Pullback Attractor for the 2D Micropolar Fluid Flows with Delay on Unbounded Domains

Abstract

In this paper, we investigate the pullback asymptotic behavior of micropolar fluid flows with delay on 2D unbounded domains. Firstly, the existence of pullback attractor for the universe given by a tempered condition is established. Then we obtain the consistency of the pullback attractor with that for the universe of fixed bounded sets. Furthermore, the tempered behavior of the pullback attractor is given. Here we develop the energy method with the technique of decomposition of spatial domain to overcome the lack of compactness due to unbounded domains.

1 Introduction

1.1 Motivation

The classical 3D incompressible micropolar fluid model was firstly derived by Eringen [13], which was used to describe the fluids consisting of randomly oriented particles suspended in a viscous medium. The model is given by

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\partial u}{\partial t} -(\nu +\nu _r)\Delta u -2\nu _r {\mathrm{rot}} \omega +(u \cdot \nabla )u +\nabla p =f, \\ \nabla \cdot u =0,\\ \displaystyle \frac{\partial \omega }{\partial t} -(c_a+c_d)\Delta \omega +4\nu _r \omega +(u \cdot \nabla )\omega -(c_0+c_d-c_a)\nabla {\mathrm{div}}\omega -2\nu _r {\mathrm{rot}} u =\tilde{f},\end{array}\right. \end{aligned}$$

(1.1)

where $u=(u_1, u_2,u_3)$ is the velocity, $\omega =(\omega _1,\omega _2,\omega _3)$ is the angular velocity field of rotation of particles, p represents the pressure, and $f=(f_1, f_2, f_3)$ and $\tilde{f}=(\tilde{f}_1, \tilde{f}_2, \tilde{f}_3)$ stand for the external force and moment, respectively. The positive parameters $\nu , \nu _r, c_0, c_a$ and $c_d$ are viscous coefficients. Actually, $\nu$ represents the usual Newtonian viscosity and $\nu _r$ is called microrotation viscosity.

Micropolar fluid model plays an important role in the fields of applied and computational mathematics. There is a wide literature on the mathematical theory of micropolar fluid model (1.1). The existence, uniqueness and regularity of solutions for the micropolar fluids have been investigated in [11, 20]. Also, lots of works are devoted to the long time behavior of solutions for the micropolar fluids. More precisely, in the case of 2D bounded domains, Chen, Chen and Dong proved the existence of $H^2$-global attractor in [6] and verified the existence of uniform attractor in [7]. Łukaszewicz and Tarasińska [22] proved the existence of $H^1$-pullback attractor. Recently, Zhao and Sun et al. [36] established the $L^2$-pullback attractor and $H^1$-pullback attractor of solutions for the universe given by a temper condition, respectively. For the case of 2D unbounded domains, Dong and Chen [9] investigated the existence and regularity of global attractors. Later, they [10] obtained the $L^2$ time decay rate for global solutions of the 2D micropolar equations via the Fourier splitting method. Zhao, Zhou and Lian [34] established the existence of $H^1$-uniform attractor and further proved the $L^2$-uniform attractor belongs to the $H^1$-uniform attractor. Also some efforts are focused on the 2D micropolar equations with partial dissipation. For example, Dong and Zhang [11] examined the microrotation viscosity, namely $c_a+c_d=0$. The global regularity problem for this partial dissipation case is not trivial due to the presence of the term $\nabla \times \omega$ in the velocity equation. Dong and Zhang overcame the difficulty by making full use of the quantity $\nabla \times u -\frac{2\nu _r}{\nu +\nu _r}\omega$, which obeys a transport–diffusion equation. When the parameters $\nu =0$ and $\nu _r\ne c_a+c_d$, the global well posedness of the micropolar fluid equations was obtained in the frame work of Besov spaces in [33]. More recently, Dong, Li and Wu [12] studied the global regularity and large time behavior of solutions to the 2D micropolar equations with only angular viscosity dissipation.

In the real world, delay terms appear naturally, for instance as effects in wind tunnel experiments (see [26]), in the equations describing the motions of the fluids. The delay situations may also occur, for example, when we want to control the system via applying a force which considers not only the present state but also the history state of the system. To the best of our knowledge, the delays for ordinary differential equations (ODE) were first studied by Hale (see [17, 18]). As regards the partial differential equations (PDE) with delays including finite delays (constant, variable, distributed, etc.) and infinite delays. Different types of delays need to be treated by different approaches. To this respect, there are lots of important foundational works, particularly in the case of random dynamical systems. For the case where the delays are finite, one can refer to [3,4,5, 8, 21]. For the other case where the delays are infinite, one can see [2, 19, 23], etc.

However, to our knowledge, there is little literature for micropolar fluid with delay. Zhao and Sun [37] established the global well posedness of the weak solutions and proved the existence of pullback attractors for the micropolar fluid flows with infinite delay on 2D bounded domains. Furthermore, Zhou et al. [38] verified the $H^2$-boundedness of the pullback attractors obtained in [37]. Nowadays, Sun [30] proved the global well posedness for the micropolar fluid flows with delay on 2D unbounded domains.

The purpose of this paper is to study the pullback asymptotic properties of the global solutions obtained by Sun [30]. The main objective is to show the existence of pullback attractor for the universe given by a tempered condition. As a consequence, the existence of pullback attractor for the universe of fixed bounded sets follows. Giving suitable assumptions for external force, the consistent relationship between two pullback attractors and the tempered property of the pullback attractors is easily obtained by the means of [16, 27].

We mention that the asymptotic compactness is needed to achieve our goal and the method we want to address here is called energy equation method. For many physical systems there are energy equations (or their analogues) in the sense that the changing rate of energy equals the rate that energy is pumped into the system minus the energy dissipation rate due to various dissipation mechanisms. As far as we know, the method was first observed in 1922 by Ball (for weakly damped, driven semilinear wave equations) that such energy equations may be used to derive the asymptotic compactness of the solution semigroup, and he then wrote it up and published in [1]. The method has now been used in a variety of applications. For example, it was applied to a weakly damped, driven Korteweg–de Vries (KdV) equation by Ghidaglia [14], to weakly damped, driven hyperbolic-type equations by Wang [31], to parabolic-type problem by Rosa [28]. At almost the same time, Moise et al. [24] used this method to derive the asymptotic compactness property of the semigroup and presented a general formulation that can handle a number of weakly damped hyperbolic equations and parabolic equations on either bounded or unbounded spatial domains. In this paper, for the lack of compactness of the usual Sobolev embedding in unbounded domains. Inspired by [15, 25, 37], we apply the technique of the decomposition of spatial domain to overcome the difficulty. It also is worth mentioning that some new techniques are needed to deal with the term of delay, which is more complex than the case without delay.

1.2 Formation of Problem

In this paper, we consider the special situation that the velocity component in the $x_3$-direction is zero and the axes of rotation of particles are parallel to the $x_3$ axis. That is, $u=(u_1, u_2,0)$, $\omega =(0,0,\omega _3)$, $f=(f_1, f_2, 0), \tilde{f}=(0,0,\tilde{f}_3)$, $g=(g_1, g_2, 0)$ and $\tilde{g}=(0,0,\tilde{g}_3)$. Let $\Omega \subseteq \mathbb {R}^2$ be an open set with boundary $\Gamma$ that is not necessarily bounded but satisfies the following Poincaré inequality:

$$\begin{aligned} \text{ There } \text{ exists } \ \lambda _1>0 \ \text{ such } \text{ that } \ \lambda _1\Vert \varphi \Vert _{L^2(\Omega )}^2 \leqslant \Vert \nabla \varphi \Vert _{L^2(\Omega )}^2, \ \ \forall \varphi \in H_0^1(\Omega ). \end{aligned}$$

(1.2)

Then, we discuss the following 2D incompressible micropolar fluid model with delay

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\partial u}{\partial t} -(\nu +\nu _r)\Delta u -2\nu _r\mathrm \nabla \times \omega +(u \cdot \nabla )u+\nabla p\\ \quad =f(t, x)+g(t,u_t), \quad \mathrm{in} \,\, (\tau , +\infty )\times \Omega ,\\ \displaystyle \frac{\partial \omega }{\partial t} -\bar{\alpha }\Delta \omega +4\nu _r \omega -2\nu _r \nabla \times u+(u \cdot \nabla )\omega \\ \quad =\tilde{f}(t, x)+\tilde{g}(t, \omega _t), \quad \mathrm{in} \,\, (\tau , +\infty )\times \Omega ,\\ \nabla \cdot u=0, \quad \mathrm{in} \,\, (\tau , +\infty )\times \Omega ,\\ u=0,\ \omega =0, \quad \mathrm{on} \,\, (\tau , +\infty )\times \Gamma ,\\ (u(\tau ),\omega (\tau ))=(u^\mathrm{in},\omega ^\mathrm{in}), \ (u(t),\omega (t))\\ \quad =(\phi _1^\mathrm{in}(t-\tau ),\phi _2^\mathrm{in}(t-\tau )), \ \ t\in (\tau -h,\tau ), \end{array} \right. \end{aligned}$$

(1.3)

where $\bar{\alpha } :=c_a+c_d$, $x := (x_1,x_2)\in \Omega$, g and $\tilde{g}$ stand for the external force containing some hereditary characteristics $u_t$ and $\omega _t$, which are defined on $(-h,0)$ as follows

$$\begin{aligned} u_t(s):=u(t+s),\ \omega _t(s):=\omega (t+s), \ \forall t\geqslant \tau ,\ s \in (-h,0). \end{aligned}$$

(1.4)

$(\phi _1^\mathrm{in},\phi _2^\mathrm{in})$ represents the initial data in the interval of time $(-h,0)$, where h is a positive fixed number, and

$$\begin{aligned} \nabla \times u:=\frac{\partial u_2}{\partial x_1}-\frac{\partial u_1}{\partial x_2}, \quad \nabla \times \omega :=\left( \frac{\partial \omega }{\partial x_2}, -\frac{\partial \omega }{\partial x_1}\right) . \end{aligned}$$

For the sake of convenience, we introduce the following useful operators

$$\begin{aligned} \left\{ \begin{array}{l} \langle Aw, \phi \rangle := (\nu +\nu _r)(\nabla u, \nabla \Phi ) +\bar{\alpha }(\nabla \omega , \nabla \phi _3), \ \forall w =(u,\omega ), \phi =(\Phi , \phi _3)\in \widehat{V}, \\ \langle B(u, w), \phi \rangle := ((u \cdot \nabla )w, \phi ), \ \forall u\in V, \, w=(u,\omega )\in \widehat{V}, \,\forall \phi \in \widehat{V}, \\ N(w) :=(-2\nu _r \nabla \times \omega , -2\nu _r \nabla \times u+4\nu _r\omega ), \ \forall w=(u, \omega )\in \widehat{V}. \end{array} \right. \end{aligned}$$

(1.5)

Then, we can formulate the weak version of Eq. (1.3) as follows

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\partial w}{\partial t} + Aw +B(u,w) + N(w) = F(t,x)+G(t,w_t),\, \mathrm{in} \ (\tau ,+\infty )\times \Omega , \\ \nabla \cdot u = 0, \, \mathrm{in} \ (\tau , +\infty )\times \Omega , \\ w = (u,\omega ) = 0, \ \ \mathrm{on} \ (\tau , +\infty )\times \Gamma , \\ w(\tau ) = w^\mathrm{in} = (u^\mathrm{in}, \omega ^\mathrm{in}), \ w(t)=\phi ^\mathrm{in}(t-\tau )\\ \quad =(\phi _1^\mathrm{in}(s),\phi _2^\mathrm{in}(s)), \ \ t\in (\tau -h,\tau ),\ s\in (-h,0), \end{array} \right. \end{aligned}$$

(1.6)

where

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle w(t,x):=(u(t,x),\omega (t,x)),\\ F(t,x):=(f(t,x), \tilde{f}(t,x))\\ G(t,w_t):=(g(t,u_t),\tilde{g}(t,\omega _t)). \end{array} \right. \end{aligned}$$

1.3 Notation

Throughout this paper, we denote the usual Lebesgue space and Sobolev space by $L^p(\Omega )$ and $W^{m,p}(\Omega )$ endowed with norms $\Vert \cdot \Vert _p$ and $\Vert \cdot \Vert _{m,p}$, respectively. Particularly, we denote $H^m(\Omega ):=W^{m,2}(\Omega )$.

$$\begin{aligned} \mathcal {V}:=&\mathcal {V}(\Omega ) :=\left\{ \varphi \in \mathcal {C}_0^\infty (\Omega ) \times \mathcal {C}_0^\infty (\Omega )|\, \varphi =(\varphi _1, \varphi _2), \nabla \cdot \varphi =0 \right\} , \\ \widehat{\mathcal {V}}:=&\widehat{\mathcal {V}}(\Omega ) :=\mathcal {V} \times \mathcal {C}_0^\infty (\Omega ),\\ H:=&H(\Omega ) := \mathrm{closure\,\, of} \,\,\mathcal {V}\,\, \mathrm{in}\,\,L^2(\Omega )\\&\times L^2(\Omega ),\,\, \mathrm{with\,\, norm}\,\, \Vert \cdot \Vert _{H} \,\, \mathrm{and\,\,\, dual\,\, space}\,\, H^*,\\ V:=&V(\Omega ) := \mathrm{closure\,\,\, of} \,\, \mathcal {V}\,\, \mathrm{in}\,\,H^1(\Omega )\\&\times H^1(\Omega ),\,\,\mathrm{with\,\, norm}\,\, \Vert \cdot \Vert _V \,\, \mathrm{and\,\, dual\,\, space}\,\, V^*,\\ \widehat{H}:=&\widehat{H}(\Omega ) := \mathrm{closure\,\, of} \,\,\widehat{\mathcal {V}}\,\, \mathrm{in}\,\,L^2(\Omega )\\&\times L^2(\Omega )\times L^2(\Omega ), \,\, \mathrm{with\,\, norm}\,\,\Vert \cdot \Vert _{\widehat{H}} \,\, \mathrm{and\,\, dual\,\, space}\,\, \widehat{H}^*,\\ \widehat{V}:=&\widehat{V}(\Omega ) := \mathrm{closure\,\, of} \,\,\widehat{\mathcal {V}}\,\, \mathrm{in}\,\,H^1(\Omega )\\&\times H^1(\Omega )\times H^1(\Omega ), \,\, \mathrm{with\,\,norm}\,\,\Vert \cdot \Vert _{\widehat{V}} \,\, \mathrm{and\,\, dual\,\, space}\,\, \widehat{V}^*. \end{aligned}$$

$(\cdot ,\cdot ) -$ the inner product in $L^2(\Omega ), H$ or $\widehat{H}$, $\langle \cdot , \cdot \rangle -$ the dual pairing between V and $V^*$ or between $\widehat{V}$ and $\widehat{V}^*$. Throughout this article, we simplify the notations $\Vert \cdot \Vert _{2}$, $\Vert \cdot \Vert _{H}$ and $\Vert \cdot \Vert _{\widehat{H}}$ by the same notation $\Vert \cdot \Vert$ if there is no confusion. Furthermore, we denote

$$\begin{aligned} L^p(I; X)&:= \text{ space } \text{ of } \text{ strongly } \text{ measurable } \text{ functions } \text{ on } \text{ the } \text{ closed } \text{ interval } I, \\&\ \ \quad \text{ with } \text{ values } \text{ in } \text{ the } \text{ Banach } \text{ space } X, \text{ endowed } \text{ with } \text{ norm }\\&\qquad \qquad \Vert \varphi \Vert _{L^p(I; X)} :=\left( \int _I \Vert \varphi \Vert ^p_X {\mathrm d}t\right) ^{1/p},\ \ \text{ for } 1\leqslant p<\infty ,\\ \mathcal {C}(I; X)&:= \text{ space } \text{ of } \text{ continuous } \text{ functions } \text{ on } \text{ the } \text{ interval } I, \\ {}&\ \ \quad \text{ with } \text{ values } \text{ in } \text{ the } \text{ Banach } \text{ space } X, \text{ endowed } \text{ with } \text{ the } \text{ usual } \text{ norm, }\\ L_{loc}^2(I; X)&:=\text{ space } \text{ of } \text{ locally } \text{ square } \text{ integrable } \text{ functions } \text{ on } \text{ the } \text{ interval } I,\\&\ \ \quad \text{ with } \text{ values } \text{ in } \text{ the } \text{ Banach } \text{ space } X, \text{ endowed } \text{ with } \text{ the } \text{ usual } \text{ norm, }\\ \hookrightarrow \hookrightarrow&- \text{ the } \text{ compact } \text{ embedding } \text{ between } \text{ spaces }. \end{aligned}$$

Following the above notations, we additionally denote

$$\begin{aligned} L_{\widehat{H}}^2 := L^2(-h,0; \widehat{H}), \quad L_{\widehat{V}}^2 := L^2(-h,0; \widehat{V}), \qquad \ \ \\ E_{\widehat{H}}^2 := \widehat{H} \times L_{\widehat{H}}^2, \ \ E_{\widehat{V}}^2 := \widehat{V} \times L_{\widehat{V}}^2, \ \ E_{\widehat{H}\times L_{\widehat{V}}^2}^2 := \widehat{H} \times L_{\widehat{V}}^2. \end{aligned}$$

The norm $\Vert \cdot \Vert _{X}$ for $X\in \{E_{\widehat{H}}^2, \, E_{\widehat{V}}^2, \, E_{\widehat{H}\times L_{\widehat{V}}^2}^2\}$ is defined as

$$\begin{aligned}&\Vert (w,v)\Vert _{E_{\widehat{H}}^2}:= (\Vert w\Vert ^2+\Vert v\Vert _{L_{\widehat{H}}^2}^2)^{1/2}, \quad \Vert (w,v)\Vert _{E_{\widehat{V}}^2} := (\Vert w\Vert ^2+\Vert v\Vert _{L_{\widehat{V}}^2}^2)^{1/2}, \\&\Vert (w,v)\Vert _{E_{\widehat{H} \times L_{\widehat{V}}^2}^2} := (\Vert w\Vert ^2 + \Vert v\Vert _{L_{\widehat{V}}^2}^2)^{1/2}. \end{aligned}$$

The rest of this paper is organized as follows. In Sect. 2, we make some preliminaries. Section 3 is devoted to proving the existence of pullback attractor for the universe given by a tempered condition. In Sect. 4, we aim at some properties of the pullback attractor obtained in Sect. 3.

2 Preliminaries

In this section, we recall some key results about the non-autonomous micropolar fluid flows and introduce some notations and definitions about pullback attractor. To begin with, we list some useful estimates and properties for the operators A, $B(\cdot )$ and $N(\cdot )$ defined in (1.5), which have been established in works [25, 30, 34, 37] .

Lemma 2.1

• 1. The operator A is linear continuous both from $\widehat{V}$ to $\widehat{V}^*$ and from D(A) to $\widehat{H}$, and so is for the operator $N(\cdot )$ from $\widehat{V}$ to $\widehat{H}$, where $D(A):=\widehat{V}\cap \left( H^2(\Omega )\right) ^3$.

• 2. The operator $B(\cdot ,\cdot )$ is continuous from $V\times \widehat{V}$ to $\widehat{V}^*$. Moreover, for any $u\in V$ and $w\in \widehat{V}$, there holds

  $$\begin{aligned} \langle B(u, \psi ), \varphi \rangle =-\langle B(u, \varphi ), \psi \rangle ,\forall \,u\in V, \forall \,\psi \in \widehat{V}, \forall \,\varphi \in \widehat{V}. \end{aligned}$$

  (2.1)

Lemma 2.2

• 1. There are two positive constants $c_1$ and $c_2$ such that

  $$\begin{aligned} c_1\langle Aw, w \rangle \leqslant \Vert w\Vert _{\widehat{V}}^2 \leqslant c_2\langle Aw, w \rangle , \forall w\in {\widehat{V}}. \end{aligned}$$

  (2.2)

• 2. There exists some positive constant $\alpha _0$ which depends only on $\Omega$, such that for any $(u, \psi , \varphi )\in V\times \widehat{V}\times \widehat{V}$ there holds

  $$\begin{aligned} |\langle B(u,\psi ),\varphi \rangle |&\leqslant \left\{ \begin{array}{ll} \alpha _0 \Vert u\Vert ^{\frac{1}{2}}\Vert \nabla u\Vert ^{\frac{1}{2}} \Vert \varphi \Vert ^{\frac{1}{2}}\Vert \nabla \varphi \Vert ^{\frac{1}{2}}\Vert \nabla \psi \Vert ,\\ \alpha _0 \Vert u\Vert ^{\frac{1}{2}}\Vert \nabla u\Vert ^{\frac{1}{2}} \Vert \psi \Vert ^{\frac{1}{2}}\Vert \nabla \psi \Vert ^{\frac{1}{2}}\Vert \nabla \varphi \Vert . \end{array} \right. \end{aligned}$$

  (2.3)

  Moreover, if $(u, \psi , \varphi )\in V\times D(A)\times D(A)$, then

  $$\begin{aligned} |\langle B(u,\psi ),A\varphi \rangle | \leqslant \alpha _0 \Vert u\Vert ^{\frac{1}{2}}\Vert \nabla u\Vert ^{\frac{1}{2}} \Vert \nabla \psi \Vert ^{\frac{1}{2}}\Vert A\psi \Vert ^{\frac{1}{2}} \Vert A\varphi \Vert . \end{aligned}$$

  (2.4)

• 3. There exists a positive constant $c(\nu _r)$ such that

  $$\begin{aligned} \Vert N(\psi )\Vert \leqslant c(\nu _r)\Vert \psi \Vert _{\widehat{V}}, \forall \psi \in \widehat{V}. \end{aligned}$$

  (2.5)

  In addition,

  $$\begin{aligned} -\langle N(\psi ), A\psi \rangle&\leqslant \frac{1}{4}\Vert A\psi \Vert ^2+c^2(\nu _r)\Vert \psi \Vert _{\widehat{V}}^2, \,\,\forall \psi \in D(A), \end{aligned}$$

  (2.6)
  $$\begin{aligned} \delta _1\Vert \psi \Vert _{\widehat{V}}^2&\leqslant \langle A\psi , \psi \rangle +\langle N(\psi ), \psi \rangle , \forall \, \psi \in \widehat{V}, \end{aligned}$$

  (2.7)

  where $\delta _1 :=\mathrm {min}\{\nu , \bar{\alpha }\}$.

Then, we recall the global well posedness of solutions established in [30].

Assumption 2.1

  Assume that $G: \mathbb {R}\times L^2(-h,0;\widehat{H}) \mapsto (L^2(\Omega ))^3$ satisfies:

• (i) For any $\xi \in L^2(-h,0;\widehat{H})$, the mapping $\mathbb {R} \ni t \mapsto G(t,\xi )\in (L^2(\Omega ))^3$ is measurable.

• (ii) $G(\cdot , 0)=(0,0,0)$.

• (iii) There exists a constant $L_G>0$ such that for any $t\in \mathbb {R}$ and any $\xi , \eta \in L^2(-h,0; \widehat{H})$,

  $$\begin{aligned} \Vert G(t,\xi )-G(t,\eta )\Vert \leqslant L_G\Vert \xi -\eta \Vert _{L^2(-h,0;\widehat{H})}. \end{aligned}$$

• (iv) There exists $C_G\in (0,\delta _1)$ such that, for any $t\geqslant \tau$ and any $w, v \in L^2(\tau -h,t; \widehat{H})$,

  $$\begin{aligned} \int _{\tau }^t \Vert G(\theta ,w_{\theta })-G(\theta ,v_{\theta })\Vert ^2 {\mathrm{d}}\theta \leqslant C_G^2\int _{\tau -h}^t \Vert w(\theta )-v(\theta )\Vert ^2 {\mathrm{d}}\theta . \end{aligned}$$

  Moreover, for any $t\geqslant \tau$, there exists a $\gamma \in (0,2\delta _1-2C_G)$ such that

  $$\begin{aligned} \int _\tau ^t e^{\gamma \theta }\Vert G(\theta ,w_{\theta })\Vert ^2 {\mathrm{d}}\theta \leqslant C_G^2\int _{\tau -h}^t e^{\gamma \theta }\Vert w(\theta )\Vert ^2 {\mathrm{d}}\theta , \ \forall \, w\in L^2(\tau -h,t;\widehat{H}). \end{aligned}$$

Theorem 2.1

  Assume $F(t,x)\in L_{loc}^2(\mathbb {R};\widehat{V}^*), \, \forall \, t\geqslant \tau , \,\tau \in \mathbb {R}$, and G satisfies Assumption 2.1. Then, for any $(w^\mathrm{in},\phi ^\mathrm{in})\in E_{\widehat{H}}^2$, there is a unique weak solution $w(\cdot ):= w(\cdot ;\tau ,w^\mathrm{in},\phi ^\mathrm{in})$ of system (1.6), which satisfies

$$\begin{aligned} w \in \mathcal {C}([\tau ,T];\widehat{H}) \cap L^2(\tau , T; \widehat{V}) \ \ \text{ and } \ \ w'\in L^2(\tau ,T;\widehat{V}^*), \ \ \forall \, T>\tau . \end{aligned}$$

Moreover, let $v(\cdot ):= v(\cdot ;\tau ,v^\mathrm{in},\psi ^\mathrm{in})$ be another weak solution corresponding to the initial value $(v^\mathrm{in},\psi ^\mathrm{in})\in E_{\widehat{H}}^2$, then, for all $t \geqslant \tau$, we have

$$\begin{aligned} \Vert w(t)-v(t)\Vert ^2 \leqslant&r_1 \Vert (w^\mathrm{in}-v^\mathrm{in}, \phi ^\mathrm{in}-\psi ^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2 \cdot e^{\sigma (w(t))}, \end{aligned}$$

(2.8)

$$\begin{aligned} \int _{\tau }^t \Vert w(\theta )-v(\theta )\Vert _{\widehat{V}}^2\,{\mathrm{d}}\theta \leqslant&\delta _1^{-1}r_1 \Vert (w^\mathrm{in}-v^\mathrm{in}, \phi ^\mathrm{in}-\psi ^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2 \nonumber \\&\quad \big ( 1+ \sigma (w(t)) \cdot e^{\sigma (w(t))} \big ), \end{aligned}$$

(2.9)

where

$$\begin{aligned} r_1 := \max \{1,2a_2C_G^2\}, \ \ \sigma (w(t)) := \int _{\tau }^t \big ( \delta _1^{-1} \alpha _0^2 \Vert w(s)\Vert _{\widehat{V}}^2 + 2a_1 + 2a_2C_G^2 \big ) {\mathrm{d}}s, \end{aligned}$$

where the positive constants $a_1, a_2$ satisfy $a_1a_2 \geqslant \frac{1}{4}$.

On the basis of Theorem 2.1, the biparametric map defined by

$$\begin{aligned} U(t,\tau ) \, : \, (w^\mathrm{in},\phi ^\mathrm{in}) \mapsto (w(t;\tau ,w^\mathrm{in},\phi ^\mathrm{in}), w_t(\cdot ;\tau ,w^\mathrm{in},\phi ^\mathrm{in})), \ \ \forall \, t \geqslant \tau , \end{aligned}$$

(2.10)

generates a continuous process in $E_{\widehat{H}}^2$ and $E_{\widehat{V}}^2$, respectively, which satisfies the following properties:

$$\begin{aligned}&\mathrm{(i)} U(\tau ,\tau ) (w^\mathrm{in},\phi ^\mathrm{in}) = (w^\mathrm{in},\phi ^\mathrm{in}),\qquad \\&\mathrm{(ii)} U(t,s)U(s,\tau ) (w^\mathrm{in},\phi ^\mathrm{in}) = U(t,\tau )(w^\mathrm{in},\phi ^\mathrm{in}). \end{aligned}$$

Finally, we end this section with some notations and definitions concerning the pullback attractors for non-autonomous dynamical systems in the following. On can refer to [16, 27, 35].

We denote by X the space $\widehat{H}$ or $\widehat{V}$ and by $\mathcal {P}(X)$ the family of all nonempty subsets of X. A universe $\mathcal {D}(X)$ in $\mathcal {P}(X)$ represents the class of families parameterized in time $\widehat{B}(X) = \{B(t) \, | \, t\in \mathbb {R}\}\subseteq \mathcal {P}(X)$.

Definition 2.1

A family of sets $\widehat{B}_0=\{B_0(t)|\, t\in {\mathbb R}\} \subseteq \mathcal {P}(X)$ is called pullback $\mathcal {D}$-absorbing for the process $\{U(t,\tau )\}_{t\geqslant \tau }$ in X if for any $t\in {\mathbb R}$ and any $\widehat{B}=\{B(t)| \, t\in {\mathbb R}\}\in \mathcal {D}$, there exists a $\tau _0(t, \widehat{B})\leqslant t$ such that $U(t,\tau )B(\tau )\subseteq B_0(t)$ for all $\tau \leqslant \tau _0(t,\widehat{B})$.

Definition 2.2

The process $\{U(t,\tau )\}_{t\geqslant \tau }$ is said to be pullback $\widehat{B}_0$-asymptotically compact in X if for any $t\in {\mathbb R}$, any sequences $\{\tau _n\}\subseteq (-\infty , t]$ and $\{x_n\}\subseteq X$ satisfying $\tau _n\rightarrow -\infty$ as $n\rightarrow \infty$ and $x_n\in B_0(\tau _n)$ for all n, the sequence $\{ U(t, \tau _n;x_n)\}$ is relatively compact in X. $\{U(t,\tau )\}_{t\geqslant \tau }$ is called pullback $\mathcal {D}$-asymptotically compact in X if it is pullback $\widehat{B}$-asymptotically compact for any $\widehat{B} \in \mathcal {D}$.

Definition 2.3

A family of sets $\widehat{\mathcal {A}}_X =\{ \mathcal {A}_X(t)|\, t \in {\mathbb R} \} \subseteq \mathcal {P} (X)$ is called a pullback $\mathcal {D}$-attractor for the process $\{U(t,\tau )\}_{t\geqslant \tau }$ on X if it has the following properties:

• Compactness: for any $t\in {\mathbb R}, \mathcal {A}_X(t)$ is a nonempty compact subset of X;

• Invariance: $U(t, \tau ) \mathcal {A}_X(\tau ) =\mathcal {A}_X(t), \, \forall \,t\geqslant \tau$;

• Pullback attracting: $\widehat{\mathcal {A}}_X$ is pullback $\mathcal {D}$ attracting in the following sense:

  $$\begin{aligned} ~~\lim \limits _{\tau \rightarrow -\infty } \mathrm {dist}_X\left( U(t, \tau )B(\tau ), \mathcal {A}_X(t) \right) =0, \, \forall \widehat{B}=\{ B(s)|\, s\in {\mathbb R}\} \in \mathcal {D},t\in {\mathbb R}, \end{aligned}$$

• Minimality: the family of sets $\widehat{\mathcal {A}}_X$ is the minimal in the sense that if $\widehat{O}=\{ O(t)|\, t\in {\mathbb R}\} \subseteq \mathcal {P}(X)$ is another family of closed sets such that

  $$\begin{aligned} \lim \limits _{\tau \rightarrow -\infty } \mathrm {dist}_X(U(t, \tau )B(\tau ), O(t))=0 ,\ \ \text{ for } \text{ any } \widehat{B}= \{ B(t)|\, t\in {\mathbb R}\} \in \mathcal {D}, \end{aligned}$$

        then $\mathcal {A}_X(t)\subseteq O(t)$ for $t\in \mathbb {R}$.

Remark 2.1

If the process U possesses a pullback $\mathcal {D}$-absorbing set $\widehat{B}_0$ and is pullback $\widehat{B}_0$-asymptotically compact in X, we can construct the pullback attractor by the standard method introduced by García-Luengo et al. [16, Proposition 9] and Marín-Rubio and Real in [27, Theorem 18]

3 Existence of Pullback Attractor for the Universe $\mathcal {D}_{\gamma }$

In the section, we investigate the pullback attractor for the universe $\mathcal {D}_{\gamma }$ given by a tempered condition in space $E_{\widehat{H}}^2$. To begin with, in order to construct the universe $\mathcal {D}_{\gamma }$, we give some useful energy estimates.

Lemma 3.1

For any $t\geqslant \tau$, assume that $F\in L^2(\tau ,t; \widehat{V}^*)$ and G satisfies Assumption 2.1. Then, for any $(w^\mathrm{in},\phi ^\mathrm{in})\in E_{\widehat{H}}^2$, we have

$$\begin{aligned}&\Vert w(t)\Vert ^2 + \beta e^{-\gamma t}\int _{\tau }^t e^{\gamma \theta }\Vert w(\theta )\Vert _{\widehat{V}}^2 {\mathrm{d}}\theta \nonumber \\ \leqslant&(1+C_G)e^{-\gamma (t-\tau )} \Vert (w^\mathrm{in},\phi ^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2 + \alpha ^{-1}e^{-\gamma t} \int _{\tau }^t e^{\gamma \theta }\Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta , \end{aligned}$$

(3.1)

where $\alpha \in (0, 2\delta _1 - 2C_G - \gamma ), \ \beta := 2\delta _1 - 2C_G - \gamma - \alpha > 0$.

Proof

Let us denote $w(\cdot )= w(\cdot ;\tau ,w^\mathrm{in},\phi ^\mathrm{in})$. Testing (1.6)$_1$ by w(t), we obtain from (2.1) and (2.7) that

$$\begin{aligned} \frac{1}{2} \frac{\mathrm{d}}{{\mathrm{d}}t} \Vert w(t)\Vert ^2 + \delta _1 \Vert w(t)\Vert _{\widehat{V}}^2 \leqslant \langle F(t),w(t) \rangle + (G(t,w_t),w(t)), \end{aligned}$$

which implies

$$\begin{aligned}&\frac{\mathrm{d}}{{\mathrm{d}}t}(e^{\gamma t}\Vert w(t)\Vert ^2) - \gamma e^{\gamma t} \Vert w(t)\Vert ^2 + 2\delta _1 e^{\gamma t} \Vert w(t)\Vert _{\widehat{V}}^2 \\&\quad \leqslant 2e^{\gamma t} \langle F(t),w(t) \rangle + 2e^{\gamma t}(G(t,w_t),w(t)). \end{aligned}$$

Let $\tau \leqslant \theta \leqslant t$. Replacing the time variable t in the above inequality with $\theta$, then integrating it with respect to $\theta$ over $[\tau ,t]$ gives

$$\begin{aligned}&e^{\gamma t} \Vert w(t)\Vert ^2 + (2\delta _1-\gamma )\int _{\tau }^t e^{\gamma \theta }\Vert w(\theta )\Vert _{\widehat{V}}^2 {\mathrm{d}}\theta \nonumber \\ \leqslant&e^{\gamma \tau }\Vert w^\mathrm{in}\Vert ^2 + 2\int _{\tau }^t e^{\gamma \theta } \langle F(\theta ),w(\theta ) \rangle {\mathrm{d}}\theta + 2\int _{\tau }^t e^{\gamma \theta } (G(\theta ,w_{\theta }),w(\theta )) {\mathrm{d}}\theta . \end{aligned}$$

(3.2)

By the Young’s inequality and Assumption 2.1, we deduce

$$\begin{aligned}&2\int _{\tau }^t e^{\gamma \theta } (G(\theta ,w_{\theta }), w(\theta )) {\mathrm{d}}\theta \leqslant 2\int _{\tau }^t e^{\gamma \theta } \Vert G(\theta ,w_{\theta }\Vert \Vert w(\theta )\Vert {\mathrm{d}}\theta \nonumber \\&\quad \leqslant 2\left( \int _{\tau }^t e^{\gamma \theta }\Vert G(\theta ,w_{\theta }\Vert ^2 {\mathrm{d}}\theta \right) ^{\frac{1}{2}} \left( \int _{\tau }^t e^{\gamma \theta }\Vert w(\theta )\Vert ^2 {\mathrm{d}}\theta \right) ^{\frac{1}{2}} \nonumber \\&\quad \leqslant C_G \int _{\tau -h}^{\tau } e^{\gamma \theta }\Vert w(\theta )\Vert ^2 {\mathrm{d}}\theta + 2C_G\int _{\tau }^t e^{\gamma \theta }\Vert w(\theta )\Vert ^2{\mathrm{d}}\theta \nonumber \\&\quad \leqslant C_G e^{\gamma \tau }\Vert \phi ^\mathrm{in}\Vert _{L_{\widehat{H}}^2}^2 + 2C_G \int _{\tau }^t e^{\gamma \theta }\Vert w(\theta )\Vert ^2 {\mathrm{d}}\theta , \end{aligned}$$

(3.3)

and

$$\begin{aligned} 2\int _{\tau }^t e^{\gamma \theta }\langle F(\theta ),w(\theta )\rangle {\mathrm{d}}\theta \leqslant&2\int _{\tau }^t e^{\gamma \theta }\Vert F(\theta )\Vert _{\widehat{V}^*} \Vert w(\theta )\Vert _{\widehat{V}} {\mathrm{d}}\theta \nonumber \\ \leqslant&\alpha ^{-1}\int _{\tau }^t e^{\gamma \theta }\Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta + \alpha \int _{\tau }^t e^{\gamma \theta }\Vert w(\theta )\Vert _{\widehat{V}}^2 {\mathrm{d}}\theta , \end{aligned}$$

(3.4)

where $\alpha \in (0, 2\delta _1-\gamma -2C_G)$. Substituting (3.3) and (3.4) into (3.2), yields (3.1). This completes the proof. $\square$

As a consequence of Lemma 3.1, we immediately have

Proposition 3.1

Under the conditions of Lemma 3.1, for any $t\geqslant T+\tau , \, T>0$ and $(w^\mathrm{in},\phi ^\mathrm{in})\in E_{\widehat{H}}^2$, it holds that

$$\begin{aligned}&\int _{t-T}^t \Vert w(\theta ;\tau ,w^\mathrm{in},\phi ^\mathrm{in})\Vert _{\widehat{V}}^2 {\mathrm{d}}\theta \nonumber \\ \leqslant&\beta ^{-1}(1+C_G) e^{-\gamma (t-T-\tau )} \Vert (w^\mathrm{in},\phi ^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2 + (\alpha \beta )^{-1} e^{-\gamma (t-T)} \int _{\tau }^t e^{\gamma \theta } \Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta , \end{aligned}$$

(3.5)

where $\alpha$ and $\beta$ come from (3.1).

Proof

For $t\geqslant T+\tau$, we have

$$\begin{aligned} \int _{\tau }^t e^{\gamma \theta } \Vert w(\theta )\Vert _{\widehat{V}}^2 {\mathrm{d}}\theta \geqslant \int _{t-T}^t e^{\gamma \theta } \Vert w(\theta )\Vert _{\widehat{V}}^2 {\mathrm{d}}\theta \geqslant e^{\gamma (t-T)} \int _{t-T}^t \Vert w(\theta )\Vert _{\widehat{V}}^2 {\mathrm{d}}\theta \end{aligned}$$

(3.6)

From (3.1), it follows that

$$\begin{aligned} \beta \int _{\tau }^t e^{\gamma \theta } \Vert w(\theta )\Vert _{\widehat{V}}^2{\mathrm{d}}\theta \leqslant (1+C_G)e^{\gamma \tau } \Vert (w^\mathrm{in},\phi ^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2 + \alpha ^{-1}\int _{\tau }^t e^{\gamma \theta }\Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta , \end{aligned}$$

which together with (3.6) implies (3.5). The proof is complete. $\square$

Now, we can construct the universe $\mathcal {D}_{\gamma }$ in the following.

Definition 3.1

(Definition of universe$\mathcal {D}_{\gamma }$)

Set

$$\begin{aligned} \mathcal {R}_{\gamma } := \{ \rho (t)\, : \, \mathbb {R} \mapsto \mathbb {R}_+ \ | \ \lim \limits _{t\rightarrow -\infty } e^{\gamma t} \rho ^2(t)=0 \}. \end{aligned}$$

We denote by $\mathcal {D}_{\gamma }$ the class of all families $\widehat{D} = \{D(t) \ | \ t\in \mathbb {R} \} \subseteq \mathcal {P}(E_{\widehat{H}}^2)$ such that

$$\begin{aligned} D(t) \subseteq \bar{B}_{E_{\widehat{H}}^2} (0,\rho _{\widehat{D}}(t)), \ \text{ for } \text{ some } \ \rho _{\widehat{D}}(t) \in \mathcal {R}_{\gamma }, \end{aligned}$$

where $\bar{B}_{E_{\widehat{H}}^2} (0,\rho _{\widehat{D}}(t))$ represents the closed ball in $E_{\widehat{H}}^2$ centered at zero with radius $\rho _{\widehat{D}}(t)$.

3.1 Pullback $\mathcal {D}_{\gamma }$-Absorbing Set

In this subsection, we prove existence of the pullback $\mathcal {D}_{\gamma }$-absorbing set.

Assumption 3.1

  Assume that $F(t,x) \in L_{loc}^2(\mathbb {R}; \widehat{V}^*), \ \forall \, t\geqslant \tau , \, \tau \in \mathbb {R}$, and

$$\begin{aligned} \int _{-\infty }^t e^{\gamma \theta } \Vert F(\theta ,x)\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta< + \infty , \quad \sup \limits _{t\in \mathbb {R}} \int _{t}^{t+1} \Vert F(\theta ,x)\Vert ^2 {\mathrm{d}}\theta < + \infty . \end{aligned}$$

(3.7)

Lemma 3.2

  (Pullback $\mathcal {D}_{\gamma }$-absorbing set)

Assume that Assumptions 2.1 and 3.1 hold. Then the family $\widehat{B} := \{B(t) \ | \ t\in \mathbb {R}\}$ with

$$\begin{aligned} B(t) := \big \{ (\varphi ,\psi ) \in E_{\widehat{H} \times L_{\widehat{V}}^2}^2 \ | \ \Vert (\varphi ,\psi )\Vert _{E_{\widehat{H} \times L_{\widehat{V}}^2}^2} \leqslant \mathcal {R}_1(t), \ \Vert \psi '(s)\Vert _{L_{\widehat{V}^*}^2} \leqslant \mathcal {R}_2(t) \big \} \end{aligned}$$

(3.8)

is a pullback $\mathcal {D}_{\gamma }$-absorbing set for the process $\{U(t,\tau )\}_{t\geqslant \tau }$, where

$$\begin{aligned} \mathcal {R}_1^2(t) :=&1 + (1+\beta ^{-1}e^{\gamma h}) \alpha ^{-1} e^{-\gamma t}\int _{-\infty }^t e^{\gamma \theta }\Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta , \end{aligned}$$

(3.9)

$$\begin{aligned} \mathcal {R}_2^2(t) :=&4\eta ^2 \beta ^{-1} \big [ 2+\lambda _1^{-1} C_G^2 e^{\gamma h} + \alpha ^{-1} e^{-\gamma (t-h)} \int _{-\infty }^t e^{\gamma \theta } \Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta \big ] \nonumber \\&\cdot \big [ 1+\alpha ^{-1} e^{-\gamma (t-h)} \int _{-\infty }^t e^{\gamma \theta } \Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta \big ] + 4\eta ^2 \int _{t-h}^t \Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta , \end{aligned}$$

(3.10)

where $\beta$ comes from (3.1), $\alpha \in (0,2\delta _1-\gamma -2C_G)$ and $\eta := \max \{ 1, \alpha _0, \lambda _1^{-\frac{1}{2}},$$c_1^{-1}+c(\nu _r)\lambda _1^{-\frac{1}{2}} \}$.

Proof

By (3.7) and (3.9), it is clear that $\lim \limits _{t\rightarrow -\infty } e^{\gamma t} \mathcal {R}_1^2(t) = 0$. Consequently,

$$\begin{aligned}&B(t) \subset \{(\varphi ,\psi ) \in E_{\widehat{H}}^2 \ | \ \Vert (\varphi ,\psi )\Vert _{E_{\widehat{H}}^2}^2 \\&\quad \leqslant \rho ^2(t), \ \lim \limits _{t\rightarrow -\infty } \rho ^2(t) = 0 \}, \ \text{ and } \text{ therefore } \ \widehat{B} \in \mathcal {D}_{\gamma }. \end{aligned}$$

Taking $T=h$ in Proposition 3.1, and for any $t\geqslant \tau +h$, we have

$$\begin{aligned}&\Vert w_t(\cdot ; \tau ,w^\mathrm{in},\phi ^\mathrm{in})\Vert _{L_{\widehat{V}}^2}^2 \\ \leqslant&\beta ^{-1}(1+C_G) e^{-\gamma (t-h-\tau )}\Vert (w^\mathrm{in},\phi ^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2 + (\alpha \beta )^{-1} e^{-\gamma (t-h)} \int _{-\infty }^t e^{\gamma \theta }\Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta , \end{aligned}$$

which together with (3.1) gives

$$\begin{aligned}&\Vert U(t,\tau )(w^\mathrm{in},\phi ^\mathrm{in})\Vert _{E_{\widehat{H}\times L_{\widehat{V}}^2}}^2 = \Vert w(t;\tau ,w^\mathrm{in},\phi ^\mathrm{in})\Vert ^2 + \Vert w_t(\cdot ;\tau ,w^\mathrm{in},\phi ^\mathrm{in})\Vert _{L_{\widehat{V}}^2}^2 \nonumber \\&\quad \leqslant (1+\beta ^{-1}e^{\gamma h}) (1+C_G) e^{-\gamma (t-\tau )} \Vert (w^\mathrm{in},\phi ^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2 \nonumber \\&\quad + (1+\beta ^{-1}e^{\gamma h}) \alpha ^{-1} e^{-\gamma t} \int _{-\infty }^t e^{\gamma \theta } \Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta , \end{aligned}$$

(3.11)

In addition, it follows from (1.2), (1.6)$_1$, (2.2), (2.3) and (2.5) that

$$\begin{aligned} |\langle w'(\theta ),v \rangle | \leqslant&|\langle Aw(\theta ),v \rangle | + |\langle B(u(\theta ),w(\theta )),v \rangle | + |\langle N(w(\theta )),v \rangle | \nonumber \\&+ |\langle F(\theta ),v \rangle | + |\langle G(\theta ,w_{\theta }),v \rangle | \nonumber \\ \leqslant&\big ( c^{-1}\Vert w(\theta )\Vert _{\widehat{V}} + \alpha _0 \Vert u(\theta )\Vert ^{\frac{1}{2}}\Vert \nabla u(\theta )\Vert ^{\frac{1}{2}} \Vert w(\theta )\Vert ^{\frac{1}{2}}\Vert \nabla w(\theta )\Vert ^{\frac{1}{2}}\nonumber \\&+ c(\nu _r)\lambda _1^{-\frac{1}{2}} \Vert w(\theta )\Vert _{\widehat{V}} \nonumber \\&+ \Vert F(\theta )\Vert _{\widehat{V}^*} + \lambda _1^{-\frac{1}{2}} \Vert G(\theta ,w_{\theta })\Vert \big ) \Vert v\Vert _{\widehat{V}}, \ \ \forall \, v \in \widehat{V}. \end{aligned}$$

(3.12)

Since $\Vert u(\theta )\Vert \leqslant \Vert w(\theta )\Vert$ and $\Vert \nabla u(\theta )\Vert \leqslant \Vert \nabla w(\theta )\Vert \leqslant \Vert w(\theta )\Vert _{\widehat{V}}$, (3.12) implies

$$\begin{aligned} \Vert w'(\theta )\Vert _{\widehat{V}^*} \leqslant \eta \big ( \Vert w(\theta )\Vert _{\widehat{V}} + \Vert w(\theta )\Vert \Vert w(\theta )\Vert _{\widehat{V}} + \Vert F(\theta )\Vert _{\widehat{V}^*} + \Vert G(\theta ,w_{\theta })\Vert \big ), \end{aligned}$$

where $\eta := \max \{ 1, \alpha _0, \lambda _1^{-\frac{1}{2}}, c_1^{-1}+c(\nu _r)\lambda _1^{-\frac{1}{2}} \}$. Integrating the above inequality and using the Cauchy inequality yield

$$\begin{aligned} \int _{t-h}^t \Vert w'(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta \leqslant&4\eta ^2 \int _{t-h}^t \big ( \Vert w(\theta )\Vert _{\widehat{V}}^2 + \Vert w(\theta )\Vert ^2 \Vert w(\theta )\Vert _{\widehat{V}}^2 \nonumber \\&+ \Vert F(\theta )\Vert _{\widehat{V}^*}^2 + \Vert G(\theta ,w_{\theta })\Vert ^2 \big ) {\mathrm{d}}\theta . \end{aligned}$$

(3.13)

Under Assumption 2.1, it holds that

$$\begin{aligned} \int _{t-h}^t \Vert G(\theta ,w_{\theta })\Vert ^2 {\mathrm{d}}\theta \leqslant C_G^2 \int _{t-2h}^t \Vert w(\theta )\Vert ^2 {\mathrm{d}}\theta \leqslant \lambda _1^{-1} C_G^2 \int _{t-2h}^t \Vert w(\theta )\Vert _{\widehat{V}}^2 {\mathrm{d}}\theta . \end{aligned}$$

(3.14)

From (3.1), we have

$$\begin{aligned} \sup \limits _{\theta \in [t-h,t]} \Vert w(\theta )\Vert ^2 \leqslant&(1+C_G)e^{-\gamma (t-h-\tau )} \Vert (w^\mathrm{in},\phi ^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2 \nonumber \\&+ \alpha ^{-1}e^{-\gamma (t-h)} \int _{-\infty }^t e^{\gamma s} \Vert F(s)\Vert _{\widehat{V}^*}^2 {\mathrm{d}}s. \end{aligned}$$

(3.15)

Taking $T=2h$ in (3.5) yields

$$\begin{aligned} \int _{t-2h}^t \Vert w(\theta )\Vert _{\widehat{V}}^2 {\mathrm{d}}\theta \leqslant&\beta ^{-1}(1+C_G) e^{-\gamma (t-2h-\tau )}\Vert (w^\mathrm{in},\phi ^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2 \nonumber \\&+ (\alpha \beta )^{-1} e^{-\gamma (t-2h)} \int _{-\infty }^t e^{\gamma \theta }\Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta . \end{aligned}$$

(3.16)

Now, taking (3.13)–(3.16) into account and writing

$$\begin{aligned} M(t,\tau ,w^\mathrm{in},\phi ^\mathrm{in}):=&(1+C_G)e^{-\gamma (t-h-\tau )} \Vert (w^\mathrm{in},\phi ^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2\\&+ \alpha ^{-1} e^{-\gamma (t-h)} \int _{-\infty }^t e^{\gamma \theta }\Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta , \end{aligned}$$

we obtain

$$\begin{aligned} \Vert w_t'(s)\Vert _{L_{\widehat{V}^*}^2} =&\int _{t-h}^t \Vert w'(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta \nonumber \\ \leqslant&4\eta ^2 \big [ 1+(1+C_G)e^{-\gamma (t-h-\tau )}\Vert (w^\mathrm{in},\phi ^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2\nonumber \\&+ \alpha ^{-1} e^{-\gamma (t-h)} \int _{-\infty }^t e^{\gamma s}\Vert F(s)\Vert _{\widehat{V}^*}^2 {\mathrm{d}}s \big ] \nonumber \\&\times \big [ \beta ^{-1}(1+C_G)e^{-\gamma (t-h-\tau )} \Vert (w^\mathrm{in},\phi ^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2 \nonumber \\&+ (\alpha \beta )^{-1} e^{-\gamma (t-h)} \int _{-\infty }^t e^{\gamma \theta }\Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta \big ] \nonumber \\&+ 4\eta ^2\lambda _1^{-1}C_G^2 \big [ \beta ^{-1}(1+C_G)e^{-\gamma (t-2h-\tau )} \Vert (w^\mathrm{in},\phi ^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2 \nonumber \\&+ (\alpha \beta )^{-1} e^{-\gamma (t-2h)} \int _{-\infty }^t e^{\gamma \theta }\Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta \big ] \nonumber \\&+ 4\eta ^2 \int _{t-h}^t \Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta \nonumber \\ =&4\eta ^2 \beta ^{-1} \big [ 1+\lambda _1^{-1}C_G^2e^{\gamma h} + M(t,\tau ,w^\mathrm{in},\phi ^\mathrm{in}) \big ] M(t,\tau ,w^\mathrm{in},\phi ^\mathrm{in}) \nonumber \\&+ 4\eta ^2 \int _{t-h}^t \Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta . \end{aligned}$$

(3.17)

Therefore, from (3.11) and (3.17), we conclude that the family $\widehat{B}$ given by (3.8) is a pullback $\mathcal {D}_{\gamma }$-absorbing set for the process $\{U(t,\tau )\}_{t\geqslant \tau }$. This completes the proof. $\square$

3.2 Pullback $\mathcal {D}_{\gamma }$-Asymptotic Compactness

This subsection is to prove the pullback $\mathcal {D}_{\gamma }$-asymptotic compactness of the process $\{U(t,\tau )\}_{t\geqslant \tau }$. Since the lack of compactness for Sobolev imbedding in unbounded domains, we first establish the tail estimate with respect to the unbounded domains.

Lemma 3.3

(Tail estimate of unbounded domains)

Assume that Assumptions 2.1 and 3.1 hold. Then, for any $\epsilon >0, t\in \mathbb {R}$ and $\widehat{D}= \{D(t) \ | \ t\in \mathbb {R}\}\in \mathcal {P}(E_{\widehat{H}}^2)$, there exist $l_0 := l_0(\epsilon ,t,\widehat{D})>0$ and $\tau _0 := \tau _0(\epsilon ,t,\widehat{D})<t$ such that, for any $l\geqslant l_0, \tau \leqslant \tau _0$ and $(w^\mathrm{in},\phi ^\mathrm{in}) \in D(\tau )$, there holds

$$\begin{aligned} \Vert w(t;\tau ,w^\mathrm{in},\phi ^\mathrm{in})\Vert _{L^2(\Omega {\setminus }\Omega _l)} \leqslant \epsilon , \end{aligned}$$

(3.18)

where $\Omega _l := \{x\in \Omega \ | \ |x| < l\}$.

Proof

Let the truncation function $\chi (\cdot ) \in \mathcal {C}^2(\mathbb {R}^2),\, \chi (x) \in [0,1]$ satisfies for some constant $c_0$

$$\begin{aligned} \chi (x) = \left\{ \begin{array}{l} 0, \ \ |x| \leqslant 1, \\ 1, \ \ |x| \geqslant 2, \end{array} \right. \qquad \Vert \nabla \chi (x)\Vert _{\mathbb {L}^{\infty }(\mathbb {R}^2)} \leqslant c_0, \ \ \Vert D^2 \chi (x)\Vert _{\mathbb {L}^{\infty }(\mathbb {R}^2)} \leqslant c_0. \end{aligned}$$

In particular, set $\chi _l(x)=\chi (\frac{x}{l})$ with $l\geqslant 1$, we have

$$\begin{aligned} \Vert \nabla \chi _l(x)\Vert _{\mathbb {L}^{\infty }(\mathbb {R}^2)} \leqslant \frac{c_0}{l}, \ \ \Vert D^2 \chi _l(x)\Vert _{\mathbb {L}^{\infty }(\mathbb {R}^2)} \leqslant \frac{c_0}{l^2}. \end{aligned}$$

(3.19)

Taking the inner product of (1.6)$_{1}$ yields

$$\begin{aligned}&\frac{1}{2} \frac{\mathrm{d}}{{\mathrm{d}}t} \Vert \chi _l w\Vert ^2 + \langle A(\chi _l w), \chi _l w \rangle - (\nu +\nu _r) \int _{\Omega } |u \nabla \chi _l|^2 {\mathrm{d}}x - \bar{\alpha } \int _{\Omega } |\omega \nabla \chi _l|^2 {\mathrm{d}}x \nonumber \\&+ ((u\cdot \nabla )w, \chi _l^2 w) + \langle N(\chi _l w), \chi _l w \rangle + (\nabla p, \chi _l^2 u) \nonumber \\ =&\langle F(t,x), \chi _l^2w \rangle + (G(t,w_t), \chi _l^2 w). \end{aligned}$$

(3.20)

It follows from (3.19) and the Hölder inequality that

$$\begin{aligned} (\nu +\nu _r)\int _{\Omega } |u \nabla \chi _l|^2 {\mathrm{d}}x \leqslant (\nu +\nu _r)\Vert \nabla \chi _l\Vert _{\mathbb {L}^{\infty }(\Omega )}^2 \Vert u\Vert ^2 \leqslant c_0^2(\nu +\nu _r)l^{-2} \Vert u\Vert ^2. \end{aligned}$$

(3.21)

Similarly, it holds that

$$\begin{aligned} \bar{\alpha }\int _{\Omega } |\omega \nabla \chi _l|^2 {\mathrm{d}}x \leqslant \bar{\alpha }\Vert \nabla \chi _l\Vert _{\mathbb {L}^{\infty }(\Omega )}^2 \Vert \omega \Vert ^2 \leqslant c_0^2 \bar{\alpha } l^{-2} \Vert \omega \Vert ^2. \end{aligned}$$

(3.22)

Using integrating by parts and the fact $\nabla \cdot u=0$, we obtain

$$\begin{aligned} ((u\cdot \nabla )w, \chi _l^2w) =&\sum \limits _{i,j=1}^2 \int _{\Omega } u_i\frac{\partial u_j}{\partial x_i}\chi _l^2u_j {\mathrm{d}}x + \sum \limits _{i=1}^2 \int _{\Omega } u_i\frac{\partial \omega }{\partial x_i}\chi _l^2\omega {\mathrm{d}}x \\ =&-\sum \limits _{i,j=1}^2 \left( \int _{\Omega } u_i u_j\chi _l^2\frac{\partial u_j}{\partial x_i}{\mathrm{d}}x + 2\int _{\Omega }u_i u_j^2\chi _l\frac{\partial \chi _l}{\partial x_i}{\mathrm{d}}x\right) \\&- \sum \limits _{i=1}^2 \left( \int _{\Omega } u_i \omega \chi _l^2\frac{\partial \omega }{\partial x_i}{\mathrm{d}}x +2\int _{\Omega }u_i\omega ^2\chi _l\frac{\partial \chi _l}{\partial x_i}{\mathrm{d}}x\right) \\ =&-((u\cdot \nabla )w,\chi _l^2 w) - 2\sum \limits _{i,j=1}^2 \int _{\Omega } u_i u_j^2\chi _l\frac{\partial \chi _l}{\partial x_i}{\mathrm{d}}x \\&- 2\sum \limits _{i=1}^2 \int _{\Omega } u_i\omega ^2\chi _l\frac{\partial \chi _l}{\partial x_i}{\mathrm{d}}x, \end{aligned}$$

which together with (3.19), the H${\ddot{\mathrm{o}}}$lder inequality, the Gagliardo–Nirenberg inequality and the Young’s inequality yield

$$\begin{aligned}&|((u\cdot \nabla )w, \chi _l^2 w)| = \big | \sum \limits _{i,j=1}^2 \int _{\Omega } u_i u_j^2\chi _l \frac{\partial \chi _l}{\partial x_i}{\mathrm{d}}x + \sum \limits _{i=1}^2 \int _{\Omega } u_i\omega ^2\chi _l\frac{\partial \chi _l}{\partial x_i}{\mathrm{d}}x \big | \nonumber \\ \leqslant&\Vert \nabla \chi _l\Vert _{\mathbb {L}^{\infty }(\Omega )} \Vert u\Vert \Vert w\Vert _{\mathbb {L}^4(\Omega )}^2 \leqslant c_0 l^{-1} \Vert u\Vert \Vert w\Vert \Vert w\Vert _{\widehat{V}} \leqslant \frac{c_0}{l}(\Vert w\Vert ^4 + \Vert w\Vert _{\widehat{V}}^2). \end{aligned}$$

(3.23)

From (3.19) and the fact $\nabla \cdot u = 0$, we also have

$$\begin{aligned} |(\nabla p, \chi _l^2 u)| =&\big | \sum \limits _{i=1}^2 \int _{\Omega } \frac{\partial p}{\partial x_i} \chi _l^2 u_i {\mathrm{d}}x \big | = \big | \sum \limits _{i=1}^2 \int _{\Omega } 2p\chi _l \frac{\partial \chi _l}{\partial x_i} u_i {\mathrm{d}}x \big | \nonumber \\ \leqslant&2\Vert p\Vert \Vert \nabla \chi _l\Vert _{\mathbb {L}^{\infty }(\Omega )} \Vert \chi _l u\Vert \leqslant 2c_0 l^{-1} \Vert p\Vert \Vert \chi _l u\Vert . \end{aligned}$$

(3.24)

Taking (1.2), (2.7), (3.20)–(3.24) and Lemma 3.2 into account, we deduce that there exists $\tau _1$ such that, for any $\tau \leqslant \tau _1$,

$$\begin{aligned}&\frac{1}{2} \frac{\mathrm{d}}{{\mathrm{d}}t} \Vert \chi _l w\Vert ^2 + \delta _1 \Vert \chi _l w\Vert _{\widehat{V}}^2 \leqslant \frac{1}{2} \frac{\mathrm{d}}{{\mathrm{d}}t} \Vert \chi _l w\Vert ^2 + \langle A(\chi _l w), \chi _l w \rangle + \langle N(\chi _l w), \chi _l w \rangle \\ \leqslant&\frac{c_0^2(\nu +\nu _r)}{l^2}\Vert u\Vert ^2 + \frac{c_0^2 \bar{\alpha }}{l^2}\Vert \omega \Vert ^2 + \frac{c_0}{l} (\Vert w\Vert ^4 + \Vert w\Vert _{\widehat{V}}^2) + \Vert \chi _l F\Vert _{\widehat{V}^*} \Vert \chi _l w\Vert _{\widehat{V}} \\&+ (G(t,w_t), \chi _l^2 w) + \frac{2c_0}{l}\Vert p\Vert \Vert \chi _l u\Vert \\ \leqslant&\frac{c_0^2\cdot \max \{\nu +\nu _l,\bar{\alpha }\}}{l^2}\Vert w\Vert ^2 + \frac{c_0}{l} \mathcal {R}_1^4 + \frac{c_0}{l} \Vert w\Vert _{\widehat{V}}^2 + \frac{1}{4\beta _1} \Vert \chi _l F\Vert _{\widehat{V}^*}^2 + \beta _1 \Vert \chi _l w\Vert _{\widehat{V}}^2 \\&+ (G(t,w_t),\chi _l^2 w) + \frac{c_0}{l} (\Vert p\Vert ^2 + \Vert \chi _l u\Vert ^2), \end{aligned}$$

where the constant $\beta _1 \in (0, \frac{2\delta _1-2C_G-\gamma }{2}]$. Hence, there exists constant $c_3>0$ such that

$$\begin{aligned}&\frac{\mathrm{d}}{{\mathrm{d}}t} \Vert \chi _l w\Vert ^2 + 2(\delta _1-\beta _1) \Vert \chi _l w\Vert ^2 \leqslant \frac{\mathrm{d}}{{\mathrm{d}}t} \Vert \chi _l w\Vert ^2 + 2(\delta _1-\beta _1) \Vert \chi _l w\Vert _{\widehat{V}}^2 \\&\quad \leqslant \frac{c_3}{l^2} \mathcal {R}_1^2 + \frac{2c_0}{l} \mathcal {R}_1^4 + \frac{2c_0}{l} \Vert w\Vert _{\widehat{V}}^2 \\&\quad + \frac{1}{2\beta _1} \Vert \chi _l F\Vert _{\widehat{V}^*}^2 + 2(G(t,w_t),\chi _l^2 w) + \frac{2c_0}{l} (\Vert p\Vert ^2 + \Vert \chi _l u\Vert ^2). \end{aligned}$$

Further, it holds that

$$\begin{aligned}&\frac{\mathrm{d}}{{\mathrm{d}}t}(e^{\gamma t}\Vert \chi _l w(t)\Vert ^2) + [2(\delta _1-\beta _1)-\gamma ] e^{\gamma t} \Vert \chi _l w(t)\Vert ^2 \\&\quad \leqslant \frac{c_3}{l^2} e^{\gamma t} \mathcal {R}_1^2(t) + \frac{2c_0}{l} e^{\gamma t} \mathcal {R}_1^4(t) + \frac{2c_0}{l} e^{\gamma t} \Vert w(t)\Vert _{\widehat{V}}^2 \\&\quad + \frac{1}{2\beta _1} e^{\gamma t} \Vert \chi _l F\Vert _{\widehat{V}^*}^2 + 2e^{\gamma t} (G(t,w_t),\chi _l^2 w(t)) \\&\quad + \frac{2c_0}{l} e^{\gamma t} (\Vert p\Vert ^2 + \Vert \chi _l w(t)\Vert ^2). \end{aligned}$$

Integrating the above inequality yields

$$\begin{aligned}&e^{\gamma t} \Vert \chi _l w(t)\Vert ^2 + [2(\delta _1-\beta _1)-\gamma ] \int _{\tau }^t e^{\gamma s} \Vert \chi _l w(s)\Vert ^2 {\mathrm{d}}s \nonumber \\&\quad \leqslant e^{\gamma \tau } \Vert \chi _l w^\mathrm{in}\Vert ^2 + \frac{c_3}{l^2} \int _{\tau }^t e^{\gamma s}\mathcal {R}_1^2(s) {\mathrm{d}}s + \frac{2c_0}{l} \int _{\tau }^t e^{\gamma s} \mathcal {R}_1^4(s) {\mathrm{d}}s \nonumber \\&\quad + \frac{2c_0}{l} \int _{\tau }^t e^{\gamma s}\Vert w(s)\Vert _{\widehat{V}}^2 {\mathrm{d}}s \nonumber \\&\quad + \frac{1}{2\beta _1} \int _{\tau }^t e^{\gamma s} \Vert \chi _l F(s)\Vert _{\widehat{V}^*}^2 {\mathrm{d}}s + 2\int _{\tau }^t e^{\gamma s} (G(s,w_s),\chi _l^2 w(s)) {\mathrm{d}}s \nonumber \\&\quad + \frac{2c_0}{l}\int _{\tau }^t e^{\gamma s} \Vert p\Vert ^2 {\mathrm{d}}s + \frac{2c_0}{l}\int _{\tau }^t e^{\gamma s} \Vert \chi _l w(s)\Vert ^2 {\mathrm{d}}s. \end{aligned}$$

(3.25)

Similar to (3.3), we have

$$\begin{aligned} 2\int _{\tau }^t e^{\gamma s}(G(s,w_s),\chi _l^2 w(s)) {\mathrm{d}}s \leqslant C_G e^{\gamma \tau } \Vert \chi _l\phi ^\mathrm{in}\Vert _{L_{\widehat{H}}^2}^2 + 2C_G \int _{\tau }^t e^{\gamma s} \Vert \chi _l w(s)\Vert ^2 {\mathrm{d}}s. \end{aligned}$$

(3.26)

Inserting (3.26) into (3.25), noting that $2(\delta _1-\beta _1-C_G)-\gamma \geqslant 0$, we deduce that

$$\begin{aligned} \Vert \chi _l w(t)\Vert ^2 \leqslant&e^{\gamma (\tau -t)} \Vert \chi _l w^\mathrm{in}\Vert ^2 + C_G e^{\gamma (\tau -t)} \Vert \chi _l \phi ^\mathrm{in}\Vert _{L_{\widehat{H}}^2}^2\nonumber \\&+ \left( \frac{c_3}{l^2}+\frac{2c_0}{l}\right) e^{-\gamma t} \int _{\tau }^t e^{\gamma s} \mathcal {R}_1^2(s) {\mathrm{d}}s \nonumber \\&+ \frac{2c_0}{l} e^{-\gamma t} \int _{\tau }^t e^{\gamma s} \mathcal {R}_1^4(s) {\mathrm{d}}s + \frac{2c_0}{l} e^{-\gamma t} \int _{\tau }^t e^{\gamma s} \Vert w(s)\Vert _{\widehat{V}}^2 {\mathrm{d}}s \nonumber \\&+ \frac{l}{2\beta _1} e^{-\gamma t} \int _{\tau }^t e^{\gamma s} \Vert \chi _l F\Vert _{\widehat{V}^*}^2 {\mathrm{d}}s + \frac{2c_0}{l} e^{-\gamma t} \int _{\tau }^t e^{\gamma s} \Vert p\Vert ^2 {\mathrm{d}}s. \end{aligned}$$

(3.27)

For any $\epsilon >0$, there exists a $\tau _2 := \tau _2(\epsilon ,t,\widehat{D})$ such that

$$\begin{aligned} e^{\gamma (\tau -t)} \Vert \chi _l w^\mathrm{in}\Vert ^2 +C_G e^{\gamma (\tau -t)} \Vert \chi _l \phi ^\mathrm{in}\Vert _{L_{\widehat{H}}^2}^2 \leqslant \frac{\epsilon }{6} \ \ \mathrm{for \,\, any} \ \tau \leqslant \tau _1. \end{aligned}$$

(3.28)

Note that (3.7) implies, see [32],

$$\begin{aligned} \lim \limits _{l\rightarrow \infty } \int _{-\infty }^t e^{\gamma \theta } \Vert F(\theta ,x)\Vert _{\widehat{V}^*(\Omega {\setminus }\Omega _l)}^2 {\mathrm{d}}\theta = 0, \ \ \forall \, t\in \mathbb {R}. \end{aligned}$$

(3.29)

Then, under Assumption 3.1, taking Lemma 3.1, Lemma 3.2 and (3.29) into account, we conclude that there exists $l_1 := l_1(\epsilon ,t,\widehat{D})$ such that for any $l \geqslant l_1$,

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \left( \frac{c_3}{l^2}+\frac{2c_0}{l}\right) e^{-\gamma t} \int _{\tau }^t e^{\gamma s} \mathcal {R}_1^2(s) {\mathrm{d}}s \leqslant \frac{\epsilon }{6}, \quad \frac{2c_0}{l} e^{-\gamma t} \int _{\tau }^t e^{\gamma s} \mathcal {R}_1^4(s) {\mathrm{d}}s \leqslant \frac{\epsilon }{6}, \\ \displaystyle \qquad \frac{2c_0}{l} e^{-\gamma t} \int _{\tau }^t e^{\gamma s} \Vert w(s)\Vert _{\widehat{V}}^2 {\mathrm{d}}s \leqslant \frac{\epsilon }{6}, \end{array} \right. \end{aligned}$$

(3.30)

and

$$\begin{aligned} \frac{1}{2\beta _1}e^{-\gamma t} \int _{\tau }^t e^{\gamma s}\Vert \chi _l F\Vert _{\widehat{V}^*} {\mathrm{d}}s = \frac{1}{2\beta _1} e^{-\gamma t} \int _{\tau }^t e^{\gamma s} \Vert F\Vert _{\widehat{V}^*(\Omega {\setminus }\Omega _l)} {\mathrm{d}}s \leqslant \frac{\epsilon }{6}. \end{aligned}$$

(3.31)

It follows from (1.3)$_1$ that $\nabla p \in L_{loc}^2(\tau , +\infty ; \mathbb {H}^{-1}(\Omega ))$, which implies $p\in L_{loc}^2(\tau , +\infty ; L^2(\Omega ))$,

$$\begin{aligned} \int _{\tau }^t e^{\gamma s} \Vert p\Vert ^2 {\mathrm{d}}s \leqslant c\int _{\tau }^t e^{\gamma s} \Vert w(s)\Vert _{\widehat{V}}^2 {\mathrm{d}}s, \ \ \text{ where }\;c\; \text{ is } \text{ a } \text{ positive } \text{ constant }. \end{aligned}$$

Consequently, we deduce that there exists $l_2 := l_2(\epsilon ,t,\widehat{D})$ such that for any $l\geqslant l_2$, it holds that

$$\begin{aligned} \frac{2c_0}{l} e^{-\gamma t} \int _{\tau }^t e^{\gamma s} \Vert p\Vert ^2 {\mathrm{d}}s \leqslant \frac{2c c_0}{l} e^{-\gamma t} \int _{\tau }^t e^{\gamma s} \Vert w(s)\Vert _{\widehat{V}}^2 {\mathrm{d}}s \leqslant \frac{\epsilon }{6}. \end{aligned}$$

(3.32)

Substituting (3.28)–(3.32) into (3.27), we immediately obtain (3.18) with $\tau _0=\min \{\tau _1, \tau _2\}$ and $l_0=\max \{l_1,l_2\}$. This completes the proof. $\square$

In order to prove the pullback asymptotic compactness, we need to improve the regularity of solutions.

Lemma 3.4

(see [29])  Let $Y_0, Y$ be two Banach spaces such that $Y_0$ is reflexive, and the inclusion $Y_0\subset Y$ is continuous. Assume that $\{w_n\}$ is a bounded sequence in $L^{\infty }(t_0,T;Y_0)$ such that $w_n\rightharpoonup w$ weakly in $L^p(t_0, T; Y_0)$ for some $p \in [1,+\infty )$ and $w\in \mathcal {C}(t_0,T;Y)$. Then $w(t)\in Y_0$ and

$$\begin{aligned} \Vert w(t)\Vert _{Y_0} \leqslant \liminf \limits _{n\rightarrow \infty } \Vert w_n\Vert _{L^{\infty }(t_0,T;Y_0)}, \ \forall \, t \in [t_0,T]. \end{aligned}$$

Lemma 3.5

(Regularity estimate) Assume that Assumptions 2.1 and 3.1 hold, then, for any $t\in \mathbb {R}$ and $\widehat{D} = \{D(s) \ | \ s\in \mathbb {R}\} \in \mathcal {D}_{\gamma }$, there exists a $\tau ^*(\widehat{D},t)$ such that, for any $\tau \leqslant \tau ^*(\widehat{D},t)$, the weak solution w(t) with initial data $(w_0^\mathrm{in},\phi _0^\mathrm{in})\subset D(\tau )$ is bounded in $\widehat{V}$.

Proof

We consider the Galerkin approximate solutions. For each integer $n\geqslant 1$, we denote by

$$\begin{aligned} w_n(t)=w_n(t;\tau _n,w_0^n,\phi _0^n) :=\sum _{i=1}^n\xi _{ni}(t)e_i, \ \ {w_n}_t(\cdot ) = w_n(t+\cdot ;\tau _n,w_0^n,\phi _0^n), \end{aligned}$$

(3.33)

the Galerkin approximation of the solution w(t) of system (1.6), where $\xi _{ni}(t)$ is the solution of the following Cauchy problem of ODEs:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\mathrm d}{{\mathrm d}t}(w_n(t), e_i) + \langle Aw_n(t)+B(u_n, w_n) + N(w_n(t)), e_i \rangle \\ \quad =\langle F(t), e_i \rangle + (G(t,{w_n}_t), e_i), \\ (w_n(\tau ), e_i) =(w_0^{n}, e_i), \ ({w_n}(s),e_i)\\ \quad =(\phi _0^{n}, e_i), \ \ s\in (\tau -h,\tau ), i=1,2, \ldots , n, \end{array} \right. \end{aligned}$$

(3.34)

where $\{e_i: i\geqslant 1\}\subseteq D(A)$, which forms a Hilbert basis of ${\widehat{V}}$ and is orthonormal in ${\widehat{H}}$. Multiplying equation (3.34)$_1$ by $A \xi _{ni}(t)$ and summing them for $i=1$ to n, we obtain

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm d}{{\mathrm d}t}\langle Aw_n(t), w_n(t) \rangle +\Vert Aw_n(t) \Vert ^2 + \langle B(u_n(t), w_n(t)), Aw_n(t) \rangle \nonumber \\&\quad +\langle N(w_n(t)), Aw_n(t) \rangle \nonumber \\&\quad = (F(t), Aw_n(t)) + (G(t,{w_n}_t), Aw_n(t)). \end{aligned}$$

(3.35)

From (2.4) and the facts $\Vert u_n\Vert ^2\leqslant \Vert w_n\Vert ^2, \ \ \Vert \nabla u_n\Vert ^2 \leqslant \Vert w_n\Vert _{\widehat{V}}^2,$ and using the Young’s inequality, we deduce that

$$\begin{aligned} -\langle B(u_n, w_n), Aw_n \rangle&\leqslant |\langle B(u_n, w_n), Aw_n\rangle | \leqslant \alpha _0 \Vert u_n\Vert ^{\frac{1}{2}}\Vert \nabla u_n\Vert ^{\frac{1}{2}} \Vert \nabla w_n\Vert ^{\frac{1}{2}}\Vert Aw_n\Vert ^{\frac{3}{2}} \\&\leqslant \frac{1}{4}\Vert Aw_n\Vert ^2 + 4^3 \alpha _0^4 \Vert w_n\Vert ^2\Vert w_n\Vert _{\widehat{V}}^4, \end{aligned}$$

which together with (2.6), (3.35) and Assumption 2.1 implies

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm d}{{\mathrm d}t}\langle Aw_n, w_n \rangle = -\Vert Aw_n\Vert ^2 +(F,Aw_n) +(G(t,{w_n}_t),Aw_n) \\&\quad -\langle B(u_n, w_n), Aw_n \rangle -\langle N(w_n), Aw_n \rangle \\&\quad \leqslant -\Vert Aw_n\Vert ^2 +\frac{1}{4}\Vert Aw_n\Vert ^2 +\Vert F(t) \Vert ^2 +\Vert G(t,{w_n}_t)\Vert ^2 +\frac{1}{4}\Vert Aw_n\Vert ^2 \\&\quad +\frac{1}{4}\Vert Aw_n\Vert ^2 +4^3 \alpha _0^4 \Vert w_n\Vert ^2\Vert w_n\Vert _{\widehat{V}}^4 \\&\quad +c^2(\nu _r) \Vert w_n\Vert _{\widehat{V}}^2 +\frac{1}{4}\Vert Aw_n\Vert ^2 \\&\quad = \Vert F(t)\Vert ^2 +\Vert G(t,{w_n}_t)\Vert ^2 +\Vert w_n\Vert _{\widehat{V}}^2 \big ( 4^3 \alpha _0^4 \Vert w_n\Vert ^2\Vert w_n\Vert _{\widehat{V}}^2 +c^2(\nu _r) \big ). \end{aligned}$$

Further, from (2.2) and the above inequality, we have

$$\begin{aligned}&\frac{\mathrm d}{{\mathrm d}t}\langle Aw_n(t), w_n(t) \rangle \nonumber \\&\quad \leqslant 2\Vert F(t)\Vert ^2 + 2\Vert G(t,{w_n}_t)\Vert ^2 + \langle Aw_n(t), w_n(t) \rangle \big ( 2^7c_2 \alpha _0^4 \Vert w_n(t)\Vert ^2\Vert w_n(t)\Vert _{\widehat{V}}^2 \nonumber \\&\quad + 2c_2c^2(\nu _r) \big ). \end{aligned}$$

(3.36)

Let us set

$$\begin{aligned} H_n(\theta )&:= \langle Aw_n(\theta ), w_n(\theta ) \rangle , \ I_n(\theta ) := 2(\Vert F(\theta )\Vert ^2+\Vert G(\theta ,{w_n}_{\theta })\Vert ^2), \\ K_n(\theta )&:=2^7c_2 \alpha _0^4 \Vert w_n(\theta )\Vert ^2\Vert w_n(\theta )\Vert _{\widehat{V}}^2 + 2c_2c^2(\nu _r). \end{aligned}$$

Replacing the variable t with $\theta$ in (3.36), we get

$$\begin{aligned} \frac{\mathrm d}{{\mathrm d}\theta }H_n(\theta ) \leqslant K_n(\theta )H_n(\theta )+I_n(\theta ). \end{aligned}$$

(3.37)

Using the Gronwall inequality to (3.37), for all $\tau \leqslant t-h \leqslant s \leqslant t$, we have

$$\begin{aligned} H_n(t)\leqslant \big ( H_n(s)+\int _{t-h}^t I_n(\theta ){\mathrm d}\theta \big ) \cdot \exp \left\{ \int _{t-h}^t K_n(\theta ){\mathrm d}\theta \right\} . \end{aligned}$$

(3.38)

Integrating (3.38) from $s=t-h$ to $s=t$, we obtain that

$$\begin{aligned} h H_n(t) \leqslant \left( \int _{t-h}^t H_n(s){\mathrm d}s + h \int _{t-h}^t I_n(\theta ){\mathrm d}\theta \right) \cdot \exp \left\{ \int _{t-h}^t K_n(\theta ){\mathrm d}\theta \right\} . \end{aligned}$$

(3.39)

In addition, it follows from (2.2), Lemma 3.2 and Assumption 2.1 that

$$\begin{aligned}&\int _{t-h}^t H_n(s){\mathrm d}s + h \int _{t-h}^t I_n(\theta ){\mathrm d}\theta = \int _{t-h}^t \langle Aw_n(s), w_n(s) \rangle {\mathrm d}s \\&\quad + h \int _{t-h}^t 2(\Vert F(\theta )\Vert ^2+\Vert G(\theta ,{w_n}_{\theta })\Vert ^2) {\mathrm d}\theta \\&\quad \leqslant {c_1}^{-1} \int _{t-h}^t \Vert w_n(s) \Vert _{\widehat{V}}^2 {\mathrm d}s + 2h\int _{t-h}^t \Vert F(\theta )\Vert ^2 {\mathrm d}\theta + 2hC_G^2 \int _{t-2h}^t \Vert w(\theta )\Vert ^2 {\mathrm d}\theta \\&\quad \leqslant c_1^{-1} \mathcal {R}_1^2(t) + 4h^2C_G^2 \mathcal {R}_1^2(t) + 2h \int _{t-h}^t \Vert F(\theta )\Vert ^2 {\mathrm{d}}\theta \leqslant c_4\big ( \mathcal {R}_1^2(t) \\&\quad + \int _{t-h}^t\Vert F(\theta )\Vert ^2 {\mathrm d}\theta \big ), \end{aligned}$$

where $c_4:=\max \{ 2h, c_1^{-1} + 4h^2C_G^2 \}$. From (3.1) and Lemma 3.2, it holds that

$$\begin{aligned}&\int _{t-h}^t K_n(\theta ) {\mathrm{d}}\theta = \int _{t-h}^t (2^7c_2 \alpha _0^4 \Vert w_n(\theta )\Vert ^2 \Vert w_n(\theta )\Vert _{\widehat{V}}^2 +2c_2 c^2(\nu _r)) {\mathrm{d}}\theta \\&\quad \leqslant 2^7c_2 \alpha _0^4 \sup \limits _{\theta \in [t-h,t]} \Vert w_n(\theta )\Vert ^2 \int _{t-h}^t \Vert w_n(\theta )\Vert _{\widehat{V}}^2 {\mathrm{d}}\theta + 2c_2 h c^2(\nu _r) \\&\quad \leqslant 2^7c_2 \alpha _0^4 \big [ (1+C_G)e^{\gamma (h+\tau -t)} \Vert (w_0^{n},\phi _0^{n})\Vert _{E_{\widehat{H}}^2}^2 \\&\quad + \alpha ^{-1} e^{\gamma (h-t)} \int _{-\infty }^t e^{\gamma \theta } \Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta \big ] \mathcal {R}_1^2(t) + 2c_2 h c^2(\nu _r). \end{aligned}$$

With the aid of (2.2), substituting the above two inequalities into (3.39), yields

$$\begin{aligned}&\Vert w_n(t)\Vert _{\widehat{V}}^2 \leqslant c_2H_n(t) \nonumber \\&\quad \leqslant c_2c_4h^{-1} \big ( \mathcal {R}_1^2(t) + \int _{t-h}^t \Vert F(\theta )\Vert ^2 {\mathrm d}\theta \big ) \nonumber \\&\quad \times \exp \big \{ c_5 \big ( e^{\gamma (\tau -t)} \Vert (w_0^{n},\phi _0^{n})\Vert _{E_{\widehat{H}}^2}^2 \nonumber \\&\quad + e^{-\gamma t} \int _{-\infty }^t e^{\gamma \theta } \Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm d}\theta \big ) \mathcal {R}_1^2(t) + 2c_2 h c^2(\nu _r) \big \}, \end{aligned}$$

(3.40)

where $c_5 := 2^7c_2 \alpha _0^4 e^{\gamma h} \cdot \max \{(1+C_G), \alpha ^{-1}\}$. Under Assumption 3.1, it is clear that there exists a constant $\bar{M}$ such that

$$\begin{aligned} \int _{t-h}^t \Vert F(\theta )\Vert ^2 {\mathrm{d}}\theta \leqslant \bar{M}, \ \ \forall \, t \in \mathbb {R}, \end{aligned}$$

which together with (3.40), Lemma 3.4 implies the boundedness of $\Vert w(t;\tau ,w_0^\mathrm{in},\phi _0^\mathrm{in})\Vert _{\widehat{V}}$ for all $\tau \leqslant \tau ^*(\widehat{D},t)$. The proof is complete. $\square$

On the basis of the above results, we are ready to prove the pullback $\mathcal {D}_{\gamma }$-asymptotic compactness of the process $\{U(t,\tau )\}_{t\geqslant \tau }$.

Lemma 3.6

(pullback$\mathcal {D}_{\gamma }$-asymptotic compactness)

Under the conditions of Lemma 3.3, the process$\{U(t,\tau )\}_{t\geqslant \tau }$generated by (2.10) is pullback$\mathcal {D}_{\gamma }$-asymptotically compact in$\widehat{H}$.

Proof

For any fixed $t\in \mathbb {R}$, and family $\widehat{D}=\{D(s)\ | \ s\in \mathbb {R}\}\in \mathcal {D}_{\gamma }$, any sequences $\{\tau _n\}\subseteq (-\infty ,t]$ satisfying $\tau _n \rightarrow -\infty$ as $n \rightarrow +\infty$ and $\{(w_n^\mathrm{in},\phi _n^\mathrm{in})\}\subset D(\tau _n)$. It suffice to show the sequence $\{(w^n(t), w^n_t(\cdot ))\}_{n\geqslant 1}$ defined by

$$\begin{aligned} (w^n(t),w^n_t(\cdot )) := U(t,\tau _n)(w_n^\mathrm{in},\phi _n^\mathrm{in}) = (w(t;\tau _n,w_n^\mathrm{in},\phi _n^\mathrm{in}), w_t(\cdot ;\tau _n,w_n^\mathrm{in},\phi _n^\mathrm{in})) \end{aligned}$$

is relatively compact in $E_{\widehat{H}}^2$.

Step 1 (the sequence $\{w^n(t)\}_{n\geqslant 1}$is relatively compact in $\widehat{H}$)

In fact, by Lemma 3.2, there exists a time $\tau _1 := \tau _1(\widehat{D},t)<t$ such that, for any $\tau \leqslant \tau _1$, $U(t,\tau )D(\tau ) \subset B(t)$. Moreover, (3.8)–(3.10) implies B(t) is uniformly bounded with respect to t. Consequently, D(t) is uniformly bounded in $E_{\widehat{H}\times L_{\widehat{V}}^2}^2$ with respect to t. Since $E_{\widehat{H}\times L_{\widehat{V}}^2}^2$ is a reflexive Banach space, we can extract a subsequence (denoting by the same symbol) $\{(w_n^\mathrm{in},\phi _n^\mathrm{in})\}_{n\geqslant 1}$ and some $(w,\phi )\in E_{\widehat{H}\times L_{\widehat{V}}^2}^2$ such that

$$\begin{aligned} U(t,\tau _n)(w_n^\mathrm{in},\phi _n^\mathrm{in}) \rightharpoonup (w,\phi ) \ \text{ weakly } \text{ in } \ E_{\widehat{H}\times L_{\widehat{V}}^2}^2 \ \text{ as } \ n \rightarrow \infty , \end{aligned}$$

(3.41)

which implies

$$\begin{aligned} w^n(t) \rightharpoonup w(t) \ \text{ weakly } \text{ in } \ \widehat{H} \ \text{ as } \ n \rightarrow \infty . \end{aligned}$$

(3.42)

Moreover, from Lemma 3.3, we conclude that, for any $\epsilon >0$, there exist $\tau _3 := \tau _3(\epsilon ,t,\widehat{D})$ and $l_3 := l_3(\epsilon ,t,\widehat{D})>0$ such that

$$\begin{aligned} \Vert w^n(t)\Vert _{\mathbb {L}^2(\Omega {\setminus }\Omega _l)} = \Vert w^n(t;\tau _n,w_n^\mathrm{in},\phi _n^\mathrm{in})\Vert _{\mathbb {L}^2(\Omega {\setminus }\Omega _l)} \leqslant \frac{\epsilon }{3}, \ \ \forall \, \tau _n \leqslant \tau _3, \ l \geqslant l_3. \end{aligned}$$

(3.43)

Observe that, for any fixed $t\in \mathbb {R}$, $w(t)\in \widehat{H}$ is fixed. Hence, for the above $\epsilon >0$, there exists $l_4>0$ such that

$$\begin{aligned} \Vert w(t)\Vert _{\mathbb {L}^2(\Omega {\setminus }\Omega _l)} \leqslant \frac{\epsilon }{3}, \ \ \forall \, \tau _n \leqslant \tau _3, \ l \geqslant l_4. \end{aligned}$$

(3.44)

Now, we define the restrictions of $w^n$ and w in $\Omega _l$, respectively, as

$$\begin{aligned} w^n(t) \big |_{\Omega _l} := \left\{ \begin{array}{l} w^n(t), \ \ x\in \Omega _l, \\ \quad 0, \qquad \ \, x \in \Omega {\setminus } \Omega _l, \end{array} \right. \qquad w(t) \big |_{\Omega _l} := \left\{ \begin{array}{l} w(t), \ \ x\in \Omega _l, \\ \ \ 0, \qquad \ x \in \Omega {\setminus } \Omega _l. \end{array} \right. \end{aligned}$$

It follows from Lemma 3.5 that, for any $l>0$, the sequence $\{w^n(t) \big |_{\Omega _l} \}_{n\geqslant 1}$ is bounded in $\widehat{V}(\Omega _l)$. Since $\widehat{V}(\Omega _l)\hookrightarrow \hookrightarrow \widehat{H}(\Omega _l)$, there exists a subsequence (denoting by the same symbol) $\{ w^n(t) \big |_{\Omega _l} \}_{n\geqslant 1}$ satisfying

$$\begin{aligned} \Vert w^n(t)-w(t)\Vert _{\widehat{H}(\Omega _l)} \rightarrow 0, \quad {as} \quad n \rightarrow \infty , \end{aligned}$$

(3.45)

which combine with (3.43) and (3.44) implies that there exists a $N_0\in \mathbb {N}$ such that, for any $n\geqslant N_0$,

$$\begin{aligned} \Vert w^n(t)-w(t)\Vert _{\widehat{H}} =&\Vert w^n(t)-w(t)\Vert _{\widehat{H}(\Omega _l)} + \Vert w^n(t)-w(t)\Vert _{\mathbb {L}^2(\Omega {\setminus }\Omega _l)} \nonumber \\ \leqslant&\Vert w^n(t)-w(t)\Vert _{\widehat{H}(\Omega _l)} + \Vert w^n(t)\Vert _{\mathbb {L}^2(\Omega {\setminus }\Omega _l)} \nonumber \\&+ \Vert w(t)\Vert _{\mathbb {L}^2(\Omega {\setminus }\Omega _l)} \leqslant \epsilon . \end{aligned}$$

(3.46)

Therefore, the sequence $\{w^n(t)\}_{n\geqslant 1}$ is relatively compact in $\widehat{H}$.

Step 2 (the sequence $\{w^n_t(\cdot )\}_{n\geqslant 1}$is relatively compact in $L_{\widehat{H}}^2$)

Let us denote $\{\theta _j\}_{j\geqslant 0}$ the sequence of all rational numbers from the interval $[-h,0]$. From the above argument, we deduce that there exists a subsequence (denoting by the same symbol) $\{(w_n^\mathrm{in},\phi _n^\mathrm{in})\}_{n\geqslant 1}$ such that for each j there exists a $w^j \in \widehat{H}$ satisfying

$$\begin{aligned} w(t+\theta _j; \tau _n,w_n^\mathrm{in},\phi _n^\mathrm{in}) \rightarrow w^j \ \ \text{ strongly } \text{ in } \ \widehat{H} \ \text{ as }\ n \rightarrow \infty . \end{aligned}$$

(3.47)

Then for any $t_1, t_2 \in [t-h,t]$ with $t_1 < t_2$, we have

$$\begin{aligned} w^n(t_2) - w^n(t_1) =&\int _{t_1}^{t_2} (w^n)'(s) {\mathrm{d}}s \\ =&\int _{t_1}^{t_2} \big [ -Aw^n(s)-B(u^n(s),w^n(s))-N(w^n(s))\\&+F(s)+G(s,w_s^n) \big ] {\mathrm{d}}s. \end{aligned}$$

Hence, from (2.2), (2.3) and (2.5), it follows that

$$\begin{aligned}&\Vert w^n(t_2)-w^n(t_1)\Vert _{\widehat{V}^*} \nonumber \\&\quad \leqslant \int _{t_1}^{t_2} \big ( \Vert Aw^n(s)\Vert _{\widehat{V}^*} + \Vert B(u^n(s),w^n(s))\Vert _{\widehat{V}^*} + \Vert N(w^n(s))\Vert _{\widehat{V}^*} + \Vert F(s)\Vert _{\widehat{V}^*}\nonumber \\&\quad + \Vert G(s,w_s^n)\Vert _{\widehat{V}^*} \big ) {\mathrm{d}}s \nonumber \\&\quad \leqslant c \int _{t_1}^{t_2} \big ( \Vert w^n(s)\Vert _{\widehat{V}} + \Vert w^n(s)\Vert \Vert w^n(s)\Vert _{\widehat{V}} + \Vert F(s)\Vert _{\widehat{V}^*} + \Vert G(s,w_s^n)\Vert \big ) {\mathrm{d}}s. \end{aligned}$$

(3.48)

Moreover, applying the Cauchy inequality and Assumption 2.1, we have

$$\begin{aligned} \int _{t_1}^{t_2} \Vert G(s,w_s^n)\Vert {\mathrm{d}}s \leqslant&C_G (t_2-t_1)^{\frac{1}{2}} \left( \int _{t_1-h}^{t_2} \Vert w^n(s)\Vert ^2 {\mathrm{d}}s \right) ^{\frac{1}{2}} \nonumber \\ \leqslant&C_G (t_2-t_1)^{\frac{1}{2}} \left( \int _{t-2h}^{t} \Vert w^n(s)\Vert _{\widehat{V}}^2 {\mathrm{d}}s \right) ^{\frac{1}{2}}. \end{aligned}$$

(3.49)

Similar to (3.49), it also holds that

$$\begin{aligned} \int _{t_1}^{t_2} \Vert F(s)\Vert _{\widehat{V}^*} {\mathrm{d}}s \leqslant&(t_2-t_1)^{\frac{1}{2}} \left( \int _{t-h}^{t} \Vert F(s)\Vert _{\widehat{V}^*}^2 {\mathrm{d}}s \right) ^{\frac{1}{2}}, \end{aligned}$$

(3.50)

$$\begin{aligned} \int _{t_1}^{t_2} \Vert w^n(s)\Vert _{\widehat{V}} {\mathrm{d}}s \leqslant&(t_2-t_1)^{\frac{1}{2}} \left( \int _{t_1}^{t_2} \Vert w^n(s)\Vert _{\widehat{V}}^2 {\mathrm{d}}s \right) ^{\frac{1}{2}}, \end{aligned}$$

(3.51)

and

$$\begin{aligned} \int _{t_1}^{t_2} \Vert w^n(s)\Vert \Vert w^n(s)\Vert _{\widehat{V}} {\mathrm{d}}s \leqslant&\sup \limits _{s\in [t-h,t]} \Vert w^n(s)\Vert \int _{t_1}^{t_2} \Vert w^n(s)\Vert _{\widehat{V}} {\mathrm{d}}s \nonumber \\ \leqslant&(t_2-t_1)^{\frac{1}{2}} \sup \limits _{s\in [t-h,t]} \Vert w^n(s)\Vert \left( \int _{t_1}^{t_2} \Vert w^n(s)\Vert _{\widehat{V}}^2 {\mathrm{d}}s \right) ^{\frac{1}{2}}. \end{aligned}$$

(3.52)

(3.48)–(3.52) imply

$$\begin{aligned}&\Vert w^n(t_2)-w^n(t_1)\Vert _{\widehat{V}^*} \leqslant c(t_2-t_1)^{\frac{1}{2}} \left[ \left( 1+\sup \limits _{s\in [t-h,t]}\Vert w^n(s)\Vert \right) \right. \nonumber \\&\quad \left. \times \left( \int _{t-2h}^t \Vert w^n(s)\Vert _{\widehat{V}}^2 {\mathrm{d}}s\right) ^{\frac{1}{2}} + \left( \int _{t-h}^t \Vert F(s)\Vert _{\widehat{V}^*}^2 {\mathrm{d}}s\right) ^{\frac{1}{2}} \right] . \end{aligned}$$

(3.53)

It follows from (3.1) that for all $s\in [t-h,t]$,

$$\begin{aligned} \Vert w^n(s)\Vert ^2 \leqslant&(1+C_G)e^{-\gamma (s-\tau _n)} \Vert (w_n^\mathrm{in},\phi _n^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2 + \alpha ^{-1} e^{-\gamma s} \int _{\tau _n}^s e^{\gamma \theta }\Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta \nonumber \\ \leqslant&(1+C_G)e^{\gamma (h+\tau _n-t)}\Vert (w_n^\mathrm{in},\phi _n^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2\nonumber \\&+ \alpha ^{-1} e^{\gamma (h-t)} \int _{-\infty }^t e^{\gamma \theta } \Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta . \end{aligned}$$

(3.54)

In addition, (3.5) gives

$$\begin{aligned} \int _{t-2h}^t \Vert w^n(s)\Vert _{\widehat{V}}^2 {\mathrm{d}}s \leqslant&\beta ^{-1}(1+C_G) e^{-\gamma (t-2h-\tau _n)} \Vert (w_n^\mathrm{in},\phi _n^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2 \nonumber \\&+ (\alpha \beta )^{-1} e^{-\gamma (t-2h)} \int _{-\infty }^t e^{\gamma \theta }\Vert F(\theta )\Vert _{\widehat{V}^*}^2 {\mathrm{d}}\theta . \end{aligned}$$

(3.55)

Since $e^{\gamma \tau _n}\Vert (w_n^\mathrm{in},\phi _n^\mathrm{in})\Vert _{E_{\widehat{H}}^2}^2$ is bounded, taking Assumption 3.1 and (3.53)–(3.55) into account, we conclude that

$$\begin{aligned} \text{ the } \text{ sequence } \ \{w^n(\cdot )\}_{n\geqslant 1} \ \text{ is } \text{ equicontinuous } \text{ in } \ \mathcal {C}([t-h,t];\widehat{V}^*). \end{aligned}$$

Next, observe that for any $r\in [t-h,t]{\setminus }\mathbb {Q}$, where $\mathbb {Q}$ represents the set of all rational number,

$$\begin{aligned} \Vert w^n(r)-w^m(r)\Vert _{\widehat{V}^*} \leqslant&\Vert w^n(r)-w^n(t+\theta _j)\Vert _{\widehat{V}^*} + \Vert w^n(t+\theta _j)-w^m(t+\theta _j)\Vert _{\widehat{V}^*} \\&+ \Vert w^m(t+\theta _j)-w^m(r)\Vert _{\widehat{V}^*}, \ \ \forall \, j \geqslant 1. \end{aligned}$$

On the one hand, by the equicontinuity of $\{w^n(s)\}_{n\geqslant 1}$, there exist a subsequence $\{{\theta _j}_k\} \subset \{\theta _j\}$ such that

$$\begin{aligned} \Vert w^n(r)-w^n(t+{\theta _j}_k)\Vert _{\widehat{V}^*} \rightarrow 0 \ \ \text{ as } \ \ k \rightarrow \infty , \\ \Vert w^m(t+{\theta _j}_k)-w^m(r)\Vert _{\widehat{V}^*} \rightarrow 0 \ \ \text{ as } \ \ k \rightarrow \infty . \end{aligned}$$

On the other hand, (3.47) implies

$$\begin{aligned} \Vert w^n(t+{\theta _j}_k)-w^m(t+{\theta _j}_k)\Vert _{\widehat{V}^*} \rightarrow 0 \ \text{ as } \ n, \, m \rightarrow \infty . \end{aligned}$$

Therefore, for any $\theta \in [t-h,t]{\setminus }\mathbb {Q}$,

$$\begin{aligned} \text{ the } \text{ sequence } \ \{w^n(r)\}_{n\geqslant 1} \ \text{ is } \text{ a } \text{ Cauchy } \text{ sequence } \text{ of } \ \widehat{V}^*. \end{aligned}$$

Thus, for each $r\in [t-h,t]$, there exists some $v(\theta )\in \widehat{V}^*$ such that

$$\begin{aligned} w^n(\theta ) \rightarrow v(\theta ) \ \text{ strongly } \text{ in } \ \widehat{V}^* \ \text{ as } \ n \rightarrow \infty . \end{aligned}$$

Based on the continuous injection of $\widehat{H}$ into $\widehat{V}^*$ and (3.54), we conclude that the sequence $\{w^n(\cdot )\}$ is bounded in $\mathcal {C}([t-h,t]; \widehat{V}^*)$. Then, applying the Lebesgue dominated convergence theorem, we obtain

$$\begin{aligned} w^n(\cdot ) \rightarrow v(\cdot ) \ \text{ strongly } \text{ in } \ L^2(t-h,t; \widehat{V}^*) \ \text{ as } \ n \rightarrow \infty . \end{aligned}$$

So

$$\begin{aligned} w^n(t+\cdot ) =: w^n_t(\cdot ; \tau _n,w_n^\mathrm{in},\phi _n^\mathrm{in}) \rightarrow v_t(\cdot ) := v(t+\cdot ) \ \text{ strongly } \text{ in } \ L_{\widehat{V}^*}^2 \ \text{ as } \ n \rightarrow \infty . \end{aligned}$$

(3.56)

From (3.41) and the uniqueness of limit, (3.56) implies

$$\begin{aligned} w^n_t(\cdot ; \tau _n,w_n^\mathrm{in},\phi _n^\mathrm{in}) \rightarrow \phi \ \text{ strongly } \text{ in } \ L_{\widehat{V}^*}^2 \ \text{ as } \ n \rightarrow \infty . \end{aligned}$$

(3.57)

Further, we conclude that

$$\begin{aligned} \int _{-h}^0 \Vert w^n_t(s)-\phi (s)\Vert ^2 {\mathrm{d}}s =&\int _{-h}^0 \langle w^n_t(s)-\phi (s), w^n_t(s)-\phi (s) \rangle {\mathrm{d}}s \\ \leqslant&\Vert w^n_t(s)-\phi (s)\Vert _{L_{\widehat{V}^*}^2} \Vert w^n_t(s)-\phi (s)\Vert _{L_{\widehat{V}}^2} \rightarrow 0 \ \ \text{ as } \ \ n \rightarrow \infty , \end{aligned}$$

which together with (3.46) gives the pullback $\mathcal {D}_{\gamma }$-asymptotic compactness. $\square$

3.3 Existence of Pullback Attractor for the Universe $\mathcal {D}_{\gamma }$

Theorem 3.1

Assume that Assumptions 2.1 and 3.1 hold, then the process $\{U(t,\tau )\}_{t\geqslant \tau }$ in (2.10) has a unique pullback $\mathcal {D}_{\gamma }$-attractor $\widehat{ \mathcal {A}}_{\mathcal {D}_{\gamma }}=\{\mathcal {A}_{\mathcal {D}_{\gamma }}(t)\ | \ t \in \mathbb {R} \}$ for the universe $\mathcal {D}_{\gamma }$.

Proof

Define

$$\begin{aligned} \mathcal {A}_{\mathcal {D}_{\gamma }}(t) = \bigcap \limits _{\tau _0\leqslant t}\overline{\bigcup \limits _{\tau \leqslant \tau _0} U(t,\tau )D(\tau )},\quad \widehat{D} = \{D(t) \ | \ t\in \mathbb {R} \}\in \mathcal {D}_{\gamma }. \end{aligned}$$

(3.58)

According to [16, Proposition 9] or [27, Theorem 18], Lemmas 3.2 and  3.6 imply $\widehat{ \mathcal {A}}_{\mathcal {D}_{\gamma }}=\{\mathcal {A}_{\mathcal {D}_{\gamma }}(t)\ | \ t \in \mathbb {R} \}$ in (3.58) is the unique pullback attractor for the universe $\mathcal {D}_{\gamma }$. $\square$

4 Some Properties of Pullback Attractor for the Universe $\mathcal {D}_{\gamma }$

In this section, we conclude some properties of pullback attractor for the universe $\mathcal {D}_{\gamma }$. The first property is that pullback attractor for the universe $\mathcal {D}_{\gamma }$ is consistent with that for the universe of fixed bounded sets. The other property is the tempered behavior.

4.1 Consistency with Pullback Attractor for the Universe of Fixed Bounded Sets

Let us denote $\mathcal {D}_F$ the class of all families

$$\begin{aligned} \widehat{D}_F = \big \{ D_F(t)=D \ | \ t\in \mathbb {R},\, D \ \text{ is } \text{ some } \text{ bounded } \text{ set } \text{ in } \ E_{\widehat{H}}^2 \big \}. \end{aligned}$$

It is clear that $\mathcal {D}_F \subset \mathcal {D}_{\gamma }$. Then we consider the universe $\mathcal {D}_F$ in $\mathcal {P}(E_{\widehat{H}}^2)$.

Theorem 4.1

Under Assumptions 2.1 and 3.1, the process $\{U(t,\tau )\}_{t\geqslant \tau }$ in (2.10) has a unique pullback $\mathcal {D}_F$-attractor $\widehat{\mathcal {A}}_{\mathcal {D}_F} =\{\mathcal {A}_{\mathcal {D}_F}(t)\ | \ t \in \mathbb {R} \}$, Moreover,

$$\begin{aligned} \mathcal {A}_{\mathcal {D}_F}(t)=\mathcal {A}_{\mathcal {D}_{\gamma }}(t), \ \forall \, t \in \mathbb {R}. \end{aligned}$$

(4.1)

Proof

The existence of pullback $\mathcal {D}_F$-attractor $\widehat{\mathcal {A}}_{\mathcal {D}_F}$ is as a consequence of Theorem 3.1. Under Assumption 3.1, (4.1) follows from  [27, Proposition 23]. $\square$

Remark 4.1

By the pullback attracting property of the pullback attractor $\widehat{\mathcal {A}}_{\mathcal {D}_{\gamma }}$ and (4.1), we can check that, for any $\widehat{D} = \{D(t) \ | \ t \in \mathbb {R}\} \in \mathcal {D}_{\gamma }$, there holds

$$\begin{aligned} \lim \limits _{\tau \rightarrow -\infty } \mathrm{{dist}}_{E_{\widehat{H}}^2} (U(t,\tau )D(\tau ),\mathcal {A}_{\mathcal {D}_F}(t)) = \lim \limits _{\tau \rightarrow -\infty } \mathrm{{dist}}_{E_{\widehat{H}}^2} (U(t,\tau )D(\tau ),\mathcal {A}_{\mathcal {D}_{\gamma }}(t)) = 0, \end{aligned}$$

which implies that $\widehat{\mathcal {A}}_{\mathcal {D}_F}$ not only attracts any bounded sets but also attracts some tempered sets in the pullback sense.

4.2 Tempered Behavior of the Pullback Attractor

Theorem 4.2

  Under the conditions of Assumptions 2.1 and 3.1, it holds that

$$\begin{aligned}&\lim \limits _{t \rightarrow - \infty } \left( e^{\gamma t} \sup \limits _{(w,\phi )\in \mathcal {A}_{\mathcal {D}_{\gamma }}(t)} \Vert (w,\phi )\Vert _{E_{\widehat{H}}^2} \right) = 0, \end{aligned}$$

(4.2)

$$\begin{aligned}&\lim \limits _{t \rightarrow - \infty } \left( e^{\gamma t} \sup \limits _{(w,\phi )\in \mathcal {A}_{\mathcal {D}_{\gamma }}(t)} \Vert (w,\phi )\Vert _{E_{\widehat{V}}^2} \right) = 0. \end{aligned}$$

(4.3)

Proof

By Definition 3.1 of universe $\mathcal {D}_{\gamma }$, we have

$$\begin{aligned} \widehat{\mathcal {A}}_{\mathcal {D}_{\gamma }} \in \mathcal {D}_{\gamma }. \end{aligned}$$

Thus, (4.2) holds.

Moreover, (4.3) is a consequence of Lemmas 3.2, 3.5 and Assumption 3.1. $\square$