 1.7. Let $S:\mathbb {R}\times \mathcal {O}\times \mathbb {R}^d\rightarrow \mathbb {R}^d$ be a continuous function such that for all $t\in \mathbb {R}$, $x\in \mathcal {O}$ and $y\in \mathbb {R}^d$,

$$\begin{aligned} |S(t,x,y)|&\le |\mathcal {S}_1(t,x)||y|^{q-1}+|\mathcal {S}_2(t,x)|, \end{aligned}$$

(1.11)

with $1\le q<r+1$, where $\mathcal {S}_1\in \textrm{L}^{\infty }_{loc }(\mathbb {R};\mathbb {L}^{\frac{r+1}{r+1-q}}(\mathcal {O}))$, $\mathcal {S}_2\in \textrm{L}^{\infty }_{loc }(\mathbb {R};\mathbb {L}^{\frac{r+1}{r}}(\mathcal {O}))$. Furthermore, we assume that S(t, x, y) is locally Lipschitz continuous with respect to the third variable.

Remark 1.10

Similar functions in Example 1.6 will work for Assumption 1.9 also. One can consider $\mathcal {S}_2\in \textrm{L}^{\infty }_{loc }(\mathbb {R};\mathbb {L}^{2}(\mathcal {O}))$ instead of $\mathcal {S}_2\in \textrm{L}^{\infty }_{loc }(\mathbb {R};\mathbb {L}^{\frac{r+1}{r}}(\mathcal {O}))$ in Assumption 1.9.

The Examples 1.4 and 1.6 show that Assumptions 1.3 and 1.5 (or Assumption 1.9) cover different classes of functions.

When we use the idea of uniform tail-estimates to obtain the asymptotic compactness of RDS associated with (1.3), we use a cut-off technique to prove that solutions to (1.3) are sufficiently small in $\mathbb {L}^2(\mathcal {O}^c_k)$ when k is large enough, where $\mathcal {O}_{k}=\{x\in \mathbb {\mathbb {R}}^d:|x|\le k\}$ and $\mathcal {O}^{c}_{k}=\mathbb {R}^d\setminus \mathcal {O}_{k}$. Unlike the parabolic or hyperbolic equations considered in [59, 66, 67], etc., the fluid equations like (1.1) (NSE also) contain the pressure term p as well. When we derive these uniform tail-estimates using cut-off technique, the pressure p does not vanish simply. However, by taking the divergence in (1.1) and using the incompressibility condition, we obtain a rigorous expression of the pressure term

$$\begin{aligned} p=(-\Delta )^{-1}\left[ \nabla \cdot \left[ \nabla \cdot \big (\varvec{u}\otimes \varvec{u}\big )\right] +\beta \nabla \cdot \left[ |\varvec{u}|^{r-1}\varvec{u}\right] -\nabla \cdot \varvec{f}-\left[ \nabla \cdot S(t,x,\varvec{u})\right] \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right] , \end{aligned}$$

(1.12)

in the weak sense, which is the most difficult term to handle in an appropriate way. Then it is possible to obtain these uniform tail-estimates with the help of the continuous embedding $\dot{\mathbb {H}}^{1}(\mathbb {R}^3)\hookrightarrow \mathbb {L}^{6}(\mathbb {R}^3)$ of the homogeneous Sobolev spaces in the whole space $\mathbb {R}^3$ (see Remark 2.4 below). Since there is no such embedding available in the case of $\mathbb {R}^2$, we are not able to prove the existence of pullback random attractors of (1.1) under Assumption 1.5. Moreover, we observe that if we restrict ourselves to the Poincaré domain $\mathcal {O}$, we can prove the existence of pullback random attractors under a more general assumption than Assumption 1.5 (see Assumption 1.9). It is also remarked that we are not able to use the idea of energy equations to prove the asymptotic compactness under Assumptions 1.5 or 1.9.

It is well-known that passing limit in nonlinear terms is not an easy task in any analysis. In this work, while proving the pullback asymptotic compactness of solutions of the system (6.1) (Lemma 6.8) or establishing the uniform compactness of random attractors of the system (6.2) (Lemma 6.14) via energy equations method, it is necessary to show the convergence of the nonlinear terms appearing in the corresponding energy equations. In our model, we have two nonlinear terms given by $(\varvec{u}\cdot \nabla )\varvec{u}$ and $\beta |\varvec{u}|^{r-1}\varvec{u}$. The required convergence is obtained by breaking down the integral containing nonlinear terms into two parts such that one part is defined in a bounded domain $\mathcal {O}_k$ with radius k, and another part is defined in the complement of $\mathcal {O}_k$. We then show that the nonlinear term is convergent in $\mathcal {O}_k$ and its tail on the complement of $\mathcal {O}_k$ is uniformly small when k is sufficiently large, from which the required convergence follows (see Corollaries 6.5 and 6.6 for handling the terms $(\varvec{u}\cdot \nabla )\varvec{u}$ and $\beta |\varvec{u}|^{r-1}\varvec{u}$, respectively).

1.4 Aims and Novelties of the Work

The major aims and novelties of this work are:

• (i) Existence of a unique pullback random attractor for the system (1.3) under Assumption 1.1on bounded domains for the following three cases:

Table 1 Values of $\mu ,\beta$ and r for $d=2,3$

• (ii) Existence of a unique pullback random attractor for the system (1.3) under Assumption 1.3on the whole space for three cases given in Table 1.

• (iii) Existence of a unique pullback random attractor for the system (1.3) under Assumption 1.5on the whole space $\mathbb {R}^3$ for the two cases given in Table 1for $d=3$ including $d=r=3$ with $2\beta \mu =1$.

• (iv) Existence of a unique pullback random attractor for the system (1.3) under Assumption 1.9on the Poincaré domain $\mathcal {O}\subset \mathbb {R}^d$ satisfying Assumption 1.7for the three cases given in Table 1including $d=r=3$ with $2\beta \mu =1$.

• (v) Existence of a unique pullback random attractor for the system (1.1) with $S(t,x,\varvec{u})=e^{\sigma t}{} {\textbf {g}}(x)$, for given $\sigma \ge 0$ and ${\textbf {g}}\in \textrm{D}(\textrm{A})$ (additive white noise) on the whole space for the three cases given in Table 1including $d=r=3$ with $2\beta \mu =1$ and excluding $d=2$ with $r=1$. Moreover, for $S(t,x,\varvec{u})=e^{\sigma t}{} {\textbf {g}}(x)$, we demonstrate the convergence of solutions and upper semicontinuity of pullback random attractors for (1.3) towards the pullback random attractor for (1.1) as $\delta \rightarrow 0$.

• (vi) For $S(t,x,\varvec{u})=\varvec{u}$, convergence of solutions and upper semicontinuity of pullback random attractors for (1.3) towards the pullback random attractor for (1.1) as $\delta \rightarrow 0$ for the three cases given in Table 1including $d=r=3$ with $2\beta \mu =1$.

• (vii) Existence of a unique pullback random attractor of the system (1.3) in two dimensions with $\alpha =\beta =0$ (cf. 2D random NSE (A.1) below) under Assumption 1.9for $r=1$ on Poincaré domains.

As it is discussed earlier that Assumptions 1.3 and 1.5 (or Assumption 1.9) cover different class of functions (cf. Examples 1.4 and 1.6). Moreover, both assumptions need different types of treatment to establish the results of this work. We use method of energy equations and method of uniform tail-estimates under Assumptions 1.3 and 1.5 (or Assumption 1.9), respectively. Also, note that the existence of pullback random attractors for the random NSE (system (1.3) when $\alpha =\beta =0$) with Assumption 1.9 for $r=1$ on $S(t,x,\varvec{u})$, has not been considered in the literature till now due to the difficulty in estimating the pressure term

$$\begin{aligned} \begin{aligned} p=(-\Delta )^{-1}\left[ \nabla \cdot \left[ \nabla \cdot \big (\varvec{u}\otimes \varvec{u}\big )\right] -\nabla \cdot {\varvec{f}}-\left[ \nabla \cdot S(t,x,\varvec{u})\right] \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right] , \end{aligned} \end{aligned}$$

which we resolve here.

1.5 Advantages of Damping Term

CBF equations are also known as NSE with damping (cf. [27]). The damping arises from the resistance to the motion of the flow or by friction effects. Due to the presence of the damping term $\alpha \varvec{u}+\beta |\varvec{u}|^{r-1}\varvec{u}$, we are able to establish better results than which are available for NSE. In particular, in this work, linear damping $\alpha \varvec{u}$ helps us to obtain results on the whole space and the nonlinear damping $\beta |\varvec{u}|^{r-1}\varvec{u}$ (for $r>1$) helps us to cover a large class of nonlinearity on $S(t,x,\varvec{u})$ (see Sect. 4.2). The authors in [23] established the existence of a unique pullback random attractor for 2D NSE on Poincaré domains with $S(t,x,\varvec{u})=e^{\sigma t}{} {\textbf {g}}(x)$, for given $\sigma >0$ and ${\textbf {g}}\in \textrm{D}(\textrm{A})$, based on the following assumption on ${\textbf {g}}(\cdot )$:

Assumption 1.11

(cf. Section 3, [23]) Assume that the function ${\textbf {g}}(\cdot )$ satisfies the following condition: there exists a strictly positive constant $\aleph$ such that

$$\begin{aligned} |b(\varvec{u},{\textbf {g}},\varvec{u})|\le \aleph \Vert \varvec{u}\Vert ^2_{\mathbb {H}}, \ \ \text { for all } \ \varvec{u}\in \mathbb {H}. \end{aligned}$$

We also point out that the requirement of Assumption 1.11 is not necessary for $r>1$ (${\textbf {g}}\in \textrm{D}(\textrm{A})$ is enough). For $r=1$, one gets linear damping, and the existence of a random attractors for the system (6.1) and upper semicontinuity of random attractors for the system (6.2) can be proved on the whole space via same arguments as it has been done for 2D stochastic NSE on Poincaré domains in [23] under Assumption 1.11. For $r>1$, due to the presence of nonlinear damping term $\beta |\varvec{u}|^{r-1}\varvec{u}$, we provide a different treatment to avoid Assumption 1.11 on ${\textbf {g}}$ (cf. Lemma 6.4).

1.6 Outline

The remaining sections are organized as follows. In the next section, we provide the necessary function spaces required to establish the results of this work, and we define linear operator, bilinear operator and nonlinear operator to obtain an abstract formulation of the system (1.3). Furthermore, we recall some well-known inequalities, properties of operators, and some properties of white and colored noises. Finally, we discuss the solvability of the system (1.3) in the same section. In Sect. 3, we prove the existence of a unique pullback random attractor for the system (1.3) under Assumption 1.1 on bounded domains (cf. Theorem 3.9) in which we prove the pullback asymptotic compactness using compact Sobolev embeddings (cf. Lemma 3.6). In Sect. 4, we prove the existence of a unique pullback random attractor for the system (1.3) under Assumptions 1.3 (for $\mathbb {R}^2$ and $\mathbb {R}^3$) and 1.5 (for $\mathbb {R}^3$ only) on the whole space (cf. Theorems 4.5 and 4.15) in which we prove the pullback asymptotic compactness using the idea of energy equations and asymptotic compactness, for Assumptions 1.3 and 1.5, respectively (cf. Lemmas 4.3 and 4.14). In Sect. 5, we prove the existence of a unique pullback random attractor for the system (1.3) under Assumptions 1.9 on Poincaré domains (cf. Theorem 5.3) in which we prove the pullback asymptotic compactness using the idea of uniform tail-estimates. Section 6 is devoted for the stochastic CBF equations (1.1) perturbed by additive white noise (that is, when $S(t,x,\varvec{u})=e^{\sigma t}{} {\textbf {g}}(x)$ with given $\sigma >0$ and ${\textbf {g}}\in \textrm{D}(\textrm{A})$). Firstly, we prove the existence of a unique pullback random attractor for stochastic CBF equations driven by additive white noise (cf. Theorem 6.9). Next, we demonstrate the convergence of solutions and upper semicontinuity of pullback random attractors for Wong–Zakai approximations of stochastic CBF equations towards the solution and the pullback random attractor of stochastic CBF equations, respectively, as the correlation time $\delta$ converges to zero, using the fundamental theory introduced in [62] (cf. Lemma 6.12 and Theorem 6.15, respectively). Since the existence of a unique pullback random attractor for stochastic CBF equations driven by linear multiplicative noise is established in [37], we prove only the convergence of solutions and upper semicontinuity of pullback random attractors for its Wong–Zakai approximations towards its solution and pullback random attractor, respectively, as the correlation time $\delta \rightarrow 0$ (cf. Lemma 7.3 and Theorem 7.5, respectively) in Sect. 7. In Appendix A, we establish the existence of a unique pullback random attractor for the system (A.1) under Assumption A.1 on Poincaré domains (bounded or unbounded) (Theorem A.4).

2 Mathematical Formulation

In this section, first we provide the necessary function spaces needed to obtain the results of this work. Next, we define some operators to set up an abstract formulation. Finally, we recall some properties of white noise as well as colored noise. We fix $\mathcal {O}$ as either a bounded subset of $\mathbb {R}^d$ with $\textrm{C}^2$-boundary or the whole space $\mathbb {R}^d$ or unbounded Poincaré domain $\mathcal {O}\subset \mathbb {R}^d$ with $\textrm{C}^3$-boundary.

2.1 Function Spaces

We define the space $\mathcal {V}:=\{\varvec{u}\in \textrm{C}_0^{\infty }(\mathcal {O};\mathbb {R}^d):\nabla \cdot \varvec{u}=0\},$ where $\textrm{C}_0^{\infty }(\mathcal {O};\mathbb {R}^d)$ denotes the space of all infinite times differentiable functions ($\mathbb {R}^d$-valued) with compact support in $\mathbb {R}^d$. Let $\mathbb {H}$, $\mathbb {V}$ and $\widetilde{\mathbb {L}}^p$ denote the completion of $\mathcal {V}$ in $\textrm{L}^2(\mathcal {O};\mathbb {R}^d)$, $\textrm{H}^1(\mathcal {O};\mathbb {R}^d)$ and $\textrm{L}^p(\mathcal {O};\mathbb {R}^d)$, $p\in (2,\infty )$, norms, respectively. The space $\mathbb {H}$ is endowed with the norm $\Vert \varvec{u}\Vert _{\mathbb {H}}^2:=\int _{\mathcal {O}}|\varvec{u}(x)|^2\textrm{d}x,$ the norm on the space $\widetilde{\mathbb {L}}^{p}$ is defined by $\Vert \varvec{u}\Vert _{\widetilde{\mathbb {L}}^p}^p:=\int _{\mathcal {O}}|\varvec{u}(x)|^p\textrm{d}x,$ for $p\in (2,\infty )$ and the norm on the space $\mathbb {V}$ is given by $\Vert \varvec{u}\Vert ^2_{\mathbb {V}}=\int _{\mathcal {O}}|\varvec{u}(x)|^2\textrm{d}x+\int _{\mathcal {O}}|\nabla \varvec{u}(x)|^2\textrm{d}x.$ The inner product in the Hilbert space $\mathbb {H}$ is denoted by $( \cdot , \cdot )$. The duality pairing between the spaces $\mathbb {V}$ and $\mathbb {V}'$, and $\widetilde{\mathbb {L}}^p$ and its dual $\widetilde{\mathbb {L}}^{\frac{p}{p-1}}$ is represented by $\langle \cdot ,\cdot \rangle .$ It should be noted that $\mathbb {H}$ can be identified with its own dual $\mathbb {H}'$. We endow the space $\mathbb {V}\cap \widetilde{\mathbb {L}}^{p}$ with the norm $\Vert \varvec{u}\Vert _{\mathbb {V}}+\Vert \varvec{u}\Vert _{\widetilde{\mathbb {L}}^{p}},$ for $\varvec{u}\in \mathbb {V}\cap \widetilde{\mathbb {L}}^p$ and its dual $\mathbb {V}'+\widetilde{\mathbb {L}}^{p'}$ (cf. [19, Subsection 2.1]) with the norm

$$\begin{aligned} \inf \left\{ \max \left( \Vert \varvec{v}_1\Vert _{\mathbb {V}'},\Vert \varvec{v}_2\Vert _{\widetilde{\mathbb {L}}^{p'}}\right) :\varvec{v}=\varvec{v}_1+\varvec{v}_2, \ \varvec{v}_1\in \mathbb {V}', \ \varvec{v}_2\in \widetilde{\mathbb {L}}^{p'}\right\} . \end{aligned}$$

Moreover, we have the following continuous embedding also:

$$\begin{aligned} \mathbb {V}\cap \widetilde{\mathbb {L}}^{p}\hookrightarrow \mathbb {V}\hookrightarrow \mathbb {H}\equiv \mathbb {H}'\hookrightarrow \mathbb {V}'\hookrightarrow \mathbb {V}'+{\widetilde{\mathbb {L}}}^{\frac{p}{p-1}}. \end{aligned}$$

One can define equivalent norms on $\mathbb {V}\cap {\widetilde{\mathbb {L}}}^{p}$ and $\mathbb {V}'+{\widetilde{\mathbb {L}}}^{\frac{p}{p-1}}$ as (cf. [19, Subsection 2.1])

$$\begin{aligned} \Vert \varvec{u}\Vert _{\mathbb {V}\cap {\widetilde{\mathbb {L}}}^{p}}=\left( \Vert \varvec{u}\Vert _{\mathbb {V}}^2+\Vert \varvec{u}\Vert _{{\widetilde{\mathbb {L}}}^{p}}^2\right) ^{\frac{1}{2}}\ \text { and } \ \Vert \varvec{u}\Vert _{\mathbb {V}'+{\widetilde{\mathbb {L}}}^{\frac{p}{p-1}}}=\inf _{\varvec{u}=\varvec{v}+\varvec{w}}\left( \Vert \varvec{v}\Vert _{\mathbb {V}'}^2+\Vert \varvec{w}\Vert _{{\widetilde{\mathbb {L}}}^{\frac{p}{p-1}}}^2\right) ^{\frac{1}{2}}. \end{aligned}$$

Remark 2.1

It is remarkable that the norms $\int _{\mathcal {O}}\left[ |\varvec{u}(x)|^2+|\nabla \varvec{u}(x)|^2\right] \textrm{d}x$ and $\int _{\mathcal {O}}|\nabla \varvec{u}(x)|^2\textrm{d}x$ are equivalent on the space $\mathbb {V}$ if the Poincaré inequality (1.10) satisfied.

Let us provide some useful definition and results which hold on the whole space $\mathbb {R}^d$ and we use them in the sequel.

Definition 2.2

([3, Definition 1.31]) Let $s\in \mathbb {R}$. The homogeneous Sobolev space $\dot{\mathbb {H}}^{s}(\mathbb {R}^d)$ is the space of tempered distributions $\varvec{v}$ over $\mathbb {R}^d$, the Fourier transform of which belongs to $\mathbb {L}^1_{\textrm{loc}}(\mathbb {R}^d)$ and satisfies

$$\begin{aligned} \Vert \varvec{u}\Vert ^2_{\dot{\mathbb {H}}^s(\mathbb {R}^d)}:= \int _{\mathbb {R}^d}|\xi |^{2s}|\hat{\varvec{v}}(\xi )|^2\textrm{d}\xi <+\infty . \end{aligned}$$

Theorem 2.3

([3, Theorem 1.38]) If $s\in [0,\frac{d}{2})$, then the space $\dot{\mathbb {H}}^{s}(\mathbb {R}^d)$ is continuously embedded in $\mathbb {L}^{\frac{2d}{d-2s}}(\mathbb {R}^d)$.

Remark 2.4

For $d=3$ and $s=1$, we have the embedding $\dot{\mathbb {H}}^{1}(\mathbb {R}^3)\hookrightarrow \mathbb {L}^{6}(\mathbb {R}^3)$ is continuous. This implies that the embedding of the dual spaces $\mathbb {L}^{\frac{6}{5}}(\mathbb {R}^3)\hookrightarrow \dot{\mathbb {H}}^{-1}(\mathbb {R}^3)$ is continuous.

2.2 Projection Operator

Let $\mathcal {P}_\mathcal {P}: \mathbb {L}^\mathcal {P}(\mathcal {O}) \rightarrow \widetilde{\mathbb {L}}^\mathcal {P}$ be the Helmholtz–Hodge (or Leray) projection (cf. [21, 47], etc.). For $p=2$, $\mathcal {P}:=\mathcal {P}_2$ becomes an orthogonal projection ([38]) and for $2<p<\infty$, it is a bounded linear operator.

2.3 Linear Operator

We define the Stokes operator

$$\begin{aligned} \textrm{A}\varvec{u}:=-\,\mathcal {P}\Delta \varvec{u},\;\varvec{u}\in \textrm{D}(\textrm{A}):=\mathbb {V}\cap \mathbb {H}^{2}(\mathcal {O}), \end{aligned}$$

where $\mathbb {H}^2(\mathcal {O}):=\textrm{H}^2(\mathcal {O};\mathbb {R}^d)$ is the second order Sobolev space.

Remark 2.5

${\textbf {Case\ I:}}$ When $\mathcal {O}$ is a bounded subset of $\mathbb {R}^d$ with $\textrm{C}^2$-boundary. Since $\mathcal {O}$ has $\textrm{C}^2$-boundary, $\mathcal {P}$ maps $\mathbb {H}^1(\mathcal {O})$ into itself (cf. Remark 1.6, [55]).

${\textbf {Case\ II:}}$ When $\mathcal {O}=\mathbb {R}^d$. The projection operator $\mathcal {P}:\mathbb {L}^2(\mathbb {R}^d) \rightarrow \mathbb {H}$ can be expressed in terms of the Riesz transform (cf. [47]). Moreover, $\mathcal {P}$ and $\Delta$ commutes, that is, $\mathcal {P}\Delta =\Delta \mathcal {P}$.

${\textbf {Case\ III:}}$ When $\mathcal {O}\subset \mathbb {R}^d$ satisfying Assumption 1.7. Since the boundary of $\mathcal {O}$ is uniformly of class $\textrm{C}^3$, we infer that $\textrm{D}(\textrm{A})=\mathbb {V}\cap \mathbb {H}^2(\mathcal {O})$ and $\Vert \textrm{A}\varvec{u}\Vert _{\mathbb {H}}$ defines a norm in $\textrm{D}(\textrm{A}),$ which is equivalent to the one in $\mathbb {H}^2(\mathcal {O})$ (cf. [28, Lemma 1]). Above argument implies that $\mathcal {P}:\mathbb {H}^2(\mathcal {O})\rightarrow \mathbb {H}^2(\mathcal {O})$ is a bounded operator.

2.4 Bilinear Operator

Let us define the trilinear form $b(\cdot ,\cdot ,\cdot ):\mathbb {V}\times \mathbb {V}\times \mathbb {V}\rightarrow \mathbb {R}$ by

$$\begin{aligned} b(\varvec{u},\varvec{v},\varvec{w})=\int _{\mathbb {R}^d}(\varvec{u}(x)\cdot \nabla )\varvec{v}(x)\cdot \varvec{w}(x)\textrm{d}x=\sum _{i,j=1}^d\int _{\mathbb {R}^d}\varvec{u}_i(x)\frac{\partial \varvec{v}_j(x)}{\partial x_i}\varvec{w}_j(x)\textrm{d}x. \end{aligned}$$

If $\varvec{u}, \varvec{v}$ are such that the linear map $b(\varvec{u}, \varvec{v}, \cdot )$ is continuous on $\mathbb {V}$, the corresponding element of $\mathbb {V}'$ is denoted by $\textrm{B}(\varvec{u}, \varvec{v})$. We also denote $\textrm{B}(\varvec{u}) = \textrm{B}(\varvec{u}, \varvec{u})=\mathcal {P}[(\varvec{u}\cdot \nabla )\varvec{u}]$. An integration by parts gives

$$\begin{aligned} \left\{ \begin{aligned} b(\varvec{u},\varvec{v},\varvec{v})&= 0,\ \text { for all }\ \varvec{u},\varvec{v}\in \mathbb {V},\\ b(\varvec{u},\varvec{v},\varvec{w})&= -b(\varvec{u},\varvec{w},\varvec{v}),\ \text { for all }\ \varvec{u},\varvec{v},\varvec{w}\in \mathbb {V}. \end{aligned} \right. \end{aligned}$$

(2.1)

Remark 2.6

The following estimate on $b(\cdot ,\cdot ,\cdot )$ is helpful in the sequel (cf. [56, Chapter 2, Section 2.3]). For all $\varvec{u}, \varvec{v}, \varvec{w}\in \mathbb {V}$,

$$\begin{aligned} |b(\varvec{u},\varvec{v},\varvec{w})|&\le C\times {\left\{ \begin{array}{ll} \Vert \varvec{u}\Vert ^{\frac{1}{2}}_{\mathbb {H}}\Vert \nabla \varvec{u}\Vert ^{\frac{1}{2}}_{\mathbb {H}}\Vert \nabla \varvec{v}\Vert _{\mathbb {H}}\Vert \varvec{w}\Vert ^{\frac{1}{2}}_{\mathbb {H}}\Vert \nabla \varvec{w}\Vert ^{\frac{1}{2}}_{\mathbb {H}},\ \ \ \text { for } d=2,\\ \Vert \varvec{u}\Vert ^{\frac{1}{4}}_{\mathbb {H}}\Vert \nabla \varvec{u}\Vert ^{\frac{3}{4}}_{\mathbb {H}}\Vert \nabla \varvec{v}\Vert _{\mathbb {H}}\Vert \varvec{w}\Vert ^{\frac{1}{4}}_{\mathbb {H}}\Vert \nabla \varvec{w}\Vert ^{\frac{3}{4}}_{\mathbb {H}}, \ \ \ \text { for } d=3. \end{array}\right. } \end{aligned}$$

(2.2)

Remark 2.7

Note that $\langle \textrm{B}(\varvec{v},\varvec{u}-\varvec{v}),\varvec{u}-\varvec{v}\rangle =0$ and it implies that

$$\begin{aligned} \langle \textrm{B}(\varvec{u})-\textrm{B}(\varvec{v}),\varvec{u}-\varvec{v}\rangle&=\langle \textrm{B}(\varvec{u}-\varvec{v},\varvec{u}),\varvec{u}-\varvec{v}\rangle \nonumber \\&=-\langle \textrm{B}(\varvec{u}-\varvec{v},\varvec{u}-\varvec{v}),\varvec{u}\rangle =-\langle \textrm{B}(\varvec{u}-\varvec{v},\varvec{u}-\varvec{v}),\varvec{v}\rangle . \end{aligned}$$

(2.3)

2.5 Nonlinear Operator

Consider the nonlinear operator $\mathcal {C}(\varvec{u}):=\mathcal {P}(|\varvec{u}|^{r-1}\varvec{u})$. It is immediate that $\langle \mathcal {C}(\varvec{u}),\varvec{u}\rangle =\Vert \varvec{u}\Vert _{\widetilde{\mathbb {L}}^{r+1}}^{r+1}$ and the map $\mathcal {C}(\cdot ):\mathbb {V}\cap \widetilde{\mathbb {L}}^{r+1}\rightarrow \mathbb {V}'+\widetilde{\mathbb {L}}^{\frac{r+1}{r}}$. For all $\varvec{u}\in \widetilde{\mathbb {L}}^{r+1}$, the map is Gateaux differentiable with Gateaux derivative

$$\begin{aligned} \mathcal {C}'(\varvec{u})\varvec{v}&=\left\{ \begin{array}{cl}\mathcal {P}(\varvec{v}),&{}\text { for }r=1,\\ \left\{ \begin{array}{cc}\mathcal {P}(|\varvec{u}|^{r-1}\varvec{v})+(r-1)\mathcal {P}\left( \frac{\varvec{u}}{|\varvec{u}|^{3-r}}(\varvec{u}\cdot \varvec{v})\right) ,&{}\text { if }\varvec{u}\ne {\textbf {0}},\\ {\textbf {0}},&{}\text { if }\varvec{u}={\textbf {0}},\end{array}\right. &{}\text { for } 1<r<3,\\ \mathcal {P}(|\varvec{u}|^{r-1}\varvec{v})+(r-1)\mathcal {P}(\varvec{u}|\varvec{u}|^{r-3}(\varvec{u}\cdot \varvec{v})), &{}\text { for }r\ge 3,\end{array}\right. \end{aligned}$$

(2.4)

for all $\varvec{v}\in \mathbb {V}\cap \widetilde{\mathbb {L}}^{r+1}$. Moreover, for any $r\in [1, \infty )$ and $\varvec{u}_{{\varvec{1}}}, \varvec{u}_{{\varvec{2}}} \in \mathbb {V}\cap \widetilde{\mathbb {L}}^{r+1}$, we have (cf. [45, Subsection 2.4])

$$\begin{aligned} \langle \mathcal {C}(\varvec{u}_{{\varvec{1}}})-\mathcal {C}(\varvec{u}_{{\varvec{2}}}),\varvec{u}_{{\varvec{1}}}-\varvec{u}_{{\varvec{2}}}\rangle \ge \frac{1}{2}\Vert |\varvec{u}_{{\varvec{1}}}|^{\frac{r-1}{2}}(\varvec{u}_{{\varvec{1}}}-\varvec{u}_{{\varvec{2}}})\Vert _{\mathbb {H}}^2+\frac{1}{2}\Vert |\varvec{u}_{{\varvec{2}}}|^{\frac{r-1}{2}}(\varvec{u}_{{\varvec{1}}}-\varvec{u}_{{\varvec{2}}})\Vert _{\mathbb {H}}^2 \ge 0. \end{aligned}$$

(2.5)

Remark 2.8

Since $\mathcal {P}$ and $\Delta$ commute on the whole space, we deduce

$$\begin{aligned} \big (\mathcal {C}(\varvec{u}),\textrm{A}\varvec{u}\big )&=\int _{\mathbb {R}^d}|\nabla \varvec{u}(x)|^2|\varvec{u}(x)|^{r-1}\textrm{d}x+4\left[ \frac{r-1}{(r+1)^2}\right] \int _{\mathbb {R}^d}|\nabla |\varvec{u}(x)|^{\frac{r+1}{2}}|^2\textrm{d}x. \end{aligned}$$

(2.6)

2.6 Inequalities

The following inequalities are frequently used in the paper.

Lemma 2.9

(Hölder’s inequality) Assume that $\frac{1}{p}+\frac{1}{p'}=1$ with $1\le p,p'\le \infty$, $\varvec{u}_{{\varvec{1}}}\in \mathbb {L}^{p}(\mathcal {O})$ and $\varvec{u}_{{\varvec{2}}}\in \mathbb {L}^{p'}(\mathcal {O})$. Then we get $\varvec{u}_{{\varvec{1}}}\varvec{u}_{{\varvec{2}}}\in \mathbb {L}^1(\mathcal {O})$ and

$$\begin{aligned} \Vert \varvec{u}_{{\varvec{1}}}\varvec{u}_{{\varvec{2}}}\Vert _{\mathbb {L}^1(\mathcal {O})}\le \Vert \varvec{u}_{{\varvec{1}}}\Vert _{\mathbb {L}^{p}(\mathcal {O})}\Vert \varvec{u}_{{\varvec{2}}}\Vert _{\mathbb {L}^{p'}(\mathcal {O})}. \end{aligned}$$

Lemma 2.10

(Interpolation inequality) Assume $1\le p_1\le p\le p_2\le \infty$, $\theta \in (0,1)$ such that $\frac{1}{p}=\frac{a}{p_1}+\frac{1-a}{p_2}$ and $\varvec{u}\in \mathbb {L}^{p_1}(\mathcal {O})\cap \mathbb {L}^{p_2}(\mathcal {O})$, then we have

$$\begin{aligned} \Vert \varvec{u}\Vert _{\mathbb {L}^p(\mathcal {O})}\le \Vert \varvec{u}\Vert _{\mathbb {L}^{p_1}(\mathcal {O})}^{a}\Vert \varvec{u}\Vert _{\mathbb {L}^{p_2}(\mathcal {O})}^{1-a}. \end{aligned}$$

Lemma 2.11

(Young’s inequality) For all $a,b,\varepsilon >0$ and for all $1<p,p'<\infty$ with $\frac{1}{p}+\frac{1}{p'}=1$, we obtain

$$\begin{aligned} ab\le \frac{\varepsilon }{p}a^p+\frac{1}{p'\varepsilon ^{p'/p}}b^{p'}. \end{aligned}$$

2.7 White Noise and Colored Noise

In this subsection, we recall some properties of white and colored noises.

Lemma 2.12

(Lemma 2.1, [23]) Assume that the correlation time $\delta \in (0,1]$. There exists a $\{\vartheta _t\}_{t\in \mathbb {R}}$-invariant subset $\widetilde{\Omega }\subseteq \Omega$ of full measure, such that for $\omega \in \widetilde{\Omega }$,

• (i)
  $$\begin{aligned} \lim \limits _{t\rightarrow \pm \infty }\frac{|\omega (t)|}{|t|}=0; \end{aligned}$$

  (2.7)

• (ii) the mapping

  $$\begin{aligned} (t,\omega )\mapsto \mathcal {Z}_{\delta }(\vartheta _t\omega )=-\frac{1}{\delta ^2}\int _{-\infty }^{0}e^{\frac{\xi }{\delta }}\vartheta _t\omega (\xi )\textrm{d}\xi \end{aligned}$$

  (2.8)

  is a stationary solution (also called a colored noise or an Ornstein–Uhlenbeck process) of one-dimensional stochastic differential equation

  

  with continuous trajectories satisfying

  $$\begin{aligned}&\lim \limits _{t\rightarrow \pm \infty }\frac{\left| \mathcal {Z}_{\delta }(\vartheta _t\omega )\right| }{|t|}=0,\quad \text { for every } 0<\delta \le 1, \end{aligned}$$

  (2.9)
  $$\begin{aligned}&\lim \limits _{t\rightarrow \pm \infty }\frac{1}{t}\int _0^t\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\textrm{d}\xi =\mathbb {E}[\mathcal {Z}_{\delta }]=0, \quad \text { uniformly for } 0<\delta \le 1; \end{aligned}$$

  (2.10)

and

• (iii) for arbitrary $T>0,$

  $$\begin{aligned} \lim _{\delta \rightarrow 0}\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\textrm{d}\xi =\omega (t) \ \text { uniformly for } \ t\in [\mathfrak {s},\mathfrak {s}+T]. \end{aligned}$$

  (2.11)

Remark 2.13

For convenience, we use $\Omega$ itself in place of $\widetilde{\Omega }$ throughout the work.

2.8 Abstract Formulation

In this subsection, we describe an abstract formulation and solution of (1.3). Taking orthogonal projection $\mathcal {P}$ to the first equation of (1.3), we obtain

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned} \frac{\text {d}\varvec{u}}{\text {d}{t}}&+\mu \text {A}\varvec{u}+\text {B}(\varvec{u})+\alpha \varvec{u}+\beta \mathcal {C}(\varvec{u})&={\varvec{f}} + S({t,x,\varvec{u}})\mathcal {Z}_{\delta }(\vartheta _t\omega ), \qquad {\varvec{t}} > \mathfrak {s}, \\ \varvec{u}|_{t=\mathfrak {s}}&=\varvec{u}_{\mathfrak {s}}, \qquad x\in \mathbb {R}^d \text{ and } \mathfrak {s}\in \mathbb {R}. \end{aligned} \right. \end{aligned} \end{aligned}$$

(2.12)

Strictly speaking, one has to write $\mathcal {P}\varvec{f}$ for $\varvec{f}$ and $\mathcal {P}S(t,x,\varvec{u})$ for $S(t,x,\varvec{u})$ in (2.12).

Definition 2.14

Let us assume that $\mathfrak {s}\in \mathbb {R},$ $\omega \in \Omega ,$ $\varvec{u}_{\mathfrak {s}}\in \mathbb {H}$, ${\varvec{f}}\in \text {L}^2_{loc }(\mathbb {R};\mathbb {V}')$, $T>0$ be any fixed time. Then, the function $\varvec{u}(\cdot ;\mathfrak {s},\omega ,\varvec{u}_{\mathfrak {s}}):=\varvec{u}(\cdot )$ is called a solution (in the weak sense) of the system (2.12) on time interval $[\mathfrak {s},\mathfrak {s}+T]$, if

$$\begin{aligned} \varvec{u}\in \textrm{L}^{\infty }(\mathfrak {s},\mathfrak {s}+T;\mathbb {H})\cap \textrm{L}^2(\mathfrak {s}, \mathfrak {s}+T;\mathbb {V})\cap \textrm{L}^{r+1}(\mathfrak {s},\mathfrak {s}+T;\widetilde{\mathbb {L}}^{r+1}), \end{aligned}$$

with $\partial _t\varvec{u}\in \textrm{L}^{2}(\mathfrak {s},\mathfrak {s}+T;\mathbb {V}')+\textrm{L}^{\frac{r+1}{r}}(\mathfrak {s},\mathfrak {s}+T;\widetilde{\mathbb {L}}^{\frac{r+1}{r}})$ satisfying:

• (i) for any $\psi \in \mathbb {V}\cap \widetilde{\mathbb {L}}^{r+1},$

  $$\begin{aligned} (\varvec{u}(t), \psi )&= (\varvec{u}_{\mathfrak {s}}, \psi ) - \mu \int \limits _{\mathfrak {s}}^{\mathfrak {s}+T}(\nabla \varvec{u}(\xi ),\nabla \psi )\textrm{d}\xi -\int \limits _{\mathfrak {s}}^{\mathfrak {s}+T}b(\varvec{u}(\xi ),\varvec{u}(\xi ),\psi )\textrm{d}\xi \\&\quad -\,\alpha \int \limits _{\mathfrak {s}}^{\mathfrak {s}+T}(\varvec{u}(\xi ),\psi )\textrm{d}\xi \\&\quad -\,\beta \int \limits _{\mathfrak {s}}^{\mathfrak {s}+T}\left\langle |\varvec{u}(\xi )|^{r-1}\varvec{u}(\xi ) +\varvec{f}(\xi ) , \psi \right\rangle \textrm{d}\xi + \int \limits _{\mathfrak {s}}^{\mathfrak {s}+T}\langle S(\xi ,x,\varvec{u}(s))\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega ), \psi \rangle \textrm{d}\xi , \end{aligned}$$

  for a.e. $t\in [\mathfrak {s},\mathfrak {s}+T]$,

• (ii) the initial data is satisfied in the following sense:

  $$\begin{aligned} \lim \limits _{t\downarrow \mathfrak {s}}\int _{\mathcal {O}}\varvec{u}(t,x)\psi (x)\textrm{d}x=\int _{\mathcal {O}}\varvec{u}_{\mathfrak {s}}(x)\psi (x)\textrm{d}x, \end{aligned}$$

  for all $\psi \in \mathbb {H}$.

By a standard Faedo–Galerkin method (cf. [44, 45, 51]), one can obtain that if any one of Assumptions 1.1 (with $\varvec{f}{\in }\textrm{L}^2_{\text {loc}}(\mathbb {R};\mathbb {V}')$), 1.3 (with $\varvec{f}{\in }\textrm{L}^2_{\text {loc}}(\mathbb {R};\mathbb {V}')$), 1.5 (with $\varvec{f}{\in }\textrm{L}^2_{\text {loc}}(\mathbb {R};\mathbb {H})$), 1.9 (with $\varvec{f}{\in } \textrm{L}^2_{\text {loc}}(\mathbb {R};\mathbb {H})$) is fulfilled. Then for all $t>\mathfrak {s}, \ \mathfrak {s}{\in }\mathbb {R},$ and for every $\varvec{u}_{\mathfrak {s}}\in \mathbb {H}$ and $\omega \in \Omega$, (2.12) has a unique solution in the sense of Definition 2.14. In addition, it follows from [14, Chapter II, Theorem 1.8] that $\varvec{u}\in \textrm{C}([\mathfrak {s},\mathfrak {s}+T];\mathbb {H})$ which gives that $\varvec{u}$ satisfies the energy equality:

$$\begin{aligned}&\Vert \varvec{u}(t)\Vert _{\mathbb {H}}^2+2\mu \int _{\mathfrak {s}}^t\Vert \nabla \varvec{u}(s)\Vert _{\mathbb {H}}^2\textrm{d}s+2\alpha \int _{\mathfrak {s}}^t\Vert \varvec{u}(s)\Vert _{\mathbb {H}}^2\textrm{d}s+2\beta \int _{\mathfrak {s}}^t\Vert \varvec{u}(s)\Vert _{\widetilde{\mathbb {L}}^{r+1}}^{r+1}\textrm{d}s\\&\quad =\Vert \varvec{u}_{\mathfrak {s}}\Vert _{\mathbb {H}}^2+2\int _0^t\langle \varvec{f}(s),\varvec{u}(s)\rangle \textrm{d}s+2\int _0^t\langle S(s,x,\varvec{u}(s))\mathcal {Z}_{\delta }(\vartheta _s\omega ),\varvec{u}(s)\rangle \textrm{d}s, \end{aligned}$$

for all $t\in [\mathfrak {s},\mathfrak {s}+T]$. Furthermore, $\varvec{u}(t;\mathfrak {s},\omega ,\varvec{u}_{\mathfrak {s}})$ is continuous with respect to the initial data $\varvec{u}_{\mathfrak {s}}$ (cf. Lemmas 3.4, 4.1 and 4.9) and $(\mathscr {F},\mathscr {B}(\mathbb {H}))$-measurable in $\omega \in \Omega .$

Now, we define a cocycle $\Phi :\mathbb {R}^+\times \mathbb {R}\times \Omega \times \mathbb {H}\rightarrow \mathbb {H}$ for the system (2.12) such that for given $t\in \mathbb {R}^+, \mathfrak {s}\in \mathbb {R}, \omega \in \Omega$ and $\varvec{u}_{\mathfrak {s}}\in \mathbb {H}$, let

$$\begin{aligned} \Phi (t,\mathfrak {s},\omega ,\varvec{u}_{\mathfrak {s}}) =\varvec{u}(t+\mathfrak {s};\mathfrak {s},\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}}). \end{aligned}$$

(2.13)

Then $\Phi$ is a continuous cocycle on $\mathbb {H}$ over $(\Omega ,\mathscr {F},\mathbb {P},\{\vartheta _{t}\}_{t\in \mathbb {R}})$, where $\{\vartheta _t\}_{t\in \mathbb {R}}$ is given by (1.2), that is,

$$\begin{aligned} \Phi (t+s,\mathfrak {s},\omega ,\varvec{u}_{\mathfrak {s}})=\Phi (t,\mathfrak {s}+s,\vartheta _{s}\omega ,\Phi (s,\mathfrak {s},\omega ,\varvec{u}_{\mathfrak {s}})). \end{aligned}$$

(2.14)

Assume that $D=\{D(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R},\omega \in \Omega \}$ is a family of non-empty subsets of $\mathbb {H}$ satisfying, for every $c>0, \mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$,

$$\begin{aligned} \lim _{t\rightarrow \infty }e^{-ct}\Vert D(\mathfrak {s}-t,\vartheta _{-t}\omega )\Vert _{\mathbb {H}}=0, \end{aligned}$$

(2.15)

where $\Vert D\Vert _{\mathbb {H}}=\sup \limits _{\varvec{x}\in D}\Vert \varvec{x}\Vert _{\mathbb {H}}.$ Let $\mathfrak {D}$ be the collection of all tempered families of bounded non-empty subsets of $\mathbb {H}$, that is,

$$\begin{aligned} \mathfrak {D}=\big \{D=\{D(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R}\text { and }\omega \in \Omega \}:D \text { satisfying } (2.15)\big \}. \end{aligned}$$

(2.16)

3 Pullback Random Attractors for Wong–Zakai Approximations: Bounded Domain

In this section, we prove the existence of a unique $\mathfrak {D}$-pullback random attractor for the system (2.12) on bounded domains with nonlinear diffusion term $S(t,x,\varvec{u})$ satisfying Assumption 1.1. Throughout the section, we assume that $\mathcal {O}$ is a bounded subset of $\mathbb {R}^d$ with $\textrm{C}^2$-boundary. In the sequel, the following assumptions are needed on the non-autonomous external forcing term $\varvec{f}$.

Assumption 3.1

For the external forcing term $\varvec{f}\in \textrm{L}^{2}_{\textrm{loc}}(\mathbb {R};\mathbb {V}')$, there exists a number $\gamma \in [0,\alpha )$ such that for every $c>0$,

(3.1)

A direct consequence of the above Assumption 3.1 is as follows.

Proposition 3.2

(Proposition 4.2, [37]) Assume that Assumption 3.1 holds. Then

(3.2)

where $\gamma$ is the same as in (3.1).

Example 3.3

Take $\varvec{f}(\cdot ,t)=t^{p}\varvec{f}_1$, for any $p\ge 0$ and $\varvec{f}_1\in \mathbb {H}$. Note that the conditions (3.2)–(3.1) do not need $\varvec{f}$ to be bounded in $\mathbb {H}$ at $\pm \infty$.

Lemma 3.4

For all the cases given in Table 1, let Assumptions 1.1 and 3.1 be satisfied. Then, the solution of (2.12) is continuous in the initial data $\varvec{u}_{\mathfrak {s}}.$

Proof

Let $\varvec{u}_{{\varvec{1}}}(\cdot )$ and $\varvec{u}_{{\varvec{2}}}(\cdot )$ be two solutions of (2.12). Then $\mathfrak {X}(\cdot )=\varvec{u}_{{\varvec{1}}}(\cdot )-\varvec{u}_{{\varvec{2}}}(\cdot )$ with $\mathfrak {X}|_{t=\mathfrak {s}}=\varvec{u}_{{\varvec{1}}}|_{t=\mathfrak {s}}-\varvec{u}_{{\varvec{2}}}|_{t=\mathfrak {s}}$ satisfies

$$\begin{aligned} \frac{\textrm{d}\mathfrak {X}(t)}{\textrm{d}t}&=-\mu \textrm{A}\mathfrak {X}(t)-\alpha \mathfrak {X}(t)-\left\{ \textrm{B}\big (\varvec{u}_{{\varvec{1}}}(t)\big )-\textrm{B}\big (\varvec{u}_{{\varvec{2}}}(t)\big )\right\} -\beta \left\{ \mathcal {C}\big (\varvec{u}_{{\varvec{1}}}(t)\big )-\mathcal {C}\big (\varvec{u}_{{\varvec{2}}}(t)\big )\right\} \nonumber \\&\quad +e^{\sigma t}\left[ \kappa \mathfrak {X}(t)+\mathcal {S}(\varvec{u}_{{\varvec{1}}}(t))-\mathcal {S}(\varvec{u}_{{\varvec{2}}}(t))\right] \mathcal {Z}_{\delta }(\vartheta _t\omega ), \end{aligned}$$

(3.3)

in $\mathbb {V}'+\widetilde{\mathbb {L}}^{\frac{r+1}{r}}$, for a.e. $t\in [0,T]$. Taking the inner product with $\mathfrak {X}(\cdot )$ to the first equation in (3.3), we obtain

$$\begin{aligned} \frac{1}{2}\frac{\textrm{d}}{\textrm{d}t} \Vert \mathfrak {X}(t)\Vert ^2_{\mathbb {H}}&=-\mu \Vert \nabla \mathfrak {X}(t)\Vert ^2_{\mathbb {H}} - \alpha \Vert \mathfrak {X}(t)\Vert ^2_{\mathbb {H}} -\left\langle \textrm{B}\big (\varvec{u}_{{\varvec{1}}}(t)\big )-\textrm{B}\big (\varvec{u}_{{\varvec{2}}}(t)\big ), \mathfrak {X}(t)\right\rangle \nonumber \\&\quad -\beta \left\langle \mathcal {C}\big (\varvec{u}_{{\varvec{1}}}(t)\big )-\mathcal {C}\big (\varvec{u}_{{\varvec{2}}}(t)\big ),\mathfrak {X}(t)\right\rangle + \kappa e^{\sigma t}\mathcal {Z}_{\delta }(\vartheta _{t}\omega )\Vert \mathfrak {X}(t)\Vert ^2_{\mathbb {H}}\nonumber \\&\quad +e^{\sigma t}\mathcal {Z}_{\delta }(\vartheta _{t}\omega )\left( \mathcal {S}(\varvec{u}_{{\varvec{1}}}(t))-\mathcal {S}(\varvec{u}_{{\varvec{2}}}(t)),\mathfrak {X}(t)\right) , \end{aligned}$$

(3.4)

for a.e. $t\in [\mathfrak {s},\mathfrak {s}+T] \text { with } T>0$. By (1.5), we get

$$\begin{aligned} \kappa&e^{\sigma t}\mathcal {Z}_{\delta }(\vartheta _{t}\omega )\Vert \mathfrak {X}\Vert ^2_{\mathbb {H}}+e^{\sigma t}\mathcal {Z}_{\delta }(\vartheta _{t}\omega )\left( \mathcal {S}(\varvec{u}_{{\varvec{1}}})-\mathcal {S}(\varvec{u}_{{\varvec{2}}}),\mathfrak {X}\right) \nonumber \\&\le \kappa e^{\sigma t}\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \mathfrak {X}\Vert ^2_{\mathbb {H}}+s_2 e^{\sigma t}\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \mathfrak {X}\Vert _{\mathbb {H}} \Vert \mathfrak {X}\Vert _{\mathbb {V}}\nonumber \\&\le \frac{\alpha }{4}\Vert \mathfrak {X}\Vert ^2_{\mathbb {H}}+ \frac{\min \{\mu ,\alpha \}}{4}\Vert \mathfrak {X}\Vert ^2_{\mathbb {V}} +Ce^{2\sigma t}\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^2\Vert \mathfrak {X}\Vert ^2_{\mathbb {H}}. \end{aligned}$$

(3.5)

Case I: When $d=2$ and $r\ge 1$. Using (2.2), (2.3) and Lemma 2.11, we obtain

$$\begin{aligned} \left| \left\langle \textrm{B}\big (\varvec{u}_{{\varvec{1}}}\big )-\textrm{B}\big (\varvec{u}_{{\varvec{2}}}\big ), \mathfrak {X}\right\rangle \right|&=\left| \left\langle \textrm{B}\big (\mathfrak {X},\mathfrak {X} \big ), \varvec{u}_{{\varvec{2}}}\right\rangle \right| \le \frac{\mu }{4}\Vert \nabla \mathfrak {X}\Vert ^2_{\mathbb {H}}+C\Vert \varvec{u}_{{\varvec{2}}}\Vert ^2_{\mathbb {H}}\Vert \nabla \varvec{u}_{{\varvec{2}}}\Vert ^2_{\mathbb {H}}\Vert \mathfrak {X}\Vert ^2_{\mathbb {H}}, \end{aligned}$$

(3.6)

and from (2.5), we have

$$\begin{aligned} \begin{aligned} -\,\beta \left\langle \mathcal {C}\big (\varvec{u}_{{\varvec{1}}}\big )-\mathcal {C}\big (\varvec{u}_{{\varvec{2}}}\big ),\mathfrak {X}\right\rangle \le - \frac{\beta }{2}\Vert |{\varvec{u}}_\textbf{2}|^{\frac{r-1}{2}} \mathfrak {X} \Vert ^2_{\mathbb {H}}\le 0. \end{aligned} \end{aligned}$$

(3.7)

Making use of (3.5)–(3.7) in (3.4), we get

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t} \Vert \mathfrak {X}(t)\Vert ^2_{\mathbb {H}} \le C\bigg \{e^{2\sigma t}\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^2+ \Vert \varvec{u}_{{\varvec{2}}}(t)\Vert ^2_{\mathbb {H}}\Vert \nabla \varvec{u}_{{\varvec{2}}}(t)\Vert ^2_{\mathbb {H}}\bigg \}\Vert \mathfrak {X}(t)\Vert ^2_{\mathbb {H}}, \end{aligned}$$

for a.e. $t\in [\mathfrak {s},\mathfrak {s}+T]$ and an application of Gronwall’s inequality completes the proof.

Case II: When $d= 3$ and $r>3$. The nonlinear term $|(\textrm{B}(\mathfrak {X},\mathfrak {X}),\varvec{u}_{{\varvec{2}}})|$ is estimated using Lemmas 2.9 and 2.11 as

$$\begin{aligned} \left| \left\langle \text {B}\big (\varvec{u}_{{\varvec{1}}}\big )-\text {B}\big (\varvec{u}_{{\varvec{2}}}\big ), \mathfrak {X}\right\rangle \right|&=\left| (\text {B}(\mathfrak {X},\mathfrak {X}),\varvec{u}_{{\varvec{2}}})\right| \nonumber \\ {}&\le \Vert |\varvec{u}_{{\varvec{2}}}|\mathfrak {X}\Vert _{\mathbb {H}}\Vert \nabla \mathfrak {X}\Vert _{\mathbb {H}}\le \frac{\mu }{4}\Vert \nabla \mathfrak {X}\Vert _{\mathbb {H}}^2+\frac{1}{\mu }\Vert |\varvec{u}_{{\varvec{2}}}|\mathfrak {X}\Vert _{\mathbb {H}}^2\nonumber \\ {}&\le \frac{\mu }{4}\Vert \nabla \mathfrak {X}\Vert _{\mathbb {H}}^2+\frac{\beta }{2}\Vert |\varvec{u}_{{\varvec{2}}}|^{\frac{r-1}{2}}\mathfrak {X} \Vert ^2_{\mathbb {H}}+C\Vert \mathfrak {X}\Vert ^2_{\mathbb {H}}. \end{aligned}$$

(3.8)

Combining (3.4), (3.5), (3.7) and (3.8), we complete the proof using Gronwall’s inequality.

Case III: When $d=3$ and $r=3$ with $2\beta \mu >1$. Since $2\beta \mu >1$, there exists $0<\theta <1$ such that

$$\begin{aligned} 2\beta \mu \ge \frac{1}{\theta }. \end{aligned}$$

(3.9)

Using (2.2) and (2.3), we have

$$\begin{aligned} \begin{aligned} \left| \left\langle \text {B}\big (\varvec{u}_{{\varvec{1}}}\big )-\text {B}\big (\varvec{u}_2\big ), \mathfrak {X}\right\rangle \right|&=\left| \left\langle \text {B}\big (\mathfrak {X},\mathfrak {X} \big ), \varvec{u}_2\right\rangle \right| \le \theta \mu \Vert \nabla \mathfrak {X}\Vert ^2_{\mathbb {H}}+\frac{1}{4\theta \mu }\Vert |\varvec{u}_2|\mathfrak {X}\Vert ^2_{\mathbb {H}}, \end{aligned} \end{aligned}$$

(3.10)

where $\theta$ is the same as in (3.9). Again by (1.5), we obtain

$$\begin{aligned} \kappa&e^{\sigma t}\mathcal {Z}_{\delta }(\vartheta _{t}\omega )\Vert \mathfrak {X}\Vert ^2_{\mathbb {H}}+e^{\sigma t}\mathcal {Z}_{\delta }(\vartheta _{t}\omega )\left( \mathcal {S}(\varvec{u}_1)-\mathcal {S}(\varvec{u}_2),\mathfrak {X}\right) \nonumber \\&\le \frac{\alpha }{2}\Vert \mathfrak {X}\Vert ^2_{\mathbb {H}}+ (1-\theta )\mu \Vert \mathfrak {X}\Vert ^2_{\mathbb {V}} +Ce^{2\sigma t}\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^2\Vert \mathfrak {X}\Vert ^2_{\mathbb {H}}. \end{aligned}$$

(3.11)

Using (3.7), (3.10) and (3.11) in (3.4) and applying Gronwall’s inequality, one can conclude the proof. $\square$

Next result supports us to show the existence of $\mathfrak {D}$-pullback random absorbing set for continuous cocycle $\Phi$.

Lemma 3.5

Let all the assumptions of Lemma 3.4 be satisfied. Then for every $0<\delta \le 1$, $\mathfrak {s}\in \mathbb {R},$ $\omega \in \Omega$ and $D=\{D(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R}, \omega \in \Omega \}\in \mathfrak {D},$ there exists $\mathcal {T}=\mathcal {T}(\delta , \mathfrak {s}, \omega , D)>0$ such that for all $t\ge \mathcal {T}$ and $\tau \ge \mathfrak {s}-t$, the solution $\varvec{u}$ of the system (2.12) with $\omega$ replaced by $\vartheta _{-\mathfrak {s}}\omega$ satisfies

$$\begin{aligned}&\Vert \varvec{u}(\tau ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}}\nonumber \\&\quad \le \frac{4}{\min \{\mu ,\alpha \}} \int _{-\infty }^{\tau -\mathfrak {s}} e^{\int _{\tau -\mathfrak {s}}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta } \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}\textrm{d}\xi \nonumber \\&\qquad +\int _{-\infty }^{\tau -\mathfrak {s}} e^{\int _{\tau -\mathfrak {s}}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta }\bigg \{2s_3e^{\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| \nonumber \\&\qquad +2s_6 e^{2\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^2+s_7\left[ e^{\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| \right] ^{\frac{2}{1-s_5}}\bigg \}\textrm{d}\xi , \end{aligned}$$

(3.12)

where $\varvec{u}_{\mathfrak {s}-t}\in D(\mathfrak {s}-t,\vartheta _{-t}\omega )$, $s_6=\frac{2}{\alpha }\Vert \varvec{h}\Vert ^2_{\mathbb {H}}$, $s_7=s_4(1-s_5)\left[ \frac{4s_4(1+s_5)}{\min \{\mu ,\alpha \}}\right] ^{\frac{1+s_5}{1-s_5}}$ and, $s_3, s_4$ and $s_5$ are the constants appearing in (1.6).

Proof

From the first equation of the system (2.12), we obtain

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t} \Vert \varvec{u}\Vert ^2_{\mathbb {H}} +2\mu \Vert \nabla \varvec{u}\Vert ^2_{\mathbb {H}} +2\alpha \Vert \varvec{u}\Vert ^2_{\mathbb {H}} + 2\beta \Vert \varvec{u}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\nonumber \\&\quad \le 2\Vert \varvec{f}\Vert _{\mathbb {V}'}\Vert \varvec{u}\Vert _{\mathbb {V}}+2\kappa e^{\sigma t}\mathcal {Z}_{\delta }(\vartheta _{t}\omega )\Vert \varvec{u}\Vert ^2_{\mathbb {H}}\nonumber \\&\qquad +2e^{\sigma t}\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \left( s_3+s_4\Vert \varvec{u}\Vert ^{1+s_5}_{\mathbb {V}}+\Vert \varvec{h}\Vert _{\mathbb {H}}\Vert \varvec{u}\Vert _{\mathbb {H}}\right) \nonumber \\&\quad \le \frac{\alpha }{2}\Vert \varvec{u}\Vert ^2_{\mathbb {H}}+ \frac{\min \{\mu ,\alpha \}}{2}\Vert \varvec{u}\Vert ^2_{\mathbb {V}}+\frac{4\Vert \varvec{f}\Vert ^2_{\mathbb {V}'}}{\min \{\mu ,\alpha \}}+2\kappa e^{\sigma t}\mathcal {Z}_{\delta }(\vartheta _{t}\omega )\Vert \varvec{u}\Vert ^2_{\mathbb {H}}+2s_3e^{\sigma t}\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \nonumber \\&\qquad +\,s_6 e^{2\sigma t}\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^2+s_7\left[ e^{\sigma t}\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \right] ^{\frac{2}{1-s_5}}, \end{aligned}$$

(3.13)

where we have used (2.1), (1.6), Lemmas 2.9 and 2.11, and the constants $s_6=\frac{2}{\alpha }\Vert \varvec{h}\Vert ^2_{\mathbb {H}}$ and $s_7=s_4(1-s_5)\left[ \frac{4s_4(1+s_5)}{\min \{\mu ,\alpha \}}\right] ^{\frac{1+s_5}{1-s_5}}$. We rewrite (3.13) as

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t} \Vert \varvec{u}\Vert ^2_{\mathbb {H}}+ \left( \alpha -2\kappa e^{\sigma t}\mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right) \Vert \varvec{u}\Vert ^2_{\mathbb {H}}\nonumber \\&\quad \le \frac{4\Vert \varvec{f}\Vert ^2_{\mathbb {V}'}}{\min \{\mu ,\alpha \}}+2s_3e^{\sigma t}\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| +s_6 e^{2\sigma t}\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^2+s_7\left[ e^{\sigma t}\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \right] ^{\frac{2}{1-s_5}}. \end{aligned}$$

(3.14)

Making use of variation of constant formula in (3.14), replacing $\omega$ by $\vartheta _{-\mathfrak {s}}\omega$ and using change of variable technique, we get

$$\begin{aligned}&\Vert \varvec{u}(\tau ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}}\nonumber \\&\quad \le e^{\int _{\tau -\mathfrak {s}}^{-t}\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta }\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^2_{\mathbb {H}}\nonumber \\&\qquad + \frac{4}{\min \{\mu ,\alpha \}}\int _{-t}^{\tau -\mathfrak {s}} e^{\int _{\tau -\mathfrak {s}}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta } \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}\textrm{d}\xi \nonumber \\&\qquad +\int _{-t}^{\tau -\mathfrak {s}} e^{\int _{\tau -\mathfrak {s}}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta }\bigg \{2s_3e^{\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| \nonumber \\&\qquad +s_6 e^{2\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^2+s_7\left[ e^{\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| \right] ^{\frac{2}{1-s_5}}\bigg \}\textrm{d}\xi . \end{aligned}$$

(3.15)

Now, we need to estimate each term on the right hand side of (3.15). Depending on $\sigma$, there are two possible cases.

Case I: When $\sigma =0$. By (2.10), we have

$$\begin{aligned} \lim _{\xi \rightarrow -\infty }\frac{1}{\xi }\int _{0}^{\xi }\left( \alpha -2\kappa \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta =\alpha -2\kappa \mathbb {E}[\mathcal {Z}_{\delta }]=\alpha . \end{aligned}$$

(3.16)

Since $\gamma <\alpha$ (see Assumption 3.1), from (3.16), we infer that there exists $\xi _0=\xi _0(\delta ,\omega )<0$ such that for all $\xi \le \xi _0$,

$$\begin{aligned} \int _{0}^{\xi }\left( \alpha -2\kappa \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta <\gamma \xi . \end{aligned}$$

(3.17)

Taking (3.2) and (3.17) into account, we obtain

$$\begin{aligned} \int _{-\infty }^{\xi _0}e^{\int _{0}^{\xi }\left( \alpha -2\kappa \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta }\Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}\textrm{d}\xi<\int _{-\infty }^{\xi _0}e^{\gamma \xi }\Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}\textrm{d}\xi <\infty , \end{aligned}$$

(3.18)

and similarly from (2.9) and (3.17), we get

$$\begin{aligned} \int _{-\infty }^{\xi _0}e^{\int _{0}^{\xi }\left( \alpha -2\kappa \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta }\bigg \{2s_3\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| +s_6\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^2+s_7\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{2}{1-s_5}}\bigg \}\textrm{d}\xi <\infty . \end{aligned}$$

(3.19)

Case II: When $\sigma >0$. From (2.9), we get

$$\begin{aligned} \lim _{\zeta \rightarrow -\infty }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) =\alpha >\gamma , \end{aligned}$$

therefore, there exists $\zeta _0=\zeta _0(\mathfrak {s},\delta ,\omega )<0$ such that for all $\zeta \le \zeta _0$,

$$\begin{aligned} \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )>\gamma . \end{aligned}$$

(3.20)

By (3.2) and (3.20), we have

$$\begin{aligned} \int _{-\infty }^{\zeta _0}e^{\int _{\zeta _0}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})} \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta }\Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}\textrm{d}\xi< e^{-\gamma \zeta _0}\int _{-\infty }^{\zeta _0}e^{\gamma \xi }\Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}\textrm{d}\xi <\infty , \end{aligned}$$

(3.21)

and similarly by (2.9) and (3.20), it is easy to obtain

$$\begin{aligned}&\int _{-\infty }^{\xi _0}e^{\int _{0}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})} \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta }\bigg \{2s_3e^{\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| +s_6e^{2\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^2\nonumber \\ {}&\qquad +s_7\left[ e^{\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| \right] ^{\frac{2}{1-s_5}}\bigg \}\textrm{d}\xi <\infty . \end{aligned}$$

(3.22)

It follows from (3.18)–(3.19) and (3.21)–(3.22) that for any $\sigma \ge 0$,

$$\begin{aligned}&\int _{-\infty }^{\tau -\mathfrak {s}} e^{\int _{\tau -\mathfrak {s}}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta } \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}\textrm{d}\xi +\int _{-\infty }^{\tau -\mathfrak {s}} e^{\int _{\tau -\mathfrak {s}}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta }\nonumber \\&\quad \times \bigg \{2s_3e^{\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| +s_6 e^{2\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^2\nonumber \\&\quad +s_7\left[ e^{\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| \right] ^{\frac{2}{1-s_5}}\bigg \}\textrm{d}\xi <\infty . \end{aligned}$$

(3.23)

Finally, since $\varvec{u}_{\mathfrak {s}-t}\in D(\mathfrak {s}-t,\vartheta _{-t}\omega )$ and $D\in \mathfrak {D}$, making use of (3.17) and (3.20), we have for $\sigma \ge 0$,

$$\begin{aligned} e^{\int _{0}^{-t}\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta }\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^2_{\mathbb {H}}\le e^{\int _{0}^{-t}\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta }\Vert D(\mathfrak {s}-t,\vartheta _{-t}\omega )\Vert ^2_{\mathbb {H}}\rightarrow 0, \end{aligned}$$

as $t\rightarrow \infty$, there exists $\mathcal {T}=\mathcal {T}(\delta ,\mathfrak {s},\omega ,D)>0$ such that for all $t\ge \mathcal {T}$,

$$\begin{aligned}&e^{\int _{\tau -\mathfrak {s}}^{-t}\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta }\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^2_{\mathbb {H}}\nonumber \\&\quad \le s_6\int _{-\infty }^{\tau -\mathfrak {s}} e^{\int _{\tau -\mathfrak {s}}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta } e^{2\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^2\textrm{d}\xi . \end{aligned}$$

(3.24)

From (3.15) along with (3.23)–(3.24), one can conclude the proof. $\square$

Now, we prove $\mathfrak {D}$-pullback asymptotic compactness using compact Sobolev embeddings on bounded domains in the next lemma.

Lemma 3.6

Let all the assumptions of Lemma 3.4 be satisfied. Then for every $0<\delta \le 1, \omega \in \Omega , \mathfrak {s}\in \mathbb {R}$ and $t>\mathfrak {s}$, the solution $\varvec{u}(t;\mathfrak {s},\omega ,\cdot ):\mathbb {H}\rightarrow \mathbb {H}$ is compact, that is, for every bounded set B in $\mathbb {H}$, the image $\varvec{u}(t;\mathfrak {s},\omega ,B)$ is precompact in $\mathbb {H}$.

Proof

Consider the solution $\varvec{u}(\tau ;\mathfrak {s},\omega ,\cdot )$ of (2.12) for $\tau \in [\mathfrak {s},\mathfrak {s}+T]$, where $T>0$. Assume that the sequence $\{\varvec{u}_{0,n}\}_{n\in \mathbb {N}}\subset B$. We know that (see the proof of Lemma 3.5)

(3.25)

From (1.4) along with (3.25), we obtain

$$\begin{aligned} \{S(\cdot ,\cdot ,\varvec{u}(\tau ;\mathfrak {s},\omega ,\varvec{u}_{0,n}))\mathcal {Z}_{\delta }(\vartheta _{\tau }\omega )\}_{n\in \mathbb {N}} \text { is bounded in } \textrm{L}^{2}(\mathfrak {s},\mathfrak {s}+T;\mathbb {H}). \end{aligned}$$

(3.26)

We also have

$$\begin{aligned} \{\textrm{A}(\varvec{u}(\tau ;\mathfrak {s},\omega ,\varvec{u}_{0,n}))\}_{n\in \mathbb {N}} \text { and } \{\textrm{B}(\varvec{u}(\tau ;\mathfrak {s},\omega ,\varvec{u}_{0,n}))\}_{n\in \mathbb {N}} \text { are bounded in } \textrm{L}^2(\mathfrak {s},\mathfrak {s}+T;\mathbb {V}'), \end{aligned}$$

(3.27)

and

$$\begin{aligned} \{\mathcal {C}(\varvec{u}(\tau ;\mathfrak {s},\omega ,\varvec{u}_{0,n}))\}_{n\in \mathbb {N}} \text { is bounded in } \textrm{L}^{\frac{r+1}{r}}(\mathfrak {s},\mathfrak {s}+T;\widetilde{\mathbb {L}}^{\frac{r+1}{r}}). \end{aligned}$$

(3.28)

It follows from (3.25)–(3.28) and (2.12) that

$$\begin{aligned} \left\{ \frac{\textrm{d}}{\textrm{d}s}(\varvec{u}(\tau ;\mathfrak {s},\omega ,\varvec{u}_{0,n}))\right\} _{n\in \mathbb {N}} \text { is bounded in } \textrm{L}^2(\mathfrak {s},\mathfrak {s}+T;\mathbb {V}')+\textrm{L}^{\frac{r+1}{r}}(\mathfrak {s},\mathfrak {s}+T;\widetilde{\mathbb {L}}^{\frac{r+1}{r}}). \end{aligned}$$

Since $\textrm{L}^2(\mathfrak {s},\mathfrak {s}+T;\mathbb {V}')+\textrm{L}^{\frac{r+1}{r}}(\mathfrak {s},\mathfrak {s}+T;\widetilde{\mathbb {L}}^{\frac{r+1}{r}})\subset \textrm{L}^{\frac{r+1}{r}}(\mathfrak {s},\mathfrak {s}+T;\mathbb {V}'+\widetilde{\mathbb {L}}^{\frac{r+1}{r}})$, the above sequence is bounded in $\textrm{L}^{\frac{r+1}{r}}(\mathfrak {s},\mathfrak {s}+T;\mathbb {V}'+\widetilde{\mathbb {L}}^{\frac{r+1}{r}})$. Note also that $\mathbb {V}\cap \widetilde{\mathbb {L}}^{\frac{r+1}{r}}\subset \mathbb {V}\subset \mathbb {H}\subset \mathbb {V}'\subset \mathbb {V}'+\widetilde{\mathbb {L}}^{\frac{r+1}{r}}$ and the embedding of $\mathbb {V}\subset \mathbb {H}$ is compact. By the Aubin-Lions compactness lemma, there exists a subsequence (keeping as it is) and $\varvec{v}\in \textrm{L}^2(\mathfrak {s},\mathfrak {s}+T;\mathbb {H})$ such that

$$\begin{aligned} \varvec{u}(\cdot ;\mathfrak {s},\omega ,\varvec{u}_{0,n})\rightarrow \varvec{v}(\cdot ) \ \text { strongly in }\ \textrm{L}^{2}(\mathfrak {s},\mathfrak {s}+T;\mathbb {H}). \end{aligned}$$

(3.29)

Along a further subsequence (again not relabeling), we infer from (3.29) that

$$\begin{aligned} \varvec{u}(\tau ;\mathfrak {s},\omega ,\varvec{u}_{0,n})\rightarrow \varvec{v}(\tau ) \text { in } \mathbb {H}\ \text { for almost all }\ \tau \in (\mathfrak {s},\mathfrak {s}+T). \end{aligned}$$

(3.30)

Since $\mathfrak {s}<t<\mathfrak {s}+T$, we obtain from (3.30) that there exists $\tau \in (\mathfrak {s},t)$ such that (3.30) holds true for this particular $\tau$. Then by Lemma 3.4, we obtain

$$\begin{aligned} \varvec{u}(t;\mathfrak {s},\omega ,\varvec{u}_{0,n})=\varvec{u}(t;\tau ,\omega ,\varvec{u}(\tau ;\mathfrak {s},\omega ,\varvec{u}_{0,n}))\rightarrow \varvec{u}(t;\tau ,\omega ,\varvec{v}(\tau )), \end{aligned}$$

which completes the proof. $\square$

In fact, Lemma 3.6 helps us to prove the $\mathfrak {D}$-pullback asymptotic compactness of $\Phi$ in $\mathbb {H}$ on bounded domains.

Corollary 3.7

Let all the assumptions of Lemma 3.4 be satisfied. Then for every $0<\delta \le 1$, $\mathfrak {s}\in \mathbb {R},$ $\omega \in \Omega ,$ $D=\{D(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R},\omega \in \Omega \}\in \mathfrak {D}$ and $t_n\rightarrow \infty ,$ $\varvec{u}_{0,n}\in D(\mathfrak {s}-t_n, \vartheta _{-t_{n}}\omega )$, the sequence $\varvec{u}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})$ of solutions of the system (2.12) has a convergent subsequence in $\mathbb {H}$.

Proof

From Lemma 3.5 with $\tau =\mathfrak {s}-1$, we have that there exists $\mathcal {T}=\mathcal {T}(\delta ,\mathfrak {s},\omega ,D)>0$ such that for all $t\ge \mathcal {T}$ and $\varvec{u}_{\mathfrak {s}-t}\in D(\mathfrak {s}-t, \vartheta _{-t}\omega )$,

$$\begin{aligned} \varvec{u}(\mathfrak {s}-1;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\in \mathbb {H}. \end{aligned}$$

(3.31)

Since $t_n\rightarrow \infty$ and $\varvec{u}_{0,n}\in D(\mathfrak {s}-t_n, \vartheta _{-t_{n}}\omega )$, from (3.31), we infer that there exists $N_1=N(\delta ,\mathfrak {s},\omega ,D)>0$ such that

$$\begin{aligned} \{\varvec{u}(\mathfrak {s}-1;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})\}_{n\ge N_1}\subset \mathbb {H}. \end{aligned}$$

(3.32)

Hence, by (3.32) and Lemma 3.6, we conclude that the sequence

$$\begin{aligned} \varvec{u}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})=\varvec{u}(\mathfrak {s};\mathfrak {s}-1,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}(\mathfrak {s}-1;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})) \end{aligned}$$

has a convergent subsequence in $\mathbb {H}$, which completes the proof. $\square$

3.1 Existence of a Unique $\mathfrak {D}$-pullback Random Attractor

In this subsection, we start with the result on the existence of a $\mathfrak {D}$-pullback random absorbing set in $\mathbb {H}$ for the non-autonomous RDS associated with the system (2.12). Then, we prove the main result of this section, that is, the existence of a unique $\mathfrak {D}$-pullback random attractor for the non-autonomous RDS associated with the system (2.12).

Lemma 3.8

Under the assumptions of Lemma 3.4, there exists a closed measurable $\mathfrak {D}$-pullback random absorbing set $\mathcal {K}=\{\mathcal {K}(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R}, \omega \in \Omega \}\in \mathfrak {D}$ for the continuous cocycle $\Phi$ associated with the system (2.12).

Proof

Let us denote, for given $\mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$, $\mathcal {K}(\mathfrak {s},\omega )=\{\varvec{u}\in \mathbb {H}:\Vert \varvec{u}\Vert ^2_{\mathbb {H}}\le \mathcal {L}(\mathfrak {s},\omega )\}$ where

$$\begin{aligned}&\mathcal {L}(\mathfrak {s},\omega )\\&\quad =\frac{4}{\min \{\mu ,\alpha \}} \int _{-\infty }^{0} e^{\int _{0}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta } \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}\textrm{d}\xi \\&\qquad +\int _{-\infty }^{0} e^{\int _{0}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta }\\&\qquad \times \bigg \{2s_3e^{\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| +2s_6 e^{2\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^2\\&\qquad +s_7\left[ e^{\sigma (\xi +\mathfrak {s})}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| \right] ^{\frac{2}{1-s_5}}\bigg \}\textrm{d}\xi . \end{aligned}$$

Since $\mathcal {L}(\mathfrak {s},\cdot ):\Omega \rightarrow \mathbb {R}$ is $(\mathscr {F},\mathscr {B}(\mathbb {R}))$-measurable for every $\mathfrak {s}\in \mathbb {R}$, $\mathcal {K}(\mathfrak {s},\cdot ):\Omega \rightarrow 2^{\mathbb {H}}$ is a measurable set-valued mapping. Furthermore, we have from Lemma 3.5 that for each $\mathfrak {s}\in \mathbb {R}$, $\omega \in \Omega$ and $D\in \mathfrak {D}$, there exists $\mathcal {T}=\mathcal {T}(\delta ,\mathfrak {s},\omega ,D)>0$ such that for all $t\ge \mathcal {T}$,

$$\begin{aligned} \Phi (t,\mathfrak {s}-t,\vartheta _{-t}\omega ,D(\mathfrak {s}-t,\vartheta _{-t}\omega ))=\varvec{u}(\mathfrak {s};\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,D(\mathfrak {s}-t,\vartheta _{-t}\omega ))\subseteq \mathcal {K}(\mathfrak {s},\omega ). \end{aligned}$$

(3.33)

Now, in order to complete the proof, we only need to prove that $\mathcal {K}\in \mathfrak {D}$, that is, for every $c>0$, $\mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$

$$\begin{aligned} \lim _{t\rightarrow -\infty }e^{ct}\Vert \mathcal {K}(\mathfrak {s} +t,\vartheta _{t}\omega )\Vert ^2_{\mathbb {H}}=0. \end{aligned}$$

For every $c>0$, $\mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$,

$$\begin{aligned}&\lim _{t\rightarrow -\infty }e^{ct}\Vert \mathcal {K}(\mathfrak {s} +t,\vartheta _{t}\omega )\Vert ^2_{\mathbb {H}}\nonumber \\&\quad =\lim _{t\rightarrow -\infty }e^{ct}\mathcal {L}(\mathfrak {s} +t,\vartheta _{t}\omega )\nonumber \\&\quad =\lim _{t\rightarrow -\infty }\frac{4e^{ct}}{\min \{\mu ,\alpha \}} \int _{-\infty }^{0} e^{\int _{0}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s}+t)}\mathcal {Z}_{\delta }(\vartheta _{\zeta +t}\omega )\right) \textrm{d}\zeta } \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s}+t)\Vert ^2_{\mathbb {V}'}\textrm{d}\xi \nonumber \\&\qquad +\lim _{t\rightarrow -\infty }e^{ct}\int _{-\infty }^{0} e^{\int _{0}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s}+t)}\mathcal {Z}_{\delta }(\vartheta _{\zeta +t}\omega )\right) \textrm{d}\zeta }\bigg \{2s_3e^{\sigma (\xi +\mathfrak {s}+t)}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi +t}\omega )\right| \nonumber \\&\qquad +2s_6 e^{2\sigma (\xi +\mathfrak {s}+t)}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi +t}\omega )\right| ^2+s_7\left[ e^{\sigma (\xi +\mathfrak {s}+t)}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi +t}\omega )\right| \right] ^{\frac{2}{1-s_5}}\bigg \}\textrm{d}\xi . \end{aligned}$$

(3.34)

Taking into account of (2.7)–(2.8) and following the same steps applied in the proof of Lemma 2.6, [23], we obtain that there exists a $\mathcal {T}_1=\mathcal {T}_1(\omega )>0$ such that for $\xi \le 0$, $\sigma >0$, $\kappa >0$, $0<\delta \le 1$ with $\sigma \delta \ne 1$ and $t\le -\mathcal {T}_1$,

$$\begin{aligned} -\,2\kappa \int _{0}^{\xi }e^{\sigma (\zeta +\mathfrak {s}+t)}\mathcal {Z}_{\delta }(\vartheta _{\zeta +t}\omega )\textrm{d}\zeta \le c_0 \left( \frac{1}{\sigma }+2-\xi -2t\right) , \end{aligned}$$

(3.35)

where $c_0=\min \left\{ \alpha -\gamma ,\frac{c}{4\kappa }\right\}$.

Let $c_1=\min \left\{ \frac{c}{2},\gamma +\sigma \right\} .$ It implies from (3.34) to (3.35) that, $\sigma >0$, $\kappa >0$, $0<\delta \le 1$ with $\sigma \delta \ne 1$ and $t\le -\mathcal {T}_1$,

$$\begin{aligned}&\lim _{t\rightarrow -\infty }e^{ct}\Vert \mathcal {K}(\mathfrak {s} +t,\vartheta _{t}\omega )\Vert ^2_{\mathbb {H}}\nonumber \\&\quad \le e^{\frac{c_0(2\sigma +1)}{\sigma }}\lim _{t\rightarrow -\infty }e^{\frac{c}{2}t}\int _{-\infty }^{0} e^{\gamma \xi }\bigg \{\frac{4}{\min \{\mu ,\alpha \}}\Vert \varvec{f}(\cdot ,\xi +\mathfrak {s}+t)\Vert ^2_{\mathbb {V}'}\nonumber \\&\qquad +2s_3e^{\sigma (\xi +\mathfrak {s}+t)}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi +t}\omega )\right| \nonumber \\&\qquad +2s_6 e^{2\sigma (\xi +\mathfrak {s}+t)}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi +t}\omega )\right| ^2+s_7\left[ e^{\sigma (\xi +\mathfrak {s}+t)}\left| \mathcal {Z}_{\delta }(\vartheta _{\xi +t}\omega )\right| \right] ^{\frac{2}{1-s_5}}\bigg \}\textrm{d}\xi \nonumber \\&\quad \le 4\frac{e^{\frac{c_0(2\sigma +1)}{\sigma }-\frac{c}{2}\mathfrak {s}}}{\min \{\mu ,\alpha \}}\lim _{t\rightarrow -\infty }e^{\frac{c}{2}t}\int _{-\infty }^{0} e^{\gamma \xi }\Vert \varvec{f}(\cdot ,\xi +t)\Vert ^2_{\mathbb {V}'}\textrm{d}\xi \nonumber \\&\qquad + e^{\frac{c_0(2\sigma +1)}{\sigma }}\lim _{t\rightarrow -\infty }\bigg [2s_3e^{\sigma (\mathfrak {s}+t)}\int _{-\infty }^{t}e^{c_1\xi } \left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| \textrm{d}\xi \nonumber \\&\qquad +2s_6e^{2\sigma (\mathfrak {s}+t)}\int _{-\infty }^{t}e^{c_1\xi }\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^2\textrm{d}\xi \nonumber \\&\qquad +s_7e^{\frac{2\sigma }{1-s_5} (\mathfrak {s}+t)}\int _{-\infty }^{t}e^{c_1\xi }\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{2}{1-s_5}}\textrm{d}\xi \bigg ]. \end{aligned}$$

(3.36)

Using (2.9), one can easily find that

$$\begin{aligned}&\int _{-\infty }^{0}e^{c_1\xi } \left\{ \left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| +\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^2+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{2}{1-s_5}}\right\} \textrm{d}\xi <\infty . \end{aligned}$$

(3.37)

Taking (3.1) and (3.37) into account, along with (3.36), we find that $\lim _{t\rightarrow -\infty }e^{ct}\Vert \mathcal {K}(\mathfrak {s} +t,\vartheta _{t}\omega )\Vert ^2_{\mathbb {H}}=0.$ Also, in the case when $\sigma =0$ or $\kappa =0$ or $\sigma \delta =1$, the above convergence holds true. In fact, in these particular cases, the proof is easier. $\square$

Theorem 3.9

Under the assumptions of Lemma 3.4, there exists a unique $\mathfrak {D}$-pullback random attractor $\mathscr {A}=\{\mathscr {A}(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R}, \omega \in \Omega \}\in \mathfrak {D},$ for the continuous cocycle $\Phi$ associated with the system (2.12) in $\mathbb {H}$.

Proof

The proof follows from Corollary 3.7 ($\mathfrak {D}$-pullback asymptotic compactness of $\Phi$), Lemma 3.8 (existence of $\mathfrak {D}$-pullback random absorbing set) and the abstract theory given in [63, Theorem 2.23]. $\square$

4 Pullback Random Attractors for Wong–Zakai Approximations: The Whole space $\mathbb {R}^d$

In this section, we prove the existence of a unique $\mathfrak {D}$-pullback random attractor for the system (2.12) on the whole space with nonlinear diffusion term $S(\cdot ,\cdot ,\cdot )$ satisfying either Assumptions 1.3 or 1.5.

4.1 Pullback Random Attractors Under Assumption 1.3

Let us fix $\mathcal {O}=\mathbb {R}^d$. In this subsection, we prove the existence of a unique $\mathfrak {D}$-pullback random attractor under Assumptions 1.3 and 3.1 on the deterministic non-autonomous forcing term $\varvec{f}(\cdot ,\cdot )$ and the nonlinear diffusion term $S(\cdot ,\cdot ,\cdot )$, respectively.

Lemma 4.1

For all the cases given in Table 1, let Assumptions 1.3 and 3.1 be satisfied. Then, the solution of (2.12) is continuous in the initial data $\varvec{u}_{\mathfrak {s}}.$

Proof

See the proof of Lemma 3.4. $\square$

4.1.1 Uniform Estimates and $\mathfrak {D}$-pullback Asymptotic Compactness of Solutions

We start by proving uniform estimates for the solutions of (2.12). In order to prove the asymptotic compactness, the method of energy equations was introduced in [4]. Due to the lack of compact Sobolev embeddings in unbounded domains, we prove $\mathfrak {D}$-pullback asymptotic compactness of solutions of the system (2.12) on unbounded domains using the idea given in [4]. The following lemma is a particular case of Lemma 3.5.

Lemma 4.2

Let all the assumptions of Lemma 4.1 be satisfied. Then, for every $0<\delta \le 1$, $\mathfrak {s}\in \mathbb {R},$ $\omega \in \Omega$ and $D=\{D(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R}, \omega \in \Omega \}\in \mathfrak {D},$ there exists $\mathcal {T}=\mathcal {T}(\delta , \mathfrak {s}, \omega , D)>0$ such that for all $t\ge \mathcal {T}$ and $\tau \ge \mathfrak {s}-t$, the solution $\varvec{u}$ of the system (2.12) with $\omega$ replaced by $\vartheta _{-\mathfrak {s}}\omega$ satisfies

$$\begin{aligned}&\Vert \varvec{u}(\tau ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}}\nonumber \\&\quad \le \frac{4}{\min \{\mu ,\alpha \}} \int _{-\infty }^{\tau -\mathfrak {s}} e^{\int _{\tau -\mathfrak {s}}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta } \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}\textrm{d}\xi \nonumber \\&\qquad +\,2s_6\int _{-\infty }^{\tau -\mathfrak {s}} e^{2\sigma (\xi +\mathfrak {s})}e^{\int _{\tau -\mathfrak {s}}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta } \left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^2\textrm{d}\xi , \end{aligned}$$

(4.1)

where $\varvec{u}_{\mathfrak {s}-t}\in D(\mathfrak {s}-t,\vartheta _{-t}\omega )$ and $s_6=\frac{2}{\alpha }\Vert \varvec{h}\Vert ^2_{\mathbb {H}}$.

Proof

When $s_3=s_4=0$, Assumptions (1.7) and (1.6) are the same. Hence, the proof is immediate from Lemma 3.5 by putting $s_3=s_4=0$. $\square$

Now, we establish the $\mathfrak {D}$-pullback asymptotic compactness of the solutions of the system (2.12).

Lemma 4.3

Let all the assumptions of Lemma 4.1 be satisfied. Then for every $0<\delta \le 1$, $\mathfrak {s}\in \mathbb {R},$ $\omega \in \Omega ,$ $D=\{D(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R},\omega \in \Omega \}\in \mathfrak {D}$ and $t_n\rightarrow \infty ,$ $\varvec{u}_{0,n}\in D(\mathfrak {s}-t_n, \vartheta _{-t_{n}}\omega )$, the sequence $\varvec{u}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})$ of solutions of the system (2.12) has a convergent subsequence in $\mathbb {H}$.

Proof

It follows from Lemma 4.2 with $\tau =\mathfrak {s}$ that there exists $\mathcal {T}=\mathcal {T}(\delta ,\mathfrak {s},\omega ,D)>0$ such that for all $t\ge \mathcal {T}$,

$$\begin{aligned} \Vert \varvec{u}(\mathfrak {s};\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}}&\le \frac{4}{\min \{\mu ,\alpha \}} \int _{-\infty }^{0} e^{\int _{0}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta } \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}\textrm{d}\xi \nonumber \\&\quad +2s_6\int _{-\infty }^{0} e^{2\sigma (\xi +\mathfrak {s})}e^{\int _{0}^{\xi }\left( \alpha -2\kappa e^{\sigma (\zeta +\mathfrak {s})}\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta } \left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^2\textrm{d}\xi \nonumber \\&=:M(\mathfrak {s},\omega ), \end{aligned}$$

(4.2)

where $\varvec{u}_{\mathfrak {s}-t}\in D(\mathfrak {s}-t,\vartheta _{-t}\omega ).$ Since $t_n\rightarrow \infty$, there exists $N_0\in \mathbb {N}$ such that $t_n\ge \mathcal {T}$ for all $n\ge N_0$. As, it is given that $\varvec{u}_{0,n}\in D(\mathfrak {s}-t_n, \vartheta _{-t_{n}}\omega )$, (4.2) implies that for all $n\ge N_0$,

$$\begin{aligned} \Vert \varvec{u}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})\Vert ^2_{\mathbb {H}} \le M(\mathfrak {s},\omega ), \end{aligned}$$

and hence $\{\varvec{u}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})\}_{n\ge N_0}\subseteq \mathbb {H}$ is a bounded sequence, which implies that there exists $\tilde{\varvec{u}}\in \mathbb {H}$ and a subsequence (keeping the same label) such that

(4.3)

By the weak lower semicontinuous property of norms and (4.3), we get

$$\begin{aligned} \Vert \tilde{\varvec{u}}\Vert _{\mathbb {H}}\le \liminf _{n\rightarrow \infty }\Vert \varvec{u}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})\Vert _{\mathbb {H}}. \end{aligned}$$

(4.4)

In order to get the desired result, we have to prove that $\varvec{u}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})\rightarrow \tilde{\varvec{u}}$ in $\mathbb {H}$ strongly, that is, we only need to show that

$$\begin{aligned} \Vert \tilde{\varvec{u}}\Vert _{\mathbb {H}}\ge \limsup _{n\rightarrow \infty }\Vert \varvec{u}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})\Vert _{\mathbb {H}}. \end{aligned}$$

(4.5)

The method of energy equations introduced in [4] will help us to prove (4.5). Applying similar arguments as in [37, Theorem 3.4 and Lemma 5.4] (we are not repeating the calculations here), one can obtain (4.5), which completes the proof. $\square$

4.1.2 Existence of $\mathfrak {D}$-pullback Random Attractors

Here, we start with the proof of the existence of $\mathfrak {D}$-pullback random absorbing set in $\mathbb {H}$ for the system (2.12). Finally, we prove the main result of this subsection, that is, the existence of a unique $\mathfrak {D}$-pullback random attractor for the system (2.12).

Lemma 4.4

Under the assumptions of Lemma 4.1, there exists a closed measurable $\mathfrak {D}$-pullback random absorbing set $\widetilde{\mathcal {K}}=\{\widetilde{\mathcal {K}}(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R}, \omega \in \Omega \}\in \mathfrak {D}$ for the continuous cocycle $\Phi$ associated with the system (2.12).

Proof

When $s_3=s_4=0$, assumptions (1.7) and (1.6) are the same. Hence, the proof is same as the proof of Lemma 3.8 by putting $s_3=s_4=0$. $\square$

Now, we are able to provide the main results of this section.

Theorem 4.5

Under the assumptions of Lemma 4.1, there exists a unique $\mathfrak {D}$-pullback random attractor $\widetilde{\mathscr {A}}=\{\widetilde{\mathscr {A}}(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R}, \omega \in \Omega \}\in \mathfrak {D},$ for the the continuous cocycle $\Phi$ associated with the system (2.12) in $\mathbb {H}$.

Proof

The proof follows from Lemma 4.3 ($\mathfrak {D}$-pullback asymptotic compactness of $\Phi$), Lemma 4.4 (existence of $\mathfrak {D}$-pullback random absorbing set) and the abstract theory given in [63, Theorem 2.23]. $\square$

Remark 4.6

It is remarkable to note that if we replace (1.5) by

$$\begin{aligned} |\left( \mathcal {S}(\varvec{u})-\mathcal {S}(\varvec{v}),\varvec{w}\right) |&\le s_2\Vert \varvec{u}-\varvec{v}\Vert _{\mathbb {H}}\Vert \varvec{w}\Vert _{\mathbb {H}}, \ \text { for all }\ \varvec{u},\varvec{v}\in \mathbb {V}\text { and }\varvec{w}\in \mathbb {H}, \end{aligned}$$

then we can also include the case $2\beta \mu =1$ for $d=r=3$ in Lemma 3.4 and hence our main result of Sect. 3 and this subsection, that is, Theorems 3.9 and 4.5 hold true for the case $d=r=3$ with $2\beta \mu =1$ also.

4.2 Pullback random attractors under Assumption 1.5

Let us fix $\mathcal {O}=\mathbb {R}^3$. In this subsection, we prove the existence of a unique $\mathfrak {D}$-pullback random attractor under Assumption 1.5 on nonlinear diffusion term $S(\cdot ,\cdot ,\cdot )$.

In order to prove the results of this subsection, we need the following assumption on non-autonomous forcing term $\varvec{f}(\cdot ,\cdot )$.

Assumption 4.7

The external forcing term $\varvec{f}\in \textrm{L}^{2}_{\textrm{loc}}(\mathbb {R};\mathbb {H})$ which satisfies

$$\begin{aligned} \lim _{s\rightarrow -\infty }e^{cs}\int _{-\infty }^{0} e^{\alpha \zeta } \Vert \varvec{f}(\cdot ,\zeta +s)\Vert ^2_{\mathbb {H}} \textrm{d}\zeta =0, \end{aligned}$$

(4.6)

for every $c>0$, where $\alpha >0$ is the Darcy coefficient. We also assume that following integration should be finite:

$$\begin{aligned} \int _{-\infty }^{0}\Vert \varvec{f}(\cdot ,\zeta +s)\Vert ^2_{\mathbb {H}}\textrm{d}\zeta + \int _{-\infty }^{0}e^{\alpha \zeta }\Vert \varvec{f}(\cdot ,\zeta +s)\Vert ^2_{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}\textrm{d}\zeta <\infty , \end{aligned}$$

(4.7)

where $\alpha >0$ is the Darcy coefficient.

Remark 4.8

The following consequence of (4.7) use in the sequel:

$$\begin{aligned} \lim _{k\rightarrow \infty }\int _{-\infty }^{0}\int \limits _{|x|\ge k} e^{\alpha \xi }|\varvec{f}(x,\xi +\mathfrak {s})|^2\textrm{d}x\textrm{d}\xi =0, \ \ \ \text { for all }\ \mathfrak {s}\in \mathbb {R}. \end{aligned}$$

(4.8)

where $\alpha >0$ is the Darcy coefficient.

Lemma 4.9

For both the cases given in Table 1 for $d=3$ including $d=r=3$ with $2\beta \mu =1$, let Assumptions 1.5 and 4.7 be satisfied. Then, the solution of (2.12) is continuous in the initial data $\varvec{u}_{\mathfrak {s}}.$

Proof

Let $\varvec{u}_{1}(\cdot )$ and $\varvec{u}_{2}(\cdot )$ be two solutions of (2.12), then $\mathfrak {X}(\cdot )=\varvec{u}_{1}(\cdot )-\varvec{u}_{2}(\cdot )$ with $\mathfrak {X}|_{t=\mathfrak {s}}=\varvec{u}_{1}|_{t=\mathfrak {s}}-\varvec{u}_{2}|_{t=\mathfrak {s}}$ satisfies

$$\begin{aligned} \frac{\textrm{d}\mathfrak {X}}{\textrm{d}t}&=-\mu \textrm{A}\mathfrak {X}-\alpha \mathfrak {X}-\left\{ \textrm{B}\big (\varvec{u}_{1}\big )-\textrm{B}\big (\varvec{u}_{2}\big )\right\} -\beta \left\{ \mathcal {C}\big (\varvec{u}_1\big )-\mathcal {C}\big (\varvec{u}_2\big )\right\} \nonumber \\&\quad +\left[ S(t,x,\varvec{u}_1)-S(t,x,\varvec{u}_2)\right] \mathcal {Z}_{\delta }(\vartheta _t\omega ), \end{aligned}$$

(4.9)

in $\mathbb {V}'+\widetilde{\mathbb {L}}^{\frac{r+1}{r}}$. Taking the inner product with $\mathfrak {X}(\cdot )$ to the Eq. (3.3), we obtain

$$\begin{aligned} \frac{1}{2}\frac{\textrm{d}}{\textrm{d}t} \Vert \mathfrak {X}(t)\Vert ^2_{\mathbb {H}}&=-\mu \Vert \nabla \mathfrak {X}(t)\Vert ^2_{\mathbb {H}} - \alpha \Vert \mathfrak {X}(t)\Vert ^2_{\mathbb {H}} -\left\langle \textrm{B}\big (\varvec{u}_1(t)\big )-\textrm{B}\big (\varvec{u}_2(t)\big ), \mathfrak {X}(t)\right\rangle \nonumber \\&\quad -\beta \left\langle \mathcal {C}\big (\varvec{u}_1(t)\big )-\mathcal {C}\big (\varvec{u}_2(t)\big ),\mathfrak {X}(t)\right\rangle \nonumber \\&\quad + \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\left\langle S(t,x,\varvec{u}_1(t))-S(t,x,\varvec{u}_2(t)),\mathfrak {X}(t)\right\rangle , \end{aligned}$$

(4.10)

for a.e. $t\in [\mathfrak {s},\mathfrak {s}+T] \text { with } T>0$. By the locally Lipschitz continuity of the nonlinear diffusion term (cf. Assumption 1.5), we get

$$\begin{aligned}&\mathcal {Z}_{\delta }(\vartheta _{t}\omega )\left\langle S(t,x,\varvec{u}_1)-S(t,x,\varvec{u}_2),\mathfrak {X}\right\rangle \nonumber \\&\quad \le \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert S(t,x,\varvec{u}_1)-S(t,x,\varvec{u}_2)\Vert _{\mathbb {H}}\Vert \mathfrak {X}\Vert _{\mathbb {H}}\le C \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \mathfrak {X}\Vert ^2_{\mathbb {H}}. \end{aligned}$$

(4.11)

From (2.5), we have

$$\begin{aligned} \begin{aligned} -\beta \left\langle \mathcal {C}\big (\varvec{u}_1\big )-\mathcal {C}\big (\varvec{u}_2\big ),\mathfrak {X}\right\rangle \le -\frac{\beta }{2}\Vert |\varvec{u}_1|^{\frac{r-1}{2}}\mathfrak {X} \Vert ^2_{\mathbb {H}} - \frac{\beta }{2}\Vert |\varvec{u}_2|^{\frac{r-1}{2}}\mathfrak {X} \Vert ^2_{\mathbb {H}}\le 0. \end{aligned} \end{aligned}$$

(4.12)

The nonlinear term $\left| \left\langle \textrm{B}\big (\varvec{u}_1\big )-\textrm{B}\big (\varvec{u}_2\big ), \mathfrak {X}\right\rangle \right|$ can be estimated using (2.3), Lemmas 2.9 and 2.11 as

$$\begin{aligned} \begin{aligned} \left| \left\langle \text {B}\big (\varvec{u}_1\big )-\text {B}\big (\varvec{u}_2\big ), \mathfrak {X}\right\rangle \right| \le {\left\{ \begin{array}{ll} \frac{1}{2\beta }\Vert \nabla \mathfrak {X}\Vert ^2_{\mathbb {H}}+\frac{\beta }{2}\Vert |\varvec{u}_1|\mathfrak {X}\Vert ^2_{\mathbb {H}}, &{}{} \text{ for } r=3,\\ \frac{\mu }{4}\Vert \nabla \mathfrak {X}\Vert _{\mathbb {H}}^2+\frac{\beta }{2}\Vert |\varvec{u}_1|^{\frac{r-1}{2}}\mathfrak {X} \Vert ^2_{\mathbb {H}}+C\Vert \mathfrak {X}\Vert ^2_{\mathbb {H}} \ \ \ {} &{}{} \text{ for } r>3. \end{array}\right. } \end{aligned} \end{aligned}$$

(4.13)

Combining (4.10)–(4.13), we get for $r\ge 3$ ($r>3$ with any $\beta ,\mu >0$ and $r=3$ with $2\beta \mu \ge 1$),

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t} \Vert \mathfrak {X}(t)\Vert ^2_{\mathbb {H}} \le C\left[ \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| +1\right] \Vert \mathfrak {X}(t)\Vert ^2_{\mathbb {H}},\ \ \ \text { for a.e. }\ \ \ t\in [\mathfrak {s},\mathfrak {s}+T]. \end{aligned}$$

(4.14)

Hence, we conclude the proof by applying Gronwall’s inequality to (4.14). $\square$

Next, we prove the existence of $\mathfrak {D}$-pullback random absorbing set for continuous cocycle $\Phi$.

Lemma 4.10

Let all the assumptions of Lemma 4.9 be satisfied. Then for every $0<\delta \le 1$, $\mathfrak {s}\in \mathbb {R},$ $\omega \in \Omega$ and $D=\{D(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R}, \omega \in \Omega \}\in \mathfrak {D},$ there exists $\mathscr {T}=\mathscr {T}(\delta , \mathfrak {s}, \omega , D)>0$ such that for all $t\ge \mathscr {T}$ and $\tau \in [\mathfrak {s}-1,\mathfrak {s}]$, the solution $\varvec{u}$ of the system (2.12) with $\omega$ replaced by $\vartheta _{-\mathfrak {s}}\omega$ satisfies

$$\begin{aligned}&\Vert \varvec{u}(\tau ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}}+\int _{\mathfrak {s}-t}^{\tau }e^{\alpha (\xi -\tau )}\bigg [\Vert \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}} \nonumber \\&\quad +\Vert \nabla \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}}+\Vert \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\bigg ]\textrm{d}\xi \nonumber \\&\quad \le \widetilde{M} \int _{-\infty }^{\tau -\mathfrak {s}}e^{\alpha (\xi +\mathfrak {s}-\tau )}\left[ \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {H}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\right] \textrm{d}\xi , \end{aligned}$$

(4.15)

where $\varvec{u}_{\mathfrak {s}-t}\in D(\mathfrak {s}-t,\vartheta _{-t}\omega )$ and $\widetilde{M}$ is a positive constant independent of $\tau , \mathfrak {s}, \omega$ and D. Moreover, for $2<k<\infty$, there exists $\widehat{\mathscr {T}}=\widehat{\mathscr {T}}(\delta ,\mathfrak {s}, \omega , D,k)>0$ such that for all $t\ge \widehat{\mathscr {T}}$ and $\tau \in [\mathfrak {s}-1,\mathfrak {s}]$, the solution $\varvec{u}$ of the system (2.12) with $\omega$ replaced by $\vartheta _{-\mathfrak {s}}\omega$ satisfies

$$\begin{aligned}&\int _{\mathfrak {s}-t}^{\tau }e^{\alpha (\zeta -\tau )} \Vert \varvec{u}(\zeta ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^k_{\mathbb {H}}\textrm{d}\zeta \nonumber \\&\quad \le \widehat{M} \bigg [\int _{-\infty }^{\tau -\mathfrak {s}}e^{\frac{2\alpha (k-1)}{k^2}\xi }\left[ \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {H}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\right] \textrm{d}\xi \bigg ]^{\frac{k}{2}}, \end{aligned}$$

(4.16)

where $\varvec{u}_{\mathfrak {s}-t}\in D(\mathfrak {s}-t,\vartheta _{-t}\omega )$ and $\widehat{M}$ is a positive constant independent of $\tau , \mathfrak {s}, \omega$ and D.

Finally, for $\mathfrak {s}\in \mathbb {R}$ and $t>0$, we obtain

$$\begin{aligned}&\int _{\mathfrak {s}-t}^{\mathfrak {s}} (\zeta -\mathfrak {s}+t) e^{\alpha (\zeta -\mathfrak {s})}\Vert \varvec{u}(\zeta ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{3(r+1)}}\textrm{d}\zeta \nonumber \\&\quad \le C(t+1)e^{-\alpha t}\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^2_{\mathbb {H}}+C(t+1)\int _{-\infty }^{0}e^{\alpha \zeta }\bigg [ \Vert \varvec{f}(\zeta +\mathfrak {s})\Vert ^2_{\mathbb {H}} +\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right| ^{\frac{r+1}{r+1-q}} \nonumber \\&\qquad +\,\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right| ^{\frac{2(r+1)}{3r+3-2q}} +\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right| ^{2}\bigg ], \end{aligned}$$

(4.17)

and

$$\begin{aligned}&\lim _{t\rightarrow \infty }\int _{\mathfrak {s}-t}^{\mathfrak {s}}e^{\alpha (\xi -\mathfrak {s})}\Vert \nabla \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^6_{\mathbb {H}} \nonumber \\&\quad \le C\bigg [\int _{-\infty }^{0}e^{\frac{10\alpha }{36}\zeta }\bigg \{\Vert \varvec{f}(\zeta +\mathfrak {s})\Vert ^2_{\mathbb {H}} +\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right| ^{\frac{2(r+1)}{3r+3-2q}} +\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right| ^{\frac{r+1}{r+1-q}}\nonumber \\&\qquad +\,\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right| ^{2}\bigg \}\textrm{d}\zeta \bigg ]^{3}. \end{aligned}$$

(4.18)

Proof

From the first equation of the system (2.12), we obtain

$$\begin{aligned}&\frac{1}{2}\frac{\textrm{d}}{\textrm{d}t} \Vert \varvec{u}\Vert ^2_{\mathbb {H}} +\mu \Vert \nabla \varvec{u}\Vert ^2_{\mathbb {H}} + \alpha \Vert \varvec{u}\Vert ^2_{\mathbb {H}} + \beta \Vert \varvec{u}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\nonumber \\&\quad \le \Vert \varvec{f}\Vert _{\mathbb {H}}\Vert \varvec{u}\Vert _{\mathbb {H}}+\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \int _{\mathbb {R}^d}\left( |\mathcal {S}_1(t,x)||\varvec{u}|^q+|\mathcal {S}_2(t,x)||\varvec{u}|\right) \textrm{d}x\nonumber \\&\quad \le \frac{\alpha }{4}\Vert \varvec{u}\Vert ^2_{\mathbb {H}}+\frac{\beta }{2}\Vert \varvec{u}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}+M_1\Vert \varvec{f}\Vert ^2_{\mathbb {H}}+M_2\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{\frac{r+1}{r+1-q}}\Vert \mathcal {S}_1(t)\Vert ^{\frac{r+1}{r+1-q}}_{\mathbb {L}^{\frac{r+1}{r+1-q}}(\mathbb {R}^3)}\nonumber \\&\qquad +\,M_3\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{2}\Vert \mathcal {S}_2(t)\Vert ^{2}_{\mathbb {L}^{2}(\mathbb {R}^3)}, \end{aligned}$$

(4.19)

where we have used (2.1), (1.8), Lemmas 2.9 and 2.11, and $M_1, M_2$ and $M_3$ are positive constants independent of $\mathfrak {s},\tau , \omega$ and D. From (4.19), we find

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t} \Vert \varvec{u}\Vert ^2_{\mathbb {H}}+ \alpha \Vert \varvec{u}\Vert ^2_{\mathbb {H}}+\min \left\{ \frac{\alpha }{2},2\mu ,\beta \right\} \left[ \Vert \varvec{u}\Vert ^2_{\mathbb {H}}+\Vert \nabla \varvec{u}\Vert ^2_{\mathbb {H}} + \Vert \varvec{u}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\right] \nonumber \\&\quad \le M_1\Vert \varvec{f}\Vert ^2_{\mathbb {H}}+M_2\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{\frac{r+1}{r+1-q}}\Vert \mathcal {S}_1(t)\Vert ^{\frac{r+1}{r+1-q}}_{\mathbb {L}^{\frac{r+1}{r+1-q}}(\mathbb {R}^3)}+M_3\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{2}\Vert \mathcal {S}_2(t)\Vert ^{2}_{\mathbb {L}^{2}(\mathbb {R}^3)}. \end{aligned}$$

(4.20)

Applying variation of constant formula to (4.20) and replacing $\omega$ by $\vartheta _{-\mathfrak {s}}\omega$, we have

$$\begin{aligned}&\Vert \varvec{u}(\tau ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}}+\min \left\{ \frac{\alpha }{2},2\mu ,\beta \right\} \int _{\mathfrak {s}-t}^{\tau }e^{\alpha (\xi -\tau )}\bigg [\Vert \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}} \nonumber \\&\quad +\Vert \nabla \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}}+\Vert \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\bigg ]\textrm{d}\xi \nonumber \\&\quad \le e^{\alpha (\mathfrak {s}-t-\tau )}\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^2_{\mathbb {H}}+M_4 \int _{-\infty }^{\tau -\mathfrak {s}}e^{\alpha (\xi +\mathfrak {s}-\tau )}\nonumber \\&\quad \left[ \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {H}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\right] \textrm{d}\xi , \end{aligned}$$

(4.21)

where $M_4$ is a positive constant independent of $\tau , \mathfrak {s}, \omega$ and D. Second term of the right hand side of (4.21) is finite due to (2.9) and (4.7). Since $\varvec{u}_{\mathfrak {s}-t}\in D(\mathfrak {s}-t,\vartheta _{-t}\omega )$ and $D\in \mathfrak {D}$, we have

$$\begin{aligned} e^{\alpha (\mathfrak {s}-t-\tau )}\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^2_{\mathbb {H}}\le e^{\alpha (\mathfrak {s}-t-\tau )}\Vert D(\mathfrak {s}-t,\vartheta _{-t}\omega )\Vert ^2_{\mathbb {H}}\rightarrow 0, \end{aligned}$$

(4.22)

as $t\rightarrow \infty$, so that there exists $\mathscr {T}=\mathscr {T}(\delta ,\mathfrak {s},\omega ,D)>0$ such that for all $t\ge \mathscr {T}$,

$$\begin{aligned} e^{\alpha (\mathfrak {s}-t-\tau )}\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^2_{\mathbb {H}}\le \int _{-\infty }^{\tau -\mathfrak {s}}e^{\alpha (\xi +\mathfrak {s}-\tau )}\left[ \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {H}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\right] \textrm{d}\xi . \end{aligned}$$

(4.23)

From (4.21) along with (4.23), we obtain (4.15). Furthermore, from (4.21), we get for any $2<k<\infty$,

$$\begin{aligned} \int _{\mathfrak {s}-t}^{\tau }&e^{\alpha (\zeta -\tau )} \Vert {\varvec{u}}(\zeta ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^k_{\mathbb {H}}\text {d}\zeta \nonumber \\ {}&\quad \le \int _{\mathfrak {s}-t}^{\tau }e^{\alpha (\zeta -\tau )} \bigg [e^{\alpha (\mathfrak {s}-t-\zeta )}\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^2_{\mathbb {H}}\nonumber \\ {}&\qquad +\,M_4 \int _{-t}^{\zeta -\mathfrak {s}}e^{\alpha (\xi +\mathfrak {s}-\zeta )}\left[ \Vert {\varvec{f}}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {H}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\right] \text {d}\xi \bigg ]^{\frac{k}{2}}\text {d}\zeta \nonumber \\ {}&\quad \le C \int _{\mathfrak {s}-t}^{\tau }e^{\alpha (\zeta -\tau )} e^{\frac{\alpha k(\mathfrak {s}-t-\zeta )}{2}}\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^k_{\mathbb {H}}\text {d}\zeta \nonumber \\ {}&\qquad +\,C \int _{\mathfrak {s}-t}^{\tau }e^{\alpha (\zeta -\tau )}\bigg [\int _{-t}^{\zeta -\mathfrak {s}}e^{\alpha (\xi +\mathfrak {s}-\zeta )}\nonumber \\ {}&\quad \qquad \times \left[ \Vert {\varvec{f}}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {H}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\right] \text {d}\xi \bigg ]^{\frac{k}{2}}\text {d}\zeta \nonumber \\ {}&\quad \le C \left[ e^{-\frac{\alpha (k-1)}{k}t}\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^k_{\mathbb {H}}\right] e^{-\alpha \tau +\frac{\alpha (k-1)}{k}\mathfrak {s}}\int _{\mathfrak {s}-t}^{\tau }e^{\frac{\alpha \zeta }{k}}\text {d}\zeta +C e^{-\alpha \tau +\frac{\alpha (k-1)}{k}\mathfrak {s}}\nonumber \\ {}&\qquad \times \! \int _{\mathfrak {s}\!-t}^{\tau }e^{\frac{\alpha \zeta }{k}}\bigg [\int _{-t}^{\zeta \!-\mathfrak {s}}e^{\frac{2\alpha (k\!-1)}{k^2}\xi }\left[ \Vert {\varvec{f}}(\cdot ,\xi \!+\mathfrak {s})\Vert ^2_{\mathbb {H}}\!+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\right] \text {d}\xi \bigg ]^{\frac{k}{2}}\text {d}\zeta \nonumber \\ {}&\quad \le C e^{-\alpha (\tau -\mathfrak {s})}\int _{-\infty }^{\tau -\mathfrak {s}}e^{\frac{\alpha \zeta }{k}}\text {d}\zeta \times \bigg \{e^{-\frac{\alpha (k-1)}{k}t}\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^k_{\mathbb {H}}\nonumber \\ {}&\qquad +\,\bigg [\int _{-\infty }^{\tau -\mathfrak {s}}e^{\frac{2\alpha (k-1)}{k^2}\xi }\left[ \Vert {\varvec{f}}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {H}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\right] \text {d}\xi \bigg ]^{\frac{k}{2}}\bigg \}. \end{aligned}$$

(4.24)

Second term of the right hand side of (4.24) is finite due to (2.9) and (4.7). Since $\varvec{u}_{\mathfrak {s}-t}\in D(\mathfrak {s}-t,\vartheta _{-t}\omega )$ and $D\in \mathfrak {D}$, from (4.22), we have that there exists $\widehat{\mathscr {T}}=\widehat{\mathscr {T}}(\delta ,\mathfrak {s},\omega ,D,k)>0$ such that for all $t\ge \widehat{\mathscr {T}}$,

$$\begin{aligned}&e^{-\frac{\alpha (k-1)}{k}t}\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^k_{\mathbb {H}}\!\le \bigg [\int _{-\infty }^{\tau -\mathfrak {s}}e^{\frac{2\alpha (k-1)}{k^2}\xi }\left[ \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {H}}\!+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}}\!+\!\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\right] \textrm{d}\xi \bigg ]^{\frac{k}{2}}, \end{aligned}$$

which along with (4.24) completes the proof of (4.16).

Our next aim is to prove the (4.17). From the Gagliardo-Nirenberg-Sobolev inequality (Theorem 1, Section 5.6.1, [17]), we have

$$\begin{aligned} \Vert \varvec{u}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{3(r+1)}}=\Vert |\varvec{u}|^{\frac{r+1}{2}}\Vert _{\textrm{L}^6(\mathbb {R}^3)}\le C \Vert \nabla |\varvec{u}|^{\frac{r+1}{2}}\Vert ^2_{\mathbb {L}^{2}(\mathbb {R}^3)}. \end{aligned}$$

(4.25)

Using Lemmas 2.9 and 2.11, we obtain

$$\begin{aligned} |b(\varvec{u},\varvec{u},\textrm{A}\varvec{u})|&\le \Vert |\varvec{u}|\nabla \varvec{u}|\Vert _{\mathbb {H}}\Vert \textrm{A}\varvec{u}\Vert _{\mathbb {H}}\nonumber \\&\le {\left\{ \begin{array}{ll} \frac{\mu }{6}\Vert \textrm{A}\varvec{u}\Vert ^2_{\mathbb {H}}+\frac{\beta }{2}\Vert |\varvec{u}|^{\frac{r-1}{2}}\nabla \varvec{u}\Vert ^2_{\mathbb {H}}+C\Vert \nabla \varvec{u}\Vert ^2_{\mathbb {H}}, &{} \text { for } r>3,\\ \frac{\mu }{6}\Vert \textrm{A}\varvec{u}\Vert ^2_{\mathbb {H}}+\frac{1}{2\mu }\Vert |\varvec{u}|\nabla \varvec{u}\Vert ^2_{\mathbb {H}}, &{} \text { for } r=3. \end{array}\right. } \end{aligned}$$

(4.26)

From the first equation of the system (2.12), we obtain for some $M_4,M_5, M_6>0$

$$\begin{aligned} \frac{1}{2}\frac{\text {d}}{\text {d}t}&\Vert \nabla \varvec{u}\Vert ^2_{\mathbb {H}}+\mu \Vert \text {A}\varvec{u}\Vert ^2_{\mathbb {H}}+\alpha \Vert \nabla \varvec{u}\Vert ^2_{\mathbb {H}}+\beta \Vert |\varvec{u}|^{\frac{{\varvec{r}}-1}{2}}\nabla \varvec{u}\Vert ^2_{\mathbb {H}}\nonumber \\&\quad +\,4\beta \left[ \frac{{\varvec{r}}-1}{(\textit{r}+1)^2}\right] \Vert \nabla |\varvec{u}|^{\frac{{\varvec{r}}+1}{2}}||^2_{\mathbb {H}}\nonumber \\&\quad \le \Vert {\varvec{f}}\Vert _{\mathbb {H}}\Vert \text {A}\varvec{u}\Vert _{\mathbb {H}}+\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \int _{\mathbb {R}^3}\left( |\mathcal {S}_1(t,x)||\varvec{u}|^{\textit{q}-1}|\text {A}\varvec{u}|\right. \nonumber \\&\quad \left. \qquad +\,|\mathcal {S}_2(t,x)||\text {A}\varvec{u}|\right) \text {d}x +|b(\varvec{u},\varvec{u},\text {A}\varvec{u})| \nonumber \\&\quad \le \Vert {\varvec{f}}\Vert _{\mathbb {H}}\Vert \text {A}\varvec{u}\Vert _{\mathbb {H}}+\left| \mathcal {Z}_{\delta }(\vartheta _{{\varvec{t}}}\omega )\right| \Vert \text {A}\varvec{u}\Vert _{\mathbb {H}}\nonumber \\&\qquad \times \left[ \Vert \mathcal {S}_1({\varvec{t}})\Vert _{\mathbb {L}^{\frac{6(r+1)}{3r+5-2q}}(\mathbb {R}^3)}\Vert \varvec{u}\Vert ^{q-1}_{\widetilde{\mathbb {L}}^{3(r+1)}}+\Vert \mathcal {S}_2(t)\Vert _{\mathbb {L}^{2}(\mathbb {R}^3)}\right] \nonumber \\&\quad \quad +\,|b(\varvec{u},\varvec{u},\text {A}\varvec{u})| \nonumber \\&\quad \le \Vert {\varvec{f}}\Vert _{\mathbb {H}}\Vert \text {A}\varvec{u}\Vert _{\mathbb {H}}+C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \text {A}\varvec{u}\Vert _{\mathbb {H}}\nonumber \\&\qquad \times \left[ \Vert \mathcal {S}_1(t)\Vert _{\mathbb {L}^{\frac{6(r+1)}{3r+5-2q}}(\mathbb {R}^3)}\Vert \nabla |\varvec{u}|^{\frac{r+1}{2}}\Vert ^{\frac{2(q-1)}{r+1}}_{\mathbb {H}}+\Vert \mathcal {S}_2(t)\Vert _{\mathbb {L}^{2}(\mathbb {R}^3)}\right] \nonumber \\&\quad \quad +\,|b(\varvec{u},\varvec{u},\text {A}\varvec{u})| \nonumber \\&\quad \le \frac{\mu }{6}\Vert \text {A}\varvec{u}\Vert ^2_{\mathbb {H}}+2\beta \left[ \frac{r-1}{(r+1)^2}\right] \Vert \nabla |\varvec{u}|^{\frac{r+1}{2}}\Vert ^{2}_{\mathbb {H}}+M_4\Vert {\varvec{f}}\Vert ^2_{\mathbb {H}}+|b(\varvec{u},\varvec{u},\text {A}\varvec{u})|\nonumber \\&\qquad +\,M_5\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{\frac{2(r+1)}{3r+3-2q}}\Vert \mathcal {S}_1(t)\Vert ^{\frac{2(r+1)}{3r+3-2q}}_{\mathbb {L}^{\frac{6(r+1)}{3r+5-2q}}(\mathbb {R}^3)}+M_6\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{2}\Vert \mathcal {S}_2(t)\Vert ^{2}_{\mathbb {L}^{2}(\mathbb {R}^3)}, \end{aligned}$$

(4.27)

where we have used the equality (2.6), condition (1.8) of Assumption 1.5, (4.25), and Lemmas 2.9 and 2.11.

Combing (4.25)–(4.27), for $\varvec{u}(\zeta ):=\varvec{u}(\zeta ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})$, we reach at

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}\zeta }\Vert \nabla \varvec{u}(\zeta )\Vert ^2_{\mathbb {H}}+2\alpha \Vert \nabla \varvec{u}(\zeta )\Vert ^2_{\mathbb {H}}+\Vert \varvec{u}(\zeta )\Vert _{\widetilde{\mathbb {L}}^{3(r+1)}}^{r+1} \nonumber \\&\quad \le C\Vert \nabla \varvec{u}(\zeta )\Vert ^2_{\mathbb {H}}+ M_4\Vert \varvec{f}(\zeta )\Vert ^2_{\mathbb {H}} +M_5\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{\frac{2(r+1)}{3r+3-2q}}\Vert \mathcal {S}_1(t)\Vert ^{\frac{2(r+1)}{3r+3-2q}}_{\mathbb {L}^{\frac{6(r+1)}{3r+5-2q}}(\mathbb {R}^3)} \nonumber \\&\quad \quad +M_6\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{2}\Vert \mathcal {S}_2(t)\Vert ^{2}_{\mathbb {L}^{2}(\mathbb {R}^3)}, \end{aligned}$$

(4.28)

for a.e. $\zeta >\mathfrak {s}-t$, which implies that

$$\begin{aligned}&(\zeta -\mathfrak {s}+t)\frac{\textrm{d}}{\textrm{d}\zeta }\left[ e^{\alpha (\zeta -\mathfrak {s})}\Vert \nabla \varvec{u}(\zeta )\Vert ^2_{\mathbb {H}}\right] +(\zeta -\mathfrak {s}+t)e^{\alpha (\zeta -\mathfrak {s})}\Vert \varvec{u}(\zeta )\Vert _{\widetilde{\mathbb {L}}^{3(r+1)}}^{r+1} \nonumber \\&\quad \le C(\zeta -\mathfrak {s}+t)e^{\alpha (\zeta -\mathfrak {s})}\bigg [\Vert \nabla \varvec{u}(\zeta )\Vert ^2_{\mathbb {H}}+ M_4\Vert \varvec{f}(\zeta )\Vert ^2_{\mathbb {H}} \nonumber \\&\qquad +M_5\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{\frac{2(r+1)}{3r+3-2q}}\Vert \mathcal {S}_1(t)\Vert ^{\frac{2(r+1)}{3r+3-2q}}_{\mathbb {L}^{\frac{6(r+1)}{3r+5-2q}}(\mathbb {R}^3)} +M_6\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{2}\Vert \mathcal {S}_2(t)\Vert ^{2}_{\mathbb {L}^{2}(\mathbb {R}^3)}\bigg ], \end{aligned}$$

(4.29)

for a.e. $\zeta \ge \mathfrak {s}-t$. We know that

$$\begin{aligned} (\zeta -s+t)\frac{\textrm{d}}{\textrm{d}\zeta }\bigg [e^{\alpha (\zeta -s)} \Vert \nabla \varvec{u}(\zeta )\Vert ^2_{\mathbb {H}}\bigg ]&=\frac{\textrm{d}}{\textrm{d}\zeta }\bigg [(\zeta -s+t) e^{\alpha (\zeta -s)} \Vert \nabla \varvec{u}(\zeta )\Vert ^2_{\mathbb {H}}\bigg ]\nonumber \\&\quad - e^{\alpha (\zeta -s)}\Vert \nabla \varvec{u}(\zeta )\Vert ^2_{\mathbb {H}}. \end{aligned}$$

(4.30)

From (4.29) and (4.30), we infer

$$\begin{aligned}&\int _{\mathfrak {s}-t}^{\mathfrak {s}}(\zeta -\mathfrak {s}+t)e^{\alpha (\zeta -\mathfrak {s})}\Vert \varvec{u}(\zeta ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert _{\widetilde{\mathbb {L}}^{3(r+1)}}^{r+1} \textrm{d}\zeta \nonumber \\&\quad \le C\int _{\mathfrak {s}-t}^{\mathfrak {s}}(\zeta -\mathfrak {s}+t+1)e^{\alpha (\zeta -\mathfrak {s})}\bigg [\Vert \nabla \varvec{u}(\zeta )\Vert ^2_{\mathbb {H}}+ M_4\Vert \varvec{f}(\zeta )\Vert ^2_{\mathbb {H}}\nonumber \\&\qquad +\,M_5\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{\frac{2(r+1)}{3r+3-2q}}\Vert \mathcal {S}_1(t)\Vert ^{\frac{2(r+1)}{3r+3-2q}}_{\mathbb {L}^{\frac{6(r+1)}{3r+5-2q}}(\mathbb {R}^3)} +M_6\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{2}\Vert \mathcal {S}_2(t)\Vert ^{2}_{\mathbb {L}^{2}(\mathbb {R}^3)}\bigg ] \nonumber \\&\quad \le C(t+1)\int _{\mathfrak {s}-t}^{\mathfrak {s}}e^{\alpha (\zeta -\mathfrak {s})}\bigg [\Vert \nabla \varvec{u}(\zeta )\Vert ^2_{\mathbb {H}}+ \Vert \varvec{f}(\zeta )\Vert ^2_{\mathbb {H}}\nonumber \\&\qquad +\,\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{\frac{2(r+1)}{3r+3-2q}} +\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{2}\bigg ]\nonumber \\&\quad \le C(t+1)e^{-\alpha t}\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^2_{\mathbb {H}}+C(t+1)\int _{-\infty }^{0}e^{\alpha \zeta }\bigg [ \Vert \varvec{f}(\zeta +\mathfrak {s})\Vert ^2_{\mathbb {L}^2(\mathbb {R}^3)} +\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right| ^{\frac{r+1}{r+1-q}} \nonumber \\&\qquad +\,\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right| ^{\frac{2(r+1)}{3r+3-2q}} +\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right| ^{2}\bigg ], \end{aligned}$$

(4.31)

where we have used (4.21) which completes the proof of (4.17).

Our final aim is to establish the inequality (4.18). In view of the variation of constants formula with respect to $\zeta \in (\xi _1,\xi )$ with $s-t<\xi _1\le \xi \le s$, we find from (4.28)

$$\begin{aligned}&\Vert \nabla \varvec{u}(\xi )\Vert ^2_{\mathbb {H}} \nonumber \\&\quad \le e^{\alpha (\xi _1-\xi )}\Vert \nabla \varvec{u}(\xi _1)\Vert ^2_{\mathbb {H}} + \int _{\mathfrak {s}-t}^{\xi }e^{\alpha (\zeta -\xi )}\bigg [\Vert \nabla \varvec{u}(\zeta )\Vert ^2_{\mathbb {H}}+ \Vert \varvec{f}(\zeta )\Vert ^2_{\mathbb {H}} +\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{\frac{2(r+1)}{3r+3-2q}}\nonumber \\&\qquad +\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{2}\bigg ]. \end{aligned}$$

(4.32)

Integrating (4.32) from $s-t$ to $\xi$ with respect to $\xi _1$ and using (4.21), we obtain

$$\begin{aligned}&\Vert \nabla \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}} \nonumber \\&\quad \le C\bigg \{\frac{1}{\xi -s+t}+1\bigg \}\bigg [e^{-\alpha (\xi -\mathfrak {s}+t)}\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^2_{\mathbb {H}}+\int _{\mathfrak {s}-t}^{\xi }e^{\alpha (\zeta -\xi )}\bigg \{ \Vert \varvec{f}(\zeta )\Vert ^2_{\mathbb {H}} +\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{\frac{2(r+1)}{3r+3-2q}}\nonumber \\&\qquad +\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{2}\bigg \}\textrm{d}\zeta \bigg ]. \end{aligned}$$

(4.33)

Now from (4.33), we estimate

$$\begin{aligned}&\int _{\mathfrak {s}-t}^{\mathfrak {s}}e^{\alpha (\xi -\mathfrak {s})}\Vert \nabla \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^6_{\mathbb {H}} \nonumber \\&\quad \le C\int _{\mathfrak {s}-t}^{\mathfrak {s}}e^{\alpha (\xi -\mathfrak {s})}\bigg \{\frac{1}{\xi -\mathfrak {s}+t}+1\bigg \}^{3}\bigg [e^{-3\alpha (\xi -\mathfrak {s}+t)}\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^6_{\mathbb {H}}+\bigg (\int _{\mathfrak {s}-t}^{\xi }e^{\alpha (\zeta -\xi )}\bigg \{ \Vert \varvec{f}(\zeta )\Vert ^2_{\mathbb {H}} \nonumber \\&\qquad +\,\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{\frac{2(r+1)}{3r+3-2q}} +\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta -\mathfrak {s}}\omega )\right| ^{2}\bigg \}\textrm{d}\zeta \bigg )^{3}\bigg ]\textrm{d}\xi \nonumber \\&\quad \le C\int _{-t}^{0}e^{\frac{\alpha }{6}\xi }\bigg \{\frac{1}{\xi +t}+1\bigg \}^{3}\textrm{d}\xi \times \bigg [e^{-\frac{5\alpha }{6}t}\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^6_{\mathbb {H}}+\bigg (\int _{-t}^{0}e^{\frac{10\alpha }{36}\zeta }\bigg \{\Vert \varvec{f}(\zeta +\mathfrak {s})\Vert ^2_{\mathbb {H}} \nonumber \\&\qquad +\,\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right| ^{\frac{2(r+1)}{3r+3-2q}} +\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right| ^{2}\bigg \}\textrm{d}\zeta \bigg )^{3}\bigg ]. \end{aligned}$$

(4.34)

Since

$$\begin{aligned} \int _{-t}^{0}e^{\frac{\alpha }{6}\xi }\bigg [\frac{1}{\xi +t}+1\bigg ]^{3}\textrm{d}\xi \rightarrow \frac{6}{\alpha }\ \text { as }\ t\rightarrow \infty , \ \ \text{ for } \text{ all } c>0\text{, } \end{aligned}$$

using (2.9) and the temperedness property of $\varvec{u}_{\mathfrak {s}-t}$, we deduce (4.18), as required. $\square$

Lemma 4.11

Under the assumptions of Lemma 4.9, the continuous cocycle $\Phi$ associated with the system (2.12) possesses a closed measurable $\mathfrak {D}$-pullback random absorbing set $\widehat{\mathcal {K}}\in \mathfrak {D}$ defined for each $\mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$ by $\widehat{\mathcal {K}}(\mathfrak {s},\omega )=\{\varvec{u}\in \mathbb {H}:\Vert \varvec{u}\Vert ^2_{\mathbb {H}}\le \widehat{\mathcal {L}}(\mathfrak {s},\omega )\},$ where

$$\begin{aligned} \widehat{\mathcal {L}}(\mathfrak {s},\omega )=\widetilde{M} \int _{-\infty }^{0}e^{\alpha \xi }\left[ \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {H}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\right] \textrm{d}\xi , \end{aligned}$$

with $\widetilde{M}$ appearing in (4.15).

Proof

Since $\widehat{\mathcal {L}}(\mathfrak {s},\cdot ):\Omega \rightarrow \mathbb {R}$ is $(\mathscr {F},\mathscr {B}(\mathbb {R}))$-measurable for every $\mathfrak {s}\in \mathbb {R}$, $\widehat{\mathcal {K}}(\mathfrak {s},\cdot ):\Omega \rightarrow 2^{\mathbb {H}}$ is a measurable set-valued mapping. Moreover, we have from Lemma 4.10 that for each $\mathfrak {s}\in \mathbb {R}$, $\omega \in \Omega$ and $D\in \mathfrak {D}$, there exists $\mathscr {T}=\mathscr {T}(\delta ,\mathfrak {s},\omega ,D)>0$ such that for all $t\ge \mathscr {T}$,

$$\begin{aligned} \Phi (t,\mathfrak {s}-t,\vartheta _{-t}\omega ,D(\mathfrak {s}-t,\vartheta _{-t}\omega ))=\varvec{u}(\mathfrak {s};\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,D(\mathfrak {s}-t,\vartheta _{-t}\omega ))\subseteq \widehat{\mathcal {K}}(\mathfrak {s},\omega ). \end{aligned}$$

(4.35)

It only remains to show that $\widehat{\mathcal {K}}\in \mathfrak {D}$, that is, for every $c>0$, $\mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$ $\lim \limits _{t\rightarrow -\infty }e^{ct}\Vert \widehat{\mathcal {K}}(\mathfrak {s} +t,\vartheta _{t}\omega )\Vert ^2_{\mathbb {H}}=0.$ For every $c>0$, $\mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$,

$$\begin{aligned}&\lim _{t\rightarrow -\infty }e^{ct}\Vert \widehat{\mathcal {K}}(\mathfrak {s} +t,\vartheta _{t}\omega )\Vert ^2_{\mathbb {H}}=\lim _{t\rightarrow -\infty }e^{ct}\widehat{\mathcal {L}}(\mathfrak {s} +t,\vartheta _{t}\omega )\nonumber \\&\quad =\widetilde{M}\lim _{t\rightarrow -\infty }e^{ct} \int _{-\infty }^{0}e^{\alpha \xi }\left[ \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s}+t)\Vert ^2_{\mathbb {H}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi +t}\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi +t}\omega )\right| ^{2}\right] \textrm{d}\xi . \end{aligned}$$

(4.36)

Let $c_2=\min \{c,\alpha \}.$ Consider,

$$\begin{aligned}&\lim _{t\rightarrow -\infty }e^{ct} \int _{-\infty }^{0}e^{\alpha \xi }\left[ \left| \mathcal {Z}_{\delta }(\vartheta _{\xi +t}\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi +t}\omega )\right| ^{2}\right] \textrm{d}\xi \nonumber \\&\quad \le \lim _{t\rightarrow -\infty } \int _{-\infty }^{t}e^{c_2\xi }\left[ \left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\right] \textrm{d}\xi =0, \end{aligned}$$

(4.37)

where we have used the fact that $\int _{-\infty }^{0}e^{c_2\xi }\left[ \left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\right] \textrm{d}\xi <\infty$ due to (2.9). Hence (4.6), (4.36) and (4.37) imply $\lim \limits _{t\rightarrow -\infty }e^{ct}\Vert \widehat{\mathcal {K}}(\mathfrak {s} +t,\vartheta _{t}\omega )\Vert ^2_{\mathbb {H}}=0,$ as required. $\square$

Next lemma plays a pivotal role to prove the $\mathfrak {D}$-pullback asymptotic compactness of $\Phi$ (see Lemma 4.14 below).

Lemma 4.12

Let all the assumptions of Lemma 4.9 be satisfied. Then, for any $\varvec{u}_{\mathfrak {s}-t}\in D(\mathfrak {s}-t,\vartheta _{-t}\omega ),$ where $D=\{D(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R}, \omega \in \Omega \}\in \mathfrak {D}$, and for any $\eta >0$, $\mathfrak {s}\in \mathbb {R}$, $\omega \in \Omega$ and $0<\delta \le 1$, there exists $\mathscr {T}^*=\mathscr {T}^*(\delta ,\mathfrak {s},\omega ,D,\eta )\ge 1$ and $P^*=P^*(\delta ,\mathfrak {s},\omega ,\eta )>0$ such that for all $t\ge \mathscr {T}^*$ and $\tau \in [\mathfrak {s}-1,\mathfrak {s}]$, the solution of (2.12) with $\omega$ replaced by $\vartheta _{-\mathfrak {s}}\omega$ satisfy

$$\begin{aligned} \int _{|x|\ge P^*}|\varvec{u}(\tau ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t}) |^2\textrm{d}x\le \eta . \end{aligned}$$

(4.38)

Proof

Let $\Psi$ be a smooth function such that $0\le \Psi (s)\le 1$ for $s\in \mathbb {R}^+$ and

$$\begin{aligned} \Psi (s)={\left\{ \begin{array}{ll} 0,\quad \text { for }0\le s\le 1,\\ 1, \quad \text { for } s\ge 2 . \end{array}\right. } \end{aligned}$$

(4.39)

Then, there exists a positive constant C such that $|\Psi '(s)|\le C,$ for all $s\in \mathbb {R}^+$. Formally, taking divergence to the first equation of (1.3), we obtain

$$\begin{aligned} -\,\Delta p=\nabla \cdot \left[ \nabla \cdot \big (\varvec{u}\otimes \varvec{u}\big )\right] +\beta \nabla \cdot \left[ |\varvec{u}|^{r-1}\varvec{u}\right] -\nabla \cdot \varvec{f}-\left[ \nabla \cdot S(t,x,\varvec{u})\right] \mathcal {Z}_{\delta }(\vartheta _{t}\omega ) \end{aligned}$$

which implies that

$$\begin{aligned} p=(-\Delta )^{-1}\left[ \nabla \cdot \left[ \nabla \cdot \big (\varvec{u}\otimes \varvec{u}\big )\right] +\beta \nabla \cdot \left[ |\varvec{u}|^{r-1}\varvec{u}\right] -\nabla \cdot \varvec{f}-\left[ \nabla \cdot S(t,x,\varvec{u})\right] \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right] , \end{aligned}$$

(4.40)

in the weak sense. Taking the inner product of the first equation of (1.3) with $\Psi \left( \frac{|x|^2}{k^2}\right) \varvec{u}$ in $\textrm{L}^2(\mathbb {R}^3;\mathbb {R}^3)$, we have

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \int _{\mathbb {R}^3}\Psi \left( \frac{|x|^2}{k^2}\right) |\varvec{u}|^2\textrm{d}x \nonumber \\&\quad = \mu \int _{\mathbb {R}^3}(\Delta \varvec{u}) \Psi \left( \frac{|x|^2}{k^2}\right) \varvec{u}\textrm{d}x-\alpha \int _{\mathbb {R}^3}\Psi \left( \frac{|x|^2}{k^2}\right) |\varvec{u}|^2\textrm{d}x-b\left( \varvec{u},\varvec{u},\Psi \left( \frac{|x|^2}{k^2}\right) \varvec{u}\right) \nonumber \\&\qquad -\beta \int _{\mathbb {R}^3}\Psi \left( \frac{|x|^2}{k^2}\right) |\varvec{u}|^{r+1}\textrm{d}x-\int _{\mathbb {R}^3}(\nabla p)\Psi \left( \frac{|x|^2}{k^2}\right) \varvec{u}\textrm{d}x+ \int _{\mathbb {R}^3}\varvec{f}(x,t)\Psi \left( \frac{|x|^2}{k^2}\right) \varvec{u}\textrm{d}x \nonumber \\&\qquad +\mathcal {Z}_{\delta }(\vartheta _{t}\omega )\int _{\mathbb {R}^3}\Psi \left( \frac{|x|^2}{k^2}\right) S(t,x,\varvec{u})\varvec{u}\textrm{d}x. \end{aligned}$$

(4.41)

Let us now estimate each term on right hand side of (4.41). Integration by parts, and Lemmas 2.9 and 2.11 help us to obtain

$$\begin{aligned}&\mu \int _{\mathbb {R}^3}(\Delta \varvec{u}) \Psi \left( \frac{|x|^2}{k^2}\right) \varvec{u}\textrm{d}x+\mu \int _{\mathbb {R}^3}|\nabla \varvec{u}|^2 \Psi \left( \frac{|x|^2}{k^2}\right) \textrm{d}x\nonumber \\&\quad = -\mu \int \limits _{k\le |x|\le \sqrt{2}k}\Psi '\left( \frac{|x|^2}{k^2}\right) \frac{2}{k^2}(x\cdot \nabla ) \varvec{u}\cdot \varvec{u}\textrm{d}x\nonumber \\&\quad \le \frac{2\sqrt{2}\mu }{k} \int \limits _{k\le |x|\le \sqrt{2}k}\left| \varvec{u}\right| \left| \Psi '\left( \frac{|x|^2}{k^2}\right) \right| \left| \nabla \varvec{u}\right| \textrm{d}x\le \frac{C}{k} \int _{\mathbb {R}^3}\left| \varvec{u}\right| \left| \nabla \varvec{u}\right| \textrm{d}x\nonumber \\&\quad \le \frac{C}{k} \left( \Vert \varvec{u}\Vert ^2_{\mathbb {H}}+\Vert \nabla \varvec{u}\Vert ^2_{\mathbb {H}}\right) , \end{aligned}$$

(4.42)

and

$$\begin{aligned} -b\left( \varvec{u},\varvec{u},\Psi \left( \frac{|x|^2}{k^2}\right) \varvec{u}\right)&=\int _{\mathbb {R}^3} \Psi '\left( \frac{|x|^2}{k^2}\right) \frac{x}{k^2}\cdot \varvec{u}|\varvec{u}|^2 \textrm{d}x\nonumber \\&= \int \limits _{k\le |x|\le \sqrt{2}k} \Psi '\left( \frac{|x|^2}{k^2}\right) \frac{x}{k^2}\cdot \varvec{u}|\varvec{u}|^2 \textrm{d}x\nonumber \\&\le \frac{\sqrt{2}}{k} \int \limits _{k\le |x|\le \sqrt{2}k} \left| \Psi '\left( \frac{|x|^2}{k^2}\right) \right| |\varvec{u}|^3 \textrm{d}x\le \frac{C}{k}\Vert \varvec{u}\Vert ^3_{\widetilde{\mathbb {L}}^3}\nonumber \\&\le \frac{C}{k}\bigg [\Vert \varvec{u}\Vert ^{2}_{\mathbb {H}}+\Vert \varvec{u}\Vert _{\widetilde{\mathbb {L}}^{r+1}}^{r+1}\bigg ], \end{aligned}$$

(4.43)

where we have used the Gagliardo–Nirenberg–Sobolev inequality also. Using integration by parts, divergence free condition and (4.40), we find

$$\begin{aligned}&-\int _{\mathbb {R}^3}(\nabla p)\Psi \left( \frac{|x|^2}{k^2}\right) \varvec{u}\textrm{d}x=\int _{\mathbb {R}^3}p\Psi '\left( \frac{|x|^2}{k^2}\right) \frac{2}{k^2}(x\cdot \varvec{u})\nonumber \\&\quad \le \frac{C}{k}\bigg [\int \limits _{\mathbb {R}^3}\left| (-\Delta )^{-1}\left[ \nabla \cdot \left[ \nabla \cdot \big (\varvec{u}\otimes \varvec{u}\big )\right] \right] \right| \cdot \left| \varvec{u}\right| \textrm{d}x \nonumber \\&\qquad +\, \int \limits _{\mathbb {R}^3}\left| (-\Delta )^{-1}\left[ \nabla \cdot \left[ |\varvec{u}|^{r-1}\varvec{u}\right] \right] \right| \cdot \left| \varvec{u}\right| \textrm{d}x \nonumber \\&\qquad +\,\int \limits _{\mathbb {R}^3}\left| (-\Delta )^{-1}\left[ \nabla \cdot \varvec{f}\right] \right| \cdot \left| \varvec{u}\right| \textrm{d}x+\int \limits _{\mathbb {R}^3}\left| (-\Delta )^{-1}\left[ \nabla \cdot S(t,x,\varvec{u})\right] \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \cdot \left| \varvec{u}\right| \textrm{d}x \bigg ]\nonumber \\&\quad =: \frac{C}{k}\left[ S_1(r)+S_2(r) +S_3(r)+S_4(r)\right] . \end{aligned}$$

(4.44)

Using Lemma 2.9, Plancherel’s theorem, and Lemmas 2.10 and 2.11, we get

$$\begin{aligned} |S_1(r)|&\le \left\| (-\Delta )^{-1}\left[ \nabla \cdot \left[ \nabla \cdot \big (\varvec{u}\otimes \varvec{u}\big )\right] \right] \right\| _{\mathbb {L}^2(\mathbb {R}^3)}\Vert \varvec{u}\Vert _{\mathbb {H}}\nonumber \\&\le C\Vert \varvec{u}\Vert ^2_{\widetilde{\mathbb {L}}^4}\Vert \varvec{u}\Vert _{\mathbb {H}}\le C\left( \Vert \varvec{u}\Vert ^{2}_{\mathbb {H}}+\Vert \varvec{u}\Vert _{\widetilde{\mathbb {L}}^{r+1}}^{r+1}\right) . \end{aligned}$$

(4.45)

Applying Lemma 2.9, Plancherel’s theorem, [3, Theorem 1.38] (see Remark 2.4 above), and Lemmas 2.10 and 2.11, we obtain

$$\begin{aligned} |S_2(r)|&\le \Vert (-\Delta )^{-1}\left[ \nabla \cdot \left[ |\varvec{u}|^{r-1}\varvec{u}\right] \right] \Vert _{\mathbb {L}^{2}(\mathbb {R}^3)}\Vert \varvec{u}\Vert _{\mathbb {H}} \nonumber \\&\le C \Vert |\varvec{u}|^{r-1}\varvec{u}\Vert _{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}\Vert \varvec{u}\Vert _{\mathbb {H}} \le C \Vert \varvec{u}\Vert ^r_{\widetilde{\mathbb {L}}^{\frac{6r}{5}}}\Vert \varvec{u}\Vert _{\mathbb {H}} \nonumber \\&\le C \Vert \varvec{u}\Vert ^{\frac{(r+1)(3r-5)}{3r+1}}_{\widetilde{\mathbb {L}}^{3(r+1)}}\Vert \varvec{u}\Vert ^{\frac{6(r+1)}{3r+1}}_{\mathbb {H}} \le C\left[ \Vert \varvec{u}\Vert ^{\frac{(r+1)(3r-5)}{3r-4}}_{\widetilde{\mathbb {L}}^{3(r+1)}}+\Vert \varvec{u}\Vert ^{2(r+1)}_{\mathbb {H}}\right] , \end{aligned}$$

(4.46)

$$\begin{aligned} \left| S_3(r)\right|&\le C\Vert (-\Delta )^{-1}\left[ \nabla \cdot \varvec{f}\right] \Vert _{\mathbb {L}^{2}(\mathbb {R}^3)}\Vert \varvec{u}\Vert _{\mathbb {H}} \nonumber \\&\le C \Vert \varvec{f}\Vert _{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}\Vert \varvec{u}\Vert _{\mathbb {H}} \le C \Vert \varvec{f}\Vert ^{2}_{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}+C\Vert \varvec{u}\Vert ^2_{\mathbb {H}}. \end{aligned}$$

(4.47)

Applying Lemma 2.9, Plancherel’s theorem, [3, Theorem 1.38] (see Remark 2.4 above), Gagliardo–Nirenberg–Sobolev inequality and Lemma 2.11, we calculate

$$\begin{aligned}&|S_4(r)|\nonumber \\&\le \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert (-\Delta )^{-1}\left[ \nabla \cdot S(t,x,\varvec{u})\right] \Vert _{\mathbb {L}^{2}}\Vert \varvec{u}\Vert _{\mathbb {H}} \nonumber \\ {}&\le \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert S(t,x,\varvec{u})\Vert _{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}\Vert \varvec{u}\Vert _{\mathbb {H}} \nonumber \\&\le \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \varvec{u}\Vert _{\mathbb {H}}\bigg [\Vert |\mathcal {S}_1(t)||\varvec{u}|^{q-1}\Vert _{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}+\Vert \mathcal {S}_2(t)\Vert _{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}\bigg ]\nonumber \\&\le C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \varvec{u}\Vert _{\mathbb {H}}\bigg [\Vert |\mathcal {S}_1(t)||\varvec{u}|^{q-1}\Vert _{\mathbb {L}^{\frac{6}{5}}(\mathbb {R}^3)}+\Vert \mathcal {S}_2(t)\Vert _{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}\bigg ] \nonumber \\&\le C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \varvec{u}\Vert _{\mathbb {H}}\bigg [\Vert \mathcal {S}_1(t)\Vert ^{\frac{6}{5}}_{\mathbb {L}^{\frac{6}{6-q}}}\Vert \varvec{u}\Vert ^{\frac{6(q-1)}{5}}_{\widetilde{\mathbb {L}}^{6}}+\Vert \mathcal {S}_2(t)\Vert _{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}\bigg ] \nonumber \\&\le C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \varvec{u}\Vert _{\mathbb {H}}\bigg [\Vert \mathcal {S}_1(t)\Vert ^{\frac{6}{5}}_{\mathbb {L}^{\frac{6}{6-q}}}\Vert \nabla \varvec{u}\Vert ^{\frac{6(q-1)}{5}}_{\mathbb {H}}+\Vert \mathcal {S}_2(t)\Vert _{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}\bigg ] \nonumber \\&\le C\bigg [\Vert \varvec{u}\Vert _{\mathbb {H}}^2+\Vert \varvec{u}\Vert _{\mathbb {H}}^{\frac{10}{6-q}}+\Vert \nabla \varvec{u}\Vert ^6_{\mathbb {H}}+\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^2\Vert \mathcal {S}_2(t)\Vert _{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}^2\nonumber \\&\quad +\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{\frac{10}{6-q}}\Vert \mathcal {S}_1(t)\Vert ^{\frac{12}{6-q}}_{\mathbb {L}^{\frac{6}{6-q}}(\mathbb {R}^3)}\bigg ]. \end{aligned}$$

(4.48)

Finally, we estimate the last two terms of (4.41) as follows:

$$\begin{aligned} \int _{\mathbb {R}^3}\varvec{f}(x,t)\Psi \left( \frac{|x|^2}{k^2}\right) \varvec{u}\textrm{d}x\le \frac{\alpha }{2} \int _{\mathbb {R}^3}\Psi \left( \frac{|x|^2}{k^2}\right) |\varvec{u}|^2\textrm{d}x +\frac{1}{2\alpha } \int _{\mathbb {R}^3}\Psi \left( \frac{|x|^2}{k^2}\right) |\varvec{f}(x,t)|^2\textrm{d}x, \end{aligned}$$

(4.49)

and

$$\begin{aligned}&\mathcal {Z}_{\delta }(\vartheta _{t}\omega )\int _{\mathbb {R}^d}\Psi \left( \frac{|x|^2}{k^2}\right) S(t,x,\varvec{u})\varvec{u}\textrm{d}x\nonumber \\&\quad \le \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \int _{\mathbb {R}^3}\Psi \left( \frac{|x|^2}{k^2}\right) \left( \mathcal {S}_1(t,x)|\varvec{u}|^q+\mathcal {S}_2(t,x)|\varvec{u}|\right) \textrm{d}x\nonumber \\&\quad \le \frac{\beta }{2} \int _{\mathbb {R}^3}\Psi \left( \frac{|x|^2}{k^2}\right) |\varvec{u}|^{r+1}\textrm{d}x+C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{\frac{r+1}{r+1-q}}\int _{\mathbb {R}^3}\Psi \left( \frac{|x|^2}{k^2}\right) \left| \mathcal {S}_1(t,x)\right| ^{\frac{r+1}{r+1-q}}\textrm{d}x\nonumber \\&\qquad +C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{2}\int _{\mathbb {R}^3}\Psi \left( \frac{|x|^2}{k^2}\right) \left| \mathcal {S}_2(t,x)\right| ^{2}\textrm{d}x. \end{aligned}$$

(4.50)

Combining (4.41)–(4.50), we get

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t}\int _{\mathbb {R}^3}\Psi \left( \frac{|x|^2}{k^2}\right) |\varvec{u}|^2\textrm{d}x+\alpha \int _{\mathbb {R}^3}\Psi \left( \frac{|x|^2}{k^2}\right) |\varvec{u}|^2\textrm{d}x\nonumber \\&\quad \le \frac{C}{k} \bigg [\Vert \varvec{u}\Vert ^{2}_{\mathbb {H}}+\Vert \varvec{u}\Vert ^{2(r+1)}_{\mathbb {H}}+\Vert \varvec{u}\Vert _{\mathbb {H}}^{\frac{10}{6-q}}+\Vert \nabla \varvec{u}\Vert ^{6}_{\mathbb {H}}+\Vert \varvec{u}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}+\Vert \varvec{u}\Vert ^{\frac{(r+1)(3r-5)}{3r-4}}_{\widetilde{\mathbb {L}}^{3(r+1)}}\bigg ]\nonumber \\&\qquad +\,\frac{1}{\alpha } \int _{|x|\ge k}|\varvec{f}(x,t)|^2\textrm{d}x +C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{\frac{r+1}{r+1-q}}\int _{|x|\ge k}\left| \mathcal {S}_1(t,x)\right| ^{\frac{r+1}{r+1-q}}\textrm{d}x \nonumber \\&\qquad +\,C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{2}\int _{|x|\ge k}\left| \mathcal {S}_2(t,x)\right| ^{2}\textrm{d}x +\frac{C}{k}\bigg [ \Vert \varvec{f}\Vert ^{2}_{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}+\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^2\Vert \mathcal {S}_2(t)\Vert _{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}^2\nonumber \\ {}&\qquad +\,\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{\frac{10}{6-q}}\Vert \mathcal {S}_1(t)\Vert ^{\frac{12}{6-q}}_{\mathbb {L}^{\frac{6}{6-q}}(\mathbb {R}^3)}\bigg ]. \end{aligned}$$

(4.51)

Applying Gronwall’s inequality to (4.51) on $(\mathfrak {s}-t,\tau )$ where $\tau \in [\mathfrak {s}-1,\mathfrak {s}]$ and replacing $\omega$ by $\vartheta _{-\mathfrak {s}}\omega$, we obtain that, for $\mathfrak {s}\in \mathbb {R}, t\ge 1$ and $\omega \in \Omega$,

$$\begin{aligned}&\int _{\mathbb {R}^d}\Psi \left( \frac{|x|^2}{k^2}\right) |\varvec{u}(\tau ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})|^2\textrm{d}x \nonumber \\&\quad \le e^{\alpha (\mathfrak {s}-t-\tau )}\int _{\mathbb {R}^d}\Psi \left( \frac{|x|^2}{k^2}\right) |\varvec{u}_{\mathfrak {s}-t}|^2\textrm{d}x\nonumber \\&\qquad +\frac{C}{k}\int _{\mathfrak {s}-t}^{\tau }e^{\alpha (\xi -\tau )} \bigg [\Vert \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}}+\Vert \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{2(r+1)}_{\mathbb {H}}\nonumber \\&\qquad +\,\Vert u(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{\frac{10}{6-q}}_{\mathbb {H}}+\Vert \nabla \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{6}_{\mathbb {H}}\nonumber \\&\qquad +\,\Vert u(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}+\Vert \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{\frac{(r+1)(3r-5)}{3r-4}}_{\widetilde{\mathbb {L}}^{3(r+1)}}\bigg ]\textrm{d}\xi \nonumber \\&\qquad +\,C\int _{-\infty }^{\tau -\mathfrak {s}}e^{\alpha (\xi +\mathfrak {s}-\tau )}\bigg [\int _{|x|\ge k}|\varvec{f}(x,\xi +\mathfrak {s})|^2\textrm{d}x+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}}\nonumber \\&\int _{|x|\ge k}\left| \mathcal {S}_1(\xi +\mathfrak {s},x)\right| ^{\frac{r+1}{r+1-q}}\textrm{d}x\nonumber \\&\qquad +\,\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\int _{|x|\ge k}\left| \mathcal {S}_2(\xi +\mathfrak {s},x)\right| ^{2}\textrm{d}x\bigg ]\textrm{d}\xi \nonumber \\&\qquad +\,\frac{C}{k}\int _{\mathfrak {s}-t}^{\tau }e^{\alpha (\xi -\tau )}\bigg [\Vert \varvec{f}(\cdot ,\xi )\Vert ^{2}_{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi -\mathfrak {s}}\omega )\right| ^2\nonumber \\&\qquad +\,\left| \mathcal {Z}_{\delta }(\vartheta _{\xi -\mathfrak {s}}\omega )\right| ^{\frac{10}{6-q}}\bigg ]\textrm{d}\xi , \end{aligned}$$

(4.52)

where we have used the fact that the assumptions on $\mathcal {S}_1$ and $\mathcal {S}_2$ from Assumption 1.5 are satisfied.

Now, we obtain from (4.17)

$$\begin{aligned}&\int _{\mathfrak {s}-t}^{\mathfrak {s}}e^{\alpha (\xi -\mathfrak {s})} \Vert \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{\frac{(r+1)(3r-5)}{3r-4}}_{\widetilde{\mathbb {L}}^{3(r+1)}}\textrm{d}\xi \nonumber \\&\quad \le \left[ \int _{\mathfrak {s}-t}^{\mathfrak {s}}\left\{ \frac{1}{\xi -s+t}\right\} ^{3r-5}e^{\alpha (\xi -\mathfrak {s})}\textrm{d}\xi \right] ^{\frac{1}{3r-4}}\nonumber \\&\qquad \times \,\left[ \int _{\mathfrak {s}-t}^{\mathfrak {s}}(\xi -\mathfrak {s}+t)e^{\alpha (\xi -\mathfrak {s})}\Vert \varvec{u}(\xi ,s-t,\vartheta _{-s}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{3(r+1)}}\textrm{d}\xi \right] ^{\frac{3r-5}{3r-4}}\nonumber \\&\quad \le C\left[ (t+1)^{3r-5}\int _{-t}^{0}\left\{ \frac{1}{\xi +t}\right\} ^{3r-5}e^{\alpha \xi }\textrm{d}\xi \right] ^{\frac{1}{3r-4}}\nonumber \\&\qquad \times \,\bigg [e^{-\alpha t}\Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^2_{\mathbb {H}}+ \int _{-\infty }^{0}e^{\alpha \xi }\bigg \{ \Vert \varvec{f}(\xi +\mathfrak {s})\Vert ^2_{\mathbb {H}} +\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}} +\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{2(r+1)}{3r+3-2q}} \nonumber \\&\qquad +\,\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\bigg \}\bigg ]^{\frac{3r-5}{3r-4}}. \end{aligned}$$

(4.53)

We also have

$$\begin{aligned} \lim _{t\rightarrow +\infty }\left[ (t+1)^{3r-5}\int _{-t}^{0}\left\{ \frac{1}{\zeta +t}\right\} ^{3r-5}e^{\frac{\alpha }{2}\zeta }\textrm{d}\zeta \right] =\frac{2}{\alpha }, \end{aligned}$$

which implies

$$\begin{aligned}&\lim _{t\rightarrow \infty }\int _{\mathfrak {s}-t}^{\mathfrak {s}}e^{\alpha (\xi -\mathfrak {s})} \Vert \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{\frac{(r+1)(3r-5)}{3r-4}}_{\widetilde{\mathbb {L}}^{3(r+1)}}\textrm{d}\xi \nonumber \\&\quad \le C\bigg [\int _{-\infty }^{0}e^{\alpha \xi }\bigg \{ \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {H}} +\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}} +\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{2(r+1)}{3r+3-2q}}\nonumber \\&\qquad +\,\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\bigg \}\bigg ]^{\frac{3r-5}{3r-4}}. \end{aligned}$$

(4.54)

Since $\varvec{u}_{\mathfrak {s}-t}\in D(\mathfrak {s}-t,\vartheta _{-t}\omega )$ and $D\in \mathfrak {D}$, from (4.22), we find that, for given $\eta >0,$ there exists $T_1=T_1(\delta ,\mathfrak {s},\omega ,D,\eta )\ge 1$ such that for all $t\ge T_1,$

$$\begin{aligned} e^{\alpha (\mathfrak {s}-t-\tau )}\int _{\mathbb {R}^d}\Psi \left( \frac{|x|^2}{k^2}\right) |\varvec{u}_{\mathfrak {s}-t}|^2\textrm{d}x\le \frac{\eta }{5}. \end{aligned}$$

(4.55)

Since $\varvec{f}\in \textrm{L}^2_{\textrm{loc}}(\mathbb {R};\mathbb {H})$ satisfies (4.8), for a given $\eta >0,$ there exists $P_1=P_1(\delta ,\mathfrak {s},\omega ,\eta )>0$ such that for all $k\ge P_1$,

$$\begin{aligned}&C\int _{-\infty }^{\tau -\mathfrak {s}}e^{\alpha (\xi +\mathfrak {s}-\tau )} \int _{|x|\ge k}|\varvec{f}(x,\xi +\mathfrak {s})|^2\textrm{d}x\textrm{d}\xi \nonumber \\&\quad \le Ce^{\alpha }\int _{-\infty }^{0}e^{\alpha \xi } \int _{|x|\ge k}|\varvec{f}(x,\xi +\mathfrak {s})|^2\textrm{d}x\textrm{d}\xi \le \frac{\eta }{5}. \end{aligned}$$

(4.56)

By (2.9) and (4.7), we have

$$\begin{aligned}&C\int _{-\infty }^{\tau -\mathfrak {s}}e^{\alpha (\xi +\mathfrak {s}-\tau )} \bigg [\Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^{2}_{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^2+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{10}{6-q}}\bigg ]\textrm{d}\xi <\infty , \end{aligned}$$

from which it follows that, for given $\eta >0,$ there exists $P_2=P_2(\delta ,\mathfrak {s},\omega ,\eta )>0$ such that for all $k\ge P_2$,

$$\begin{aligned} \frac{C}{k}\int _{\mathfrak {s}-t}^{\tau }e^{\alpha (\xi -\tau )} \bigg [\Vert \varvec{f}(\cdot ,\xi )\Vert ^{2}_{\dot{\mathbb {H}}^{-1}(\mathbb {R}^3)}+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi -\mathfrak {s}}\omega )\right| ^2+\left| \mathcal {Z}_{\delta }(\vartheta _{\xi -\mathfrak {s}}\omega )\right| ^{\frac{10}{6-q}}\bigg ]\textrm{d}\xi \le \frac{\eta }{5}, \end{aligned}$$

(4.57)

and there exists $P_3=P_3(\delta ,\mathfrak {s},\omega ,\eta )>0$ such that for all $k\ge P_3$,

$$\begin{aligned}&C\int _{-\infty }^{\tau -\mathfrak {s}}e^{\alpha (\xi +\mathfrak {s}-\tau )}\bigg [\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{r+1}{r+1-q}}\int _{|x|\ge k}\left| \mathcal {S}_1(\xi +\mathfrak {s},x)\right| ^{\frac{r+1}{r+1-q}}\textrm{d}x\nonumber \\&\quad +\left| \mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\right| ^{2}\int _{|x|\ge k}\left| \mathcal {S}_2(\xi +\mathfrak {s},x)\right| ^{2}\textrm{d}x\bigg ]\textrm{d}\xi \le \frac{\eta }{5}. \end{aligned}$$

(4.58)

Due to (4.15)–(4.18), (4.54), (4.57) and Assumption 4.7, there exists $T_2=T_2(\delta ,\mathfrak {s},\omega ,D,\eta )>0$ and $P_4=P_4(\delta ,\mathfrak {s},\omega ,\eta )>0$ such that for all $t\ge T_2$ and $k\ge P_4$,

$$\begin{aligned} \frac{C}{k}\int _{\mathfrak {s}-t}^{\tau }&e^{\alpha (\xi -\tau )} \bigg [\Vert \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}}+\Vert \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{2(r+1)}_{\mathbb {H}}\nonumber \\ {}&\quad +\Vert u(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{\frac{10}{6-q}}_{\mathbb {H}}+\Vert \nabla \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{6}_{\mathbb {H}}\nonumber \\ {}&\quad +\Vert u(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}+\Vert \varvec{u}(\xi ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^{\frac{(r+1)(3r-5)}{3r-4}}_{\widetilde{\mathbb {L}}^{3(r+1)}}\bigg ]\textrm{d}\xi \le \frac{\eta }{5}, \end{aligned}$$

(4.59)

Let $\mathscr {T}^*=\mathscr {T}^*(\delta ,\mathfrak {s},\omega ,D,\eta )=\max \{T_1,T_2\}, P^*=P^*(\delta ,\mathfrak {s},\omega ,\eta )=\max \{P_1,P_2,P_3,P_4\}$. Then it implies from (4.52) to (4.59) that, for all $t\ge \mathscr {T}^*$ and $k\ge P^*$, we obtain that for all $\tau \in [\mathfrak {s}-1,\mathfrak {s}]$,

$$\begin{aligned} \int \limits _{|x|\ge P^*}|\varvec{u}(\tau ;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})|^2\textrm{d}x\le \eta , \end{aligned}$$

which completes the proof. $\square$

Lemma 4.13

Let $\{\varvec{u}(\cdot ;\mathfrak {s},\omega ,\varvec{u}^n)\}_{n\in \mathbb {N}}$ be a bounded sequence of solutions of (2.12) in $\mathbb {H}$ and all the assumptions of Lemma 4.9 be satisfied. Then for every $0<\delta \le 1, \omega \in \Omega , \mathfrak {s}\in \mathbb {R}$ and $t>\mathfrak {s}$, there exists $\varvec{u}^0\in \textrm{L}^2(\mathfrak {s},\mathfrak {s}+T;\mathbb {H})$ with $T>0$ and a subsequence $\{\varvec{u}(\cdot ;\mathfrak {s},\omega ,\varvec{u}^{n_m})\}_{m\in \mathbb {N}}$ of $\{\varvec{u}(\cdot ;\mathfrak {s},\omega ,\varvec{u}^n)\}_{n\in \mathbb {N}}$ such that $\varvec{u}(\tau ;\mathfrak {s},\omega ,\varvec{u}^{n_m})\rightarrow \varvec{u}^0(\tau )$ in $\mathbb {L}^2(\mathcal {O}_k)$ as $m\rightarrow \infty$ for every $k\in \mathbb {N}$ and for almost all $\tau \in (\mathfrak {s},\mathfrak {s}+T)$, where

$$\begin{aligned} \mathcal {O}_k=\{x\in \mathbb {R}^d:|x|<k\}. \end{aligned}$$

Proof

Since embedding $\mathbb {H}^1_0(\mathcal {O}_k)\hookrightarrow \mathbb {L}^2(\mathcal {O}_k)$ is compact, the proof can be completed analogously as in the proof of Lemma 3.6. $\square$

The following theorem proves the $\mathfrak {D}$-pullback asymptotic compactness of $\Phi$.

Lemma 4.14

Let all the assumptions of Lemma 4.9 be satisfied. Then for every $0<\delta \le 1$, $\mathfrak {s}\in \mathbb {R},$ $\omega \in \Omega ,$ $D=\{D(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R},\omega \in \Omega \}\in \mathfrak {D}$ and $t_n\rightarrow \infty ,$ $\varvec{u}_{0,n}\in D(\mathfrak {s}-t_n, \vartheta _{-t_{n}}\omega )$, the sequence $\Phi (t_n,\mathfrak {s}-t_n,\vartheta _{-t_n}\omega ,\varvec{u}_{0,n})$ or $\varvec{u}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})$ of solutions of the system (2.12) has a convergent subsequence in $\mathbb {H}$.

Proof

Lemma 4.10 implies that there exists $\mathscr {T}=\mathscr {T}(\delta ,\mathfrak {s},\omega ,D)>0$ and $R(\mathfrak {s},\omega )$ such that for all $t\ge \mathscr {T}$,

$$\begin{aligned} \Vert \varvec{u}(\mathfrak {s}-1;\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}} \le R(\mathfrak {s},\omega ), \end{aligned}$$

(4.60)

where $\varvec{u}_{\mathfrak {s}-t}\in D(\mathfrak {s}-t,\vartheta _{-t}\omega ).$ Since $t_n\rightarrow \infty$, there exists $N_2\in \mathbb {N}$ such that $t_n\ge \mathscr {T},$ for all $n\ge N_2$. Since $\varvec{u}_{0,n}\in D(\mathfrak {s}-t_n, \vartheta _{-t_{n}}\omega )$, (4.60) implies that for all $n\ge N_2$,

$$\begin{aligned} \Vert \varvec{u}(\mathfrak {s}-1;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})\Vert ^2_{\mathbb {H}} \le R(\mathfrak {s},\omega ), \end{aligned}$$

and hence

$$\begin{aligned} \{\varvec{u}(\mathfrak {s}-1;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})\}_{n\ge N_2} \text { is a bounded sequence in }\mathbb {H}. \end{aligned}$$

(4.61)

It yields from (4.61) and Lemma 4.13 that there exists $\tau \in (\mathfrak {s}-1,\mathfrak {s})$, $\varvec{u}_0\in \mathbb {H}$ and a subsequence (not relabeled) such that for every $k\in \mathbb {N}$ as $n\rightarrow \infty$

$$\begin{aligned}&\varvec{u}(\tau ;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})=\varvec{u}(\tau ;\mathfrak {s}-1,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}(\mathfrak {s}-1;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n}))\nonumber \\&\quad \rightarrow \varvec{u}_0 \ \text { in }\ \mathbb {L}^2(\mathcal {O}_k). \end{aligned}$$

(4.62)

Since $\varvec{u}_0\in \mathbb {H}$, for given $\eta >0$, there exists $K_1=K_1(\delta ,\mathfrak {s},\omega ,\eta )>0$ such that for all $k\ge K_1$,

$$\begin{aligned} \int \limits _{|x|\ge k}|\varvec{u}_0|^2\textrm{d}x\le \frac{\eta }{3}. \end{aligned}$$

(4.63)

Also, it follows from Lemma 4.12 that there exists $N_3=N_3(\delta ,\mathfrak {s},\omega ,D,\eta )\ge 1$ and $K_2=K_2(\delta ,\mathfrak {s},\omega ,\eta )\ge K_1$ such that for all $n\ge N_3$ and $k\ge K_2$,

$$\begin{aligned} \int \limits _{|x|\ge k}|\varvec{u}(\tau ;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})|^2\textrm{d}x\le \frac{\eta }{3}. \end{aligned}$$

(4.64)

From (4.62), we have that there exists $N_4=N_4(\delta ,\mathfrak {s},\omega ,D,\eta )>N_3$ such that for all $n\ge N_4$,

$$\begin{aligned} \int \limits _{|x|< K_2}|\varvec{u}(\tau ;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})-\varvec{u}_0|^2\textrm{d}x\le \frac{\eta }{3}. \end{aligned}$$

(4.65)

Finally, Lemma 4.9 implies

$$\begin{aligned}&\Vert \varvec{u}(\mathfrak {s};\tau ,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}(\tau ;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n}))-\varvec{u}(\mathfrak {s};\tau ,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_0)\Vert ^2_{\mathbb {H}}\nonumber \\&\quad \le C\Vert \varvec{u}(\tau ;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})-\varvec{u}_0\Vert ^2_{\mathbb {H}}\nonumber \\&\quad \le C\left[ \int \limits _{|x|<K_2}|\varvec{u}(\tau ;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})-\varvec{u}_0|^2\textrm{d}x\right. \nonumber \\&\qquad \left. +\int \limits _{|x|\ge K_2}|\varvec{u}(\tau ;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})-\varvec{u}_0|^2\textrm{d}x\right] \nonumber \\&\quad \le C\left[ \int \limits _{|x|<K_2}|\varvec{u}(\tau ;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})-\varvec{u}_0|^2\textrm{d}x\right. \nonumber \\&\qquad \left. +\int \limits _{|x|\ge K_2}(|\varvec{u}(\tau ;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})|^2+|\varvec{u}_0|^2)\textrm{d}x\right] . \end{aligned}$$

(4.66)

Hence, (4.66) along with (4.63)–(4.65) conclude the proof. $\square$

The following theorem is the main results of this section, that is, the existence of a unique $\mathfrak {D}$-pullback random attractor for $\Phi$.

Theorem 4.15

Under the assumptions of Lemma 4.9, there exists a unique $\mathfrak {D}$-pullback random attractor $\widehat{\mathscr {A}}=\{\widehat{\mathscr {A}}(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R}, \omega \in \Omega \}\in \mathfrak {D},$ for the the continuous cocycle $\Phi$ associated with the system (2.12) in $\mathbb {H}$.

Proof

Note that Lemmas 4.11 and 4.14 provide the existence of a closed measurable $\mathfrak {D}$-pullback random absorbing set for $\Phi$ and asymptotic compactness of $\Phi$, respectively. Hence, Lemmas 4.11 and 4.14 together with the abstract theory given in [63, Theorem 2.23] complete the proof. $\square$

5 Pullback Random Attractors for Wong–Zakai Approximations: Unbounded Poincaré Domain $\mathcal {O}\subset \mathbb {R}^d$

In this section, we prove the existence of a unique $\mathfrak {D}$-pullback random attractor for the system (2.12) on the Poincaré domain $\mathcal {O}\subset \mathbb {R}^d$ satisfying Assumption 1.7 with nonlinear diffusion term $S(t,x,\varvec{u})$ satisfying Assumption 1.9.

In order to prove the results of this subsection, we need the following assumption on the non-autonomous forcing term $\varvec{f}(\cdot ,\cdot )$.

Assumption 5.1

The external forcing term $\varvec{f}\in \textrm{L}^{2}_{\textrm{loc}}(\mathbb {R};\mathbb {H})$ which satisfies

$$\begin{aligned} \lim _{s\rightarrow -\infty }e^{cs}\int _{-\infty }^{0} e^{\alpha \zeta }\Vert \varvec{f}(\cdot ,\zeta +s)\Vert ^2_{\mathbb {H}}\textrm{d}\zeta =0, \end{aligned}$$

(5.1)

for every $c>0$, where $\alpha >0$ is the Darcy coefficient. We also assume that the following integration should be finite:

$$\begin{aligned} \int _{-\infty }^{0}\Vert \varvec{f}(\cdot ,\zeta +s)\Vert ^2_{\mathbb {H}}\textrm{d}\zeta <\infty . \end{aligned}$$

(5.2)

The proof of the main result of this section is similar to the main result obtained in Sect. 4. Therefore, we provide a proof of the following Lemma, which differ from Lemma 4.12. As we have seen in Sect. 4, we are facing difficulties to prove the uniform tail-estimates due to a non-availability of continuous embedding of first order homogeneous Sobolev space in $\mathbb {R}^2$ (see Theorem 2.3).

Lemma 5.2

For all the cases given in Table 1 including $d=r=3$ with $2\beta \mu =1$, let Assumptions 1.7, 1.9 and 5.1 be satisfied. Then, for any $\varvec{u}_{\mathfrak {s}-t}\in D(\mathfrak {s}-t,\vartheta _{-t}\omega ),$ where $D=\{D(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R},\ \omega \in \Omega \}\in \mathfrak {D}$, and for any $\eta >0$, $\mathfrak {s}\in \mathbb {R}$, $\omega \in \Omega$ and $0<\delta \le 1$, there exists $\mathscr {T}^*=\mathscr {T}^*(\delta ,\mathfrak {s},\omega ,D,\eta )\ge 1$ and $P^*=P^*(\delta ,\mathfrak {s},\omega ,\eta )>0$ such that for all $t\ge \mathscr {T}^*$ and $\tau \in [\mathfrak {s}-1,\mathfrak {s}]$, the solution of (2.12) with $\omega$ replaced by $\vartheta _{-\mathfrak {s}}\omega$ satisfy

$$\begin{aligned} \int _{|x|\ge P^*}|\varvec{u}(\tau ;\vartheta _{-\mathfrak {s}}\omega ,\mathfrak {s}-t,\varvec{u}_{\mathfrak {s}-t}) |^2\textrm{d}x\le \eta . \end{aligned}$$

(5.3)

Proof

As some estimates of the proof are similar to the proof of Lemma 4.12, we are not repeating here. We provide only those estimates, where the main difference is occurring. For a detailed calculation, see the proof of Lemma 4.12.

Let us now estimate each term on the right hand side of (4.41). Integration by parts, and Lemmas 2.9 and 2.11 help us to obtain

$$\begin{aligned}&\mu \int _{\mathcal {O}}(\Delta \varvec{u}) \Psi \left( \frac{|x|^2}{k^2}\right) \varvec{u}\textrm{d}x+\mu \int _{\mathcal {O}}|\nabla \varvec{u}|^2 \Psi \left( \frac{|x|^2}{k^2}\right) \textrm{d}x \le \frac{C}{k}\Vert \varvec{u}\Vert ^2_{\mathbb {V}}, \end{aligned}$$

(5.4)

and

$$\begin{aligned} -b\left( \varvec{u},\varvec{u},\Psi \left( \frac{|x|^2}{k^2}\right) \varvec{u}\right)&\le \frac{C}{k}\Vert \varvec{u}\Vert ^3_{\widetilde{\mathbb {L}}^3}\le \frac{C}{k}\Vert \varvec{u}\Vert ^{\frac{6-d}{2}}_{\mathbb {H}}\Vert \nabla \varvec{u}\Vert _{\mathbb {H}}^{\frac{d}{2}} \le \frac{C}{k}\bigg [\Vert \nabla \varvec{u}\Vert ^{2}_{\mathbb {H}}+\Vert \varvec{u}\Vert _{\mathbb {H}}^{\frac{2(6-d)}{4-d}}\bigg ], \end{aligned}$$

(5.5)

where we have used the Gagliardo–Nirenberg–Sobolev inequality also. From (4.44), we have

$$\begin{aligned}&-\int _{\mathcal {O}}(\nabla p)\Psi \left( \frac{|x|^2}{k^2}\right) \varvec{u}\textrm{d}x\nonumber \\&\quad \le \frac{C}{k}\bigg [\int \limits _{\mathcal {O}}\left| (-\Delta )^{-1}\left[ \nabla \cdot \left[ \nabla \cdot \big (\varvec{u}\otimes \varvec{u}\big )\right] \right] \right| \cdot \left| \varvec{u}\right| \textrm{d}x + \int \limits _{\mathcal {O}}\left| (-\Delta )^{-1}\left[ \nabla \cdot \left[ |\varvec{u}|^{r-1}\varvec{u}\right] \right] \right| \cdot \left| \varvec{u}\right| \textrm{d}x \nonumber \\&\qquad +\int \limits _{\mathcal {O}}\left| (-\Delta )^{-1}\left[ \nabla \cdot \varvec{f}\right] \right| \cdot \left| \varvec{u}\right| \textrm{d}x+\int \limits _{\mathcal {O}}\left| (-\Delta )^{-1}\left[ \nabla \cdot S(t,x,\varvec{u})\right] \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \cdot \left| \varvec{u}\right| \textrm{d}x\nonumber \\&=: \frac{C}{k}\left[ \widetilde{S}_1(d,r)+\widetilde{S}_2(d,r) +\widetilde{S}_3(d,r)+\widetilde{S}_4(d,r)\right] . \end{aligned}$$

(5.6)

Estimate of $\widetilde{S}_1(d,r)$: Using Hölder’s inequality, elliptic regularity, and Ladyzhenskaya’s and Young’s inequalities, respectively, we get

$$\begin{aligned} \widetilde{S}_1(d,r)&\le \left\| (-\Delta )^{-1}\left[ \nabla \cdot \left[ \nabla \cdot \big (\varvec{u}\otimes \varvec{u}\big )\right] \right] \right\| _{\mathbb {L}^2(\mathcal {O})}\Vert \varvec{u}\Vert _{\mathbb {H}}\nonumber \\&\le C\Vert \varvec{u}\Vert ^2_{\widetilde{\mathbb {L}}^4}\Vert \varvec{u}\Vert _{\mathbb {H}}\le C\Vert \varvec{u}\Vert ^{\frac{6-d}{2}}_{\mathbb {H}}\Vert \nabla \varvec{u}\Vert _{\mathbb {H}}^{\frac{d}{2}}\le C\bigg [\Vert \nabla \varvec{u}\Vert ^{2}_{\mathbb {H}}+\Vert \varvec{u}\Vert _{\mathbb {H}}^{\frac{2(6-d)}{4-d}}\bigg ]. \end{aligned}$$

(5.7)

Estimate of $\widetilde{S}_2(d,r)$: Applying Hölder’s inequality, elliptic regularity (for $d=2$ with $r\ge 1$ and $d=3$ with $r\in [3,5]$), the Gagliardo–Nirenberg–Sobolev inequality (for $d=3$ with $r>5$), Sobolev’s embedding ($\mathbb {H}^1(\mathcal {O})\subset \mathbb {L}^{r+1}(\mathcal {O})$, for $d=2$ with $r\ge 1$ and $d=3$ with $r\in [3,5]$), Interpolation inequality (for $d=3$ with $r>5$) and Young’s inequality, we obtain

$$\begin{aligned} \widetilde{S}_2(d,r)&\le {\left\{ \begin{array}{ll} \Vert (-\Delta )^{-1}\left[ \nabla \cdot \left[ |\varvec{u}|^{r-1}\varvec{u}\right] \right] \Vert _{\mathbb {L}^{2}(\mathcal {O})}\Vert \varvec{u}\Vert _{\mathbb {H}} ,\ \ {} &{} \text { for } d=2\text { and }r\in [1,\infty ),\\ \Vert (-\Delta )^{-1}\left[ \nabla \cdot \left[ |\varvec{u}|^{r-1}\varvec{u}\right] \right] \Vert _{\mathbb {L}^{2}(\mathcal {O})}\Vert \varvec{u}\Vert _{\mathbb {H}} ,\ \ {} &{} \text { for } d=3\text { and } r\in [3,5], \\ \Vert (-\Delta )^{-1}\left[ \nabla \cdot \left[ |\varvec{u}|^{r-1}\varvec{u}\right] \right] \Vert _{\mathbb {L}^{\frac{3(r+1)}{2r-1}}(\mathcal {O})}\Vert \varvec{u}\Vert _{\widetilde{\mathbb {L}}^{\frac{3(r+1)}{r+4}}}, \ \ {} &{} \text { for } d=3\text { and } r\in (5,\infty ), \end{array}\right. } \nonumber \\ {}&\le C \times {\left\{ \begin{array}{ll} \Vert |\varvec{u}|^{r-1}\varvec{u}\Vert _{\mathbb {H}^{-1}(\mathcal {O})}\Vert \varvec{u}\Vert _{\mathbb {H}} ,\ \ {} &{} \text { for } d=2\text { and }r\in [1,\infty ),\\ \Vert \Vert \varvec{u}|^{r-1}\varvec{u}\Vert _{\mathbb {H}^{-1}(\mathcal {O})}\Vert \varvec{u}\Vert _{\mathbb {H}} ,\ \ \hspace{26.5mm} &{} \text { for } d=3\text { and } r\in [3,5], \\ \Vert |\varvec{u}|^{r-1}\varvec{u}\Vert _{\mathbb {L}^{\frac{r+1}{r}}(\mathcal {O})}\Vert \varvec{u}\Vert _{\widetilde{\mathbb {L}}^{\frac{3(r+1)}{r+4}}}, \ \ {} &{} \text { for } d=3\text { and } r\in (5,\infty ), \end{array}\right. } \nonumber \\&\le C \times {\left\{ \begin{array}{ll} \Vert |\varvec{u}|^{r-1}\varvec{u}\Vert _{\mathbb {L}^{\frac{r+1}{r}}(\mathcal {O})}\Vert \varvec{u}\Vert _{\mathbb {H}} ,\ \ {} &{} \text { for } d=2\text { and }r\in [1,\infty ),\\ \Vert |\varvec{u}|^{r-1}\varvec{u}\Vert _{\mathbb {L}^{\frac{r+1}{r}}(\mathcal {O})}\Vert \varvec{u}\Vert _{\mathbb {H}} ,\ \ \hspace{26.5mm} &{} \text { for } d=3\text { and } r\in [3,5], \\ \Vert \varvec{u}\Vert ^{r}_{\widetilde{\mathbb {L}}^{r+1}}\Vert \varvec{u}\Vert _{\widetilde{\mathbb {L}}^{\frac{3(r+1)}{r+4}}}, \ \ {} &{} \text { for } d=3\text { and } r\in (5,\infty ), \end{array}\right. } \nonumber \\ {}&\le C \times {\left\{ \begin{array}{ll} \Vert \varvec{u}\Vert ^r_{\widetilde{\mathbb {L}}^{r+1}}\Vert \varvec{u}\Vert _{\mathbb {H}},\ \ {} &{} \text { for } d=2\text { and }r\in [2,\infty ),\\ \Vert \varvec{u}\Vert ^r_{\widetilde{\mathbb {L}}^{r+1}}\Vert \varvec{u}\Vert _{\mathbb {H}} ,\hspace{26.5mm} \ \ {} &{} \text { for } d=3\text { and } r\in [3,5] ,\\ \Vert \varvec{u}\Vert ^{\frac{(r+1)(3r-5)}{3(r-1)}}_{\widetilde{\mathbb {L}}^{r+1}}\Vert \varvec{u}\Vert ^{\frac{2(r+1)}{3(r-1)}}_{\mathbb {H}}, \ \ {} &{} \text { for } d=3\text { and } r\in (5,\infty ), \end{array}\right. } \nonumber \\&\le C\left[ \Vert \varvec{u}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}+\Vert \varvec{u}\Vert ^{r+1}_{\mathbb {H}}\right] . \end{aligned}$$

(5.8)

Estimate of $\widetilde{S}_3(d,r)$: Applying Hölder’s inequality, elliptic regularity and Young’s inequality, we find

$$\begin{aligned} \widetilde{S}_3(d,r)&\le \left\| (-\Delta )^{-1}\left[ \nabla \cdot \varvec{f}\right] \right\| _{\mathbb {L}^2(\mathcal {O})}\Vert \varvec{u}\Vert _{\mathbb {H}}\nonumber \\&\le C\left\| \varvec{f}\right\| _{\mathbb {H}^{-1}(\mathcal {O})}\Vert \varvec{u}\Vert _{\mathbb {H}} \le C\left\| \varvec{f}\right\| _{\mathbb {H}}\Vert \varvec{u}\Vert _{\mathbb {H}} \le C\left\| \varvec{f}\right\| ^2_{\mathbb {H}}+C\Vert \varvec{u}\Vert ^2_{\mathbb {H}} . \end{aligned}$$

(5.9)

Estimate of $\widetilde{S}_4(d,r)$: In view of Hölder’s inequality, elliptic regularity, Sobolev’s embedding, and the Gagliardo-Nirenberg-Sobolev, interpolation and Young’s inequalities, we estimate $\widetilde{S}_4(d,r)$ as follows:

When $d=2$ with $r\ge 1$ and $d=3$ with $r\in [3,5]$. We get

$$\begin{aligned} \widetilde{S}_4(d,r)&\le \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert (-\Delta )^{-1}\left[ \nabla \cdot S(t,x,\varvec{u})\right] \Vert _{\mathbb {L}^{2}(\mathcal {O})}\Vert \varvec{u}\Vert _{\mathbb {H}}\nonumber \\&\le C \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert S(t,x,\varvec{u})\Vert _{\mathbb {H}^{-1}(\mathcal {O})}\Vert \varvec{u}\Vert _{\mathbb {H}}\nonumber \\ {}&\le C \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert S(t,x,\varvec{u})\Vert _{\mathbb {L}^{\frac{r+1}{r}}}\Vert \varvec{u}\Vert _{\mathbb {H}} \nonumber \\&\le C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \mathcal {S}_1(t)|\varvec{u}|^{q-1}\Vert _{\mathbb {L}^{\frac{r+1}{r}}}\Vert \varvec{u}\Vert _{\mathbb {H}} + C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \mathcal {S}_2(t)\Vert _{\mathbb {L}^{\frac{r+1}{r}}}\Vert \varvec{u}\Vert _{\mathbb {H}}\nonumber \\&\le C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \mathcal {S}_1(t)\Vert _{\mathbb {L}^{\frac{r+1}{r+1-q}}}\Vert \varvec{u}\Vert _{\mathbb {H}}\Vert \varvec{u}\Vert ^{q-1}_{\widetilde{\mathbb {L}}^{r+1}} + C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \mathcal {S}_2(t)\Vert _{\mathbb {L}^{\frac{r+1}{r}}}\Vert \varvec{u}\Vert _{\mathbb {H}}\nonumber \\&\le C\left[ \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{\frac{r+1}{r+1-q}}\Vert \mathcal {S}_1(t)\Vert ^{\frac{r+1}{r+1-q}}_{\mathbb {L}^{\frac{r+1}{r+1-q}}} + \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{\frac{r+1}{r}}\Vert \mathcal {S}_2(t)\Vert ^{\frac{r+1}{r}}_{\mathbb {L}^{\frac{r+1}{r}}} \right. \nonumber \\&\quad \left. + \Vert \varvec{u}\Vert ^{r+1}_{\mathbb {H}}+ \Vert \varvec{u}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\right] . \end{aligned}$$

(5.10)

When $d=3$ and $r\in [5,\infty )$. We obtain

$$\begin{aligned} \widetilde{S}_4(d,r)&\le \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert (-\Delta )^{-1}\left[ \nabla \cdot S(t,x,\varvec{u})\right] \Vert _{\mathbb {L}^{\frac{3(r+1)}{2r-1}}}\Vert \varvec{u}\Vert _{\widetilde{\mathbb {L}}^{\frac{3(r+1)}{r+4}}}\nonumber \\&\le C \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert S(t,x,\varvec{u})\Vert _{\mathbb {L}^{\frac{r+1}{r}}}\Vert \varvec{u}\Vert _{\widetilde{\mathbb {L}}^{\frac{3(r+1)}{r+4}}} \nonumber \\&\le C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \mathcal {S}_1(t)|\varvec{u}|^{q-1}\Vert _{\mathbb {L}^{\frac{r+1}{r}}}\Vert \varvec{u}\Vert ^{\frac{2(r+1)}{3(r-1)}}_{\mathbb {H}}\Vert \varvec{u}\Vert ^{\frac{r-5}{3(r-1)}}_{\widetilde{\mathbb {L}}^{r+1}}\nonumber \\&\quad + C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \mathcal {S}_2(t)\Vert _{\mathbb {L}^{\frac{r+1}{r}}}\Vert \varvec{u}\Vert ^{\frac{2(r+1)}{3(r-1)}}_{\mathbb {H}}\Vert \varvec{u}\Vert ^{\frac{r-5}{3(r-1)}}_{\widetilde{\mathbb {L}}^{r+1}}\nonumber \\ {}&\le C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \mathcal {S}_1(t)\Vert _{\mathbb {L}^{\frac{r+1}{r+1-q}}}\Vert \varvec{u}\Vert ^{\frac{2(r+1)}{3(r-1)}}_{\mathbb {H}}\Vert \varvec{u}\Vert ^{\frac{3q(r-1)-2(r+1)}{3(r-1)}}_{\widetilde{\mathbb {L}}^{r+1}}\nonumber \\&\quad + C\left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| \Vert \mathcal {S}_2(t)\Vert _{\mathbb {L}^{\frac{r+1}{r}}}\Vert \varvec{u}\Vert ^{\frac{2(r+1)}{3(r-1)}}_{\mathbb {H}}\Vert \varvec{u}\Vert ^{\frac{r-5}{3(r-1)}}_{\widetilde{\mathbb {L}}^{r+1}}\nonumber \\&\le C\left[ \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{\frac{r+1}{r+1-q}}\Vert \mathcal {S}_1(t)\Vert ^{\frac{r+1}{r+1-q}}_{\mathbb {L}^{\frac{r+1}{r+1-q}}} + \left| \mathcal {Z}_{\delta }(\vartheta _{t}\omega )\right| ^{\frac{r+1}{r}}\Vert \mathcal {S}_2(t)\Vert ^{\frac{r+1}{r}}_{\mathbb {L}^{\frac{r+1}{r}}} \right. \nonumber \\&\quad \left. + \Vert \varvec{u}\Vert ^{r+1}_{\mathbb {H}} + \Vert \varvec{u}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\right] . \end{aligned}$$

(5.11)

For further steps, one can see the proof of Lemma 4.12. $\square$

The following theorem is the main result of this section, that is, the existence of a unique $\mathfrak {D}$-pullback random attractor for $\Phi$ under Assumptions 1.9 and 5.1.

Theorem 5.3

For all the cases given in Table 1 including $d=r=3$ with $2\beta \mu =1$, let Assumptions 1.7, 1.9 and 5.1 be satisfied. Then, there exists a unique $\mathfrak {D}$-pullback random attractor for the the continuous cocycle $\Phi$ associated with the system (2.12) in $\mathbb {H}$.

6 Convergence of Attractors: Additive White Noise

In this section, we examine the approximations of solutions of the following stochastic CBF equations with additive white noise. For given $\sigma \ge 0$ and ${\textbf {g}}\in \textrm{D}(\textrm{A})$,

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned} \frac{\partial \varvec{u}}{\partial t}+\mu \text {A}\varvec{u}+\text {B}(\varvec{u})+\alpha \varvec{u}+\beta \mathcal {C}(\varvec{u})&={\varvec{f}}+{\varvec{e}}^{\sigma t}{} {\varvec{g}}(x)\frac{{d}\text {W}}{\text {d}t}, \qquad \text {in } \mathbb {R}^d\times (\mathfrak {s},\infty ), \\ \varvec{u}|_{t=\mathfrak {s}}&=\varvec{u}_{\mathfrak {s}}, \qquad \varvec{x}\in \mathbb {R}^d \text{ and } \mathfrak {s}\in \mathbb {R}. \end{aligned} \right. \end{aligned}\end{aligned}$$

(6.1)

For $\delta >0$, consider the pathwise random system:

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}\frac{\partial \varvec{u}_{\delta }}{\partial t}+\mu \text {A}\varvec{u}_{\delta }+\text {B}(\varvec{u}_{\delta })+\alpha \varvec{u}_{\delta }+\beta \mathcal {C}(\varvec{u}_{\delta })&={\varvec{f}}+{\varvec{e}}^{\sigma t}{} {\varvec{g}}(x)\mathcal {Z}_{\delta }(\vartheta _{t}\omega ), \qquad \text {in } \mathbb {R}^{\varvec{d}}\times (\mathfrak {s},\infty ), \\ \varvec{u}_{\delta }|_{{\varvec{t}}=\mathfrak {s}}&=\varvec{u}_{\delta ,\mathfrak {s}}, \qquad x\in \mathbb {R}^d \text{ and } \mathfrak {s}\in \mathbb {R}. \end{aligned} \right. \end{aligned} \end{aligned}$$

(6.2)

Throughout this section, we prove results for 2D stochastic CBF with $r>1$ and 3D stochastic CBF with $r\ge 3$ ($r>3$ for any $\beta ,\mu >0$ and $r=3$ for $2\beta \mu \ge 1$). Since, 2D stochastic CBF equations are linear perturbation of 2D stochastic NSE for $r=1$, one can prove the results of this section for $d=2$ and $r=1$ by using the same arguments as it is carried for 2D stochastic NSE on Poincaré domains in [23, Section 3]. In [23], authors imposed an extra condition (Assumption 1.11) on ${\textbf {g}}(\cdot ),$ which was used to prove the existence of a $\mathfrak {D}$-pullback random absorbing set. We observe that there is no need to impose Assumption 1.11 on ${\textbf {g}}(\cdot )$ for $r>1$.

Assumption 6.1

The external forcing term $\varvec{f}\in \textrm{L}^{2}_{\textrm{loc}}(\mathbb {R};\mathbb {V}')$ satisfies

$$\begin{aligned} \lim _{s\rightarrow -\infty }e^{cs}\int _{-\infty }^{0} e^{\alpha \zeta }\Vert \varvec{f}(\cdot ,\zeta +s)\Vert ^2_{\mathbb {V}'}\textrm{d}\zeta =0, \end{aligned}$$

(6.3)

for every $c>0$, where $\alpha >0$ is the Darcy coefficient.

A direct consequence of the above Assumption 6.1 is as follows:

Proposition 6.2

(Proposition 4.2, [37]) Assume that Assumption 6.1 holds. Then

$$\begin{aligned} \int _{-\infty }^{\mathfrak {s}} e^{\alpha \xi }\Vert \varvec{f}(\cdot ,\xi )\Vert ^2_{\mathbb {V}'}\textrm{d}\xi <\infty , \ \ \text { for all }\ \mathfrak {s}\in \mathbb {R}. \end{aligned}$$

(6.4)

6.1 $\mathfrak {D}$-pullback Random Attractors for Stochastic CBF Equations with Additive White Noise

Consider, for some $\ell >0$

$$\begin{aligned} \varvec{y}(\vartheta _{t}\omega ) = \int _{-\infty }^{t} e^{-\ell (t-\mathfrak {s})}{\textrm{dW}}(\mathfrak {s}), \ \ \omega \in \Omega , \end{aligned}$$

(6.5)

which is the stationary solution of the one-dimensional Ornstein–Uhlenbeck equation

Let us recall from [15, 18] that there exists a $\vartheta$-invariant subset of $\Omega$ (will be denoted by $\Omega$ itself) of full measure such that $\varvec{y}(\vartheta _t\omega )$ is continuous in t for every $\omega \in \Omega ,$ and

$$\begin{aligned} \lim _{t\rightarrow \pm \infty } \frac{|\varvec{y}(\vartheta _t\omega )|}{|t|}= \lim _{t\rightarrow \pm \infty } \frac{1}{t} \int _{0}^{t} \varvec{y}(\vartheta _{\xi }\omega )\textrm{d}\xi =0. \end{aligned}$$

(6.6)

Define

$$\begin{aligned} \varvec{v}(t;\mathfrak {s},\omega ,\varvec{v}_{\mathfrak {s}})=\varvec{u}(t;\mathfrak {s},\omega ,\varvec{u}_{\mathfrak {s}})-e^{\sigma t}{} {\textbf {g}}(x)\varvec{y}(\vartheta _{t}\omega ). \end{aligned}$$

(6.7)

Then, from (6.1), we obtain

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}\frac{\partial {v}}{\partial } t&+\mu {\text {A}}{v}+{\text {B}}({v}+e^{\sigma t}{} {{\textbf {g}}}{y})+\alpha {v}+\beta \mathcal {C}({v}+e^{\sigma t}{} {{\textbf {g}}}{y})\\ {}&\qquad \qquad ={f}+\left( \ell -\sigma -\alpha \right) e^{\sigma t}{} {{\textbf {g}}}{y}-\mu e^{\sigma t}{y}\text {A}{{\textbf {g}}}, \qquad \text {in } \mathbb {R}^d\times (\mathfrak {s},\infty ), \\ {v}|_{t=\mathfrak {s}}&={v}_{\mathfrak {s}}={u}_{\mathfrak {s}}-e^{\sigma \mathfrak {s}}{} {{\textbf {g}}}(x){y}(\omega ), \qquad x\in \mathbb {R}^d \text{ and } \mathfrak {s}\in \mathbb {R}. \end{aligned} \right. \end{aligned} \end{aligned}$$

(6.8)

For all $\mathfrak {s}\in \mathbb {R},$ $t>\mathfrak {s},$ and for every $\varvec{v}_{\mathfrak {s}}\in \mathbb {H}$ and $\omega \in \Omega$, (6.8) has a unique solution $\varvec{v}(\cdot ;\mathfrak {s},\omega ,\varvec{v}_{\mathfrak {s}})\in \textrm{C}([\mathfrak {s},\mathfrak {s}+T];\mathbb {H})\cap \textrm{L}^2(\mathfrak {s}, \mathfrak {s}+T;\mathbb {V})\cap \textrm{L}^{r+1}(\mathfrak {s},\mathfrak {s}+T;\widetilde{\mathbb {L}}^{r+1})$. Moreover, $\varvec{v}(t;\mathfrak {s},\omega ,\varvec{v}_{\mathfrak {s}})$ is continuous with respect to initial data $\varvec{v}_{\mathfrak {s}}$ (Lemma 6.3) and $(\mathscr {F},\mathscr {B}(\mathbb {H}))$-measurable in $\omega \in \Omega .$ Define a cocycle $\Phi _0:\mathbb {R}^+\times \mathbb {R}\times \Omega \times \mathbb {H}\rightarrow \mathbb {H}$ for the system (6.1) such that for given $t\in \mathbb {R}^+, \mathfrak {s}\in \mathbb {R}, \omega \in \Omega$ and $\varvec{u}_{\mathfrak {s}}\in \mathbb {H}$,

$$\begin{aligned} \Phi _0(t,\mathfrak {s},\omega ,\varvec{u}_{\mathfrak {s}})&=\varvec{u}(t+\mathfrak {s};\mathfrak {s},\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}})=\varvec{v}(t+\mathfrak {s};\mathfrak {s},\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}})+e^{\sigma (t+\mathfrak {s})}{} {\textbf {g}}\varvec{y}(\vartheta _{t}\omega ). \end{aligned}$$

(6.9)

Lemma 6.3

For all the cases given in Table 1 including $d=r=3$ with $2\beta \mu =1$ and excluding $d=2$ with $r=1$, assume that $\varvec{f}\in \textrm{L}^2_{loc }(\mathbb {R};\mathbb {V}')$. Then, the solution of (6.8) is continuous in initial data $\varvec{v}_{\mathfrak {s}}.$

Proof

The proof will go through with similar arguments as in the proof of [32, Theorem 3.9]. $\square$

Next, we prove the existence of $\mathfrak {D}$-pullback random absorbing set of $\Phi _0.$

Lemma 6.4

For all the cases given in Table 1 including $d=r=3$ with $2\beta \mu =1$ and excluding $d=2$ with $r=1$, let the Assumption 6.1 be satisfied. Then $\Phi _0$ possesses a closed measurable $\mathfrak {D}$-pullback random absorbing set $\mathcal {K}_0=\{\mathcal {K}^1_0(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R},\ \omega \in \Omega \}\in \mathfrak {D}$ in $\mathbb {H}$ given by

$$\begin{aligned} \mathcal {K}_0(\mathfrak {s},\omega )=\{\varvec{u}\in \mathbb {H}:\Vert \varvec{u}\Vert ^2_{\mathbb {H}}\le \mathcal {R}_0(\mathfrak {s},\omega )\}, \end{aligned}$$

(6.10)

where $\mathcal {R}_0(\mathfrak {s},\omega )$ is defined by

$$\begin{aligned} \mathcal {R}_0(\mathfrak {s},\omega )&=3\Vert {\textbf {g}}\Vert ^2_{\mathbb {H}}\left| e^{\sigma \mathfrak {s}}\varvec{y}(\omega )\right| ^2+2R_8 \int _{-\infty }^{0}e^{\alpha \xi }\bigg [\Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\right| ^2\nonumber \\&\quad +\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\right| ^{r+1}+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\right| ^{\frac{2(r+1)}{r-1}}\bigg ]\textrm{d}\xi . \end{aligned}$$

(6.11)

Here $R_8$ is a positive constant which does not depend on $\mathfrak {s}$ and $\omega$.

Proof

We infer from (6.8) that

$$\begin{aligned} \frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\Vert \varvec{v}\Vert ^2_{\mathbb {H}}=&-\mu \Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}-\alpha \Vert \varvec{v}\Vert ^2_{\mathbb {H}}-\beta \Vert \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\nonumber \\&\quad +b(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y},\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y},e^{\sigma t}{} {\textbf {g}}\varvec{y})\nonumber \\&\quad +\beta \left\langle \mathcal {C}(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}),e^{\sigma t}{} {\textbf {g}}\varvec{y}\right\rangle +\left\langle \varvec{f},\varvec{v}\right\rangle +e^{\sigma t}\varvec{y}\left( \left( \ell -\sigma -\alpha \right) {\textbf {g}}-\mu \textrm{A}{\textbf {g}},\varvec{v}\right) . \end{aligned}$$

(6.12)

Applying Lemmas 2.9 and 2.11, we deduce that there exist constants $R_1,R_2,R_3>0$ such that

$$\begin{aligned}&\beta \left\langle \mathcal {C}(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}),e^{\sigma t}{} {\textbf {g}}\varvec{y}\right\rangle \le \beta \left| e^{\sigma t}\varvec{y}\right| \Vert \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\Vert ^{r}_{\widetilde{\mathbb {L}}^{r+1}}\Vert {\textbf {g}}\Vert _{\widetilde{\mathbb {L}}^{r+1}}\nonumber \\&\quad \le \frac{\beta }{4}\Vert \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}} + R_1\left| e^{\sigma t}\varvec{y}\right| ^{r+1}, \end{aligned}$$

(6.13)

$$\begin{aligned}&\left\langle \varvec{f},\varvec{v}\right\rangle \le \Vert \varvec{f}\Vert _{\mathbb {V}'}\Vert \varvec{v}\Vert _{\mathbb {V}}\le \frac{\min \{\alpha ,\mu \}}{6}\Vert \varvec{v}\Vert ^2_{\mathbb {V}}+R_2\Vert \varvec{f}\Vert ^{2}_{\mathbb {V}'}, \end{aligned}$$

(6.14)

$$\begin{aligned}&e^{\sigma t}\varvec{y}\left( \left( \ell -\sigma -\alpha \right) {\textbf {g}}-\mu \textrm{A}{\textbf {g}},\varvec{v}\right) = \left( \ell -\sigma -\alpha \right) e^{\sigma t}\varvec{y}\left( {\textbf {g}},\varvec{v}\right) +\mu e^{\sigma t}\varvec{y}\left( \nabla {\textbf {g}},\nabla \varvec{v}\right) \nonumber \\&\quad \le \frac{\alpha }{6}\Vert \varvec{v}\Vert ^2_{\mathbb {H}}+\frac{\mu }{6}\Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}+R_3\left| e^{\sigma t}\varvec{y}\right| ^2. \end{aligned}$$

(6.15)

Case I: When $d=2$ and $r>1$. Using (2.1), Lemmas 2.9 and 2.11, and Sobolev’s inequality, we obtain that there exists a constant $R_4>0$ such that

$$\begin{aligned} \left| b(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y},\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y},e^{\sigma t}{} {\textbf {g}}\varvec{y})\right|&= \left| b(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y},e^{\sigma t}{} {\textbf {g}}\varvec{y},\varvec{v})\right| \nonumber \\&\le \left| e^{\sigma t}\varvec{y}\right| \Vert \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\Vert _{\widetilde{\mathbb {L}}^{r+1}}\Vert \nabla {\textbf {g}}\Vert _{\mathbb {H}}\Vert \varvec{v}\Vert _{\widetilde{\mathbb {L}}^{\frac{2(r+1)}{r-1}}}\nonumber \\&\le \frac{\beta }{4}\Vert \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}} +\frac{\min \{\alpha ,\mu \}}{6}\Vert \varvec{v}\Vert ^{2}_{\mathbb {V}}+R_4\left| e^{\sigma t}\varvec{y}\right| ^{\frac{2(r+1)}{r-1}}. \end{aligned}$$

(6.16)

Combining (6.12)–(6.16), we finally get

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\Vert \varvec{v}\Vert ^2_{\mathbb {H}}+\alpha \Vert \varvec{v}\Vert ^2_{\mathbb {H}}\le R_5\left[ \Vert \varvec{f}\Vert ^{2}_{\mathbb {V}'}+\left| e^{\sigma t}\varvec{y}\right| ^2+\left| e^{\sigma t}\varvec{y}\right| ^{r+1}+\left| e^{\sigma t}\varvec{y}\right| ^{\frac{2(r+1)}{r-1}}\right] , \end{aligned}$$

(6.17)

where $R_5=\max \{2R_1,2R_2,2R_3,2R_4\}$.

Case II: When $d= 3$ and $r\ge 3$ ($r>3$ with any $\beta ,\mu >0$ and $r=3$ with $2\beta \mu \ge 1$). Using (2.1), Lemmas 2.9, 2.10 and 2.11, we obtain that there exist two constants $R_6,R_7>0$ such that

$$\begin{aligned} \left| b(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y},\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y},e^{\sigma t}{} {\textbf {g}}\varvec{y})\right|&\le \left| e^{\sigma t}\varvec{y}\right| \Vert \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\Vert _{\widetilde{\mathbb {L}}^{r+1}}\Vert \nabla {\textbf {g}}\Vert _{\mathbb {H}}\Vert \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\Vert _{\widetilde{\mathbb {L}}^{\frac{2(r+1)}{r-1}}}\nonumber \\&\le \left| e^{\sigma t}\varvec{y}\right| \Vert \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\Vert ^{\frac{r+1}{r-1}}_{\widetilde{\mathbb {L}}^{r+1}}\Vert \nabla {\textbf {g}}\Vert _{\mathbb {H}}\Vert \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\Vert ^{\frac{r-3}{r-1}}_{\mathbb {H}}\nonumber \\&\le \frac{\beta }{4}\Vert \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}} +\frac{\alpha }{6}\Vert \varvec{v}\Vert ^{2}_{\mathbb {H}}+R_6\left| e^{\sigma t}\varvec{y}\right| ^{2},\nonumber \\&\quad \ \ \text { for } r>3, \end{aligned}$$

(6.18)

and

$$\begin{aligned} \left| b(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y},\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y},e^{\sigma t}{} {\textbf {g}}\varvec{y})\right|&\le \left| e^{\sigma t}\varvec{y}\right| \Vert \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\Vert ^2_{\widetilde{\mathbb {L}}^{4}}\Vert \nabla {\textbf {g}}\Vert _{\mathbb {H}}\nonumber \\&\le \frac{\beta }{4}\Vert \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\Vert ^{4}_{\widetilde{\mathbb {L}}^{4}} +R_7\left| e^{\sigma t}\varvec{y}\right| ^{2}, \ \ \text { for } r=3. \end{aligned}$$

(6.19)

Combining (6.12)–(6.15) and (6.18)–(6.19), we get

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\Vert \varvec{v}\Vert ^2_{\mathbb {H}}+\alpha \Vert \varvec{v}\Vert ^2_{\mathbb {H}}&\le R_8\left[ \Vert \varvec{f}\Vert ^{2}_{\mathbb {V}'}+\left| e^{\sigma t}\varvec{y}\right| ^2+\left| e^{\sigma t}\varvec{y}\right| ^{r+1}\right] , \end{aligned}$$

(6.20)

where $R_8=\max \{2R_1,2R_2,2(R_3+R_6),2(R_3+R_7),R_5\}$. Therefore, for both Cases I and II, we have

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\Vert \varvec{v}\Vert ^2_{\mathbb {H}}+\alpha \Vert \varvec{v}\Vert ^2_{\mathbb {H}}\le R_8\left[ \Vert \varvec{f}\Vert ^{2}_{\mathbb {V}'}+\left| e^{\sigma t}\varvec{y}\right| ^2+\left| e^{\sigma t}\varvec{y}\right| ^{r+1}+\left| e^{\sigma t}\varvec{y}\right| ^{\frac{2(r+1)}{r-1}}\right] . \end{aligned}$$

(6.21)

Applying the variation of constant formula to (6.21) over $(\mathfrak {s}-t,\tau )$ with $t\ge 0$, $\mathfrak {s}\in \mathbb {R}$ and $\tau \ge \mathfrak {s}-t$, and replacing $\omega$ by $\vartheta _{-\mathfrak {s}}\omega$, we obtain

$$\begin{aligned}&\Vert \varvec{v}(\tau ,\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}} \nonumber \\&\quad \le e^{\alpha (\mathfrak {s}-t-\tau )}\Vert \varvec{v}_{\mathfrak {s}-t}\Vert ^2_{\mathbb {H}}+R_5 \int _{-\infty }^{\tau -\mathfrak {s}}e^{\alpha (\xi +\mathfrak {s}-\tau )}\bigg [\Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}\nonumber \\&\qquad +\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\right| ^2+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\right| ^{r+1}+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\right| ^{\frac{2(r+1)}{r-1}}\bigg ]\textrm{d}\xi . \end{aligned}$$

(6.22)

From (6.9), we have

$$\begin{aligned} \varvec{u}(\mathfrak {s},\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})=\varvec{v}(\mathfrak {s},\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{\mathfrak {s}-t})+e^{\sigma \mathfrak {s}}\varvec{y}(\omega ){\textbf {g}}, \end{aligned}$$

with $\varvec{v}_{\mathfrak {s}-t}=\varvec{u}_{\mathfrak {s}-t}-e^{\sigma (\mathfrak {s}-t)}\varvec{y}(\vartheta _{-t}\omega ){\textbf {g}}$, $\varvec{u}_{\mathfrak {s}-t}\in D(\mathfrak {s}-t,\vartheta _{-t}\omega )$ and $D\in \mathfrak {D}$, which together with (6.22) gives

$$\begin{aligned}&\Vert \varvec{u}(\mathfrak {s},\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}}\nonumber \\&\quad \le 2\Vert \varvec{v}(\mathfrak {s},\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{\mathfrak {s}-t})\Vert ^2_{\mathbb {H}}+2\Vert {\textbf {g}}\Vert ^2_{\mathbb {H}}\left| e^{\sigma \mathfrak {s}}\varvec{y}(\omega )\right| ^2 \nonumber \\&\quad \le 4e^{-\alpha t}\left( \Vert \varvec{u}_{\mathfrak {s}-t}\Vert ^2_{\mathbb {H}}+\Vert {\textbf {g}}\Vert ^2_{\mathbb {H}}\left| e^{\sigma (\mathfrak {s}-t)}\varvec{y}(\vartheta _{-t}\omega )\right| ^2\right) +2\Vert {\textbf {g}}\Vert ^2_{\mathbb {H}}\left| e^{\sigma \mathfrak {s}}\varvec{y}(\omega )\right| ^2+2R_5 \int _{-\infty }^{0}e^{\alpha \xi }\nonumber \\&\qquad \times \bigg [\Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\right| ^2+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\right| ^{r+1}\nonumber \\&\qquad +\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\right| ^{\frac{2(r+1)}{r-1}}\bigg ]\textrm{d}\xi . \end{aligned}$$

(6.23)

Using (6.4) and (6.6), and arguing similarly as in the proofs of Lemmas 4.10 and 4.11, one can conclude the proof. $\square$

The next two Corollaries 6.5 and 6.6 help us to prove the asymptotic compactness of $\Phi _0$ in this subsection (cf. Lemma 6.8) as well as the uniform compactness of pullback random attractors in the next subsection (cf. Lemma 6.14).

Corollary 6.5

Let $\{\varvec{v}_m\}_{m\in \mathbb {N}}\subset \textrm{L}^{\infty }(0, T; \mathbb {H})\cap \textrm{L}^2(0, T;\mathbb {V})\cap \textrm{L}^{r+1}(0, T;\widetilde{\mathbb {L}}^{r+1})$ be a bounded sequence and $\varvec{v}\in \textrm{L}^{\infty }(0, T; \mathbb {H})\cap \textrm{L}^2(0, T;\mathbb {V})\cap \textrm{L}^{r+1}(0, T;\widetilde{\mathbb {L}}^{r+1})$ be such that

(6.24)

Then, for all the cases given in Table 1 including $d=r=3$ with $2\beta \mu =1$ and excluding $d=2$ with $r=1$,

$$\begin{aligned} \int _{0}^{T} b(\varvec{v}_m(t), \varvec{v}_m(t), \uprho (t){\textbf {g}}) \textrm{d}t \rightarrow \int _{0}^{T} b(\varvec{v}(t), \varvec{v}(t), \uprho (t){\textbf {g}}) \textrm{d}t\ \ \text { as }\ \ m\rightarrow \infty , \end{aligned}$$

(6.25)

where $\uprho (t)$ is a continuous function of t on [0, T].

Proof

Since $\varvec{v}_m, \varvec{v}\in \textrm{L}^{\infty }(0, T; \mathbb {H})\cap \textrm{L}^2(0, T;\mathbb {V})\cap \textrm{L}^{r+1}(0, T;\widetilde{\mathbb {L}}^{r+1})$ and $\uprho (t)$ is continuous, there exists a constant $\varpi >0$ such that

$$\begin{aligned}&\sup _{t \in [0, T]} \Vert \varvec{v}_m(t)\Vert _{\mathbb {H}} +\sup _{t \in [0, T]} \Vert \varvec{v}(t)\Vert _{\mathbb {H}} + \left( \int _{0}^{T}\Vert \varvec{v}_m(t)\Vert ^2_{\mathbb {V}} \textrm{d}t\right) ^{\frac{1}{2}} + \left( \int _{0}^{T}\Vert \varvec{v}(t)\Vert ^2_{\mathbb {V}}\textrm{d}t\right) ^{\frac{1}{2}}\\&\quad +\left( \int _{0}^{T}\Vert \varvec{v}_m(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}} \textrm{d}t\right) ^{\frac{1}{r+1}} + \left( \int _{0}^{T}\Vert \varvec{v}(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\textrm{d}t\right) ^{\frac{1}{r+1}} + \left( \int _{0}^{T}\left| \uprho (t)\right| ^2\textrm{d}t\right) ^{\frac{1}{2}} \le \varpi . \end{aligned}$$

Since ${\textbf {g}}\in \textrm{D}(\textrm{A})$, for given $\varepsilon >0$, we can say that there exists $k=k(\varepsilon ,{\textbf {g}})>0$ such that

$$\begin{aligned} \int _{\mathbb {R}^d\backslash \mathcal {O}_k}\left( |{\textbf {g}}(x)|^2+|\nabla {\textbf {g}}(x)|^2\right) \textrm{d}x \le \frac{\varepsilon ^2}{256 \varpi ^6}, \ \ \ \text { for } d=2, \end{aligned}$$

(6.26)

and

$$\begin{aligned} \int _{\mathbb {R}^d\backslash \mathcal {O}_k}\left( |{\textbf {g}}(x)|^2+|\nabla {\textbf {g}}(x)|^2\right) \textrm{d}x \le \frac{\varepsilon ^2}{64 \varpi ^6}, \ \ \ \text { for } d=3, \end{aligned}$$

(6.27)

where $\mathcal {O}_k=\{x\in \mathbb {R}^d:|x|<k\}$. Let us consider, for $\varvec{u},\varvec{v},\varvec{w}\in \mathbb {V}$,

$$\begin{aligned} b(\varvec{u},\varvec{v},\varvec{w})=b_1(\varvec{u},\varvec{v},\varvec{w})+b_2(\varvec{u},\varvec{v},\varvec{w}) \end{aligned}$$

where

$$\begin{aligned} b_1(\varvec{u},\varvec{v},\varvec{w})=\sum _{i,j=1}^d\int _{\mathcal {O}_k}\varvec{u}_i\frac{\partial \varvec{v}_j}{\partial x_i}\varvec{w}_j\textrm{d}x \ \ \text { and } \ \ b_2(\varvec{u},\varvec{v},\varvec{w})=\sum _{i,j=1}^d\int _{\mathbb {R}^d\backslash \mathcal {O}_k}\varvec{u}_i\frac{\partial \varvec{v}_j}{\partial x_i}\varvec{w}_j\textrm{d}x. \end{aligned}$$

Consider,

$$\begin{aligned}&\int _{0}^{T} b(\varvec{v}_m(t), \varvec{v}_m(t), \uprho (t){\textbf {g}}) \textrm{d}t-\int _{0}^{T} b(\varvec{v}(t), \varvec{v}(t), \uprho (t){\textbf {g}}) \textrm{d}t \nonumber \\&\quad =\int _{0}^{T} b_1(\varvec{v}_m(t)-\varvec{v}(t), \varvec{v}_m(t), \uprho (t){\textbf {g}}) \textrm{d}t +\int _{0}^{T} b_2(\varvec{v}_m(t)-\varvec{v}(t), \varvec{v}_m(t), \uprho (t){\textbf {g}}) \textrm{d}t \nonumber \\&\qquad +\int _{0}^{T} b_1(\varvec{v}(t), \varvec{v}_m(t)-\varvec{v}(t), \uprho (t){\textbf {g}}) \textrm{d}t +\int _{0}^{T} b_2(\varvec{v}(t), \varvec{v}_m(t)-\varvec{v}(t), \uprho (t){\textbf {g}}) \textrm{d}t . \end{aligned}$$

(6.28)

In order to prove (6.25), it is enough to show that each term on the right hand side of (6.28) converges to 0 as $m\rightarrow \infty$.

Case I: $d=2$ and $r>1$. By (2.2) and (6.26), we obtain

$$\begin{aligned}&\left| \int _{0}^{T} b_2( \varvec{v}_m(t)-\varvec{v}(t),\varvec{v}_m(t), \uprho (t){\textbf {g}}) \textrm{d}t\right| \nonumber \\&\quad \le \frac{\varepsilon }{8\varpi ^3}\int _{0}^{T}\left| \uprho (t)\right| \left[ \Vert \varvec{v}_m(t)\Vert _{\mathbb {H}}\Vert \nabla \varvec{v}_m(t)\Vert _{\mathbb {H}}+\Vert \varvec{v}(t)\Vert ^{\frac{1}{2}}_{\mathbb {H}}\Vert \nabla \varvec{v}(t)\Vert ^{\frac{1}{2}}_{\mathbb {H}}\Vert \varvec{v}_m(t)\Vert ^{\frac{1}{2}}_{\mathbb {H}}\Vert \nabla \varvec{v}_m(t)\Vert ^{\frac{1}{2}}_{\mathbb {H}}\right] \textrm{d}t \nonumber \\&\quad \le \frac{\varepsilon }{8\varpi ^3}\sup _{t \in [0, T]} \Vert \varvec{v}_m(t)\Vert _{\mathbb {H}}\left( \int _{0}^{T}\left| \uprho (t)\right| ^2\textrm{d}t\right) ^{\frac{1}{2}}\nonumber \\&\quad \left( \int _{0}^{T}\Vert \nabla \varvec{v}_m(t)\Vert ^2_{\mathbb {H}}\textrm{d}t\right) ^{\frac{1}{2}}+ \frac{\varepsilon }{8\varpi ^3}\sup _{t \in [0, T]} \Vert \varvec{v}(t)\Vert ^{\frac{1}{2}}_{\mathbb {H}}\nonumber \\&\quad \qquad \times \sup _{t \in [0, T]} \Vert \varvec{v}_m(t)\Vert ^{\frac{1}{2}}_{\mathbb {H}}\left( \int _{0}^{T}\left| \uprho (t)\right| ^2\textrm{d}t\right) ^{\frac{1}{2}}\left( \int _{0}^{T}\Vert \nabla \varvec{v}(t)\Vert ^2_{\mathbb {H}}\textrm{d}t\right) ^{\frac{1}{4}} \left( \int _{0}^{T}\Vert \nabla \varvec{v}_m(t)\Vert ^2_{\mathbb {H}}\textrm{d}t\right) ^{\frac{1}{4}}\nonumber \\&\quad \le \frac{\varepsilon }{4}. \end{aligned}$$

(6.29)

By (2.2) and continuity of $\uprho (t)$, we have

$$\begin{aligned}&\left| \int _{0}^{T} b_1( \varvec{v}_m(t)-\varvec{v}(t),\varvec{v}_m(t), \uprho (t){\textbf {g}}) \textrm{d}t\right| \nonumber \\&\quad \le C\Vert \nabla {\textbf {g}}\Vert _{\mathbb {H}}\int _{0}^{T}\left| \uprho (t)\right| \Vert \varvec{v}_m(t)-\varvec{v}(t)\Vert ^{\frac{1}{2}}_{\mathbb {L}^2(\mathcal {O}_k)}\Vert \nabla \left( \varvec{v}_m(t)\right. \nonumber \\&\qquad \left. -\varvec{v}(t)\right) \Vert ^{\frac{1}{2}}_{\mathbb {H}}\Vert \varvec{v}_m(t)\Vert ^{\frac{1}{2}}_{\mathbb {H}} \Vert \nabla \varvec{v}_m(t)\Vert ^{\frac{1}{2}}_{\mathbb {H}}\textrm{d}t\nonumber \\&\quad \le CT^{\frac{1}{4}}\sup _{t \in [0, T]}\left[ \left| \uprho (t)\right| \Vert \varvec{v}_m(t)\Vert ^{\frac{1}{2}}_{\mathbb {H}}\right] \left( \int _{0}^{T}\Vert \varvec{v}_m(t)-\varvec{v}(t)\Vert ^2_{\mathbb {L}^2(\mathcal {O}_k)}\textrm{d}t\right) ^{\frac{1}{4}}\nonumber \\&\qquad \bigg [\left( \int _{0}^{T}\Vert \nabla \varvec{v}_m(t)\Vert ^2_{\mathbb {H}}\textrm{d}t\right) ^{\frac{1}{2}}\nonumber \\&\qquad +\left( \int _{0}^{T}\Vert \nabla \varvec{v}(t)\Vert ^2_{\mathbb {H}}\textrm{d}t\right) ^{\frac{1}{4}}\left( \int _{0}^{T}\Vert \nabla \varvec{v}_m(t)\Vert ^2_{\mathbb {H}}\textrm{d}t\right) ^{\frac{1}{4}}\bigg ]\nonumber \\&\quad \rightarrow 0 \ \text { as }\ m\rightarrow \infty . \end{aligned}$$

(6.30)

Case II: $d= 3$ and $r\ge 3$ ($r>3$ with any $\beta ,\mu >0$ and $r=3$ with $2\beta \mu \ge 1$). By (6.27), Lemmas 2.9 and 2.10, we have

$$\begin{aligned}&\left| \int _{0}^{T} b_2( \varvec{v}_m(t)-\varvec{v}(t),\varvec{v}_m(t), \uprho (t){\textbf {g}}) \textrm{d}t\right| \nonumber \\&\quad \le \frac{\varepsilon }{8\varpi ^3}\int _{0}^{T}\left| \uprho (t)\right| \left[ \Vert \varvec{v}_m(t)\Vert _{\widetilde{\mathbb {L}}^{r+1}}\Vert \varvec{v}_m(t)\Vert _{\widetilde{\mathbb {L}}^{\frac{2(r+1)}{r-1}}}+\Vert \varvec{v}(t)\Vert _{\widetilde{\mathbb {L}}^{r+1}}\Vert \varvec{v}_m(t)\Vert _{\widetilde{\mathbb {L}}^{\frac{2(r+1)}{r-1}}}\right] \textrm{d}t\nonumber \\&\quad \le \frac{\varepsilon }{8\varpi ^3}\int _{0}^{T}\left| \uprho (t)\right| \left[ \Vert \varvec{v}_m(t)\Vert ^{\frac{r+1}{r-1}}_{\widetilde{\mathbb {L}}^{r+1}}\Vert \varvec{v}_m(t)\Vert ^{\frac{r-3}{r-1}}_{\mathbb {H}}+\Vert \varvec{v}(t)\Vert _{\widetilde{\mathbb {L}}^{r+1}}\Vert \varvec{v}_m(t)\Vert ^{\frac{2}{r-1}}_{\widetilde{\mathbb {L}}^{r+1}}\Vert \varvec{v}_m(t)\Vert ^{\frac{r-3}{r-1}}_{\mathbb {H}}\right] \textrm{d}t \nonumber \\&\quad \le \frac{\varepsilon }{8\varpi ^3}\left( \int _{0}^{T}\left| \uprho (t)\right| ^2\textrm{d}t\right) ^{\frac{1}{2}}\left( \int _{0}^{T}\Vert \varvec{v}_m(t)\Vert ^2_{\mathbb {H}}\textrm{d}t\right) ^{\frac{r-3}{2(r-1)}}\nonumber \\&\qquad \bigg [\left( \int _{0}^{T}\Vert \varvec{v}_m(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\textrm{d}t\right) ^{\frac{1}{r-1}}\nonumber \\&\qquad + \left( \int _{0}^{T}\Vert \varvec{v}(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\textrm{d}t\right) ^{\frac{1}{r+1}}\left( \int _{0}^{T}\Vert \varvec{v}_m(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\textrm{d}t\right) ^{\frac{2}{(r+1)(r-1)}}\bigg ]\le \frac{\varepsilon }{4}, \end{aligned}$$

(6.31)

for $r>3$ and

$$\begin{aligned}&\left| \int _{0}^{T} b_2( \varvec{v}_m(t)-\varvec{v}(t),\varvec{v}_m(t), \uprho (t){\textbf {g}}) \textrm{d}t\right| \nonumber \\&\quad \le \frac{\varepsilon }{8\varpi ^3}\int _{0}^{T}\left| \uprho (t)\right| \Vert \varvec{v}_m(t)\Vert ^2_{\widetilde{\mathbb {L}}^{4}}+ \frac{\varepsilon }{8\varpi ^3}\int _{0}^{T}\left| \uprho (t)\right| \Vert \varvec{v}(t)\Vert _{\widetilde{\mathbb {L}}^{4}}\Vert \varvec{v}_m(t)\Vert _{\widetilde{\mathbb {L}}^4}\textrm{d}t\nonumber \\&\quad \le \frac{\varepsilon }{8\varpi ^3}\left( \int _{0}^{T}\left| \uprho (t)\right| ^2\textrm{d}t\right) ^{\frac{1}{2}} \left[ \left( \int _{0}^{T}\Vert \varvec{v}_m(t)\Vert ^{4}_{\widetilde{\mathbb {L}}^{4}}\textrm{d}t\right) ^{\frac{1}{2}} \right. \nonumber \\&\qquad \left. +\,\left( \int _{0}^{T}\Vert \varvec{v}(t)\Vert ^{4}_{\widetilde{\mathbb {L}}^{4}}\textrm{d}t\right) ^{\frac{1}{4}}\left( \int _{0}^{T}\Vert \varvec{v}_m(t)\Vert ^{4}_{\widetilde{\mathbb {L}}^{4}}\textrm{d}t\right) ^{\frac{1}{4}}\right] \nonumber \\&\quad \le \frac{\varepsilon }{4}, \end{aligned}$$

(6.32)

for $r=3$. By (2.2) and continuity of $\uprho (t)$, we have

$$\begin{aligned}&\left| \int _{0}^{T} b_1( \varvec{v}_m(t)-\varvec{v}(t),\varvec{v}_m(t), \uprho (t){\textbf {g}}) \textrm{d}t\right| \nonumber \\&\quad \le C\Vert \nabla {\textbf {g}}\Vert _{\mathbb {H}}\int _{0}^{T}\left| \uprho (t)\right| \Vert \varvec{v}_m(t)-\varvec{v}(t)\Vert ^{\frac{1}{4}}_{\mathbb {L}^2(\mathcal {O}_k)}\Vert \nabla \left( \varvec{v}_m(t)-\varvec{v}(t)\right) \Vert ^{\frac{3}{4}}_{\mathbb {H}}\Vert \varvec{v}_m(t)\Vert ^{\frac{1}{4}}_{\mathbb {H}}\Vert \nabla \varvec{v}_m(t)\Vert ^{\frac{3}{4}}_{\mathbb {H}}\textrm{d}t\nonumber \\&\quad \le CT^{\frac{1}{8}}\sup _{t \in [0, T]}\left[ \left| \uprho (t)\right| \cdot \Vert \varvec{v}_m(t)\Vert ^{\frac{1}{4}}_{\mathbb {H}}\right] \left( \int _{0}^{T}\Vert \varvec{v}_m(t)-\varvec{v}(t)\Vert ^2_{\mathbb {L}^2(\mathcal {O}_k)}\textrm{d}t\right) ^{\frac{1}{8}}\nonumber \\&\qquad \bigg [\left( \int _{0}^{T}\Vert \nabla \varvec{v}_m(t)\Vert ^2_{\mathbb {H}}\textrm{d}t\right) ^{\frac{3}{4}}\nonumber \\&\qquad +\left( \int _{0}^{T}\Vert \nabla \varvec{v}(t)\Vert ^2_{\mathbb {H}}\textrm{d}t\right) ^{\frac{3}{8}}\left( \int _{0}^{T}\Vert \nabla \varvec{v}_m(t)\Vert ^2_{\mathbb {H}}\textrm{d}t\right) ^{\frac{3}{8}}\bigg ]\rightarrow 0 \ \text { as }\ m\rightarrow \infty . \end{aligned}$$

(6.33)

Calculations similar to (6.29) (for $d=2$ and $r>1$), (6.31) (for $d=3$ and $r>3$) and (6.32) (for $d=r=3$), and (6.30) (for $d=2$ and $r>1$) and (6.33) (for $d=3$ and $r\ge 3$), we obtain

$$\begin{aligned}&\left| \int _{0}^{T} b_2( \varvec{v}(t),\varvec{v}_m(t)-\varvec{v}(t), \uprho (t){\textbf {g}}) \textrm{d}t\right| \le \frac{\varepsilon }{4}, \end{aligned}$$

(6.34)

and

$$\begin{aligned} \left| \int _{0}^{T} b_1(\varvec{v}(t), \varvec{v}_m(t)-\varvec{v}(t), \uprho (t){\textbf {g}}) \textrm{d}t\right| \rightarrow 0 \ \text { as }\ m\rightarrow \infty , \end{aligned}$$

(6.35)

respectively. For given $\varepsilon >0$, we infer from (6.30) (for $d=2$ and $r>1$) and (6.33) (for $d=3$ and $r\ge 3$), and (6.35) that there exists $M(\varepsilon )\in \mathbb {N}$ such that

$$\begin{aligned} \left| \int _{0}^{T} b_1( \varvec{v}_m(t)-\varvec{v}(t),\varvec{v}_m(t), \uprho (t){\textbf {g}}) \textrm{d}t\right| \le \frac{\varepsilon }{4}, \end{aligned}$$

(6.36)

and

$$\begin{aligned} \left| \int _{0}^{T} b_1(\varvec{v}(t), \varvec{v}_m(t)-\varvec{v}(t), \uprho (t){\textbf {g}}) \textrm{d}t\right| \le \frac{\varepsilon }{4}, \end{aligned}$$

(6.37)

respectively, for all $m\ge M(\varepsilon ).$ Hence, (6.28), (6.29) (for $d=2$ and $r>1$), (6.31) (for $d=3$ and $r>3$), (6.32) (for $d=r=3$), (6.34) and (6.36)–(6.37) imply that, for a given $\varepsilon >0$, there exists $M(\varepsilon )\in \mathbb {N}$ such that

$$\begin{aligned} \left| \int _{0}^{T} b(\varvec{v}_m(t), \varvec{v}_m(t), \uprho (t){\textbf {g}}) \textrm{d}t-\int _{0}^{T} b(\varvec{v}(t), \varvec{v}(t), \uprho (t){\textbf {g}}) \textrm{d}t \right| \le \varepsilon , \end{aligned}$$

for all $m\ge M(\varepsilon ),$ which completes the proof. $\square$

Corollary 6.6

Let $\{\varvec{v}_m\}_{m\in \mathbb {N}}\subset \textrm{L}^{\infty }(0, T; \mathbb {H})\cap \textrm{L}^2(0, T;\mathbb {V})\cap \textrm{L}^{r+1}(0, T;\widetilde{\mathbb {L}}^{r+1})$ be a bounded sequence and $\varvec{v}\in \textrm{L}^{\infty }(0, T; \mathbb {H})\cap \textrm{L}^2(0, T;\mathbb {V})\cap \textrm{L}^{r+1}(0, T;\widetilde{\mathbb {L}}^{r+1})$ such that (6.24) be satisfied. Then, for all the cases given in Table 1 including $d=r=3$ with $2\beta \mu =1$ and excluding $d=2$ with $r=1$,

$$\begin{aligned} \int _{0}^{T} \big \langle \mathcal {C}(\varvec{v}_m(t)) ,\uprho (t){\textbf {g}} \big \rangle \textrm{d}t \rightarrow \int _{0}^{T} \big \langle \mathcal {C}(\varvec{v}(t)) ,\uprho (t){\textbf {g}} \big \rangle \textrm{d}t,\ \text { as }\ m\rightarrow \infty , \end{aligned}$$

where $\uprho (t)$ is a continuous function of t on [0, T].

Proof

Since $\varvec{v}_m, \varvec{v}\in \textrm{L}^{\infty }(0, T; \mathbb {H})\cap \textrm{L}^2(0, T;\mathbb {V})\cap \textrm{L}^{r+1}(0, T;\widetilde{\mathbb {L}}^{r+1})$ and $\uprho (t)$ is continuous, there exists a constant $\varpi >0$ such that

$$\begin{aligned} \left( \int _{0}^{T}\Vert \varvec{v}_m(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}} \textrm{d}t\right) ^{\frac{1}{r+1}} + \left( \int _{0}^{T}\Vert \varvec{v}(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\textrm{d}t\right) ^{\frac{1}{r+1}} + \left( \int _{0}^{T}\left| \uprho (t)\right| ^{r+1}\textrm{d}t\right) ^{\frac{1}{r+1}} \le \varpi . \end{aligned}$$

Since ${\textbf {g}}\in \textrm{D}(\textrm{A})$, Sobolev’s embeddings ($\mathbb {V}\subset \widetilde{\mathbb {L}}^{r+1}$ for $d=2$ and $\textrm{D}(\textrm{A})\subset \widetilde{\mathbb {L}}^{r+1}$ for $d=3$) imply that ${\textbf {g}}\in \widetilde{\mathbb {L}}^{r+1}$, for all $r\ge 1$. Hence, for given $\varepsilon >0$, there exists $k=k(\varepsilon ,{\textbf {g}})>0$ such that

$$\begin{aligned} \left( \int _{\mathbb {R}^d\backslash \mathcal {O}_k}|{\textbf {g}}(x)|^{r+1}\textrm{d}x\right) ^{\frac{1}{r+1}} \le \frac{\varepsilon }{4 \varpi ^{r+1}}, \end{aligned}$$

(6.38)

where $\mathcal {O}_k=\{x\in \mathbb {R}^d:|x|<k\}$. Consider,

$$\begin{aligned}&\int _{0}^{T} \big \langle \mathcal {C}(\varvec{v}_m(t)) ,\uprho (t){\textbf {g}} \big \rangle \textrm{d}t- \int _{0}^{T} \big \langle \mathcal {C}(\varvec{v}(t)) ,\uprho (t){\textbf {g}} \big \rangle \textrm{d}t\nonumber \\&\quad =\int _{0}^{T}\uprho (t) \left[ \int _{\mathcal {O}_k}\left( \left| \varvec{v}_m(x,t)\right| ^{r-1}\varvec{v}_m(x,t)-\left| \varvec{v}(x,t)\right| ^{r-1}\varvec{v}(x,t)\right) {\textbf {g}}(x) \textrm{d}x\right] \textrm{d}t\nonumber \\&\qquad +\int _{0}^{T}\uprho (t) \left[ \int _{\mathbb {R}^d\backslash \mathcal {O}_k}\left( \left| \varvec{v}_m(x,t)\right| ^{r-1}\varvec{v}_m(x,t)-\left| \varvec{v}(x,t)\right| ^{r-1}\varvec{v}(x,t)\right) {\textbf {g}}(x) \textrm{d}x\right] \textrm{d}t. \end{aligned}$$

(6.39)

From Lemma 2.9 and (6.38), we infer that

$$\begin{aligned}&\left| \int _{0}^{T}\uprho (t) \left[ \int _{\mathbb {R}^d\backslash \mathcal {O}_k}\left( \left| \varvec{v}_m(x,t)\right| ^{r-1}\varvec{v}_m(x,t)-\left| \varvec{v}(x,t)\right| ^{r-1}\varvec{v}(x,t)\right) {\textbf {g}}(x) \textrm{d}x\right] \textrm{d}t\right| \nonumber \\&\quad \le \frac{\varepsilon }{4\varpi ^{r+1}}\int _{0}^{T}\left| \uprho (t)\right| \Vert \varvec{v}_m(t)\Vert ^r_{\widetilde{\mathbb {L}}^{r+1}} \textrm{d}t +\frac{\varepsilon }{4\varpi ^{r+1}}\int _{0}^{T}\left| \uprho (t)\right| \Vert \varvec{v}(t)\Vert ^r_{\widetilde{\mathbb {L}}^{r+1}} \textrm{d}t\nonumber \\&\quad \le \frac{\varepsilon }{4\varpi ^{r+1}} \left( \int _{0}^{T}\left| \uprho (t)\right| ^{r+1}\textrm{d}t\right) ^{\frac{1}{r+1}}\nonumber \\&\quad \left[ \left( \int _{0}^{T}\Vert \varvec{v}_m(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\textrm{d}t\right) ^{\frac{r}{r+1}}+\left( \int _{0}^{T}\Vert \varvec{v}(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\textrm{d}t\right) ^{\frac{r}{r+1}}\right] \le \frac{\varepsilon }{2}. \end{aligned}$$

(6.40)

Using Taylor’s formula, (2.4) and Lemma 2.9, we achieve

$$\begin{aligned}&\left| \int _{0}^{T}\uprho (t) \left[ \int _{\mathcal {O}_k}\left( \left| \varvec{v}_m(x,t)\right| ^{r-1}\varvec{v}_m(x,t)-\left| \varvec{v}(x,t)\right| ^{r-1}\varvec{v}(x,t)\right) {\textbf {g}}(x) \textrm{d}x\right] \textrm{d}t\right| \\&\quad \le C\int _{0}^{T}\left| \uprho (t)\right| \left[ \int _{\mathcal {O}_k}\left( \left| \varvec{v}_m(x,t)\right| ^{r-1}+\left| \varvec{v}(x,t)\right| ^{r-1}\right) \left| \varvec{v}_m(x,t)-\varvec{v}(x,t)\right| \left| {\textbf {g}}(x)\right| \textrm{d}x\right] \textrm{d}t\\&\quad \le C\int _{0}^{T}\left| \uprho (t)\right| \left( \Vert \varvec{v}_m(t)\Vert ^{r-1}_{\widetilde{\mathbb {L}}^{r+1}}+\Vert \varvec{v}(t)\Vert ^{r-1}_{\widetilde{\mathbb {L}}^{r+1}}\right) \Vert \varvec{v}_m(t)-\varvec{v}(t)\Vert _{\mathbb {L}^2(\mathcal {O}_k)} \Vert {\textbf {g}}\Vert _{\widetilde{\mathbb {L}}^{\frac{2(r+1)}{3-r}}}\textrm{d}t\\&\quad \le CT^{\frac{3-r}{2(r+1)}}\sup _{t \in [0, T]}\left| \uprho (t)\right| \left( \Vert \varvec{v}_m\Vert ^{r-1}_{\textrm{L}^{r+1}(0,T;\widetilde{\mathbb {L}}^{r+1})}+\Vert \varvec{v}\Vert ^{r-1}_{\textrm{L}^{r+1}(0,T;\widetilde{\mathbb {L}}^{r+1})}\right) \Vert \varvec{v}_m-\varvec{v}\Vert _{\textrm{L}^2(0,T;\mathbb {L}^2(\mathcal {O}_k))}\nonumber \\ {}&\rightarrow 0 \ \text { as }\ m\rightarrow \infty , \end{aligned}$$

for $1<r<3$, and

$$\begin{aligned}&\left| \int _{0}^{T}\uprho (t) \left[ \int _{\mathcal {O}_k}\left( \left| \varvec{v}_m(x,t)\right| ^{r-1}\varvec{v}_m(x,t)-\left| \varvec{v}(x,t)\right| ^{r-1}\varvec{v}(x,t)\right) {\textbf {g}}(x) \textrm{d}x\right] \textrm{d}t\right| \\&\quad \le C\int _{0}^{T}\left| \uprho (t)\right| \left[ \int _{\mathcal {O}_k}\left( \left| \varvec{v}_m(x,t)\right| ^{r-1}+\left| \varvec{v}(x,t)\right| ^{r-1}\right) \left| \varvec{v}_m(x,t)-\varvec{v}(x,t)\right| \left| {\textbf {g}}(x)\right| \textrm{d}x\right] \textrm{d}t\\&\quad \le C\int _{0}^{T}\left| \uprho (t)\right| \left( \Vert \varvec{v}_m(t)\Vert ^{r-1}_{\widetilde{\mathbb {L}}^{r+1}}+\Vert \varvec{v}(t)\Vert ^{r-1}_{\widetilde{\mathbb {L}}^{r+1}}\right) \Vert \varvec{v}_m(t)-\varvec{v}(t)\Vert ^{\frac{1}{r-1}}_{\mathbb {L}^{2}(\mathcal {O}_k)}\nonumber \\ {}&\qquad \qquad \times \Vert \varvec{v}_m(t)-\varvec{v}(t)\Vert ^{\frac{r-2}{r-1}}_{\widetilde{\mathbb {L}}^{r+1}} \Vert {\textbf {g}}\Vert _{\widetilde{\mathbb {L}}^{2(r+1)}}\textrm{d}t\\&\quad \le CT^{\frac{1}{2(r+1)}}\sup _{t \in [0, T]}\left| \uprho (t)\right| \left( \Vert \varvec{v}_m\Vert ^{r-1}_{\textrm{L}^{r+1}(0,T;\widetilde{\mathbb {L}}^{r+1})}+\Vert \varvec{v}\Vert ^{r-1}_{\textrm{L}^{r+1}(0,T;\widetilde{\mathbb {L}}^{r+1})}\right) \nonumber \\ {}&\qquad \qquad \times \Vert \varvec{v}_m-\varvec{v}\Vert ^{\frac{1}{r-1}}_{\textrm{L}^2(0,T;\mathbb {L}^2(\mathcal {O}_k))}\Vert \varvec{v}_m-\varvec{v}\Vert ^{\frac{r-2}{r-1}}_{\textrm{L}^{r+1}(0,T;\widetilde{\mathbb {L}}^{r+1})} \rightarrow 0 \ \text { as }\ m\rightarrow \infty , \end{aligned}$$

for $r\ge 3$, which implies that, for given $\varepsilon >0$, there exists $M(\varepsilon )\in \mathbb {N}$ such that

$$\begin{aligned} \left| \int _{0}^{T}\uprho (t) \left[ \int _{\mathcal {O}_k}\left( \left| \varvec{v}_m(x,t)\right| ^{r-1}\varvec{v}_m(x,t)-\left| \varvec{v}(x,t)\right| ^{r-1}\varvec{v}(x,t)\right) {\textbf {g}}(x) \textrm{d}x\right] \textrm{d}t\right| \le \frac{\varepsilon }{2}. \end{aligned}$$

(6.41)

Finally, from (6.39) to (6.41), we infer that, for a given $\varepsilon >0$, there exists $M(\varepsilon )\in \mathbb {N}$ such that

$$\begin{aligned} \left| \int _{0}^{T} \big \langle \mathcal {C}(\varvec{v}_m(t)) ,\uprho (t){\textbf {g}} \big \rangle \textrm{d}t- \int _{0}^{T} \big \langle \mathcal {C}(\varvec{v}(t)) ,\uprho (t){\textbf {g}} \big \rangle \textrm{d}t\right| \le \varepsilon , \end{aligned}$$

for all $m\ge M(\varepsilon ),$ which completes the proof. $\square$

Lemma 6.7

Let all the assumptions of Lemma 6.3 be satisfied. Let $\mathfrak {s}\in \mathbb {R}, \omega \in \Omega$ and $\varvec{v}_{\mathfrak {s}}^0, \varvec{v}_{\mathfrak {s}}^n\in \mathbb {H}$ for all $n\in \mathbb {N}.$ If in $\mathbb {H}$, then the solution $\varvec{v}$ of the system (6.8) satisfies the following convergences:

• (i) in $\mathbb {H}$ for all $\xi \ge \mathfrak {s}$.

• (ii) in $\textrm{L}^2((\mathfrak {s},\mathfrak {s}+T);\mathbb {V})\cap \textrm{L}^{r+1}((\mathfrak {s},\mathfrak {s}+T);\widetilde{\mathbb {L}}^{r+1})$ for every $T>0$.

• (iii) $\varvec{v}(\cdot ;\mathfrak {s},\omega ,\varvec{v}_{\mathfrak {s}}^n)\rightarrow \varvec{v}(\cdot ;\mathfrak {s},\omega ,\varvec{v}_{\mathfrak {s}}^0)$ in $\textrm{L}^2((\mathfrak {s},\mathfrak {s}+T);\mathbb {L}^2(\mathcal {O}_k))$ for every $T>0$ and $k>0$, where $\mathcal {O}_k=\{x\in \mathbb {R}^d:|x|<k\}$.

Proof

Using the standard method as in [33] (cf. [33, Lemmas 5.2 and 5.3]), one can complete the proof. Note that, due to compactness of Sobolev embedding in bounded domains, we get the strong convergence in $\textrm{L}^2((\mathfrak {s},\mathfrak {s}+T);\mathbb {L}^2(\mathcal {O}_k))$. $\square$

Lemma 6.8

Let all the assumptions of Lemma 6.4 be satisfied. Then for every $\mathfrak {s}\in \mathbb {R},$ $\omega \in \Omega ,$ $D=\{D(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R},\omega \in \Omega \}\in \mathfrak {D}$ and $t_n\rightarrow \infty ,$ $\varvec{u}_{0,n}\in D(\mathfrak {s}-t_n, \vartheta _{-t_{n}}\omega )$, the sequence $\varvec{u}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})$ of solutions of the system (6.1) has a convergent subsequence in $\mathbb {H}$.

Proof

We infer from (6.23) that there exists $\mathfrak {T}=\mathfrak {T}(\mathfrak {s},\omega ,D)>0$ such that for all $t\ge \mathfrak {T}$, $\varvec{u}(\mathfrak {s};\mathfrak {s}-t,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\mathfrak {s}-t})\in \mathbb {H}$ with $\varvec{u}_{\mathfrak {s}-t}\in D(\mathfrak {s}-t,\vartheta _{-t}\omega ).$ Since $t_n\rightarrow \infty$, there exists $N_5\in \mathbb {N}$ such that $t_n\ge \mathfrak {T}$ for all $n\ge N_5$. It is given that $\varvec{u}_{0,n}\in D(\mathfrak {s}-t_n, \vartheta _{-t_{n}}\omega )$, we get $\{\varvec{u}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})\}_{n\ge N_5}$ is a bounded sequence in $\mathbb {H}$. Due to

$$\begin{aligned} \varvec{u}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})=\varvec{v}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{0,n}) +e^{\sigma \mathfrak {s}}\varvec{y}(\omega ){\textbf {g}}, \end{aligned}$$

(6.42)

with $\varvec{v}_{0,n}=\varvec{u}_{0,n}-e^{\sigma (\mathfrak {s}-t)}\varvec{y}(\vartheta _{-t}\omega ){\textbf {g}}$, we have that $\{\varvec{v}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{0,n})\}_{n\ge N_5}$ is also a bounded sequence in $\mathbb {H}$, which implies that there exists $\tilde{\varvec{v}}\in \mathbb {H}$ and a subsequence (not relabeling) such that

(6.43)

which gives

$$\begin{aligned} \Vert \tilde{\varvec{v}}\Vert _{\mathbb {H}}\le \liminf _{n\rightarrow \infty }\Vert \varvec{v}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{0,n})\Vert _{\mathbb {H}}. \end{aligned}$$

(6.44)

For our purpose, we need to show that $\varvec{v}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{0,n})\rightarrow \tilde{\varvec{v}}$ in $\mathbb {H}$ strongly. Along with the above expression, it is enough to prove that

$$\begin{aligned} \Vert \tilde{\varvec{v}}\Vert _{\mathbb {H}}\ge \limsup _{n\rightarrow \infty }\Vert \varvec{v}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{0,n})\Vert _{\mathbb {H}}. \end{aligned}$$

(6.45)

Now, for a given $j\in \mathbb {N}$ ($j\le t_n$), we have

$$\begin{aligned} \varvec{v}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{0,n})=\varvec{v}(\mathfrak {s};\mathfrak {s}-j,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}(\mathfrak {s}-j;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{0,n})). \end{aligned}$$

(6.46)

For each j, let $N_j$ be sufficiently large such that $t_n\ge \mathfrak {T}+j$ for all $n\ge N_j$. From (6.22) with $\tau =\mathfrak {s}-j$, we have that the sequence $\{\varvec{v}(\mathfrak {s}-j;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{0,n})\}_{n\ge N_j}$ is bounded in $\mathbb {H}$, for each $j\in \mathbb {N}$. By a diagonal argument, there exists a subsequence (denoting by same label) and $\tilde{\varvec{u}}_{j}\in \mathbb {H}$ for each $j\in \mathbb {N}$ such that

(6.47)

From (6.46) to (6.47) together with Lemma 6.7, we have that for $j\in \mathbb {N}$,

(6.48)

(6.49)

(6.50)

and

$$\begin{aligned}&\varvec{v}(\cdot ;\mathfrak {s}-j,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}(\mathfrak {s}-j;\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{0,n}))\rightarrow \varvec{v}(\cdot ;\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\tilde{\varvec{v}}_{j}) \ \text { in } \nonumber \\&\quad \textrm{L}^2((\mathfrak {s}-j,\mathfrak {s});\mathbb {L}^2(\mathcal {O}_k)), \end{aligned}$$

(6.51)

where $\mathcal {O}_k=\{x\in \mathbb {R}^d:|x|<k\}$. Clearly, (6.43) and (6.48) imply that

$$\begin{aligned} \varvec{v}(\mathfrak {s};\mathfrak {s}-j,\vartheta _{-\mathfrak {s}}\omega ,\tilde{\varvec{v}}_{j})=\tilde{\varvec{v}}. \end{aligned}$$

(6.52)

From (6.12) we have

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t} \Vert \varvec{v}\Vert ^2_{\mathbb {H}} + \alpha \Vert \varvec{v}\Vert ^2_{\mathbb {H}}\nonumber \\&\quad =-2\mu \Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}-\alpha \Vert \varvec{v}\Vert ^2_{\mathbb {H}} - 2\beta \Vert \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}} +2\beta \left\langle \mathcal {C}(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}),e^{\sigma t}{} {\textbf {g}}\varvec{y}\right\rangle \nonumber \\&\qquad +2b(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y},\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y},e^{\sigma t}{} {\textbf {g}}\varvec{y})+2\left\langle \varvec{f},\varvec{v}\right\rangle +2e^{\sigma t}\varvec{y}\left( \left( \ell -\sigma -\alpha \right) {\textbf {g}}-\mu \textrm{A}{\textbf {g}},\varvec{v}\right) . \end{aligned}$$

(6.53)

Now, following the similar arguments (that is, the method of energy equation) as in Lemma 6.14 below together with the convergence in (6.48)–(6.51), and Corollaries 6.5 and 6.6, we obtain that $\varvec{v}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{0,n})\rightarrow \tilde{\varvec{v}}$ strongly in $\mathbb {H}$. Hence, from (6.42), we infer that $\varvec{u}(\mathfrak {s};\mathfrak {s}-t_n,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{0,n})$ has a convergent subsequence in $\mathbb {H}$, as required. $\square$

The following theorem demonstrates the existence of a unique $\mathfrak {D}$-pullback random attractor for the system (6.1).

Theorem 6.9

Let all the assumptions of Lemma 6.4 be satisfied. Then there exists a unique $\mathfrak {D}$-pullback random attractor

$$\begin{aligned} \mathscr {A}_0=\{\mathscr {A}_0(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R}, \omega \in \Omega \}\in \mathfrak {D}, \end{aligned}$$

for the the continuous cocycle $\Phi _0$ associated with the system (6.1), in $\mathbb {H}$.

Proof

The proof follows from Lemmas 4.3, 4.4 and the abstract theory available in [63, Theorem 2.23]. $\square$

6.2 Upper Semicontinuity of Pullback Random Attractors for Stochastic CBF Equations with Additive Colored Noise

Consider the random equation driven by colored noise

$$\begin{aligned} \frac{\textrm{d}\varvec{z} _{\delta }}{\textrm{d}t}=-\ell \varvec{z} _{\delta } + \mathcal {Z}_{\delta }(\vartheta _t\omega ). \end{aligned}$$

(6.54)

From (2.9), it is clear that for all $\omega \in \Omega$, the integral

$$\begin{aligned} \varvec{z} _{\delta }(\omega )=\int _{-\infty }^{0}e^{\ell \tau }\mathcal {Z}_{\delta }(\vartheta _{\tau }\omega )\textrm{d}\tau \end{aligned}$$

(6.55)

is finite, and hence $\varvec{z} _{\delta }:\Omega \rightarrow \mathbb {R}$ is a well defined random variable. Let us recall some properties of $\varvec{z} _{\delta }$ from [23].

Lemma 6.10

(Lemma 3.2, [23]) Let $\varvec{z} _{\delta }$ be the random variable given by (6.55). Then the mapping

$$\begin{aligned} (t,\omega )\mapsto \varvec{z} _{\delta }(\vartheta _{t}\omega )=\int _{-\infty }^{t}e^{-\ell ( t-\tau )}\mathcal {Z}_{\delta }(\vartheta _{\tau }\omega )\textrm{d}\tau \end{aligned}$$

(6.56)

is a stationary solution of (6.54) with continuous paths. Moreover, $\mathbb {E}[\varvec{z} _{\delta }]=0$ and for every $\omega \in \Omega$,

$$\begin{aligned}&\lim _{\delta \rightarrow 0}\varvec{z} _{\delta }(\vartheta _{t}\omega )=\varvec{y}(\vartheta _{t}\omega ) \text { uniformly on } [\mathfrak {s},\mathfrak {s}+T]\text { with } \mathfrak {s}\in \mathbb {R}\text { and } T>0; \end{aligned}$$

(6.57)

$$\begin{aligned}&\lim _{t\rightarrow \pm \infty }\frac{\left| \varvec{z} _{\delta }(\vartheta _{t}\omega )\right| }{|t|}=0 \text { uniformly for } 0<\delta \le \tilde{\ell }; \end{aligned}$$

(6.58)

where $\tilde{\ell }=\min \{1,\frac{1}{2\ell }\}$.

Define

$$\begin{aligned} \varvec{v}_{\delta }(t,\mathfrak {s},\omega ,\varvec{v}_{\delta ,\mathfrak {s}})=\varvec{u}_{\delta }(t,\mathfrak {s},\omega ,\varvec{u}_{\delta ,\mathfrak {s}})-e^{\sigma t}{} {\textbf {g}}(x)\varvec{z} _{\delta }(\vartheta _{t}\omega ). \end{aligned}$$

(6.59)

Then, from (6.2), we obtain

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned} \frac{\partial {\varvec{v}}_{\delta }}{\partial t}&+\mu \text {A}{\varvec{v}}_{\delta }+\text {B}({\varvec{v}}_{\delta }+e^{\sigma t}{} {\textit{\textbf{g z}}} _{\delta })+\alpha {\varvec{v}}_{\delta }+\beta \mathcal {C}({\varvec{v}}_{\delta }+e^{\sigma t}{} {\varvec{g z}} _{\delta })\\ {}&={\varvec{f}}+\left( \ell -\sigma -\alpha \right) e^{\sigma t}{} {\varvec{g z}} _{\delta }-\mu e^{\sigma t}{\varvec{z}} _{\delta }\text {A}{{\textbf {g}}}, \qquad \text {in } \mathbb {R}^d\times (\mathfrak {s},\infty ), \\ {v}_{\delta }(x,\mathfrak {s})&={v}_{\delta ,\mathfrak {s}}(x)=\varvec{u}_{\delta ,\mathfrak {s}}(x)-e^{\sigma \mathfrak {s}}{} { {{\textbf {g}}}}(x){z} _{\delta }(\omega ), \qquad x\in \mathbb {R}^d \text{ and } \mathfrak {s}\in \mathbb {R}. \end{aligned} \right. \end{aligned} \end{aligned}$$

(6.60)

For all $\mathfrak {s}\in \mathbb {R},$ $t>\mathfrak {s},$ and for every $\varvec{v}_{\delta ,\mathfrak {s}}\in \mathbb {H}$ and $\omega \in \Omega$, (6.60) has a unique solution $\varvec{v}_{\delta }(\cdot ;\mathfrak {s},\omega ,\varvec{v}_{\delta ,\mathfrak {s}})\in \textrm{C}([\mathfrak {s},\mathfrak {s}+T];\mathbb {H})\cap \textrm{L}^2(\mathfrak {s}, \mathfrak {s}+T;\mathbb {V})\cap \textrm{L}^{r+1}(\mathfrak {s},\mathfrak {s}+T;\widetilde{\mathbb {L}}^{r+1})$. Moreover, $\varvec{v}_{\delta }(t;\mathfrak {s},\omega ,\varvec{v}_{\delta ,\mathfrak {s}})$ is continuous with respect to the initial data $\varvec{v}_{\delta ,\mathfrak {s}}(x)$ and $(\mathscr {F},\mathscr {B}(\mathbb {H}))$-measurable in $\omega \in \Omega .$ Define a cocycle $\Phi :\mathbb {R}^+\times \mathbb {R}\times \Omega \times \mathbb {H}\rightarrow \mathbb {H}$ for the system (6.2) such that for given $t\in \mathbb {R}^+, \mathfrak {s}\in \mathbb {R}, \omega \in \Omega$ and $\varvec{u}_{\delta ,\mathfrak {s}}\in \mathbb {H}$,

$$\begin{aligned} \Phi (t,\mathfrak {s},\omega ,\varvec{u}_{\delta ,\mathfrak {s}})&=\varvec{u}_{\delta }(t+\mathfrak {s};\mathfrak {s},\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\delta ,\mathfrak {s}})=\varvec{v}_{\delta }(t+\mathfrak {s};\mathfrak {s},\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{\delta ,\mathfrak {s}})+e^{\sigma (t+\mathfrak {s})}{} {\textbf {g}}\varvec{z} _{\delta }(\vartheta _{t}\omega ). \end{aligned}$$

(6.61)

Lemma 6.11

Let all the assumptions of Lemma 6.4 be satisfied. Then $\Phi _\delta$ possesses a closed measurable $\mathfrak {D}$-pullback random absorbing set $\mathcal {K}_\delta =\{\mathcal {K}_\delta (\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R}, \omega \in \Omega \}\in \mathfrak {D}$ in $\mathbb {H}$ given by

$$\begin{aligned} \mathcal {K}_\delta (\mathfrak {s},\omega )=\{\varvec{u}\in \mathbb {H}:\Vert \varvec{u}\Vert ^2_{\mathbb {H}}\le \mathcal {R}_\delta (\mathfrak {s},\omega )\}, \end{aligned}$$

(6.62)

where $\mathcal {R}_\delta (\mathfrak {s},\omega )$ is defined by

$$\begin{aligned} \mathcal {R}_\delta (\mathfrak {s},\omega )&=3\Vert {\textbf {g}}\Vert ^2_{\mathbb {H}}\left| e^{\sigma \mathfrak {s}}\varvec{z} _{\delta }(\omega )\right| ^2+2R_{8} \int _{-\infty }^{0}e^{\alpha \xi }\bigg [\Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^2\nonumber \\&\quad +\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^{r+1}+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{2(r+1)}{r-1}}\bigg ]\textrm{d}\xi . \end{aligned}$$

(6.63)

Here $R_{8}$ is the same as in Lemma 6.4. Furthermore, for every $\mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$,

$$\begin{aligned} \lim _{\delta \rightarrow 0}\mathcal {R}_\delta (\mathfrak {s},\omega )=\mathcal {R}_0(\mathfrak {s},\omega ), \end{aligned}$$

(6.64)

where $\mathcal {R}_0(\mathfrak {s},\omega )$ is given by (6.11).

Proof

Since, the systems (6.8) and (6.60) have similar terms, the existence of closed measurable $\mathfrak {D}$-pullback random absorbing set $\mathcal {K}_\delta (\mathfrak {s},\omega )$ is confirmed from the similar calculations as in Lemma 6.4. Now, it is left to prove (6.64) only.

Using (6.58), we can find $\xi _0<0$ such that for all $0<\delta \le \tilde{\ell }$,

$$\begin{aligned} \left| \varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| \le |\xi |, \ \ \ \text { for all } \ \xi \le \xi _0. \end{aligned}$$

(6.65)

From (6.65), we get that for all $\xi \le \xi _0$ and $0<\delta \le \tilde{\ell }$,

$$\begin{aligned}&\int _{-\infty }^{0}e^{\alpha \xi }\left[ \left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^2+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^{r+1}+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{2(r+1)}{r-1}}\right] \textrm{d}\xi \nonumber \\&\quad \le \int _{-\infty }^{0}e^{\alpha \xi }\left[ \left| e^{\sigma (\xi +\mathfrak {s})}\xi \right| ^2+\left| e^{\sigma (\xi +\mathfrak {s})}\xi \right| ^{r+1}+\left| e^{\sigma (\xi +\mathfrak {s})}\xi \right| ^{\frac{2(r+1)}{r-1}}\right] \textrm{d}\xi <\infty . \end{aligned}$$

(6.66)

Consider,

$$\begin{aligned}&\int _{-\infty }^{0}e^{\alpha \xi }\left[ \left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^2+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^{r+1}+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{2(r+1)}{r-1}}\right] \textrm{d}\xi \nonumber \\&\quad =\int _{\xi _0}^{0}e^{\alpha \xi }\left[ \left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^2+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^{r+1}+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{2(r+1)}{r-1}}\right] \textrm{d}\xi \nonumber \\&\qquad +\int _{-\infty }^{\xi _0}e^{\alpha \xi }\left[ \left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^2+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^{r+1}+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{2(r+1)}{r-1}}\right] \textrm{d}\xi . \end{aligned}$$

(6.67)

Using (6.57), (6.66) and the Lebesgue Dominated Convergence Theorem, we get

$$\begin{aligned}&\lim _{\delta \rightarrow 0}\int _{-\infty }^{\xi _0}e^{\alpha \xi }\left[ \left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^2+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^{r+1}+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{2(r+1)}{r-1}}\right] \textrm{d}\xi \nonumber \\&\quad =\int _{-\infty }^{\xi _0}e^{\alpha \xi }\left[ \left| e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\right| ^2+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\right| ^{r+1}+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\right| ^{\frac{2(r+1)}{r-1}}\right] \textrm{d}\xi . \end{aligned}$$

(6.68)

From (6.57), we obtain

$$\begin{aligned}&\lim _{\delta \rightarrow 0}\int _{\xi _0}^{0}e^{\alpha \xi }\left[ \left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^2+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^{r+1}+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta }(\vartheta _{\xi }\omega )\right| ^{\frac{2(r+1)}{r-1}}\right] \textrm{d}\xi \nonumber \\&\quad =\int _{\xi _0}^{0}e^{\alpha \xi }\left[ \left| e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\right| ^2+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\right| ^{r+1}+\left| e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\right| ^{\frac{2(r+1)}{r-1}}\right] \textrm{d}\xi . \end{aligned}$$

(6.69)

Hence, from (6.57), (6.68) and (6.69), we deduce the desired convergence. $\square$

Next result demonstrates the convergence of solution of (6.2) to the solution of (6.1) in $\mathbb {H}$ as $\delta \rightarrow 0$.

Lemma 6.12

Let all the assumptions of Lemma 6.3 be satisfied. Suppose that $\{\delta _n\}_{n\in \mathbb {N}}$ be a sequence such that $\delta _n\rightarrow 0$. Let $\varvec{u}_{\delta _n}$ and $\varvec{u}$ be the solutions of (6.2) and (6.1) with initial data $\varvec{u}_{\delta _n,\mathfrak {s}}$ and $\varvec{u}_{\mathfrak {s}}$, respectively. If $\Vert \varvec{u}_{\delta _n,\mathfrak {s}}-\varvec{u}_{\mathfrak {s}}\Vert _{\mathbb {H}}\rightarrow 0$ as $n\rightarrow \infty$, then for every $\mathfrak {s}\in \mathbb {R}$, $\omega \in \Omega$ and $t>\mathfrak {s}$,

$$\begin{aligned} \Vert \varvec{u}_{\delta _n}(t;\mathfrak {s},\omega ,\varvec{u}_{\delta _n,\mathfrak {s}})-\varvec{u}(t;\mathfrak {s},\omega ,\varvec{u}_{\mathfrak {s}})\Vert _{\mathbb {H}} \rightarrow 0 \ \ \text { as }\ \ n\rightarrow \infty . \end{aligned}$$

Proof

Let $\mathfrak {F}=\varvec{v}_{\delta _n}-\varvec{v}$, where $\varvec{v}_{\delta _n}$ and $\varvec{v}$ are the solutions of (6.60) and (6.8), respectively. Also, let $\mathfrak {z}=\varvec{z} _{\delta _n}-\varvec{y},$ where $\varvec{y}$ is defined in (6.5). Then we get from (6.60) and (6.8) that

$$\begin{aligned} \frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\Vert \mathfrak {F}\Vert ^2_{\mathbb {H}}&=-\mu \Vert \nabla \mathfrak {F}\Vert ^2_{\mathbb {H}}-\alpha \Vert \mathfrak {F}\Vert ^2_{\mathbb {H}}+e^{\sigma t}\mathfrak {z}\left( \left( \ell -\sigma -\alpha \right) {\textbf {g}}-\mu \textrm{A}{\textbf {g}},\mathfrak {F}\right) \nonumber \\&\quad -\left\langle \textrm{B}(\varvec{v}_{\delta _n}+e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n})-\textrm{B}(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}), (\varvec{v}_{\delta _n}+e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n})-(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y})\right\rangle \nonumber \\&\quad -\beta \left\langle \mathcal {C}(\varvec{v}_{\delta _n}+e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n})-\mathcal {C}(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}),(\varvec{v}_{\delta _n}+e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n})-(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y})\right\rangle \nonumber \\&\quad +e^{\sigma t}\mathfrak {z}b(\varvec{v}_{\delta _n},\varvec{v}_{\delta _n},{\textbf {g}})+e^{2\sigma t}\mathfrak {z}\varvec{z} _{\delta _n}b({\textbf {g}},\varvec{v}_{\delta _n},{\textbf {g}})-e^{\sigma t}\mathfrak {z}b(\varvec{v},\varvec{v},{\textbf {g}})-e^{2\sigma t}\mathfrak {z}\varvec{y}b({\textbf {g}},\varvec{v},{\textbf {g}})\nonumber \\&\quad +\beta e^{\sigma t}\mathfrak {z}\left\langle \mathcal {C}(\varvec{v}_{\delta _n}+e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n})-\mathcal {C}(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}),{\textbf {g}}\right\rangle . \end{aligned}$$

(6.70)

From (2.5), we write

$$\begin{aligned}&-\beta \left\langle \mathcal {C}(\varvec{v}_{\delta _n}+e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n})-\mathcal {C}(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}),\mathfrak {F}+e^{\sigma t}{} {\textbf {g}}\mathfrak {z}\right\rangle \nonumber \\ {}&\quad \le -\frac{\beta }{2}\Vert \left| \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\right| ^{\frac{r-1}{2}}\left| \mathfrak {F}+e^{\sigma t}{} {\textbf {g}}\mathfrak {z}\right| \Vert ^2_{\mathbb {H}}\le 0. \end{aligned}$$

(6.71)

Applying Lemmas 2.9 and 2.11, we obtain

$$\begin{aligned}&\left| \beta e^{\sigma t}\mathfrak {z}\left\langle \mathcal {C}(\varvec{v}_{\delta _n}+e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n})-\mathcal {C}(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}),{\textbf {g}}\right\rangle \right| \nonumber \\&\quad \le \beta e^{\sigma t}\left| \mathfrak {z}\right| \left( \Vert \varvec{v}_{\delta _n}+e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n}\Vert ^r_{\widetilde{\mathbb {L}}^{r+1}}+\Vert \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\Vert ^r_{\widetilde{\mathbb {L}}^{r+1}}\right) \Vert {\textbf {g}}\Vert _{\widetilde{\mathbb {L}}^{r+1}}\nonumber \\&\quad \le C e^{\sigma t}\left| \mathfrak {z}\right| \left( 1+\left| e^{\sigma t}\varvec{z} _{\delta _n}\right| ^r+\left| e^{\sigma t}\varvec{y}\right| ^r+\Vert \varvec{v}_{\delta _n}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}} +\Vert \varvec{v}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\right) , \end{aligned}$$

(6.72)

and

$$\begin{aligned} \left| e^{\sigma t}\mathfrak {z}\left( \left( \ell -\sigma -\alpha \right) {\textbf {g}}-\mu \textrm{A}{\textbf {g}},\mathfrak {F}\right) \right| \le \frac{\alpha }{2}\Vert \mathfrak {F}\Vert ^2_{\mathbb {H}} + Ce^{2\sigma t}\left| \mathfrak {z}\right| ^2. \end{aligned}$$

(6.73)

Case I: $d= 2$ and $r>1$. Using (2.2), (2.3), Lemmas 2.9 and 2.11, we get

$$\begin{aligned}&\left| \left\langle \textrm{B}(\varvec{v}_{\delta _n}+e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n})-\textrm{B}(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}), (\varvec{v}_{\delta _n}+e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n})-(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y})\right\rangle \right| \nonumber \\&\quad \le \frac{\mu }{2}\Vert \nabla \mathfrak {F}\Vert ^2_{\mathbb {H}}+C\left( e^{2\sigma t}\left| \varvec{y}\right| ^2+\Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}\right) \Vert \mathfrak {F}\Vert ^2_{\mathbb {H}}+ Ce^{2\sigma t}\left| \mathfrak {z}\right| ^2\left( 1+e^{2\sigma t}\left| \varvec{y}\right| ^2+\Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}\right) . \end{aligned}$$

(6.74)

Again, from (2.2), Lemmas 2.9 and 2.11, we have

$$\begin{aligned} \left| e^{\sigma t}\mathfrak {z}b(\varvec{v}_{\delta _n},\varvec{v}_{\delta _n},{\textbf {g}})\right|&\le Ce^{\sigma t}\left| \mathfrak {z}\right| \Vert \varvec{v}_{\delta _n}\Vert _{\mathbb {H}}\Vert \nabla \varvec{v}_{\delta _n}\Vert _{\mathbb {H}}\Vert \nabla {\textbf {g}}\Vert _{\mathbb {H}}\nonumber \\&\le Ce^{\sigma t}\left| \mathfrak {z}\right| \left( 1+\Vert \varvec{v}_{\delta _n}\Vert ^2_{\mathbb {H}}\Vert \nabla \varvec{v}_{\delta _n}\Vert ^2_{\mathbb {H}}\right) , \end{aligned}$$

(6.75)

$$\begin{aligned} \left| e^{2\sigma t}\mathfrak {z}\varvec{z} _{\delta _n}b({\textbf {g}},\varvec{v}_{\delta _n},{\textbf {g}})\right|&\le Ce^{2\sigma t}\left| \mathfrak {z}\right| \left| \varvec{z} _{\delta _n}\right| \Vert {\textbf {g}}\Vert _{\mathbb {H}}\Vert \nabla {\textbf {g}}\Vert _{\mathbb {H}}\Vert \nabla \varvec{v}_{\delta _n}\Vert _{\mathbb {H}}\nonumber \\&\le Ce^{2\sigma t}\left| \mathfrak {z}\right| \left| \varvec{z} _{\delta _n}\right| \left( 1+\Vert \nabla \varvec{v}_{\delta _n}\Vert ^2_{\mathbb {H}}\right) , \end{aligned}$$

(6.76)

$$\begin{aligned} \left| e^{\sigma t}\mathfrak {z}b(\varvec{v},\varvec{v},{\textbf {g}})\right|&\le Ce^{\sigma t}\left| \mathfrak {z}\right| \Vert \varvec{v}\Vert _{\mathbb {H}}\Vert \nabla \varvec{v}\Vert _{\mathbb {H}}\Vert \nabla {\textbf {g}}\Vert _{\mathbb {H}}\le Ce^{\sigma t}\left| \mathfrak {z}\right| \left( 1+\Vert \varvec{v}\Vert ^2_{\mathbb {H}}\Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}\right) , \end{aligned}$$

(6.77)

$$\begin{aligned} \left| e^{2\sigma t}\mathfrak {z}\varvec{y}b({\textbf {g}},\varvec{v},{\textbf {g}})\right|&\le Ce^{2\sigma t}\left| \mathfrak {z}\right| \left| \varvec{y}\right| \Vert {\textbf {g}}\Vert _{\mathbb {H}}\Vert \nabla {\textbf {g}}\Vert _{\mathbb {H}}\Vert \nabla \varvec{v}\Vert _{\mathbb {H}}\le Ce^{2\sigma t}\left| \mathfrak {z}\right| \left| \varvec{y}\right| \left( 1+\Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}\right) . \end{aligned}$$

(6.78)

Combining (6.70)–(6.78), we obtain

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\Vert \mathfrak {F}\Vert ^2_{\mathbb {H}}\le C\left[ Q_1(t)\Vert \mathfrak {F}\Vert ^2_{\mathbb {H}} + \left| \mathfrak {z}(\vartheta _{t}\omega )\right| Q_2(t)\right] , \end{aligned}$$

(6.79)

for a.e. $t\in [\mathfrak {s},\mathfrak {s}+T]$, where

$$\begin{aligned} Q_1(t)&=e^{2\sigma t}\left| \varvec{y}(\vartheta _{t}\omega )\right| ^2+\Vert \nabla \varvec{v}(t)\Vert ^2_{\mathbb {H}}\ \ \text { and }\\ Q_2(t)&=e^{2\sigma t}\left| \mathfrak {z}(\vartheta _{t}\omega )\right| \bigg [1+e^{2\sigma t}\left| \varvec{y}(\vartheta _{t}\omega )\right| ^2+\Vert \nabla \varvec{v}(t)\Vert ^2_{\mathbb {H}}\bigg ]\\&\quad +e^{\sigma t}\bigg [1+\Vert \varvec{v}_{\delta _n}(t)\Vert ^2_{\mathbb {H}}\Vert \nabla \varvec{v}_{\delta _n}(t)\Vert ^2_{\mathbb {H}}+\Vert \varvec{v}(t)\Vert ^2_{\mathbb {H}}\Vert \nabla \varvec{v}(t)\Vert ^2_{\mathbb {H}}+\left| e^{\sigma t}\varvec{z} _{\delta _n}(\vartheta _{t}\omega )\right| ^r\\&\quad +\left| e^{\sigma t}\varvec{y}(\vartheta _{t}\omega )\right| ^r+\Vert \varvec{v}_{\delta _n}(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}} +\Vert \varvec{v}(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\bigg ]\\&\quad +e^{2\sigma t}\left| \varvec{z} _{\delta _n}(\vartheta _{t}\omega )\right| \bigg [1+\Vert \nabla \varvec{v}_{\delta _n}(t)\Vert ^2_{\mathbb {H}}\bigg ]+e^{2\sigma t}\left| \varvec{y}(\vartheta _{t}\omega )\right| \bigg [1+\Vert \nabla \varvec{v}(t)\Vert ^2_{\mathbb {H}}\bigg ]. \end{aligned}$$

Due to continuity of $e^{\sigma t},$ $\varvec{y}(\vartheta _{t}\omega )$ and $\varvec{z} _{\delta _n}(\vartheta _{t}\omega )$, and $\varvec{v}_{\delta _n}, \varvec{v}\in \textrm{C}([\mathfrak {s},\mathfrak {s}+T];\mathbb {H})\cap \textrm{L}^2(\mathfrak {s}, \mathfrak {s}+T;\mathbb {V})\cap \textrm{L}^{r+1}(\mathfrak {s},\mathfrak {s}+T;\widetilde{\mathbb {L}}^{r+1})$, we infer that

$$\begin{aligned} \int _{\mathfrak {s}}^{\mathfrak {s}+T}Q_1(\xi )\textrm{d}\xi<\infty \ \text { and }\ \int _{\mathfrak {s}}^{\mathfrak {s}+T}Q_2(\xi )\textrm{d}\xi <\infty . \end{aligned}$$

(6.80)

An application of Gronwall’s inequality yields

$$\begin{aligned}&\Vert \varvec{v}_{\delta _n}(t;\mathfrak {s},\omega ,\varvec{v}_{\delta _n,\mathfrak {s}})-\varvec{v}(t;\mathfrak {s},\omega ,\varvec{v}_{\mathfrak {s}})\Vert ^2_{\mathbb {H}}\nonumber \\&\quad \le \left[ \Vert \varvec{v}_{\delta _n,\mathfrak {s}}-\varvec{v}_{\mathfrak {s}}\Vert ^2_{\mathbb {H}}+C\sup _{\xi \in [\mathfrak {s},\mathfrak {s}+ T]}\left| \varvec{z} _{\delta _n}(\vartheta _{\xi }\omega )-\varvec{y}(\vartheta _{\xi }\omega )\right| \int _{\mathfrak {s}}^{\mathfrak {s}+T}Q_1(\xi )\textrm{d}\xi \right] e^{C\int \limits _{\mathfrak {s}}^{\mathfrak {s}+T}Q_2(\xi )\textrm{d}\xi }. \end{aligned}$$

(6.81)

By (6.7), (6.59) and (6.81), we get

$$\begin{aligned}&\Vert \varvec{u}_{\delta _n}(t;\mathfrak {s},\omega ,\varvec{u}_{\delta _n,\mathfrak {s}})-\varvec{u}(t;\mathfrak {s},\omega ,\varvec{u}_{\mathfrak {s}})\Vert ^2_{\mathbb {H}}\nonumber \\&\quad \le 2\bigg [ 2\Vert \varvec{u}_{\delta _n,\mathfrak {s}}-\varvec{u}_{\mathfrak {s}}\Vert ^2_{\mathbb {H}} +2e^{\sigma \mathfrak {s}}\left| \varvec{z} _{\delta _n}(\omega )-\varvec{y}(\omega )\right| \Vert {\textbf {g}}\Vert ^2_{\mathbb {H}}\nonumber \\&\qquad +C\sup _{\xi \in [\mathfrak {s},\mathfrak {s}+ T]}\left| \varvec{z} _{\delta _n}(\vartheta _{\xi }\omega )-\varvec{y}(\vartheta _{\xi }\omega )\right| \int \limits _{\mathfrak {s}}^{\mathfrak {s}+T}Q_1(\xi )\textrm{d}\xi \bigg ]e^{C\int \limits _{\mathfrak {s}}^{\mathfrak {s}+T}Q_2(\xi )\textrm{d}\xi } \nonumber \\&\qquad + 2e^{\sigma t}\left| \varvec{z} _{\delta _n}(\vartheta _{t}\omega )-\varvec{y}(\vartheta _{t}\omega )\right| \Vert {\textbf {g}}\Vert ^2_{\mathbb {H}}, \end{aligned}$$

(6.82)

for all $t\in [\mathfrak {s},\mathfrak {s}+T]$. Hence, due to (6.57) and (6.80), we achieve the required convergence from (6.82).

Case II: $d= 3$ and $r\ge 3$ ($r>3$ with any $\beta ,\mu >0$ and $r=3$ with $2\beta \mu \ge 1$). Using (2.3), Lemmas 2.9 and 2.11, we get

$$\begin{aligned}&\left| \left\langle \textrm{B}(\varvec{v}_{\delta _n}+e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n})-\textrm{B}(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}), (\varvec{v}_{\delta _n}+e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n})-(\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y})\right\rangle \right| \nonumber \\&\quad \le \left| b(\mathfrak {F}+e^{\sigma t}{} {\textbf {g}}\mathfrak {z},\mathfrak {F}+e^{\sigma t}{} {\textbf {g}}\mathfrak {z},\varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y})\right| \nonumber \\&\quad \le {\left\{ \begin{array}{ll} \frac{1}{2\beta }\Vert \nabla \left( \mathfrak {F}+e^{\sigma t}{} {\textbf {g}}\mathfrak {z}\right) \Vert ^2_{\mathbb {H}}+\frac{\beta }{2}\Vert \left| \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\right| \left| \mathfrak {F}+e^{\sigma t}{} {\textbf {g}}\mathfrak {z}\right| \Vert ^2_{\mathbb {H}}, \text { for } r=3,\\ \frac{\mu }{4}\Vert \nabla \left( \mathfrak {F}+e^{\sigma t}{} {\textbf {g}}\mathfrak {z}\right) \Vert ^2_{\mathbb {H}}+\frac{\beta }{2}\Vert \left| \varvec{v}+e^{\sigma t}{} {\textbf {g}}\varvec{y}\right| ^{\frac{r-1}{2}}\left| \mathfrak {F}+e^{\sigma t}{} {\textbf {g}}\mathfrak {z}\right| \Vert ^2_{\mathbb {H}}+C\Vert \mathfrak {F}+e^{\sigma t}{} {\textbf {g}}\mathfrak {z}\Vert ^2_{\mathbb {H}}, \text { for } r>3, \end{array}\right. }\nonumber \\&\quad \le {\left\{ \begin{array}{ll} \frac{1}{2\beta }\Vert \nabla \mathfrak {F}\Vert ^2_{\mathbb {H}}\!+C\left| e^{\sigma t}\mathfrak {z}\right| ^2+\!C\left| e^{\sigma t}\mathfrak {z}\right| \left( 1\!+\!\Vert \nabla \varvec{v}_{\delta _n}\Vert ^2_{\mathbb {H}}+\!\Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}\right) \!+\frac{\beta }{2}\Vert |\varvec{v}\!+e^{\sigma t}{} {\textbf {g}}\varvec{y}||\mathfrak {F}\!+e^{\sigma t}{} {\textbf {g}}\mathfrak {z}|\Vert ^2_{\mathbb {H}},\\ \text { for } r\!=3,\\ \frac{\mu }{2}\Vert \nabla \mathfrak {F}\Vert ^2_{\mathbb {H}}\!+C\Vert \mathfrak {F}\Vert ^2_{\mathbb {H}}\!+ Ce^{2\sigma t}\left| \mathfrak {z}\right| ^2\!+\frac{\beta }{2}\Vert \left| \varvec{v}\!+e^{\sigma t}{} {\textbf {g}}\varvec{y}\right| ^{\frac{r-\!1}{2}}\left| \mathfrak {F}\!+e^{\sigma t}{} {\textbf {g}}\mathfrak {z}\right| \Vert ^2_{\mathbb {H}}, \text { for } r>3. \end{array}\right. } \end{aligned}$$

(6.83)

From (2.2), Lemmas 2.9 and 2.11, we have

$$\begin{aligned} \left| e^{\sigma t}\mathfrak {z}b(\varvec{v}_{\delta _n},\varvec{v}_{\delta _n},{\textbf {g}})\right|&\le Ce^{\sigma t}\left| \mathfrak {z}\right| \Vert \varvec{v}_{\delta _n}\Vert ^{\frac{1}{2}}_{\mathbb {H}}\Vert \nabla \varvec{v}_{\delta _n}\Vert ^{\frac{3}{2}}_{\mathbb {H}}\Vert \nabla {\textbf {g}}\Vert _{\mathbb {H}}\nonumber \\&\le Ce^{\sigma t}\left| \mathfrak {z}\right| \left( 1+\Vert \varvec{v}_{\delta _n}\Vert ^{\frac{2}{3}}_{\mathbb {H}}\Vert \nabla \varvec{v}_{\delta _n}\Vert ^2_{\mathbb {H}}\right) , \end{aligned}$$

(6.84)

$$\begin{aligned} \left| e^{2\sigma t}\mathfrak {z}\varvec{z} _{\delta _n}b({\textbf {g}},\varvec{v}_{\delta _n},{\textbf {g}})\right|&\le Ce^{2\sigma t}\left| \mathfrak {z}\right| \left| \varvec{z} _{\delta _n}\right| \Vert {\textbf {g}}\Vert ^{\frac{1}{2}}_{\mathbb {H}}\Vert \nabla {\textbf {g}}\Vert ^{\frac{3}{2}}_{\mathbb {H}}\Vert \nabla \varvec{v}_{\delta _n}\Vert _{\mathbb {H}}\nonumber \\&\le Ce^{2\sigma t}\left| \mathfrak {z}\right| \left| \varvec{z} _{\delta _n}\right| \left( 1+\Vert \nabla \varvec{v}_{\delta _n}\Vert ^2_{\mathbb {H}}\right) , \end{aligned}$$

(6.85)

$$\begin{aligned} \left| e^{\sigma t}\mathfrak {z}b(\varvec{v},\varvec{v},{\textbf {g}})\right|&\le Ce^{\sigma t}\left| \mathfrak {z}\right| \Vert \varvec{v}\Vert ^{\frac{1}{2}}_{\mathbb {H}}\Vert \nabla \varvec{v}\Vert ^{\frac{3}{2}}_{\mathbb {H}}\Vert \nabla {\textbf {g}}\Vert _{\mathbb {H}}\le Ce^{\sigma t}\left| \mathfrak {z}\right| \left( 1+\Vert \varvec{v}\Vert ^{\frac{2}{3}}_{\mathbb {H}}\Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}\right) , \end{aligned}$$

(6.86)

$$\begin{aligned} \left| e^{2\sigma t}\mathfrak {z}\varvec{y}b({\textbf {g}},\varvec{v},{\textbf {g}})\right|&\le Ce^{2\sigma t}\left| \mathfrak {z}\right| \left| \varvec{y}\right| \Vert {\textbf {g}}\Vert ^{\frac{1}{2}}_{\mathbb {H}}\Vert \nabla {\textbf {g}}\Vert ^{\frac{3}{2}}_{\mathbb {H}}\Vert \nabla \varvec{v}\Vert _{\mathbb {H}}\le Ce^{2\sigma t}\left| \mathfrak {z}\right| \left| \varvec{y}\right| \left( 1+\Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}\right) . \end{aligned}$$

(6.87)

Combining (6.70)–(6.73) and (6.83)–(6.87), we obtain

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\Vert \mathfrak {F}\Vert ^2_{\mathbb {H}}&\le C\times {\left\{ \begin{array}{ll} \left| \mathfrak {z}(\vartheta _{t}\omega )\right| Q_3(t), &{}\text { for } r=3 ,\\ \Vert \mathfrak {F}\Vert ^2_{\mathbb {H}} + \left| \mathfrak {z}(\vartheta _{t}\omega )\right| Q_4(t), &{}\text { for } r>3, \end{array}\right. } \end{aligned}$$

(6.88)

for a.e. $t\in [\mathfrak {s},\mathfrak {s}+T]$, where

$$\begin{aligned} Q_3(t)&=e^{2\sigma t}\left| \mathfrak {z}(\vartheta _{t}\omega )\right| +e^{\sigma t}\bigg [1+\Vert \nabla \varvec{v}_{\delta _n}(t)\Vert ^2_{\mathbb {H}}+\Vert \nabla \varvec{v}(t)\Vert ^2_{\mathbb {H}}+\Vert \varvec{v}_{\delta _n}(t)\Vert ^{\frac{2}{3}}_{\mathbb {H}}\Vert \nabla \varvec{v}_{\delta _n}(t)\Vert ^2_{\mathbb {H}}\nonumber \\ {}&\quad +\Vert \varvec{v}(t)\Vert ^{\frac{2}{3}}_{\mathbb {H}}\Vert \nabla \varvec{v}(t)\Vert ^2_{\mathbb {H}}+\left| e^{\sigma t}\varvec{z} _{\delta _n}(\vartheta _{t}\omega )\right| ^3\\&\quad +\left| e^{\sigma t}\varvec{y}(\vartheta _{t}\omega )\right| ^3+\Vert \varvec{v}_{\delta _n}(t)\Vert ^{4}_{\widetilde{\mathbb {L}}^{4}} +\Vert \varvec{v}(t)\Vert ^{4}_{\widetilde{\mathbb {L}}^{4}}\bigg ]\\ {}&\quad +e^{2\sigma t}\left| \varvec{z} _{\delta _n}(\vartheta _{t}\omega )\right| \bigg [1+\Vert \nabla \varvec{v}_{\delta _n}(t)\Vert ^2_{\mathbb {H}}\bigg ]+e^{2\sigma t}\left| \varvec{y}(\vartheta _{t}\omega )\right| \bigg [1+\Vert \nabla \varvec{v}(t)\Vert ^2_{\mathbb {H}}\bigg ] \end{aligned}$$

and

$$\begin{aligned} Q_4(t)&=e^{2\sigma t}\left| \mathfrak {z}(\vartheta _{t}\omega )\right| +e^{\sigma t}\bigg [1+\Vert \varvec{v}_{\delta _n}(t)\Vert ^{\frac{2}{3}}_{\mathbb {H}}\Vert \nabla \varvec{v}_{\delta _n}(t)\Vert ^2_{\mathbb {H}}+\Vert \varvec{v}(t)\Vert ^{\frac{2}{3}}_{\mathbb {H}}\Vert \nabla \varvec{v}(t)\Vert ^2_{\mathbb {H}}\\&\quad +\left| e^{\sigma t}\varvec{z} _{\delta _n}(\vartheta _{t}\omega )\right| ^r+\left| e^{\sigma t}\varvec{y}(\vartheta _{t}\omega )\right| ^r+\Vert \varvec{v}_{\delta _n}(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}} +\Vert \varvec{v}(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\bigg ]\\&\quad +e^{2\sigma t}\left| \varvec{z} _{\delta _n}(\vartheta _{t}\omega )\right| \bigg [1+\Vert \nabla \varvec{v}_{\delta _n}(t)\Vert ^2_{\mathbb {H}}\bigg ]\\&\quad +e^{2\sigma t}\left| \varvec{y}(\vartheta _{t}\omega )\right| \bigg [1+\Vert \nabla \varvec{v}(t)\Vert ^2_{\mathbb {H}}\bigg ]. \end{aligned}$$

Hence, arguing similarly as in Case I, we obtain the desired convergence. $\square$

Lemma 6.13

Let all the assumptions of Lemma 6.3 be satisfied. Suppose that $\{\delta _n\}_{n\in \mathbb {N}}$ be a sequence such that $\delta _n\rightarrow 0$. Let $\varvec{v}_{\delta _n}$ and $\varvec{v}$ be the solutions of (6.2) and (6.1) with initial data $\varvec{v}_{\delta _n,\mathfrak {s}}$ and $\varvec{v}_{\mathfrak {s}}$, respectively. If in $\mathbb {H}$ as $n\rightarrow \infty$, then for every $\mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$,

• (i) in $\mathbb {H}$ for all $\xi \ge \mathfrak {s}$.

• (ii) in $\textrm{L}^2((\mathfrak {s},\mathfrak {s}+T);\mathbb {V})\cap \textrm{L}^{r+1}((\mathfrak {s},\mathfrak {s}+T);\widetilde{\mathbb {L}}^{r+1})$ for every $T>0$.

• (iii) $\varvec{v}_{\delta _n}(\cdot ;\mathfrak {s},\omega ,\varvec{v}_{\delta _n,\mathfrak {s}})\rightarrow \varvec{v}(\cdot ;\mathfrak {s},\omega ,\varvec{v}_{\mathfrak {s}})$ in $\textrm{L}^2((\mathfrak {s},\mathfrak {s}+T);\mathbb {L}^2(\mathcal {O}_k))$ for every $T>0$ and $k>0$, where $\mathcal {O}_k=\{x\in \mathbb {R}^d:|x|<k\}$.

Proof

The proof is straightforward, see the proof of [32, Lemmas 4.2 and 4.3]. $\square$

The existence of a unique $\mathfrak {D}$-pullback random attractor (denote here by $\mathscr {A}_\delta$) for continuous cocycle $\Phi _\delta$ follows from Theorem 4.5. The following Lemma demonstrates the uniform compactness of family of random attractors $\mathscr {A}_{\delta }$.

Lemma 6.14

Let all the assumptions of Lemma 6.4 be satisfied. Let $\mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$ be fixed. If $\delta _n\rightarrow 0$ as $n\rightarrow \infty$ and $\varvec{u}_n\in \mathscr {A}_{\delta _n}(\mathfrak {s},\omega )$, then the sequence $\{\varvec{u}_n\}_{n\in \mathbb {N}}$ has a convergent subsequence in $\mathbb {H}$.

Proof

Since, $\delta _n\rightarrow 0$, it follows from (6.64) that for every $\mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$, there exists $\mathcal {N}_1=\mathcal {N}_1(\mathfrak {s},\omega )$ such that for all $n\ge \mathcal {N}_1$,

$$\begin{aligned} \mathcal {R}_{\delta _n}(\mathfrak {s},\omega )\le 2\mathcal {R}_0(\mathfrak {s},\omega ). \end{aligned}$$

(6.89)

It is given that $\varvec{u}_n\in \mathscr {A}_{\delta _n}(\mathfrak {s},\omega )$ and due to the property of attractors, we have that $\mathscr {A}_{\delta _n}(\mathfrak {s},\omega )$ is a subset of absorbing set, by Lemma 6.11 and (6.89) we have, for all $n\ge \mathcal {N}_1$,

$$\begin{aligned} \Vert \varvec{u}_n\Vert ^2_{\mathbb {H}}\le 2\mathcal {R}_{0}(\mathfrak {s},\omega ). \end{aligned}$$

(6.90)

It is clear that $\{\varvec{u}_n:n\in \mathbb {N}\}\subset \mathbb {H}$ and therefore, there exists a subsequence (not relabeling) and $\hat{\varvec{u}}\in \mathbb {H}$ such that

(6.91)

Our next aim to prove that $\varvec{u}_n\rightarrow \hat{\varvec{u}}$ in $\mathbb {H}$. Since $\varvec{u}_n\in \mathscr {A}_{\delta _n}(\mathfrak {s},\omega )$, by the invariance property of $\mathscr {A}_{\delta _n}(\mathfrak {s},\omega )$, for all $l\ge 1$, there exists $\varvec{u}_{n,l}\in \mathscr {A}_{\delta _n}(\mathfrak {s}-l,\vartheta _{-l}\omega )$ such that

$$\begin{aligned} \varvec{u}_n=\Phi _{\delta _n}(l,\mathfrak {s}-l,\vartheta _{-l}\omega ,\varvec{u}_{n,l})=\varvec{u}_{\delta _n}(\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{n,l}). \end{aligned}$$

(6.92)

Since $\varvec{u}_{n,l}\in \mathscr {A}_{\delta _n}(\mathfrak {s}-l,\vartheta _{-l}\omega )$ and $\mathscr {A}_{\delta _n}(\mathfrak {s}-l,\vartheta _{-l}\omega )\subseteq \mathcal {K}_{\delta _n}(\mathfrak {s}-l,\vartheta _{-l}\omega ),$ by Lemma 6.11 and (6.89), we find that for each $l\ge 1$ and $n\ge \mathcal {N}_1(\mathfrak {s}-l,\vartheta _{-l}\omega )$,

$$\begin{aligned} \Vert \varvec{u}_{n,l}\Vert ^2_{\mathbb {H}}\le 2\mathcal {R}_{0}(\mathfrak {s}-l,\vartheta _{-l}\omega ). \end{aligned}$$

(6.93)

Due to (6.59), we get

$$\begin{aligned} \varvec{v}_{\delta _n}(\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})=\varvec{u}_{\delta _n}(\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{u}_{n,l})-e^{\sigma \mathfrak {s}}{} {\textbf {g}}(x)\varvec{z} _{\delta _n}(\omega ), \end{aligned}$$

(6.94)

where

$$\begin{aligned} \varvec{v}_{n,l}=\varvec{u}_{n,l}-e^{\sigma (\mathfrak {s}-l)}{} {\textbf {g}}(x)\varvec{z} _{\delta _n}(\vartheta _{-l}\omega ). \end{aligned}$$

(6.95)

From, (6.92) and (6.94), we obtain

$$\begin{aligned} \varvec{u}_n=\varvec{v}_{\delta _n}(\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})+e^{\sigma \mathfrak {s}}{} {\textbf {g}}(x)\varvec{z} _{\delta _n}(\omega ). \end{aligned}$$

(6.96)

Using (6.93) and (6.95), we find that for $n\ge \mathcal {N}_1(\mathfrak {s}-l,\vartheta _{-l}\omega )$,

$$\begin{aligned} \Vert \varvec{v}_{n,l}\Vert ^2_{\mathbb {H}}\le 4\mathcal {R}_{0}(\mathfrak {s}-l,\vartheta _{-l}\omega )+2e^{2\sigma (\mathfrak {s}-l)}\left| {\textbf {g}}(x)\varvec{z} _{\delta _n}(\vartheta _{-l}\omega )\right| ^2. \end{aligned}$$

(6.97)

Thanks to (6.57) and (6.97), we can find an $\mathcal {N}_2=\mathcal {N}_2(\mathfrak {s},\omega ,l)\ge \mathcal {N}_1$ such that for every $l\ge 1$ and $n\ge \mathcal {N}_2$,

$$\begin{aligned} \Vert \varvec{v}_{n,l}\Vert ^2_{\mathbb {H}}\le 4\mathcal {R}_{0}(\mathfrak {s}-l,\vartheta _{-l}\omega )+4e^{2\sigma (\mathfrak {s}-l)}\left| {\textbf {g}}(x)\right| ^2(1+\left| \varvec{y}(\vartheta _{-l}\omega )\right| ^2). \end{aligned}$$

(6.98)

By (6.91) and (6.96) along with (6.57), we get as $n\rightarrow \infty$,

(6.99)

It follows from (6.97) that for each $l\ge 1$, the sequence $\{\varvec{v}_{n,l}\}_{n\in \mathbb {N}}\subset \mathbb {H}$, and hence, we get a subsequence (not relabeling) by a diagonal argument such that for every $l\ge 1$, there exists $\tilde{\varvec{v}}_l\in \mathbb {H}$ such that

(6.100)

It yields from Lemma 6.13 and (6.100) that as $n\rightarrow \infty$,

(6.101)

(6.102)

(6.103)

$$\begin{aligned}&\varvec{v}_{\delta _n}(\cdot ;\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})\rightarrow \varvec{v}(\cdot ;\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\tilde{\varvec{v}}_{l}) \ \text { in } \ \textrm{L}^{2}(\mathfrak {s}-l,\mathfrak {s};\mathbb {L}^{2}(\mathcal {O}_k)), \end{aligned}$$

(6.104)

where $\mathcal {O}_k=\{x\in \mathbb {R}^d:|x|<k\}$. Now, (6.99) and (6.101) imply

$$\begin{aligned} \hat{\varvec{v}}=\varvec{v}(\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\tilde{\varvec{v}}_{l}). \end{aligned}$$

(6.105)

From (6.60) together with (2.1), we obtain

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t}\Vert \varvec{v}_{\delta _n}\Vert ^2_{\mathbb {H}}+2\alpha \Vert \varvec{v}_{\delta _n}\Vert ^2_{\mathbb {H}}+2\mu \Vert \nabla \varvec{v}_{\delta _n}\Vert ^2_{\mathbb {H}}+2\beta \Vert \varvec{v}_{\delta _n}+e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}} \nonumber \\&\quad =2b(\varvec{v}_{\delta _n}+e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n},\varvec{v}_{\delta _n},e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n})+2\beta \left\langle \mathcal {C}(\varvec{v}_{\delta _n}+e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n}),e^{\sigma t}{} {\textbf {g}}\varvec{z} _{\delta _n}\right\rangle +2\left\langle \varvec{f}(\cdot ,t),\varvec{v}_{\delta _n}\right\rangle \nonumber \\&\quad + 2e^{\sigma t}\varvec{z} _{\delta _n}\left( \left( \ell -\sigma -\alpha \right) {\textbf {g}}-\mu \textrm{A}{\textbf {g}},\varvec{v}_{\delta _n}\right) . \end{aligned}$$

(6.106)

An application of variation of constant formula with $\omega$ being replaced by $\vartheta _{-\mathfrak {s}}\omega$ yields

$$\begin{aligned}&\Vert \varvec{v}_{\delta _n}(\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})\Vert ^2_{\mathbb {H}}\nonumber \\&\quad =I_1(n,l)+I_2(n,l)+I_3(n,l)+I_4(n,l)+I_5(n,l)+I_6(n,l)+I_7(n,l), \end{aligned}$$

(6.107)

where

$$\begin{aligned} I_1(n,l)&=e^{-2\alpha l}\Vert \varvec{v}_{n,l}\Vert ^2_{\mathbb {H}},\\ I_2(n,l)&=-2\mu \int _{-l}^{0}e^{2\alpha \xi }\Vert \nabla \varvec{v}_{\delta _n}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})\Vert ^2_{\mathbb {H}}\textrm{d}\xi ,\\ I_3(n,l)&=-2\beta \int _{-l}^{0}e^{2\alpha \xi }\Vert \varvec{v}_{\delta _n}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})+e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{z} _{\delta _n}(\vartheta _{\xi }\omega )\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\textrm{d}\xi ,\\ I_4(n,l)&= 2 \int _{-l}^{0} e^{2\alpha \xi }b(\varvec{v}_{\delta _n}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})+e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{z} _{\delta _n}(\vartheta _{\xi }\omega ),\\&\quad \varvec{v}_{\delta _n}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l}),e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{z} _{\delta _n}(\vartheta _{\xi }\omega ))\textrm{d}\xi ,\\ I_5(n,l)&=2\beta \int _{-l}^{0} e^{2\alpha \xi }\left\langle \mathcal {C}(\varvec{v}_{\delta _n}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})\right. \\&\quad \left. +e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{z} _{\delta _n}(\vartheta _{\xi }\omega )),e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{z} _{\delta _n}(\vartheta _{\xi }\omega )\right\rangle \textrm{d}\xi ,\\ I_6(n,l)&=2 \int _{-l}^{0} e^{2\alpha \xi }\left\langle \varvec{f}(\cdot ,\xi +\mathfrak {s}),\varvec{v}_{\delta _n}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})\right\rangle \textrm{d}\xi , \end{aligned}$$

and

$$\begin{aligned} I_7(n,l)&=2 \int _{-l}^{0} e^{2\alpha \xi } e^{\sigma (\xi +\mathfrak {s})}\varvec{z} _{\delta _n}(\vartheta _{\xi }\omega )\big ((\ell -\sigma -\alpha ){\textbf {g}}-\mu \textrm{A}{\textbf {g}},\varvec{v}_{\delta _n}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})\big )\textrm{d}\xi . \end{aligned}$$

Similar to (6.107), by (6.8) and (6.105), it can also be obtained that

(6.108)

From (6.98), we have

$$\begin{aligned} \limsup _{n\rightarrow \infty }I_1(n,l)\le 4e^{-2\alpha l}\left[ \mathcal {R}_{0}(\mathfrak {s}-l,\vartheta _{-l}\omega )+e^{2\sigma (\mathfrak {s}-l)}\left| {\textbf {g}}(x)\right| ^2(1+\left| \varvec{y}(\vartheta _{-l}\omega )\right| ^2)\right] . \end{aligned}$$

(6.109)

Similarly, from (6.102), we obtain

$$\begin{aligned} \limsup _{n\rightarrow \infty }I_2(n,l)\le -2\mu \int _{-l}^{0}e^{2\alpha \xi }\Vert \nabla \varvec{v}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\tilde{\varvec{v}}_l)\Vert ^2_{\mathbb {H}}\textrm{d}\xi , \end{aligned}$$

(6.110)

and

$$\begin{aligned} \lim _{n\rightarrow \infty }I_6(n,l)=2 \int _{-l}^{0} e^{2\alpha \xi }\left\langle \varvec{f}(\cdot ,\xi +\mathfrak {s}),\varvec{v}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\tilde{\varvec{v}}_l)\right\rangle \textrm{d}\xi . \end{aligned}$$

(6.111)

By (6.57) and (6.102), we have

$$\begin{aligned} \lim _{n\rightarrow \infty }I_7(n,l)&=2 \int _{-l}^{0} e^{2\alpha \xi } e^{\sigma (\xi +\mathfrak {s})}\varvec{y}(\vartheta _{\xi }\omega )\big ((\ell -\sigma -\alpha ){\textbf {g}}-\mu \textrm{A}{\textbf {g}},\nonumber \\&\quad \varvec{v}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\tilde{\varvec{v}}_l)\big )\textrm{d}\xi . \end{aligned}$$

(6.112)

From (6.103), we infer that

$$\begin{aligned} \limsup _{n\rightarrow \infty }I_3(n,l)\le -2\beta \int _{-l}^{0}e^{2\alpha \xi }\Vert \varvec{v}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\tilde{\varvec{v}}_l)+e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{y}(\vartheta _{\xi }\omega )\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\textrm{d}\xi . \end{aligned}$$

(6.113)

Using (2.1), we obtain

$$\begin{aligned} I_4(n,l)&= 2 \int _{-l}^{0} e^{2\alpha \xi }b(\varvec{v}_{\delta _n}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})+e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{z} _{\delta _n}(\vartheta _{\xi }\omega ),\\&\quad \varvec{v}_{\delta _n}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})+e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{z} _{\delta _n}(\vartheta _{\xi }\omega ),e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{y}(\vartheta _{\xi }\omega ))\textrm{d}\xi \\&\quad + 2 \int _{-l}^{0} e^{2\alpha \xi }\left( \varvec{z} _{\delta _n}(\vartheta _{\xi }\omega )-\varvec{y}(\vartheta _{\xi }\omega )\right) \\&\quad \times \, b(\varvec{v}_{\delta _n}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})+e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{z} _{\delta _n}(\vartheta _{\xi }\omega ),\\&\quad \varvec{v}_{\delta _n}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})+e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{z} _{\delta _n}(\vartheta _{\xi }\omega ),e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}})\textrm{d}\xi \\&=:I_{8}(n,l)+I_9(n,l). \end{aligned}$$

Making use of (6.57) and Corollary 6.5, we get that $\lim \limits _{n\rightarrow \infty }I_9(n,l)=0$, and hence, by Corollary 6.5, we arrive at

$$\begin{aligned} \lim _{n\rightarrow \infty }I_4(n,l)&= 2 \int _{-l}^{0} e^{2\alpha \xi }b(\varvec{v}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\tilde{\varvec{v}}_l)+e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{y}(\vartheta _{\xi }\omega ),\nonumber \\&\quad \varvec{v}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\tilde{\varvec{v}}_l),e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{y}(\vartheta _{\xi }\omega ))\textrm{d}\xi . \end{aligned}$$

(6.114)

Finally, we have

$$\begin{aligned}&I_5(n,l)\\ {}&\quad =2\beta \int _{-l}^{0} e^{2\alpha \xi }\left\langle \mathcal {C}(\varvec{v}_{\delta _n}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})+e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{z} _{\delta _n}(\vartheta _{\xi }\omega )),e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{y}(\vartheta _{\xi }\omega )\right\rangle \textrm{d}\xi \\&\qquad +2\beta \int _{-l}^{0} e^{2\alpha \xi }\left( \varvec{z} _{\delta _n}(\vartheta _{\xi }\omega )-\varvec{y}(\vartheta _{\xi }\omega )\right) \big \langle \mathcal {C}(\varvec{v}_{\delta _n}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})\\ {}&\qquad +e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{z} _{\delta _n}(\vartheta _{\xi }\omega )), e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\big \rangle \textrm{d}\xi =:I_{10}(n,l)+I_{11}(n,l). \end{aligned}$$

By (6.57) and Corollary 6.6, we get that $\lim \limits _{n\rightarrow \infty }I_{11}(n,l)=0$, and hence, by Corollary 6.6, we deduce

$$\begin{aligned} \lim _{n\rightarrow \infty }I_5(n,l)&=2\beta \int _{-l}^{0} e^{2\alpha \xi }\left\langle \mathcal {C}(\varvec{v}(\xi +\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\tilde{\varvec{v}}_l)\right. \nonumber \\&\quad \left. +e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{y}(\vartheta _{\xi }\omega )),e^{\sigma (\xi +\mathfrak {s})}{} {\textbf {g}}\varvec{y}(\vartheta _{\xi }\omega )\right\rangle \textrm{d}\xi . \end{aligned}$$

(6.115)

Combining (6.107)–(6.115), we obtain

$$\begin{aligned}&\limsup _{n\rightarrow \infty }\Vert \varvec{v}_{\delta _n}(\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})\Vert ^2_{\mathbb {H}}\nonumber \\&\quad \le 4e^{-2\alpha l}\left[ \mathcal {R}_{0}(\mathfrak {s}-l,\vartheta _{-l}\omega )+e^{2\sigma (\mathfrak {s}-l)}\left| {\textbf {g}}(x)\right| ^2(1+\left| \varvec{y}(\vartheta _{-l}\omega )\right| ^2)\right] + \Vert \tilde{\varvec{v}}\Vert ^2_{\mathbb {H}}- e^{-2\alpha l}\Vert \tilde{\varvec{v}}_{l}\Vert ^2_{\mathbb {H}}\nonumber \\&\quad \le 4e^{-2\alpha l}\left[ \mathcal {R}_{0}(\mathfrak {s}-l,\vartheta _{-l}\omega )+e^{2\sigma (\mathfrak {s}-l)}\left| {\textbf {g}}(x)\right| ^2(1+\left| \varvec{y}(\vartheta _{-l}\omega )\right| ^2)\right] + \Vert \tilde{\varvec{v}}\Vert ^2_{\mathbb {H}}. \end{aligned}$$

(6.116)

Passing $l\rightarrow \infty$, we get

$$\begin{aligned} \limsup _{n\rightarrow \infty }\Vert \varvec{v}_{\delta _n}(\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l})\Vert ^2_{\mathbb {H}}\le \Vert \tilde{\varvec{v}}\Vert ^2_{\mathbb {H}}, \end{aligned}$$

which together with (6.99) gives

$$\begin{aligned} \varvec{v}_{\delta _n}(\mathfrak {s};\mathfrak {s}-l,\vartheta _{-\mathfrak {s}}\omega ,\varvec{v}_{n,l}) \rightarrow \hat{\varvec{v}} \text { in } \mathbb {H}. \end{aligned}$$

(6.117)

From (6.57), (6.96), (6.105) and (6.117), we get

$$\begin{aligned} \varvec{u}_n\rightarrow \tilde{\varvec{u}} \text { in } \mathbb {H}, \end{aligned}$$

as required, and the proof is completed. $\square$

The next Theorem demonstrates the upper semicontinuity of $\mathfrak {D}$-pullback random attractors as $\delta \rightarrow 0$, using the abstract theory given in [62, Theorem 3.2].

Theorem 6.15

For $0<\delta \le 1$, assume that all the assumptions of Lemma 6.4 are satisfied. Then for every $\omega \in \Omega$ and $\mathfrak {s}\in \mathbb {R}$,

$$\begin{aligned} \lim _{\delta \rightarrow 0}{} dist _{\mathbb {H}}\left( \mathscr {A}_{\delta }(\mathfrak {s},\omega ),\mathscr {A}_0(\mathfrak {s},\omega )\right) =0. \end{aligned}$$

(6.118)

Proof

By Lemma 6.11, we have for every $\mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$,

$$\begin{aligned} \limsup _{\delta \rightarrow 0}\Vert \mathcal {K}_{\delta }(\mathfrak {s},\omega )\Vert ^2_{\mathbb {H}}\le \limsup _{\delta \rightarrow 0}\mathcal {R}_{\delta }(\mathfrak {s},\omega )=\mathcal {R}_0(\mathfrak {s},\omega ). \end{aligned}$$

(6.119)

Consider a sequence $\delta \rightarrow 0$ and $\varvec{u}_{n,\mathfrak {s}}\rightarrow \varvec{u}_{\mathfrak {s}}$ in $\mathbb {H}$. By Lemma 6.12, we get that for every $t\ge 0, \mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$,

$$\begin{aligned} \Phi (t,\mathfrak {s},\omega ,\varvec{u}_{n,\mathfrak {s}}) \rightarrow \Phi _0(t,\mathfrak {s},\omega ,\varvec{u}_{\mathfrak {s}}) \ \text { in } \ \mathbb {H}. \end{aligned}$$

(6.120)

Hence, by (6.119), (6.120) and Lemma 6.14 together with Theorem 3.2 in [62], one can conclude the proof. $\square$

7 Convergence of Attractors: Multiplicative White Noise

In this section, we examine the approximations of solutions of the following stochastic CBF equations with multiplicative white noise,

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}\frac{\partial \varvec{u}}{\partial t}+\mu \text {A}\varvec{u}+\text {B}(\varvec{u})+\alpha \varvec{u}+\beta \mathcal {C}(\varvec{u})&={\varvec{f}}+\varvec{u}\circ \frac{\text {d}\text {W}}{\text {d}t}, \qquad \text {in } \mathbb {R}^d\times (\mathfrak {s},\infty ), \\ \varvec{u}|_{t=\mathfrak {s}}&=\varvec{u}_{\mathfrak {s}}, \qquad x\in \mathbb {R}^d \text{ and } \mathfrak {s}\in \mathbb {R}. \end{aligned} \right. \end{aligned} \end{aligned}$$

(7.1)

For $\delta >0$, consider the pathwise random system:

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}\frac{\partial \varvec{u}_{\delta }}{\partial t}+\mu \text {A}\varvec{u}_{\delta }+\text {B}(\varvec{u}_{\delta })+\alpha \varvec{u}_{\delta }+\beta \mathcal {C}(\varvec{u}_{\delta })&={\varvec{f}}+\mathcal {Z}_{\delta }(\vartheta _{t}\omega )\varvec{u}_{\delta }, \qquad \text {in } \mathbb {R}^d\times (\mathfrak {s},\infty ), \\ \varvec{u}_{\delta }|_{t=\mathfrak {s}}&=\varvec{u}_{\delta ,\mathfrak {s}}, \qquad x\in \mathbb {R}^d \text{ and } \mathfrak {s}\in \mathbb {R}. \end{aligned} \right. \end{aligned} \end{aligned}$$

(7.2)

The existence of a unique $\mathfrak {D}$-pullback random attractor for (7.1) and (7.2) is established in [37, Theorem 5.8] and Sect. 3, respectively. In this section, we prove the upper semicontinuity of attractors as $\delta \rightarrow 0$. Let us denote by $\widehat{\Phi }_0$ and $\widehat{\Phi }_{\delta }$, the continuous cocycles for the systems (7.1) and (7.2), respectively, and $\widehat{\mathscr {A}}_0$ and $\widehat{\mathscr {A}}_{\delta }$, $\mathfrak {D}$-pullback random attractors for the systems (7.1) and (7.2), respectively. Define

$$\begin{aligned} \begin{aligned} {v}(t;\mathfrak {s},\omega ,{v}_{\mathfrak {s}})&=e^{-\omega (t)}\varvec{u}(t;\mathfrak {s},\omega ,\varvec{u}_{\mathfrak {s}}) \ \text{ and } \ \\ {}{v}_{\delta }(t;\mathfrak {s},\omega ,{v}_{\delta ,\mathfrak {s}})&=e^{-\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }\varvec{u}_{\delta }(t;\mathfrak {s},\omega ,\varvec{u}_{\delta ,\mathfrak {s}}). \end{aligned} \end{aligned}$$

Then, from (7.1) and (7.2) (formally), we obtain

$$\begin{aligned} \left\{ \begin{aligned} \frac{\partial {v}}{\partial t}+\mu \text {A}{v}+e^{\omega (t)}\text {B}({v})+\alpha {v}+\beta e^{(r-1)\omega (t)}\mathcal {C}({v})&=e^{-\omega (t)}{\varvec{f}}, \qquad \text {in } \mathbb {R}^d\times (\mathfrak {s},\infty ), \\ {v}|_{t=\mathfrak {s}}={v}_{\mathfrak {s}}&=e^{-\omega (\mathfrak {s})}\varvec{u}_{\mathfrak {s}},\qquad x\in \mathbb {R}^d \text{ and } \mathfrak {s}\in \mathbb {R}. \end{aligned} \right. \nonumber \\ \end{aligned}$$

(7.3)

$$\begin{aligned}{} & {} \begin{aligned}{}&{} \left\{ \begin{aligned}\frac{\partial {v}_\delta }{\partial t}&+\mu \text {A}{v}_\delta +e^{\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }\text {B}({v}_\delta )+\alpha {v}_\delta +\beta e^{(r-1)\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }\mathcal {C}({v}_\delta )\\ {}&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad =e^{-\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }{\varvec{f}}, \qquad \text{ in } \mathbb {R}^d\times (\mathfrak {s},\infty ), \\ {v}_\delta |_{t=\mathfrak {s}}&={v}_{\delta ,\mathfrak {s}} =e^{-\int _{0}^{\mathfrak {s}}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }\varvec{u}_{\delta ,\mathfrak {s}}, \qquad x\in \mathbb {R}^d \text{ and } \mathfrak {s}\in \mathbb {R}. \end{aligned} \right. \nonumber \\ \end{aligned}\nonumber \\ \end{aligned}$$

(7.4)

For all $\mathfrak {s}\in \mathbb {R},$ $t>\mathfrak {s},$ and for every initial data in $\mathbb {H}$ and $\omega \in \Omega$, (7.3) and (7.4) have unique solutions in $\textrm{C}([\mathfrak {s},\mathfrak {s}+T];\mathbb {H})\cap \textrm{L}^2(\mathfrak {s}, \mathfrak {s}+T;\mathbb {V})\cap \textrm{L}^{r+1}(\mathfrak {s},\mathfrak {s}+T;\widetilde{\mathbb {L}}^{r+1}).$ Furthermore, the solution is continuous with respect to the initial data and $(\mathscr {F},\mathscr {B}(\mathbb {H}))$-measurable in $\omega \in \Omega .$

Lemma 7.1

Let Assumption 3.1 holds. Then, for all the cases given in Table 1 including $d=r=3$ with $2\beta \mu =1$, $\widehat{\Phi }_0$ possesses a closed measurable $\mathfrak {D}$-pullback random absorbing set $\widehat{\mathcal {K}}_0=\{\widehat{\mathcal {K}}_0(\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R}, \omega \in \Omega \}\in \mathfrak {D}$ in $\mathbb {H}$ given by

$$\begin{aligned} \widehat{\mathcal {K}}_0(\mathfrak {s},\omega )=\{\varvec{u}\in \mathbb {H}:\Vert \varvec{u}\Vert ^2_{\mathbb {H}}\le \widehat{\mathcal {R}}_0(\mathfrak {s},\omega )\}, \end{aligned}$$

(7.5)

where $\widehat{\mathcal {R}}_0(\mathfrak {s},\omega )$ is defined by

$$\begin{aligned} \widehat{\mathcal {R}}_0(\mathfrak {s},\omega )&=\frac{4}{\min \{\mu ,\alpha \}} \int _{-\infty }^{0} e^{\alpha \xi -2\omega (\xi )} \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}\textrm{d}\xi . \end{aligned}$$

(7.6)

Proof

Since the existence of $\mathfrak {D}$-pullback random absorbing set for $\widehat{\Phi }_0$ is proved in [37, Lemma 5.6], the rest of the proof follows in a similar way and hence we omit it here. $\square$

Lemma 7.2

Let Assumption 3.1 holds. Then, for all the cases given in Table 1 including $d=r=3$ with $2\beta \mu =1$, $\widehat{\Phi }_\delta$ possesses a closed measurable $\mathfrak {D}$-pullback random absorbing set $\widehat{\mathcal {K}}_\delta =\{\widehat{\mathcal {K}}_\delta (\mathfrak {s},\omega ):\mathfrak {s}\in \mathbb {R}, \omega \in \Omega \}\in \mathfrak {D}$ in $\mathbb {H}$ given by

$$\begin{aligned} \widehat{\mathcal {K}}_\delta (\mathfrak {s},\omega )=\{\varvec{u}\in \mathbb {H}:\Vert \varvec{u}\Vert ^2_{\mathbb {H}}\le \widehat{\mathcal {R}}_\delta (\mathfrak {s},\omega )\}, \end{aligned}$$

(7.7)

where $\widehat{\mathcal {R}}_\delta (\mathfrak {s},\omega )$ is defined by

$$\begin{aligned} \widehat{\mathcal {R}}_\delta (\mathfrak {s},\omega )=\frac{4}{\min \{\mu ,\alpha \}} \int _{-\infty }^{0} e^{\int _{0}^{\xi }\left( \alpha -2\mathcal {Z}_{\delta }(\vartheta _{\zeta }\omega )\right) \textrm{d}\zeta } \Vert \varvec{f}(\cdot ,\xi +\mathfrak {s})\Vert ^2_{\mathbb {V}'}\textrm{d}\xi . \end{aligned}$$

(7.8)

Furthermore, for every $\mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$,

$$\begin{aligned} \lim _{\delta \rightarrow 0}\widehat{\mathcal {R}}_\delta (\mathfrak {s},\omega )=\widehat{\mathcal {R}}_{0}(\mathfrak {s},\omega ). \end{aligned}$$

(7.9)

Proof

Since the existence of $\mathfrak {D}$-pullback random absorbing set $\widehat{\mathcal {K}}_\delta (\mathfrak {s},\omega )$ for $\widehat{\Phi }_\delta$ is proved in Lemma 3.8 and the convergence in (7.9) is proved in [24] (cf. [24, Lemma 3.7]), hence we omit the proof here. $\square$

Lemma 7.3

For all the cases given in Table 1 including $d=r=3$ with $2\beta \mu =1$, assume that $\varvec{f}\in \textrm{L}^2_{loc }(\mathbb {R};\mathbb {V}')$. Suppose that $\{\delta _n\}_{n\in \mathbb {N}}$ is a sequence such that $\delta _n\rightarrow 0$. Let $\varvec{u}_{\delta _n}$ and $\varvec{u}$ be the solutions of (7.2) and (7.1) with initial data $\varvec{u}_{\delta _n,\mathfrak {s}}$ and $\varvec{u}_{\mathfrak {s}}$, respectively. If $\Vert \varvec{u}_{\delta _n,\mathfrak {s}}-\varvec{u}_{\mathfrak {s}}\Vert _{\mathbb {H}}\rightarrow 0$ as $n\rightarrow \infty$, then for every $\mathfrak {s}\in \mathbb {R}$, $\omega \in \Omega$ and $t>\mathfrak {s}$,

$$\begin{aligned} \Vert \varvec{u}_{\delta _n}(t;\mathfrak {s},\omega ,\varvec{u}_{\delta _n,\mathfrak {s}})-\varvec{u}(t;\mathfrak {s},\omega ,\varvec{u}_{\mathfrak {s}})\Vert _{\mathbb {H}} \rightarrow 0 \ \text { as }\ n\rightarrow \infty . \end{aligned}$$

Proof

Let $\Upsilon =\varvec{v}_{\delta _n}-\varvec{v}$ and $\mathfrak {Z}(t)=\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\textrm{d}\xi -\omega (t)$. Then from (7.4) and (7.3), we find

$$\begin{aligned} \frac{1}{2}\frac{\text {d}}{\text {d}t}\Vert \Upsilon \Vert ^2_{\mathbb {H}}&=-\mu \Vert \nabla \Upsilon \Vert ^2_{\mathbb {H}}-\alpha \Vert \Upsilon \Vert ^2_{\mathbb {H}}-e^{\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }b(\Upsilon ,{v},\Upsilon )\nonumber \\ {}&\quad -e^{\omega (t)}(e^{\mathfrak {Z}(t)}-1)b({v},{v},\Upsilon ) \nonumber \\ {}&\quad -e^{(r-1)\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }\left\langle \mathcal {C}({v}_{\delta _n})-\mathcal {C}({v}),{v}_{\delta _n}-{v}\right\rangle \nonumber \\ {}&\quad -e^{(r-1)\omega (t)}\nonumber (e^{(r-1)\mathfrak {Z}(t)}-1)\left\langle \mathcal {C}({v}),\Upsilon \right\rangle \nonumber \\ {}&\quad +e^{-\omega (t)}(e^{-\mathfrak {Z}(t)}-1)\left\langle {\varvec{f}},\Upsilon \right\rangle .\end{aligned}$$

(7.10)

From (2.5), we obtain

$$\begin{aligned} -&e^{(r-1)\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }\left\langle \mathcal {C}({v}_{\delta _n})-\mathcal {C}({v}),{v}_{\delta _n}-{v}\right\rangle \nonumber \\ {}&\le -\frac{\beta }{2}e^{(r-1)\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }\Vert |{v}_{\delta _n}|^{\frac{r-1}{2}}\Upsilon \Vert ^2_{\mathbb {H}}-\frac{\beta }{2}e^{(r-1)\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }\Vert |{v}|^{\frac{r-1}{2}}\Upsilon \Vert ^2_{\mathbb {H}}\le 0. \end{aligned}$$

(7.11)

Case I: $d=2$ and $r\ge 1$. Applying (2.2), Lemmas 2.9 and 2.11, we get

$$\begin{aligned} \left| e^{\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\textrm{d}\xi }b(\Upsilon ,\varvec{v},\Upsilon )\right|&\le Ce^{2\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\textrm{d}\xi }\Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}\Vert \Upsilon \Vert ^2_{\mathbb {H}}+\frac{\mu }{4}\Vert \nabla \Upsilon \Vert ^2_{\mathbb {H}}, \end{aligned}$$

(7.12)

$$\begin{aligned} \left| e^{\omega (t)}(e^{\mathfrak {Z}(t)}-1)b(\varvec{v},\varvec{v},\Upsilon )\right|&\le C\Vert \Upsilon \Vert ^2_{\mathbb {H}}\Vert \nabla \Upsilon \Vert ^2_{\mathbb {H}} + Ce^{\frac{4}{3}\omega (t)}\left| e^{\mathfrak {Z}(t)}-1\right| ^{4/3}\nonumber \\&\quad \Vert \varvec{v}\Vert ^{2/3}_{\mathbb {H}}\Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}\nonumber \\&\le C\Vert \nabla \varvec{v}_{\delta _n}\Vert ^2_{\mathbb {H}}\Vert \Upsilon \Vert ^2_{\mathbb {H}}+C\Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}\Vert \Upsilon \Vert ^2_{\mathbb {H}} \nonumber \\&\quad + Ce^{\frac{4}{3}\omega (t)}|e^{\mathfrak {Z}(t)}-1|^{4/3} \Vert \varvec{v}\Vert ^{2/3}_{\mathbb {H}}\Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}, \end{aligned}$$

(7.13)

$$\begin{aligned} \left| e^{(r-1)\omega (t)}(e^{(r-1)\mathfrak {Z}(t)}-1)\left\langle \mathcal {C}(\varvec{v}),\Upsilon \right\rangle \right|&\le C e^{(r-1)\omega (t)}\left| e^{(r-1)\mathfrak {Z}(t)}-1\right| \left[ \Vert \varvec{v}\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}+\Vert \Upsilon \Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\right] , \end{aligned}$$

(7.14)

$$\begin{aligned} e^{-\omega (t)}(e^{-\mathfrak {Z}(t)}-1)\left\langle \varvec{f},\Upsilon \right\rangle&\le Ce^{-2\omega (t)}\left| e^{-\mathfrak {Z}(t)}-1\right| ^2\Vert \varvec{f}\Vert ^2_{\mathbb {V}'}+\frac{\min \{\mu ,\alpha \}}{4}\Vert \Upsilon \Vert ^2_{\mathbb {V}}. \end{aligned}$$

(7.15)

Combining (7.10)–(7.15), we get

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\Vert \Upsilon (t)\Vert ^2_{\mathbb {H}}&\le P_1(t)\Vert \Upsilon (t)\Vert ^2_{\mathbb {H}} + P_2(t), \end{aligned}$$

(7.16)

for a.e. $t\in [\mathfrak {s},\mathfrak {s}+T]$, where

$$\begin{aligned} P_1(t)&= C\left[ (e^{2\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\textrm{d}\xi }+1)\Vert \nabla \varvec{v}(t)\Vert ^2_{\mathbb {H}}+\Vert \nabla \varvec{v}_{\delta _n}(t)\Vert ^2_{\mathbb {H}}\right] ,\\ P_2(t)&=Ce^{(r-1)\omega (t)}|e^{(r-1)\mathfrak {Z}(t)}-1|\left[ \Vert \varvec{v}(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}+\Vert \varvec{v}_{\delta _n}(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\right] \\&\quad +Ce^{\frac{4}{3}\omega (t)}|e^{\mathfrak {Z}(t)}-1|^{4/3}\\&\quad \Vert \varvec{v}(t)\Vert ^{2/3}_{\mathbb {H}}\Vert \nabla \varvec{v}(t)\Vert ^2_{\mathbb {H}}+Ce^{-2\omega (t)}|e^{-\mathfrak {Z}(t)}-1|^2\Vert \varvec{f}(t)\Vert ^2_{\mathbb {V}'}. \end{aligned}$$

Case II: $d= 3$ and $r\ge 3$ ($r>3$ with any $\beta ,\mu >0$ and $r=3$ with $2\beta \mu \ge 1$). An application of Lemmas 2.9 and 2.11 yield

$$\begin{aligned}&\left| e^{\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }b(\Upsilon ,\varvec{v},\Upsilon )\right| \nonumber \\ {}&\quad \le {\left\{ \begin{array}{ll} \frac{1}{2\beta }\Vert \nabla \Upsilon \Vert _{\mathbb {H}}^2+\frac{\beta }{2}e^{2\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }\Vert |\varvec{v}|\Upsilon \Vert ^2_{\mathbb {H}}, \text{ for } r=3,\\ \frac{\mu }{4}\Vert \nabla \Upsilon \Vert _{\mathbb {H}}^2+\frac{\beta }{4}e^{(r-1)\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }\Vert |\varvec{v}|^{\frac{r-1}{2}}\Upsilon \Vert ^2_{\mathbb {H}}+C\Vert \Upsilon \Vert ^2_{\mathbb {H}}, \text{ for } r > 3, \end{array}\right. } \end{aligned}$$

(7.17)

and

$$\begin{aligned}&\left| e^{\omega (t)}(e^{\mathfrak {Z}(t)}-1)b(\varvec{v},\varvec{v},\Upsilon )\right| \nonumber \\ {}&\quad \le \left| 1-e^{-\mathfrak {Z}(t)}\right| e^{\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }\Vert \nabla \varvec{v}\Vert _{\mathbb {H}}\Vert |\varvec{v}|\Upsilon \Vert _{\mathbb {H}}\nonumber \\ {}&\quad \le {\left\{ \begin{array}{ll} \frac{1}{2\beta }\left| 1-e^{-\mathfrak {Z}(t)}\right| ^2\Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}+\frac{\beta }{2}e^{2\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }\Vert |\varvec{v}|\Upsilon \Vert ^2_{\mathbb {H}}, \text{ for } r=3, \\ \frac{1}{2\beta }\left| 1-e^{-\mathfrak {Z}(t)}\right| ^2\Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}+\frac{\beta }{4}e^{(r-1)\int _{0}^{t}\mathcal {Z}_{\delta }(\vartheta _{\xi }\omega )\text {d}\xi }\Vert |\varvec{v}|^{\frac{r-1}{2}}\Upsilon \Vert ^2_{\mathbb {H}}+C\Vert \Upsilon \Vert ^2_{\mathbb {H}}, \text{ for } r > 3. \end{array}\right. } \end{aligned}$$

(7.18)

Combining (7.10)–(7.11), (7.14)–(7.15) and (7.17)–(7.18), we obtain

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\Vert \Upsilon (t)\Vert ^2_{\mathbb {H}}&\le C\Vert \Upsilon (t)\Vert ^2_{\mathbb {H}} + P(t), \end{aligned}$$

(7.19)

for a.e. $t\in [\mathfrak {s},\mathfrak {s}+T]$, where

$$\begin{aligned} P(t)&=Ce^{(r-1)\omega (t)}|e^{(r-1)\mathfrak {Z}(t)}-1|\left[ \Vert \varvec{v}(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}+\Vert \varvec{v}_{\delta _n}(t)\Vert ^{r+1}_{\widetilde{\mathbb {L}}^{r+1}}\right] \nonumber \\ {}&\quad +\frac{1}{\beta }\left| 1-e^{-\mathfrak {Z}(t)}\right| ^2\Vert \nabla \varvec{v}\Vert ^2_{\mathbb {H}}+Ce^{-2\omega (t)}|e^{-\mathfrak {Z}(t)}-1|^2\Vert \varvec{f}(t)\Vert ^2_{\mathbb {V}'}. \end{aligned}$$

Now, applying Gronwall’s inequality in (7.16) and (7.19), and calculating similarly as in Lemma 6.12, one can conclude the proof with the help of the convergence (2.11). $\square$

The following result shows the uniform compactness of family of random attractors $\widehat{\mathscr {A}}_{\delta }$, which can be obtained by similar calculations of Lemma 6.14. In fact, the proof is much easier here and hence we omit here.

Lemma 7.4

Let Assumption 3.1 holds. Let $\mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$ be fixed. If $\delta _n\rightarrow 0$ as $n\rightarrow \infty$ and $\varvec{u}_n\in \widehat{\mathscr {A}}_{\delta _n}(\mathfrak {s},\omega )$. Then, for all the cases given in Table 1 including $d=r=3$ with $2\beta \mu =1$, the sequence $\{\varvec{u}_n\}_{n\in \mathbb {N}}$ has a convergent subsequence in $\mathbb {H}$.

Finally, we are in the position of giving main result of this section.

Theorem 7.5

For $0<\delta \le 1$, assume that Assumption 3.1 is satisfied. Then, for all the cases given in Table 1 including $d=r=3$ with $2\beta \mu =1$, for every $\omega \in \Omega$ and $\mathfrak {s}\in \mathbb {R}$,

$$\begin{aligned} \lim _{\delta \rightarrow 0}{} dist _{\mathbb {H}}\left( \widehat{\mathscr {A}}_{\delta }(\mathfrak {s},\omega ),\widehat{\mathscr {A}}_0(\mathfrak {s},\omega )\right) =0. \end{aligned}$$

(7.20)

Proof

By Lemma 7.2, we have for every $\mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$,

$$\begin{aligned} \limsup _{\delta \rightarrow 0}\Vert \widehat{\mathcal {K}}_{\delta }(\mathfrak {s},\omega )\Vert ^2_{\mathbb {H}}\le \limsup _{\delta \rightarrow 0}\widehat{\mathcal {R}}_{\delta }(\mathfrak {s},\omega )=\widehat{\mathcal {R}}_0(\mathfrak {s},\omega ). \end{aligned}$$

(7.21)

Consider a sequence $\delta \rightarrow 0$ and $\varvec{u}_{n,\mathfrak {s}}\rightarrow \varvec{u}_{\mathfrak {s}}$ in $\mathbb {H}$. By Lemma 7.3, we get that for every $t\ge 0, \mathfrak {s}\in \mathbb {R}$ and $\omega \in \Omega$,

$$\begin{aligned} \widehat{\Phi }_{\delta }(t,\mathfrak {s},\omega ,\varvec{u}_{n,\mathfrak {s}}) \rightarrow \widehat{\Phi }_0(t,\mathfrak {s},\omega ,\varvec{u}_{\mathfrak {s}}) \ \text { in } \ \mathbb {H}. \end{aligned}$$

(7.22)

Hence, by (7.21), (7.22) and Lemma 7.4 together with Theorem 3.2 in [62], we conclude the proof. $\square$

Pullback Random Attractors for Wong–Zakai Approximations of 2D Stochastic NSE on Unbounded Poincaré Domains

In this appendix, we discuss the existence and uniqueness of $\mathfrak {D}$-pullback random attractors for Wong–Zakai approximations of 2D stochastic NSE on Poincaré domains $\mathcal {O}$ (that is, satisfying Assumption 1.7) with nonlinear diffusion term. Consider the Wong–Zakai approximations of 2D NSE on $\mathcal {O}$ as

$$\begin{aligned} \left\{ \begin{aligned}\frac{\partial \varvec{u}}{\partial t}-\nu \Delta \varvec{u}+(\varvec{u}\cdot \nabla )\varvec{u}+\nabla p&=\varvec{f}(t) + S(t,x,\varvec{u})\mathcal {Z}_{\delta }(\vartheta _t\omega ),{} & {} \text{ in } \ \mathcal {O}\times (\mathfrak {s},\infty ), \\ \nabla \cdot \varvec{u}&=0,{} & {} \text{ in } \ \mathcal {O}\times (\mathfrak {s},\infty ), \\ \varvec{u}&={{\textbf {0}}}{} & {} \text{ on } \ \partial \mathcal {O}\times (\mathfrak {s},\infty ), \\ \varvec{u}|_{t=\mathfrak {s}}&=\varvec{u}_{\mathfrak {s}},{} & {} x\in \mathcal {O} \text{ and } \mathfrak {s}\in \mathbb {R}, \end{aligned} \right. \end{aligned}$$

(A.1)

where $\nu$ is the coefficient of kinematic viscosity of the fluid. In the work [23], authors proved the existence of a unique $\mathfrak {D}$-pullback random random attractor for the system (A.1) under Assumption 1.3. Here, we assume that the following conditions are satisfied:

Assumption A.1

Let $S:\mathbb {R}\times \mathcal {O}\times \mathbb {R}^2\rightarrow \mathbb {R}^2$ be a continuous function such that for all $t\in \mathbb {R}$, $x\in \mathcal {O}$ and ${\textbf {u}}\in \mathcal {O}$

$$\begin{aligned} |S(t,x,{\varvec{u}})|&\le |\mathcal {S}_1(t,x)||{\varvec{u}}|^{q-1}+|\mathcal {S}_2(t,x)|, \end{aligned}$$

where $1\le q<2$, $\mathcal {S}_1\in \text {L}^{\infty }_{loc }(\mathbb {R};\mathbb {L}^{\frac{2}{2-q}}(\mathcal {O}))$ and $\mathcal {S}_2\in \text {L}^{\infty }_{loc }(\mathbb {R};\mathbb {L}^{2}(\mathcal {O}))$. Furthermore, suppose that $S(t,x,\varvec{u})$ is locally Lipschitz continuous with respect to $\varvec{u}$.

Remark A.2

Note that Assumption A.1 is clearly different from Assumption 1.3. Therefore, it is worth to prove the results under Assumption A.1.

Assumption A.3

We assume that the external forcing term $\varvec{f}\in \text {L}^2_{\text {loc}}(\mathbb {R};\mathbb {L}^2(\mathcal {O}))$ satisfies

$$\begin{aligned} \int _{-\infty }^{\mathfrak {s}} e^{\nu \lambda \xi }\Vert \varvec{f}(\cdot ,\xi )\Vert ^2_{\mathbb {L}^2(\mathcal {O})}\textrm{d}\xi <\infty , \ \ \text { for all } \mathfrak {s}\in \mathbb {R}. \end{aligned}$$

and for every $c>0$

$$\begin{aligned} \lim _{\tau \rightarrow -\infty }e^{c\tau }\int _{-\infty }^{0} e^{\nu \lambda \xi }\Vert \varvec{f}(\cdot ,\xi +\tau )\Vert ^2_{\mathbb {L}^2(\mathcal {O})}\textrm{d}\xi =0, \end{aligned}$$

where $\lambda$ and $\nu$ are given in (1.10) and (A.1), respectively.

Since, 2D CBF equations with $r=1$ is a linear perturbation of 2D NSE, one can prove the next theorem using the same arguments as it is carried for 2D random CBF equations (2.12) with $d=2$ and $r=1$ in Sect. 4.2. Moreover, there are only minor changes, hence we omit the proof here.

Theorem A.4

Assume that Assumptions A.1 and A.3 hold true. Then there exists a unique $\mathfrak {D}$-pullback random attractor for the system (A.1), in $\mathbb {L}^2(\mathcal {O})$.