Stability Analysis of Random Attractors for Stochastic Modified Swift–Hohenberg Equations with Delays

Abstract

A new type of random attractors is introduced to study dynamics of a stochastic modified Swift–Hohenberg equation with a general delay. A compact, pullback attracting and dividedly invariant set is called a backward attractor, while the criteria for its existence are established in terms of increasing dissipation and backward asymptotic compactness of a cocycle. If the delay term in the equation is Lipschitz continuous such that the Lipschitz bound and the external force are backward limitable, then we prove the existence of a backward attractor, which further leads to the longtime stability as well as the existence of a pullback attractor, where the pullback attractor and the backward attractor are shown to be random and dividedly random, respectively. Two examples of the delay term are provided to illustrate variable and distributed delays without restricting the upper bound of Lipschitz bounds.

1 Introduction

We study the backward-pullback dynamics for non-autonomous stochastic modified Swift–Hohenberg equations with general delays on a bounded 2D-domain ${\mathcal {O}}$:

$$\begin{aligned} \left\{ \begin{array}{l} du+(\Delta ^{2}u+2\Delta u+au+b|\nabla u|^{2}+u^{3})dt=f(t, u_{t})dt+g(t, x)dt+h(x)dW, \\ u(t)=\Delta u(t)=0, \ \text { on } \partial {\mathcal {O}}, \ t>\tau , \\ u(\tau +\sigma , x):=\phi (\sigma , x), \ \sigma \in [-\varrho , 0], x\in {\mathcal {O}},\ \tau \in {\mathbb {R}}, \end{array}\right. \end{aligned}$$

(1.1)

where $a,b\in {\mathbb {R}}$, the delay shift is defined by $u_{t}(\sigma )=u(t+\sigma )$ for $\sigma \in [-\varrho , 0]$, and $\varrho >0$ is the memory time.

If $b=\varrho =0$ and $h=0$, then the equation is reduced to the classical Swift–Hohenberg (SH) equation introduced by [24], which has a connection with Rayleigh–Benard convection. It describes the pattern formation in fluid layers confined between horizontal well-conducting boundaries, and it is also a model for the thermal convection in a thin layer of fluid heated from below. The global dynamics of the SH equation has been investigated by [8, 12, 20] in the deterministic case ($h=0$) and by [9, 16] in the stochastic case ($h\ne 0$).

If $b\ne 0$ and $\varrho =h=0$, the equation is called a modified SH equation [11]. The modified term $b|\nabla u|^{2}$ just describes the connection between nonlinear energy optimization and instantons (see [13]). The long-term behavior for the modified SH equation has been studied in [21,22,23].

To the best of our knowledge, there is no paper dealing with the (modified or original) stochastic SH equation with a delay ($\varrho \ne 0$), although other stochastic delayed equations have been studied in [4, 5, 14, 15, 19, 27,28,29,30, 32, 34]. While, the delay term f just describes the dynamics influenced by events from the past.

For a non-autonomous random dynamical system (cocycle), the dynamics can be described in terms of pullback random attractors which was first introduced by Caraballo et al. in [1] and later was studied, e.g., in [6, 25].

In this article, a basic subject is to prove that the cocycle generated by Eq. (1.1) possesses a PRA ${\mathcal {A}}=\{{\mathcal {A}}(t, \omega ): t\in {\mathbb {R}}, \omega \in \Omega \}$ in the state space $C([-\varrho ,0], H_0^2({\mathcal {O}}))$ over a probability space $(\Omega , {\mathfrak {F}}, P)$. A further subject is to consider the longtime stability of the PRA, which means that for each $\omega \in \Omega$ there is a minimal compact set (this set is called a backward controller and denoted by ${\mathcal {A}}_{-\infty }(\omega )$) such that

$$\begin{aligned} \lim _{t\rightarrow -\infty }\text {dist}_{C([-\varrho ,0], H_0^2({\mathcal {O}}))}({\mathcal {A}}(t, \omega ), {\mathcal {A}}_{-\infty }(\omega ))=0. \end{aligned}$$

(1.2)

An abstract criterion for (1.2) in terms of backward compactness of the PRA has been established in [18, 26]. We will establish an alternative criterion for the longtime stability (1.2) from a completely different point of view. Our idea is to introduce a new type of a non-autonomous random attractor rather than the pullback attractor.

By a backward attractor, we mean a compact, dividedly invariant and pullback attracting set, where the backward union of an invariant set is called a dividedly invariant set (instead of the invariance in a pullback attractor). We then show that such a backward attractor must be pullback attracting not only in the current but also in the past. This property is called backward attraction, which is stronger than the pullback attraction.

More importantly, we show that the existence of a backward attractor implies the longtime stability [like (1.2)] as well as the existence of a pullback attractor, see Theorem 2.6. Meanwhile, criteria for the existence of a backward attractor are stated in terms of increasing dissipation and backward asymptotic compactness of the cocycle (see Theorem 2.9).

To apply the abstract results to equation (1.1), we need to assume that the delay force f is pointwise Lipschitz continuous such that the Lipschitz bound $L_{f}(\cdot )$ is backward limitable:

$$\begin{aligned} \lim _{\gamma \rightarrow +\infty }\sup _{s\le \tau }\int _{-\infty }^{0}e^{\gamma r}L_{f}^{2}(r+s)dr=0, \ \forall \tau \in {\mathbb {R}}, \end{aligned}$$

(1.3)

and the external force $g(\cdot )$ is backward limitable in ${\mathbb {R}}\times L^2({\mathcal {O}})$ [i.e. $\Vert g(\cdot )\Vert$ instead of $L_{f}(\cdot )$ in (1.3)]. We also assume that the integral of f is Lipschitz continuous with a bound $c_f$.

In order to obtain an attractor for a delayed equation, one often restricts the upper bound of $L_f$ or $c_f$ by a function of the Poincaré constant $\lambda _1$, see [2, 3, 10, 27]. In this paper, we do not need this restriction condition by using a new method for the asymptotic estimates as in Lemma  4.1. This allows us to consider more delayed forces such as variable delay and distributed delay without the restriction of the upper bound, see Examples 5.4 and 5.5.

We prove that the bounds of the solutions in $L^2({\mathcal {O}})$ and $H^2_0({\mathcal {O}})$ can be preserved in the past. By using the method of spectrum decomposition, we show that the solution is pointwise backward asymptotically compact and equi-continuous from $[-\varrho ,0]$ into $H^2_0({\mathcal {O}})$. Then, by using the Ascoli–Arzelá theorem, we can prove the backward asymptotic compactness in $C([-\varrho ,0], H_0^2({\mathcal {O}}))$ and thus obtain a backward attractor, which further implies the longtime stability (1.2) as well as the existence of a PRA.

2 Backward Attractors and Longtime Stability of Pullback Attractors

2.1 Backward Attractors

Let $(\Omega , {\mathfrak {F}}, P)$ be a probability space with a group of measure-preserving transforms $\theta _{t}:\Omega \rightarrow \Omega$ ($t\in {\mathbb {R}}$) and $(X, \Vert \cdot \Vert _{X})$ a separable Banach space with Borel algebra ${\mathfrak {B}}(X)$.

A measurable mapping $\Phi : {\mathbb {R}}^{+}\times {\mathbb {R}}\times \Omega \times X\rightarrow X$ is called a non-autonomous random dynamical system (briefly, a cocycle, see [6, 25]) if $\Phi (0, \tau , \omega )=id_{X}$ and

$$\begin{aligned} \Phi (t+s, \tau , \omega )=\Phi (t, \tau +s, \theta _{s}\omega )\Phi (s, \tau , \omega ), \ t, s\in {\mathbb {R}}^{+}, \tau \in {\mathbb {R}}, \omega \in \Omega . \end{aligned}$$

The cocycle $\Phi$ is called continuous in X if the operator $\Phi (t, \tau , \omega ):X\rightarrow X$ is continuous.

A family of nonempty subsets $D(\tau , \omega )$ of X parameterized by $\tau \in {\mathbb {R}}$ and $\omega \in \Omega$ is called a bi-parametric set in X over ${\mathbb {R}}\times \Omega$. Such a bi-parametric set $D(\cdot ,\cdot )$ is called

• random if for each $\tau \in {\mathbb {R}}$, the subset ${\mathcal {D}}(\tau , \cdot )$ of $\Omega \times X$ is ${\mathfrak {F}}\times {\mathfrak {B}}(X)$-measurable;

• compact, closed, bounded if every component $D(\tau , \omega )$ is compact, closed and bounded, respectively;

• invariant under a cocycle $\Phi$ if

  $$\begin{aligned} \Phi (t, \tau , \omega )D(\tau , \omega )=D(\tau +t, \theta _{t}\omega ), \ \forall t\ge 0, \ \tau \in {\mathbb {R}},\ \omega \in \Omega . \end{aligned}$$

  (2.1)

• dividedly invariant if there is an invariant set $E(\cdot ,\cdot )$ such that

  $$\begin{aligned} D(\tau , \omega )=\overline{\cup _{s\le \tau }E(s, \omega )}=:E_b(\tau ,\omega ), \ \forall \tau \in {\mathbb {R}},\ \omega \in \Omega . \end{aligned}$$

  (2.2)

We need to consider a universe ${\mathfrak {D}}$ formed from some bi-parametric sets in X over ${\mathbb {R}}\times \Omega$.

Definition 2.1

${\mathcal {A}}\in {\mathfrak {D}}$ is called a ${\mathfrak {D}}$-pullback attractor for a cocycle $\Phi$ if it is compact, invariant and ${\mathfrak {D}}$-pullback attracting, where the ${\mathfrak {D}}$-pullback attraction means that for each $D\in {\mathfrak {D}}$ and $\omega \in \Omega$,

$$\begin{aligned} \lim _{t\rightarrow +\infty } \textrm{dist}_{X}(\Phi (t, \tau -t, \theta _{-t}\omega )D(\tau -t, \theta _{-t}\omega ), {\mathcal {A}}(\tau , \omega ))=0,\ \forall \tau \in {\mathbb {R}}. \end{aligned}$$

(2.3)

Such a ${\mathfrak {D}}$-pullback attractor ${\mathcal {A}}$ is called a ${\mathfrak {D}}$-pullback random attractor if it is random.

We introduce a new type of attractor for a cocycle, where we just replace the invariance of the pullback random attractor by the dividedly invariance.

Definition 2.2

${\mathcal {B}}\in {\mathfrak {D}}$ is called a ${\mathfrak {D}}$-backward attractor if ${\mathcal {B}}(\cdot , \cdot )$ is compact, dividedly invariant and ${\mathfrak {D}}$-pullback attracting.

We do not stipulate the measurability of a backward attractor. But we will prove in the application part that the backward attractor ${\mathcal {B}}$ is dividedly random, which means ${\mathcal {B}}$ is the backward union $E_b$ of a random set $E(\cdot ,\cdot )$.

Definition 2.3

${\mathcal {B}}\in {\mathfrak {D}}$ is called ${\mathfrak {D}}$-backward attracting if for each $D\in {\mathfrak {D}}$, $\tau \in {\mathbb {R}}$ and $\omega \in \Omega$,

$$\begin{aligned} \lim _{t\rightarrow +\infty } \textrm{dist}_{X}(\Phi (t, s-t, \theta _{-t}\omega )D(s-t, \theta _{-t}\omega ), {\mathcal {B}}(\tau , \omega ))=0,\ \forall s\le \tau . \end{aligned}$$

(2.4)

Such a backward attraction means that the pullback attraction holds in the whole past, including the current time $s=\tau$ as in (2.3).

The following result implies that a backward attractor in the sense of Definition 2.2 must be backward attracting (this is the reason why we call it a backward attractor). We need two properties of the universe. A cocycle ${\mathfrak {D}}$ is called:

• backward closed if $E\in {\mathfrak {D}}$ whenever $D\in {\mathfrak {D}}$ and $\emptyset \ne E(\tau ,\omega )\subset \overline{\cup _{s\le \tau }D(s, \omega )}$ for all $\tau \in {\mathbb {R}}, \omega \in \Omega$;

• truncation-closed if for each $\tau \in {\mathbb {R}}$, $D_\tau \in {\mathfrak {D}}$ whenever $D\in {\mathfrak {D}}$, where the truncation family $D_\tau (\cdot ,\cdot )$ is defined by

  $$\begin{aligned} D_\tau (s, \omega )=D(\tau , \omega ) \text { if } s\le \tau \text { and } D_\tau (s, \omega )=D(s, \omega ) \text { if } s> \tau . \end{aligned}$$

  (2.5)

Proposition 2.4

Let $\Phi$ be a cocycle and ${\mathfrak {D}}$ a backward closed and truncation-closed universe. Then the following assertions are equivalent:

• (i) ${\mathcal {B}}$ is a ${\mathfrak {D}}$-backward attractor for $\Phi$ in the sense of Definition 2.2;

• (ii) ${\mathcal {B}}$ is the minimum among all compact ${\mathfrak {D}}$-backward attracting sets.

Proof

(ii) $\Rightarrow$ (i) Suppose ${\mathcal {B}}$ is the minimum amongst all compact ${\mathfrak {D}}$-backward attracting sets. Given $\tau \in {\mathbb {R}}$, we define the truncation ${\mathcal {B}}_\tau$ of ${\mathcal {B}}$ by

$$\begin{aligned} {\mathcal {B}}_\tau (s, \omega )\equiv {\mathcal {B}}(\tau , \omega ) \ \text {if} \ s\le \tau , \ \text {and} \ {\mathcal {B}}_\tau (s, \omega )={\mathcal {B}}(s, \omega ) \ \text {if} \ s>\tau . \end{aligned}$$

By the compactness of ${\mathcal {B}}(\cdot ,\cdot )$, it is easy to see that ${\mathcal {B}}_\tau (\cdot ,\cdot )$ is compact too. Since ${\mathcal {B}}$ is ${\mathfrak {D}}$-backward attracting, by Definition 2.3, ${\mathcal {B}}\in {\mathfrak {D}}$, and thus by the truncation-closedness of ${\mathfrak {D}}$ we know ${\mathcal {B}}_\tau \in {\mathfrak {D}}$. The ${\mathfrak {D}}$-backward attraction of ${\mathcal {B}}$ also implies that for each $D\in {\mathfrak {D}}$ and $s\le \tau$,

$$\begin{aligned}&\text {dist}_{X}(\Phi (t, s-t, \theta _{-t}\omega )D(s-t, \theta _{-t}\omega ), {\mathcal {B}}_\tau (s, \omega )) \\&\quad =\text {dist}_{X}(\Phi (t, s-t, \theta _{-t}\omega )D(s-t, \theta _{-t}\omega ), {\mathcal {B}}(\tau , \omega ))\rightarrow 0 \ \text {as} \ t\rightarrow +\infty , \end{aligned}$$

and for $s>\tau$,

$$\begin{aligned}&\text {dist}_{X}(\Phi (t, s-t, \theta _{-t}\omega )D(s-t, \theta _{-t}\omega ), {\mathcal {B}}_\tau (s, \omega )) \\&\quad =\text {dist}_{X}(\Phi (t, s-t, \theta _{-t}\omega )D(s-t, \theta _{-t}\omega ), {\mathcal {B}}(s, \omega ))\rightarrow 0 \ \text {as} \ t\rightarrow +\infty . \end{aligned}$$

Hence, ${\mathcal {B}}_\tau (\cdot ,\cdot )$ is ${\mathfrak {D}}$-pullback attracting (at all $s\in {\mathbb {R}}$). In a word, $\Phi$ has a compact ${\mathfrak {D}}$-pullback attracting set ${\mathcal {B}}_\tau \in {\mathfrak {D}}$. By [7, Lemma 2.5], there is a unique ${\mathfrak {D}}$-pullback attractor ${\mathcal {A}}\in {\mathfrak {D}}$. In particular, ${\mathcal {A}}$ is an invariant set and thus we see from the ${\mathfrak {D}}$-pullback attraction of ${\mathcal {B}}_\tau$ that for all $s\le \tau$,

$$\begin{aligned}&\text {dist}_{X}({\mathcal {A}}(s, \omega ), {\mathcal {B}}(\tau , \omega )) \\&\quad =\text {dist}_{X}(\Phi (t, s-t, \theta _{-t}\omega ){\mathcal {A}}(s-t, \theta _{-t}\omega ), {\mathcal {B}}_\tau (s, \omega ))\rightarrow 0 \ \text {as} \ t\rightarrow +\infty , \end{aligned}$$

which further implies that ${\mathcal {A}}(s, \omega )\subset {\mathcal {B}}(\tau , \omega )$ for all $s\le \tau$ and thus

$$\begin{aligned} {\mathcal {B}}(\tau , \omega )\supset \overline{\cup _{s\le \tau }{\mathcal {A}}(s, \omega )}=:{\mathcal {A}}_b(\tau , \omega ), \forall \tau \in {\mathbb {R}}. \end{aligned}$$

(2.6)

Since ${\mathfrak {D}}$ is backward closed, it follows that ${\mathcal {A}}_b\in {\mathfrak {D}}$ and, by the compactness of ${\mathcal {B}}$, we see from (2.6) that ${\mathcal {A}}_b$ is compact too. By the ${\mathfrak {D}}$-pullback attraction of ${\mathcal {A}}$, we have for all $D\in {\mathfrak {D}}$ and $s\le \tau$,

$$\begin{aligned}&\text {dist}_{X}(\Phi (t, s-t, \theta _{-t}\omega ){\mathcal {D}}(s-t, \theta _{-t}\omega ), {\mathcal {A}}_b(\tau , \omega )) \\&\quad \le \text {dist}_{X}(\Phi (t, s-t, \theta _{-t}\omega ){\mathcal {D}}(s-t, \theta _{-t}\omega ), {\mathcal {A}}(s, \omega )) \rightarrow 0 \ \text {as} \ t\rightarrow +\infty , \end{aligned}$$

which means that ${\mathcal {A}}_b$ is ${\mathfrak {D}}$-backward attracting. By the minimality of ${\mathcal {B}}$ amongst all compact ${\mathfrak {D}}$-backward attracting sets, we have ${\mathcal {B}}\subset {\mathcal {A}}_b$ which, together with (2.6) implies that

$$\begin{aligned} {\mathcal {B}}(\tau , \omega )=\overline{\cup _{s\le \tau }{\mathcal {A}}(s, \omega )},\ \forall \tau \in {\mathbb {R}}, \omega \in \Omega . \end{aligned}$$

(2.7)

By the invariance of ${\mathcal {A}}$ again, ${\mathcal {B}}$ is dividedly invariant and thus ${\mathcal {B}}$ is a backward attractor in the sense of Definition 2.2.

(i) $\Rightarrow$ (ii) Suppose ${\mathcal {B}}$ is a backward attractor, that is, ${\mathcal {B}}\in {\mathfrak {D}}$ is compact, dividedly invariant and ${\mathfrak {D}}$-pullback attracting. Then, from the divided invariance of ${\mathcal {B}}$, there is an invariant set $E(\cdot ,\cdot )$ such that

$$\begin{aligned} {\mathcal {B}}(\tau , \omega )=\overline{\cup _{r\le \tau }E(r, \omega )}=E_b(\tau , \omega ),\ {\forall \tau \in {\mathbb {R}}, \omega \in \Omega .} \end{aligned}$$

By definition, the backward closedness of ${\mathfrak {D}}$ implies the inclusion closedness and thus $E\in {\mathfrak {D}}$. We also have for all $s\le \tau$,

$$\begin{aligned} {\mathcal {B}}(s, \omega )=\overline{\cup _{r\le s}E(r, \omega )}\subset \overline{\cup _{r\le \tau }E(r, \omega )}={\mathcal {B}}(\tau , \omega ). \end{aligned}$$

(2.8)

This monotonicity together with the ${\mathfrak {D}}$-pullback attraction of ${\mathcal {B}}$ implies that for each $D\in {\mathfrak {D}}$ and $s\le \tau$,

$$\begin{aligned}&\text {dist}_{X}(\Phi (t, s-t, \theta _{-t}\omega )D(s-t, \theta _{-t}\omega ), {\mathcal {B}}(\tau , \omega )) \\&\quad \le \text {dist}_{X}(\Phi (t, s-t, \theta _{-t}\omega )D(s-t, \theta _{-t}\omega ), {\mathcal {B}}(s, \omega ))\rightarrow 0 \ \text {as} \ t\rightarrow +\infty , \end{aligned}$$

which means that ${\mathcal {B}}$ is ${\mathfrak {D}}$-backward attracting.

Note that ${\mathcal {B}}(\cdot ,\cdot )$ is compact, it suffices to prove the minimality. If $K\in {\mathfrak {D}}$ is compact and ${\mathfrak {D}}$-backward attracting, then K backward attracts $E\in {\mathfrak {D}}$ and thus, by the invariance of E, we have for all $s\le \tau$,

$$\begin{aligned} \text {dist}_{X}(E(s, \omega ), K(\tau , \omega )) =\text {dist}_{X}(\Phi (t, s-t, \theta _{-t}\omega )E(s-t, \theta _{-t}\omega ), K(\tau , \omega ))\rightarrow 0 \end{aligned}$$

as $t\rightarrow +\infty$, which further implies $E(s, \omega )\subset K(\tau , \omega )$ for all $s\le \tau$ and thus

$$\begin{aligned} {\mathcal {B}}(\tau , \omega )=\overline{\cup _{s\le \tau }E(s, \omega )}\subset K(\tau , \omega ). \end{aligned}$$

Hence ${\mathcal {B}}$ is the minimum among all compact ${\mathfrak {D}}$-backward attracting sets. $\square$

From the minimality in (ii) of Proposition 2.4, a ${\mathfrak {D}}$-backward attractor (if exists) must be unique.

2.2 Connection with Longtime Stability of a Pullback Attractor

In fact, the longtime stability can be defined with respect to any bi-parametric set.

Definition 2.5

A bi-parametric set $D(\cdot ,\cdot )$ in X over ${\mathbb {R}}\times \Omega$ is called backward stable if for each $\omega \in \Omega$ there is a nonempty compact set $F(\omega )$ such that

$$\begin{aligned} \lim _{\tau \rightarrow -\infty } \textrm{dist}_{X}(D(\tau , \omega ), F(\omega ))=0. \end{aligned}$$

(2.9)

If there is a minimum among those nonempty compact sets $F(\omega )$ with the property (2.9), then the minimum is called a backward controller of $D(\cdot ,\cdot )$ and denoted by $D_{-\infty }(\omega )$.

Note that the backward controller $D_{-\infty }(\omega )$ must be nonempty compact as a limit set in (2.9). Such a backward controller for a pullback attractor in the deterministic case had been studied in [33]. We also remark that a pullback attractor is not always backward stable even in the deterministic case, a counterexample can be found in [17].

Theorem 2.6

Suppose a cocycle $\Phi$ has a ${\mathfrak {D}}$-backward attractor ${\mathcal {B}}\in {\mathfrak {D}}$, where ${\mathfrak {D}}$ is backward closed and truncation-closed, then we have the following arguments.

• (a) ${\mathcal {B}}$ is (always) backward stable with a backward controller ${\mathcal {B}}_{-\infty }(\cdot )$. Moreover, the upper semi-convergence in (2.9) holds with respect to the (symmetrical) Hausdorff metric:

  $$\begin{aligned} \textrm{dist}_{X}^{\text {sym}}({\mathcal {B}}(\tau , \omega ), {\mathcal {B}}_{-\infty }(\omega )):=\max \{ \textrm{dist}_{X}({\mathcal {B}}(\tau , \omega ), {\mathcal {B}}_{-\infty }(\omega )), \textrm{dist}_{X}({\mathcal {B}}_{-\infty }(\omega ), {\mathcal {B}}(\tau , \omega ))\rightarrow 0 \end{aligned}$$

  (2.10)

  as $\tau \rightarrow -\infty$ for all $\omega \in \Omega$.

• (b) There is a ${\mathfrak {D}}$-pullback attractor ${\mathcal {A}}\in {\mathfrak {D}}$ such that ${\mathcal {A}}$ is backward stable with a backward controller ${\mathcal {A}}_{-\infty }(\cdot )$ given by

  $$\begin{aligned} {\mathcal {A}}_{-\infty }(\omega )=\cap _{\tau \le 0}\overline{\cup _{s\le \tau }{\mathcal {A}}(s, \omega )} \text { and } {\mathcal {A}}_{-\infty }(\omega )={\mathcal {B}}_{-\infty }(\omega ). \end{aligned}$$

  (2.11)

Proof

(a) Denote the $\Upsilon$-limit set of ${\mathcal {B}}(\tau ,\cdot )$ as $\tau \rightarrow -\infty$ by

$$\begin{aligned} \Upsilon _{{\mathcal {B}}}(\omega ):=\cap _{\tau \le 0}\overline{\cup _{s\le \tau }{\mathcal {B}}(s, \omega )} ,\ \forall \omega \in \Omega . \end{aligned}$$

By (2.8), $\tau \mapsto {\mathcal {B}}(\tau ,\omega )$ is increasing and thus $\Upsilon _{{\mathcal {B}}}(\omega )=\cap _{\tau \le 0}{\mathcal {B}}(\tau ,\omega )$. By the theorem of nested compact sets, $\Upsilon _{{\mathcal {B}}}(\omega )$ is nonempty compact. We prove

$$\begin{aligned} \lim _{\tau \rightarrow -\infty }\text {dist}_{X}({\mathcal {B}}(\tau , \omega ), \Upsilon _{{\mathcal {B}}}(\omega ))=0,\ \forall \omega \in \Omega . \end{aligned}$$

(2.12)

If (2.12) is not true, then there are $\delta >0$, $\omega \in \Omega$ and $x_n\in {\mathcal {B}}(\tau _n, \omega )$ with $0\ge \tau _n\rightarrow -\infty$ such that

$$\begin{aligned} \text {dist}_{X}(x_n, \Upsilon _{{\mathcal {B}}}(\omega ))\ge \delta , \ \forall n\in {\mathbb {N}}. \end{aligned}$$

(2.13)

By the monotonicity of $\tau \mapsto {\mathcal {B}}(\tau ,\omega )$ again, $x_n\in {\mathcal {B}}(0, \omega )$, which means $\{x_n\}$ is pre-compact. Passing to a subsequence, we have $x_{n_k}\rightarrow x\in \Upsilon _{{\mathcal {B}}}(\omega )$, which contradicts (2.13). Therefore (2.12) holds true, which means ${\mathcal {B}}$ is backward stable.

We then show that $\Upsilon _{{\mathcal {B}}}(\omega )$ is just the backward controller of ${\mathcal {B}}$, that is, ${\mathcal {B}}_{-\infty }(\omega )=\Upsilon _{{\mathcal {B}}}(\omega )$. It suffices to prove the minimality. Suppose $F(\omega )$ is nonempty compact such that

$$\begin{aligned} \lim _{\tau \rightarrow -\infty } \textrm{dist}_{X}({\mathcal {B}}(\tau , \omega ), F(\omega ))=0,\ \forall \omega \in \Omega . \end{aligned}$$

Given $x\in \Upsilon _{{\mathcal {B}}}(\omega )$, there are $\tau _n\rightarrow -\infty$ and $x_n\in {\mathcal {B}}(\tau _n, \omega )$ such that $x_n\rightarrow x$. We further choose $y_n\in F(\omega )$ such that

$$\begin{aligned} \Vert x_n-y_n\Vert =\text {dist}_{X}(x_n, F(\omega ))\le \text {dist}_{X}({\mathcal {B}}(\tau _n, \omega ), F(\omega ))\rightarrow 0. \end{aligned}$$

By the compactness of $F(\omega )$, there is a subsequence such that $y_{n_k}\rightarrow y\in F(\omega )$. Hence $x=y\in F(\omega )$ and thus $\Upsilon _{{\mathcal {B}}}(\omega )\subset F(\omega )$ as desired.

Since $\Upsilon _{{\mathcal {B}}}(\omega )=\cap _{\tau \le 0}{\mathcal {B}}(\tau ,\omega )\subset {\mathcal {B}}(s, \omega )$ for all $s\le 0$ and thus $\text {dist}_{X}(\Upsilon _{{\mathcal {B}}}(\omega ), {\mathcal {B}}(s, \omega ))\equiv 0$ for all $s\le 0$, which together with (2.12) implies (2.10).

(b) By Definition 2.2, the backward attractor ${\mathcal {B}}\in {\mathfrak {D}}$ must be compact and ${\mathfrak {D}}$-pullback attracting. Applying [7, Lemma 2.5], we obtain a unique ${\mathfrak {D}}$-pullback attractor ${\mathcal {A}}\in {\mathfrak {D}}$ such that ${\mathcal {A}}\subset {\mathcal {B}}$. Set

$$\begin{aligned} \Upsilon _{{\mathcal {A}}}(\omega ):=\cap _{\tau \le 0}\overline{\cup _{s\le \tau }{\mathcal {A}}(s, \omega )} ,\ \forall \omega \in \Omega . \end{aligned}$$

Since ${\mathcal {B}}$ is dividedly invariant, it follows from (2.8) that $\tau \mapsto {\mathcal {B}}(\tau ,\omega )$ is increasing and thus

$$\begin{aligned} \overline{\cup _{s\le \tau }{\mathcal {A}}(s, \omega )}\subset \overline{\cup _{s\le \tau }{\mathcal {B}}(s, \omega )}={\mathcal {B}}(\tau , \omega ). \end{aligned}$$

Hence, $\overline{\cup _{s\le \tau }{\mathcal {A}}(s, \omega )}$ is a nest of compact sets as $\tau \rightarrow -\infty$ and so $\Upsilon _{{\mathcal {A}}}(\omega )$ is nonempty compact. We then prove

$$\begin{aligned} \lim _{\tau \rightarrow -\infty }\text {dist}_{X}({\mathcal {A}}(\tau , \omega ), \Upsilon _{{\mathcal {A}}}(\omega ))=0,\ \forall \omega \in \Omega . \end{aligned}$$

(2.14)

If (2.14) is not true, then there are $\delta >0$, $\omega \in \Omega$ and $y_n\in {\mathcal {A}}(\tau _n, \omega )$ with $0\ge \tau _n\rightarrow -\infty$ such that

$$\begin{aligned} \text {dist}_{X}(y_n, \Upsilon _{{\mathcal {A}}}(\omega ))\ge \delta , \ \forall n\in {\mathbb {N}}. \end{aligned}$$

(2.15)

By the monotonicity of $\tau \mapsto {\mathcal {B}}(\tau ,\omega )$ again, we know $y_n\in {\mathcal {A}}(\tau _n, \omega )\subset {\mathcal {B}}(\tau _n, \omega )\subset {\mathcal {B}}(0, \omega )$ and so $\{y_n\}$ is pre-compact. Hence, there is a subsequence $y_{n_k}\rightarrow y\in X$. Since $\tau _{n_k}\rightarrow -\infty$ and $y_{n_k}\in {\mathcal {A}}(\tau _{n_k}, \omega )$, it follows that $y\in \Upsilon _{{\mathcal {A}}}(\omega )$, which contradicts (2.15). Therefore (2.14) holds true, which means ${\mathcal {A}}$ is backward stable.

We prove the minimality. Suppose $F(\omega )$ is nonempty compact such that

$$\begin{aligned} \lim _{\tau \rightarrow -\infty } \textrm{dist}_{X}({\mathcal {A}}(\tau , \omega ), F(\omega ))=0,\ \forall \omega \in \Omega . \end{aligned}$$

Given $x\in \Upsilon _{{\mathcal {A}}}(\omega )$, it is easy to show that there are $\tau _n\rightarrow -\infty$ and $x_n\in {\mathcal {A}}(\tau _n, \omega )$ such that $x_n\rightarrow x$. We further choose $z_n\in F(\omega )$ such that

$$\begin{aligned} \Vert x_n-z_n\Vert =\text {dist}_{X}(x_n, F(\omega ))+\frac{1}{n}\le \text {dist}_{X}({\mathcal {B}}(\tau _n, \omega ), F(\omega ))+\frac{1}{n}\rightarrow 0. \end{aligned}$$

By the compactness of $F(\omega )$, there is a subsequence $z_{n_k}$ of $z_n$ such that $z_{n_k}\rightarrow z\in F(\omega )$. Hence $x=z\in F(\omega )$ and thus $\Upsilon _{{\mathcal {A}}}(\omega )\subset F(\omega )$. Therefore, $\Upsilon _{{\mathcal {A}}}(\cdot )$ is just the unique backward controller of ${\mathcal {A}}$, that is, ${\mathcal {A}}_{-\infty }(\omega )=\Upsilon _{{\mathcal {A}}}(\omega )$.

Finally, we prove both backward controllers for ${\mathcal {A}}$ and ${\mathcal {B}}$ are equal. Indeed, by (2.7) in the proof of Proposition 2.4, the backward attractor is just the backward union of the pullback attractor:

$$\begin{aligned} {\mathcal {B}}(\tau , \omega )=\overline{\cup _{s\le \tau }{\mathcal {A}}(s, \omega )},\ \forall \tau \in {\mathbb {R}}, \omega \in \Omega , \end{aligned}$$

which together with the monotonicity of ${\mathcal {B}}$ implies that

$$\begin{aligned} \Upsilon _{{\mathcal {B}}}(\omega )=\cap _{\tau \le 0}{\mathcal {B}}(\tau , \omega )=\cap _{\tau \le 0}\overline{\cup _{s\le \tau }{\mathcal {A}}(s, \omega )}=\Upsilon _{{\mathcal {A}}}(\omega ) \end{aligned}$$

and thus ${\mathcal {B}}_{-\infty }(\omega )={\mathcal {A}}_{-\infty }(\omega )$ as desired. $\square$

2.3 Criteria for the Existence of a Backward Attractor

Definition 2.7

A cocycle $\Phi$ is called ${\mathfrak {D}}$-backward asymptotically compact if for each $(D, \tau , \omega )\in {\mathfrak {D}}\times {\mathbb {R}}\times \Omega$, the sequence $\{\Phi (t_{n}, s_{n}-t_{n}, \theta _{-t_{n}}\omega )x_{n}\}_{n\in {\mathbb {N}}}$ is pre-compact in X, whenever $s_{n}\le \tau$, $t_{n}\rightarrow +\infty$, and $x_{n}\in D(s_{n}-t_{n}, \theta _{-t_{n}}\omega )$.

Recall that $K\in {\mathfrak {D}}$ is said to be ${\mathfrak {D}}$-pullback absorbing if for each $(D, \tau , \omega )\in {\mathfrak {D}}\times {\mathbb {R}}\times \Omega$, there exists $T=T(D, \tau , \omega )>0$ such that

$$\begin{aligned} \Phi (t, \tau -t, \theta _{-t}\omega )D(\tau -t, \theta _{-t}\omega )\subset K(\tau , \omega ), \ \forall t\ge T. \end{aligned}$$

Definition 2.8

A cocycle $\Phi$ is called increasingly dissipative if $\Phi$ has an increasing, closed and ${\mathfrak {D}}$-pullback absorbing set $K\in {\mathfrak {D}}$.

Theorem 2.9

Suppose a cocycle $\Phi$ is ${\mathfrak {D}}$-backward asymptotically compact, where the universe ${\mathfrak {D}}$ is backward closed and truncation-closed. Then the following two assertions are equivalent:

• (i) $\Phi$ has a unique backward attractor ${\mathcal {B}}\in {\mathfrak {D}}$.

• (ii) $\Phi$ is increasingly dissipative with respect to ${\mathfrak {D}}$. We also need the neighborhood-closedness [25] of ${\mathfrak {D}}$ for (i) $\Rightarrow$ (ii).

Proof

(ii) $\Rightarrow$ (i) Suppose $K\in {\mathfrak {D}}$ is an increasing closed ${\mathfrak {D}}$-pullback absorbing set. Note that $\Phi$ is ${\mathfrak {D}}$-backward asymptotically compact and so ${\mathfrak {D}}$-pullback asymptotically compact. It follows from [25, Theorem 2.23] that $\Phi$ has a ${\mathfrak {D}}$-pullback attractor ${\mathcal {A}}\in {\mathfrak {D}}$. Let ${\mathcal {B}}$ be the backward union of ${\mathcal {A}}$, that is,

$$\begin{aligned} {\mathcal {B}}(\tau , \omega )=\overline{\cup _{s\le \tau }{\mathcal {A}}(s, \omega )}, \ \forall \tau \in {\mathbb {R}}, \omega \in \Omega . \end{aligned}$$

Since ${\mathfrak {D}}$ is backward closed, it follows that ${\mathcal {B}}\in {\mathfrak {D}}$. Since ${\mathcal {A}}$ is invariant, it follows that ${\mathcal {B}}$ is dividedly invariant. Since ${\mathcal {A}}$ is ${\mathfrak {D}}$-pullback attracting and ${\mathcal {A}}\subset {\mathcal {B}}$, it follows that ${\mathcal {B}}$ is ${\mathfrak {D}}$-pullback attracting too. It suffices to prove ${\mathcal {B}}$ is compact.

We need to prove that $\cup _{s\le \tau }{\mathcal {A}}(s, \omega )$ is pre-compact, where $\tau \in {\mathbb {R}}$ and $\omega \in \Omega$ are fixed. Given $x_n\in \cup _{s\le \tau }{\mathcal {A}}(s, \omega )$, there is $s_n\le \tau$ such that $x_n\in {\mathcal {A}}(s_n, \omega )$. We take a sequence $t_n\rightarrow +\infty$. By the invariance of ${\mathcal {A}}$ again, there is $y_n\in {\mathcal {A}}(s_n-t_n, \theta _{-t_n}\omega )$ such that

$$\begin{aligned} x_n=\Phi (t_n, s_n-t_n, \theta _{-t_n}\omega )y_n. \end{aligned}$$

Since $\Phi$ is ${\mathfrak {D}}$-backward asymptotically compact, it follows that $\{\Phi (t_n, s_n-t_n, \theta _{-t_n}\omega )y_n\}$ is pre-compact and so is $\{x_n\}$. Hence ${\mathcal {B}}(\tau , \omega )$ is compact. So far, we have proved that ${\mathcal {B}}$ is a backward attractor in the sense of Definition 2.2.

(i) $\Rightarrow$ (ii) Suppose $\Phi$ has a backward attractor ${\mathcal {B}}\in {\mathfrak {D}}$. By the neighborhood-closedness of ${\mathfrak {D}}$, there is a $\delta >0$ such that $N_\delta ({\mathcal {B}})\in {\mathfrak {D}}$, where the closed $\delta$-neighborhood of ${\mathcal {B}}$ is defined by

$$\begin{aligned} N_\delta ({\mathcal {B}}(\tau , \omega ))=\{x\in X: \text {dist}_X(x, {\mathcal {B}}(\tau , \omega ))\le \delta \}, \ \forall \tau \in {\mathbb {R}}, \omega \in \Omega . \end{aligned}$$

By (2.8) in Proposition 2.4, we know the backward attractor is increasing, that is, ${\mathcal {B}}(s, \omega )\subset {\mathcal {B}}(\tau , \omega )$ for all $s\le \tau$. Hence $\tau \mapsto N_\delta ({\mathcal {B}}(\tau , \omega ))$ is increasing too.

By the ${\mathfrak {D}}$-pullback attraction of ${\mathcal {B}}$, for each $D\in {\mathfrak {D}}$, $\tau \in {\mathbb {R}}$ and $\omega \in \Omega$,

$$\begin{aligned} \text {dist}_X(\Phi (t, \tau -t, \theta _{-t}\omega )D(\tau -t, \theta _{-t}\omega ), {\mathcal {B}}(\tau , \omega ))\rightarrow 0\ \text { as } t\rightarrow \infty . \end{aligned}$$

Hence there is a $T=T(\delta , D,\tau , \omega )>0$ such that

$$\begin{aligned} \Phi (t, \tau -t, \theta _{-t}\omega )D(\tau -t, \theta _{-t}\omega )\subset N_\delta ({\mathcal {B}}(\tau , \omega )), \ \forall t\ge T. \end{aligned}$$

Therefore, $N_\delta ({\mathcal {B}})$ is ${\mathfrak {D}}$-pullback absorbing and so $\Phi$ is increasingly dissipative. $\square$

3 A Stochastic Modified SH Equation with Delay

The previous abstract results will be illustrated in the context of a non-autonomous stochastic modified Swift–Hohenberg equations with variable delays:

$$\begin{aligned} \left\{ \begin{array}{l} du+(\Delta ^{2}u+2\Delta u+au+b|\nabla u|^{2}+u^{3})dt=f(t, u_{t})dt+g(t, x)dt+h(x)dW, \\ u(t)=\Delta u(t)=0, \ \text { on } \partial {\mathcal {O}}, \ t>\tau , \\ u(\tau +\sigma , x):=\phi (\sigma , x), \ \sigma \in [-\varrho , 0], x\in {\mathcal {O}}, \end{array}\right. \end{aligned}$$

(3.1)

where $a,b, \tau \in {\mathbb {R}}$, $\varrho >0$, $h\in H_{0}^{2}({\mathcal {O}}) \cap H^{4}({\mathcal {O}})$ and ${\mathcal {O}}$ is a bounded smooth domain in ${\mathbb {R}}^2$.

We identify the Wiener process $W(\cdot , \omega )$ with $\omega (\cdot )$ on the measurable dynamical system $(\Omega , {\mathfrak {F}}, P, \theta )$, where $\Omega =\{\omega \in C({\mathbb {R}}, {\mathbb {R}}): \omega (0)=0\}$ with the compact-open topology, ${\mathfrak {F}}$ is the Borel $\sigma$-algebra, P is the Wiener measure and $\theta _{t}\omega (\cdot )=\omega (\cdot +t)-\omega (t)$ for $(\omega , t)\in \Omega \times {\mathbb {R}}$. In this realization, the unique stationary solution of $dz+zdt=dW(t)$ can be denoted by $z(t,\omega )=z(\theta _{t}\omega )$. In addition, there exists a $\theta$-invariant full-measure set ${\tilde{\Omega }}$ (we do not distinguish ${\tilde{\Omega }}$ and $\Omega$) such that for each $\omega \in {\tilde{\Omega }}$ the mapping $t\rightarrow z(\theta _{t}\omega )$ is continuous and

$$\begin{aligned} \lim _{t\rightarrow \pm \infty }\frac{z(\theta _{t}\omega )}{t}=0, \ \ \lim _{t\rightarrow \pm \infty }\frac{1}{t}\int _{0}^{t}z(\theta _{s}\omega )ds=0. \end{aligned}$$

(3.2)

Consider the change of variables:

$$\begin{aligned} v(t, \tau , \omega , \psi )=u(t, \tau , \omega , \phi )-h z(\theta _{t}\omega ), \ \ t\ge \tau . \end{aligned}$$

(3.3)

Then we can transform the stochastic equation (3.1) into a random equation:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial v}{\partial t}+\Delta ^{2}v+2\Delta v+av+b|\nabla v+z\nabla h|^{2}+(v+zh)^{3} \\ =f(t, v_{t}+hz(\theta _{t+\cdot }\omega ))+g(t, x)+(h-\Delta ^{2}h-2\Delta h-ah)z, \\ v(t)=\Delta v(t)=0, \ \text { on } \partial {\mathcal {O}}, \ t>\tau , \\ v(\tau +\sigma , x)=\phi (\sigma , x)-hz(\theta _{\tau +\sigma }\omega ):=\psi (\sigma , x), \ \sigma \in [-\varrho , 0], \ x\in {\mathcal {O}}. \end{array}\right. \end{aligned}$$

(3.4)

Denote the norm in $L^{p}({\mathcal {O}})$ by $\Vert \cdot \Vert _{p}$ (the subscript will be omitted if $p=2$). Consider both state spaces:

$$\begin{aligned}&C({L^2}):=C([-\varrho , 0], L^{2}({\mathcal {O}})) \text { with } \Vert \eta \Vert _{C({L^2})}=\sup _{\sigma \in [-\varrho , 0]}\Vert \eta (\sigma )\Vert \\&C({H_0^2}):=C([-\varrho , 0], H_0^{2}({\mathcal {O}})) \text { with } \Vert \eta \Vert _{C(H_0^2)}=\sup _{\sigma \in [-\varrho , 0]}\Vert \eta (\sigma )\Vert _{H_0^2}. \end{aligned}$$

We assume the delay force f: ${\mathbb {R}}\times C({L^2})\rightarrow L^{2}({\mathcal {O}})$ satisfying

(F1):

$f(\cdot , \eta )$ is Borel-measurable for each $\eta \in C({L^2})$, and $f(t, 0)=0$ for all $t\in {\mathbb {R}}$;

(F2):

There are $m_{f}>0$ and $c_{f}>0$ such that for $m\in [0, m_{f}]$ and $u, v\in C([\tau -\varrho , t], L^{2}({\mathcal {O}}))$,

$$\begin{aligned} \int _{\tau }^{t}e^{mr}\Vert f(r, u_{r})-f(r, v_{r})\Vert ^{2}dr\le c_{f}^{2}\int _{\tau -\varrho }^{t}e^{mr}\Vert u(r)-v(r)\Vert ^{2}dr, \ \forall t\ge \tau ; \end{aligned}$$

(3.5)

(F3):

There is a continuous function $L_{f}(\cdot )$ such that for all $\xi , \eta \in C({L^2})$ and $t\in {\mathbb {R}}$,

$$\begin{aligned} \Vert f(t, \xi )-f(t, \eta )\Vert \le L_{f}(t)\Vert \xi -\eta \Vert _{C({L^2})}, \end{aligned}$$

(3.6)

where the Lipschtz bound $L_{f}(\cdot )$ is backward limitable:

$$\begin{aligned} \lim _{\gamma \rightarrow +\infty }\sup _{s\le \tau }\int _{-\infty }^{0}e^{\gamma r}L_{f}^{2}(r+s)dr=0, \ \forall \tau \in {\mathbb {R}}. \end{aligned}$$

(3.7)

We also assume the external force $g: {\mathbb {R}}\times {\mathcal {O}}\rightarrow {\mathbb {R}}$ satisfying

(G) $g\in L^{2}_{\text {loc}}({\mathbb {R}}, L^{2}({\mathcal {O}}))$ is backward limitable:

$$\begin{aligned} \lim _{\gamma \rightarrow +\infty }\sup _{s\le \tau }\int _{-\infty }^{0}e^{\gamma r}\Vert g(r+s)\Vert ^{2}dr=0, \ \forall \tau \in {\mathbb {R}}. \end{aligned}$$

(3.8)

Lemma 3.1

If $g\in L^{2}_{\text {loc}}({\mathbb {R}}, L^{2}({\mathcal {O}}))$ is backward limitable, then it is backward tempered:

$$\begin{aligned} \sup _{s\le \tau }\int _{-\infty }^{0}e^{\gamma r}\Vert g(r+s)\Vert ^{2}dr<+\infty , \ \forall \gamma >0, \ \tau \in {\mathbb {R}}. \end{aligned}$$

(3.9)

The same argument is true for $L_{f}(\cdot )$.

Proof

Suppose (3.8) holds, then, for each $\tau \in {\mathbb {R}}$, there is a $\gamma _0>0$ such that

$$\begin{aligned} G(\gamma _0):=\sup _{s\le \tau }\int _{-\infty }^{s}e^{\gamma _0 (r-s)}\Vert g(r)\Vert ^{2}dr=\sup _{s\le \tau }\int _{-\infty }^{0}e^{\gamma _0 r}\Vert g(r+s)\Vert ^{2}dr<+\infty . \end{aligned}$$

If $\gamma \ge \gamma _0$, then $G(\gamma )\le G(\gamma _0)<+\infty$. If $\gamma \in (0,\gamma _0)$, then

$$\begin{aligned} G(\gamma )&=\sup _{s\le \tau }\sum _{n=0}^\infty \int ^{s-n}_{s-(n+1)}e^{\gamma (r-s)}\Vert g(r)\Vert ^2dr\\&\le \sum _{n=0}^\infty e^{-n \gamma } \sup _{s\le \tau } \int ^{s-n}_{(s-n)-1}\Vert g(r)\Vert ^{2}dr \\&\le \frac{1}{1-e^{-\gamma }} \sup _{\rho \le \tau } \int ^{\rho }_{\rho -1}\Vert g(r)\Vert ^{2}dr\\&\le \frac{e^{\gamma _0}}{1-e^{-\gamma }} \sup _{\rho \le \tau } \int ^{\rho }_{\rho -1}e^{\gamma _0 (r-\rho )}\Vert g(r)\Vert ^{2}dr\\&\le \frac{e^{\gamma _0}G(\gamma _0)}{1-e^{-\gamma }}<\infty . \end{aligned}$$

So, g is backward tempered with arbitrary rate $\gamma >0$. $\square$

The authors in [31] ever assumed that g is backward limitable and backward tempered simultaneously, but the former actually implies the latter as in Lemma 3.1. We then provide two sufficient conditions for (G).

Lemma 3.2

$g\in L^{2}_{\text {loc}}({\mathbb {R}}, L^{2}({\mathcal {O}}))$ is backward limitable if g is strongly backward tempered:

$$\begin{aligned} \int _{-\infty }^{0}e^{\gamma _0 r}\sup _{s\le \tau }\Vert g(r+s)\Vert ^{2}dr<+\infty , \ \forall \tau \in {\mathbb {R}}, \text { for some } \gamma _0>0, \end{aligned}$$

(3.10)

or, if g is backward tempered and backward absolutely continuous:

$$\begin{aligned} \lim _{\epsilon \rightarrow 0^+}\sup _{s\le \tau }\int _{s-\epsilon }^{s}\Vert g(r)\Vert ^{2}dr=0, \ \forall \tau \in {\mathbb {R}}. \end{aligned}$$

Proof

Suppose (3.10) holds. Then the Lebesgue dominated convergence theorem yields

$$\begin{aligned} \lim _{\gamma \rightarrow +\infty }\sup _{s\le \tau }\int _{-\infty }^{0}e^{\gamma r}\Vert g(r+s)\Vert ^{2}dr&\le \lim _{\gamma \rightarrow +\infty }\int _{-\infty }^{0}e^{\gamma r}\sup _{s\le \tau }\Vert g(r+s)\Vert ^{2}dr\\&=\int _{-\infty }^{0} \lim _{\gamma \rightarrow +\infty }e^{\gamma r}\sup _{s\le \tau }\Vert g(r+s)\Vert ^{2}dr= 0, \end{aligned}$$

which means g is backward limitable. Another criterion can be similarly proved as in [31, Proposition 3.2]. $\square$

Since Eq. (3.4) can be regarded as a deterministic equation parameterized by $\omega \in \Omega$, by a standard method as in [21, 22], one can prove the well-posedness of solutions.

Lemma 3.3

Assume (F1)–(F3) and $g\in L^{2}_{\text {loc}}({\mathbb {R}}, L^{2}({\mathcal {O}}))$. Then for all $\tau \in {\mathbb {R}}$ and $\psi \in C(H_0^2)$, Eq. (3.4) has a unique weak solution

$$\begin{aligned} v\in C([\tau -\varrho , +\infty ); H^{2}_{0}({\mathcal {O}})), \end{aligned}$$

such that $v(\tau +\sigma , \tau , \omega , \psi )=\psi (\sigma )$ for all $\sigma \in [-\varrho , 0]$, that is, $v_\tau (\cdot , \tau , \omega , \psi )=\psi$ in $C(H_0^2)$.

It is easy to show the solution mapping $\omega \rightarrow v(t, \tau , \omega , \psi )$ is $({\mathfrak {F}}, {\mathfrak {B}}(C(H_0^2)))$ measurable. We define a mapping $\Phi : {\mathbb {R}}^{+}\times {\mathbb {R}}\times \Omega \times C(H_0^2)\rightarrow C(H_0^2)$ given by

$$\begin{aligned} \Phi (t, \tau , \omega )\phi =u_{t+\tau }(\cdot , \tau , \theta _{-\tau }\omega , \phi ), \ t\ge 0, \ \tau \in {\mathbb {R}}, \ \omega \in \Omega . \end{aligned}$$

(3.11)

Note that $v(t, \tau , \omega , \psi )=u(t, \tau , \omega , \phi )-h z(\theta _{t}\omega )$. Then, by Lemma 3.3 we have $\Phi$ is a continuous cocycle.

Denoted by ${\mathfrak {D}}$ the universe of all backward tempered sets in $C(H_0^2)$, where $D(\cdot , \cdot )$ is called backward tempered if

$$\begin{aligned} \lim _{t\rightarrow +\infty }e^{-\gamma t}\sup _{s\le \tau }\Vert D(s-t, \theta _{-t}\omega )\Vert ^{2}_{C(H_0^2)}=0, \ \forall \gamma >0,\ \tau \in {\mathbb {R}}, \omega \in \Omega . \end{aligned}$$

(3.12)

Denote by ${\mathfrak {U}}$ the usual universe of all tempered sets in $C(H_0^2)$, where $U\in {\mathfrak {U}}$ if and only if

$$\begin{aligned} \lim _{t\rightarrow +\infty }e^{-\gamma t}\Vert U(s-t, \theta _{-t}\omega )\Vert ^{2}_{C(H_0^2)}=0, \ \forall \gamma >0,\ \tau \in {\mathbb {R}}, \omega \in \Omega . \end{aligned}$$

(3.13)

It is easy to show that ${\mathfrak {D}}$ is backward closed and truncation-closed. ${\mathfrak {U}}$ may not be backward closed, although it is inclusion-closed and truncation-closed.

In the last two sections, we will prove the existence of a ${\mathfrak {D}}$-backward attractor ${\mathcal {B}}$, a ${\mathfrak {D}}$-pullback attractor ${\mathcal {A}}$ and a ${\mathfrak {U}}$-pullback attractor ${\mathcal {A}}_{{\mathfrak {U}}}$. For the measurability of the attractors, we will prove ${\mathcal {A}}={\mathcal {A}}_{{\mathfrak {U}}}$, then the measurability of ${\mathcal {A}}_{{\mathfrak {U}}}$ implies that ${\mathcal {B}}$ is dividedly random.

4 Backward Uniform Estimates

We will frequently use the Gagliardo-Nirenberg inequality:

$$\begin{aligned} \Vert D^{j}u\Vert _{p}\le c\Vert u\Vert _{r}^{1-\gamma }\Vert D^{l}u\Vert _{q}^{\gamma }, \ \forall u\in L^r({\mathcal {O}})\cap W^{l,q}({\mathcal {O}}), \end{aligned}$$

(4.1)

where $p, q\in [1,\infty ]$, $r\in [1,\infty )$, $\gamma \in (0,1]$ and

$$\begin{aligned} j-\frac{2}{p}\le \gamma \left( l-\frac{2}{q}\right) +(1-\gamma )\frac{2}{r}. \end{aligned}$$

Lemma 4.1

Assume (F1)–(F3) and (G). Then the solutions of Eq. (3.4) have the following properties:

• (i) For each $U\in {\mathfrak {U}}$, $\tau \in {\mathbb {R}}$, $\omega \in \Omega$ and $\phi \in U(\tau -t, \theta _{-t}\omega )$, there exists a ${\widetilde{T}}={\widetilde{T}}(U,\tau , \omega )>2\varrho +4$ such that for all $t\ge {\widetilde{T}}$ ,

  $$\begin{aligned} \sup _{\sigma \in [-2\varrho -4, 0]}\Vert u(\tau +\sigma , \tau -t, \theta _{-\tau }\omega , \phi )\Vert ^{2} \le c\sup _{\sigma \in [-2\varrho -4, 0]}|z(\theta _{\sigma }\omega )|^{2}+c{\widetilde{\chi }}(\tau , \omega ), \end{aligned}$$

  (4.2)

  where

  $$\begin{aligned}&{\widetilde{\chi }}(\tau , \omega )=1+\int _{-\infty }^{0}e^{m_f r}|z(\theta _{r}\omega )|^{6}dr+\int _{-\infty }^{0}e^{m_f r}\Vert g(r+\tau )\Vert ^{2}dr<+\infty , \end{aligned}$$

  (4.3)

  note that $m_f>0$ is given as in (F3);

• (ii) For each $D\in {\mathfrak {D}}$, $\tau \in {\mathbb {R}}$, $\omega \in \Omega$ and $\phi \in D(s-t, \theta _{-t}\omega )$ with $s\le \tau$, there exists a $T=T(\tau , \omega , D)>2\varrho +4$ such that for all $t\ge T$,

  $$\begin{aligned} \sup _{s\le \tau }\sup _{\sigma \in [-2\varrho -4, 0]}\Vert u(s+\sigma , s-t, \theta _{-s}\omega , \phi )\Vert ^{2} \le c\sup _{\sigma \in [-2\varrho -4, 0]}|z(\theta _{\sigma }\omega )|^{2}+c\chi (\tau , \omega ), \end{aligned}$$

  (4.4)

  where

  $$\begin{aligned} \chi (\tau , \omega )=\sup _{s\le \tau }{\widetilde{\chi }}(s, \omega )<+\infty . \end{aligned}$$

  (4.5)

Proof

Taking the inner product of (3.4) with $v(r, s-t, \theta _{-s}\omega , \psi )$ in $L^{2}({\mathcal {O}})$, we obtain

$$\begin{aligned}&\frac{d}{dr}\Vert v(r)\Vert ^{2}+2\Vert \Delta v\Vert ^{2}=4\Vert \nabla v\Vert ^{2}-2a \Vert v\Vert ^{2}-2((v+hz(\theta _{r-s}\omega ))^{3}, v)\nonumber \\&\quad -2b(|\nabla v+z(\theta _{r-s}\omega )\nabla h|^{2},v) \nonumber \\&\quad +2(f(r,v_{r}+hz(\theta _{r-s+\cdot }\omega )),v)+2(g(r),v)\nonumber \\&\quad +2((h-\Delta ^{2}h-2\Delta h-ah)z(\theta _{r-s}\omega ),v). \end{aligned}$$

(4.6)

By (3.3) and $h\in H_{0}^{2}({\mathcal {O}})\hookrightarrow L^p({\mathcal {O}})$ for any $p\ge 2$, we have

$$\begin{aligned}&-2((v+hz(\theta _{r-s}\omega ))^{3}, v) \\&\quad =\ -2\Vert v\Vert _{4}^{4}-6(v^{2}hz(\theta _{r-s}\omega ), v) -6(vh^{2}z^{2}(\theta _{r-s}\omega ), v)-2(h^{3}z^{3}(\theta _{r-s}\omega ), v) \\&\quad \le \ -\frac{3}{2}\Vert v\Vert _{4}^{4}+\Vert v\Vert ^{2}+c(|z(\theta _{r-s}\omega )|^{4}+|z(\theta _{r-s}\omega )|^{6}). \end{aligned}$$

By $h\in H_{0}^{2}({\mathcal {O}})\cap H^4({\mathcal {O}})$ and the Young inequality,

$$\begin{aligned}&2((h-\Delta ^{2}h-2\Delta h-ah)z(\theta _{r-s}\omega ),v)\le \Vert v(r)\Vert ^{2}+c|z(\theta _{r-s}\omega )|^{2}. \end{aligned}$$

By the Young inequality again, we obtain

$$\begin{aligned} 2(f(r,v_{r}+hz(\theta _{r-s+\cdot }\omega )),v)\le c_f \Vert v(r)\Vert ^{2}+ \frac{1}{c_f} \Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}, \end{aligned}$$

and

$$\begin{aligned} 2(g(r),v)\le \Vert v(r)\Vert ^{2}+\Vert g(r)\Vert ^{2}. \end{aligned}$$

Using the interpolation inequality that $|y|^p\le \epsilon |y|^q+c(\epsilon )$ for $p<q$, we have

$$\begin{aligned} \Vert v\Vert ^2\le \epsilon \Vert v\Vert ^4_4+c(\epsilon ) \text { and } |z(\theta _{r-s}\omega )|^{2}+|z(\theta _{r-s}\omega )|^{4}\le |z(\theta _{r-s}\omega )|^{6}+c. \end{aligned}$$

In particular, we can rewrite all terms containing $\Vert v\Vert ^2$ as follows:

$$\begin{aligned} (-2a+3+c_f)\Vert v\Vert ^2&=-(m_f+2c_f)\Vert v\Vert ^2+(m_f-2a+3+3c_f)\Vert v\Vert ^2\\&\le -(m_f+2c_f)\Vert v\Vert ^2+\frac{1}{2}\Vert v\Vert ^4_4+c, \end{aligned}$$

where $a\in {\mathbb {R}}$. We substitute all estimates into (4.6) to find

$$\begin{aligned} \frac{d}{dr}\Vert v\Vert ^{2}+m_f\Vert v\Vert ^2+2\Vert \Delta v\Vert ^{2}&\le -2c_f\Vert v\Vert ^2-\Vert v\Vert ^4_4+ 4\Vert \nabla v\Vert ^{2}\nonumber \\&\quad -2b(|\nabla v+z(\theta _{r-s}\omega )\nabla h|^{2},v) \nonumber \\&\quad +\Vert g(r)\Vert ^2+c(|z(\theta _{r-s}\omega )|^{6}+1)\nonumber \\&\quad + \frac{1}{c_f} \Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}. \end{aligned}$$

(4.7)

Using the Gagliardo-Nirenberg inequality on the two terms containing $\nabla v$. Let $j=1, p=l=q=r=2, \gamma =\frac{1}{2}$ in (4.1), we have

$$\begin{aligned} 4\Vert \nabla v\Vert ^{2}\le c\Vert \Delta v\Vert \Vert v\Vert \le \frac{1}{2}\Vert \Delta v\Vert ^{2}+c\Vert v\Vert ^{2}\le \frac{1}{2}\Vert \Delta v\Vert ^{2}+\frac{1}{2}\Vert v\Vert _{4}^{4}+c. \end{aligned}$$

Let $j=1, p=4, l=2, q=2, r=4, \gamma =\frac{1}{8}$ in (4.1), we have

$$\begin{aligned} \Vert \nabla v\Vert _{4}^{2}\le c\Vert \Delta v\Vert ^{\frac{1}{4}}\Vert v\Vert _{4}^{\frac{7}{4}} \text { and } \Vert \nabla h\Vert _{4}^{2}\le c\Vert \Delta h\Vert ^{\frac{1}{4}}\Vert h\Vert _{4}^{\frac{7}{4}}<+\infty , \end{aligned}$$

which, together with the continuous embedding $L^4({\mathcal {O}})\hookrightarrow L^2({\mathcal {O}})$, implies that

$$\begin{aligned} -2b(|\nabla v+z(\theta _{r-s}\omega )\nabla h|^2,v)&\le 2|b|\Vert \nabla v+z(\theta _{r-s}\omega )\nabla h\Vert ^2\Vert v\Vert \\&\le c\Vert \nabla v+z(\theta _{r-s}\omega )\nabla h\Vert ^2_{4}\Vert v\Vert \\&\le c\Vert \nabla v\Vert _{4}^{2}\Vert v\Vert +c\Vert \nabla h\Vert _{4}^{2}|z(\theta _{r-s}\omega )|^4\Vert v\Vert \\&\le c\Vert \Delta v\Vert ^{\frac{1}{4}}\Vert v\Vert _{4}^{\frac{11}{4}}+\Vert v\Vert ^2+c|z(\theta _{r-s}\omega )|^4\\&\le \frac{1}{2}\Vert \Delta v\Vert ^{2}+c\Vert v\Vert _{4}^{\frac{22}{7}}+\Vert v\Vert ^2+c|z(\theta _{r-s}\omega )|^4\\&\le \frac{1}{2}\Vert \Delta v\Vert ^{2}+\frac{1}{2}\Vert v\Vert _{4}^4+c(|z(\theta _{r-s}\omega )|^6+1). \end{aligned}$$

We substitute them into (4.7) to obtain

$$\begin{aligned} \frac{d}{dr}\Vert v\Vert ^{2}+m_f\Vert v\Vert ^2+\Vert \Delta v\Vert ^{2}&\le -2c_f\Vert v\Vert ^2+\frac{1}{c_{f}}\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2} \nonumber \\&\quad +\Vert g(r)\Vert ^2+c(|z(\theta _{r-s}\omega )|^6+1). \end{aligned}$$

(4.8)

Multiplying (4.8) by $e^{m_f r}$, and then integrating the result on $[s-t, s+\sigma ]$ with $\sigma \in [-2\varrho -4, 0]$ and $t\ge 2\varrho +4$ yield

$$\begin{aligned}&e^{m_f(s+\sigma )}\Vert v(s+\sigma , s-t, \theta _{-s}\omega , \psi )\Vert ^{2}+\int _{s-t}^{s+\sigma }e^{m_f r}\Vert \Delta v(r, s-t, \theta _{-s}\omega , \psi )\Vert ^{2}dr \nonumber \\&\quad \le e^{m_f(s-t)}\Vert \psi \Vert _{C(L^2)}^{2}-2c_f\int _{s-t}^{s+\sigma }e^{m_fr}\Vert v(r)\Vert ^{2}dr\nonumber \\&\qquad +\frac{1}{c_{f}}\int _{s-t}^{s+\sigma }e^{m_fr}\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}dr \nonumber \\&\qquad +\int _{s-t}^{s+\sigma }e^{m_fr}\Vert g(r)\Vert ^{2}dr +c\int _{s-t}^{s+\sigma }e^{m_fr}(|z(\theta _{r-s}\omega )|^{6}+1)dr. \end{aligned}$$

(4.9)

We now deal with the delay term. By (3.5), we have

$$\begin{aligned}&\frac{1}{c_{f}}\int _{s-t}^{s+\sigma }e^{m_fr}\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}dr\\&\quad \le \frac{1}{c_{f}}c_{f}^{2}\int _{s-t-\varrho }^{s+\sigma } e^{m_fr} \Vert v(r)+hz(\theta _{r-s}\omega )\Vert ^{2}dr \\&\quad \le 2c_{f}\int _{s-t-\varrho }^{s+\sigma }e^{m_fr}\Vert v(r)\Vert ^{2}dr +c\int _{s-t-\varrho }^{s+\sigma }e^{m_fr}|z(\theta _{r-s}\omega )|^{2}dr \\&\quad \le 2c_{f}\int _{s-t}^{s+\sigma }e^{m_fr}\Vert v(r)\Vert ^{2}dr + ce^{m_f(s-t)}\Vert \psi \Vert _{C(L^2)}^{2}\\&\qquad +c\int _{s-t-\varrho }^{s+\sigma }e^{m_fr}|z(\theta _{r-s}\omega )|^{2}dr. \end{aligned}$$

Substituting it into (4.9), multiplying by $e^{-m_f(s+\sigma )}$ and using the embedding $C([-\varrho ,0], H_0^2({\mathcal {O}}))\hookrightarrow C([-\varrho ,0], L^2({\mathcal {O}}))$, we obtain for all $s\in {\mathbb {R}}$, $\sigma \in [-2\varrho -4, 0]$ and $t\ge 2\varrho +4$,

$$\begin{aligned}&\Vert v(s+\sigma , s-t, \theta _{-s}\omega , \psi )\Vert ^{2}+\int _{s-t}^{s+\sigma }e^{m_f (r-s-\sigma )}\Vert \Delta v(r, s-t, \theta _{-s}\omega , \psi )\Vert ^{2}dr\nonumber \\&\quad \le ce^{-m_ft}\Vert \psi \Vert _{C(L^2)}^{2} +c\int _{-t-\varrho }^{0}e^{m_fr}(1+|z(\theta _{r}\omega )|^{6}+\Vert g(r+s)\Vert ^{2})dr\nonumber \\&\quad \le ce^{-m_ft}\Vert \psi \Vert _{C(H_0^2)}^{2}+c{\widetilde{\chi }}(s,\omega ), \end{aligned}$$

(4.10)

which, along with $v(s+\sigma , s-t, \theta _{-s}\omega , \psi )=u(s+\sigma , s-t, \theta _{-s}\omega , \phi )-hz(\theta _{\sigma }\omega )$ with $\psi =\phi -hz(\theta _{-t}\omega )$ and $\sigma \in [-2\varrho -4, 0]$, implies

$$\begin{aligned}&\Vert u(s+\sigma , s-t, \theta _{-s}\omega , \phi )\Vert ^{2}+\int _{s-t}^{s+\sigma }e^{m_f (r-s-\sigma )}\Vert \Delta u(r, s-t, \theta _{-s}\omega , \phi )\Vert ^{2}dr \nonumber \\&\quad \le 2\Vert v(s+\sigma , s-t, \theta _{-s}\omega , \psi )\Vert ^{2}+2\int _{s-t}^{s+\sigma }e^{m_f (r-s-\sigma )}\Vert \Delta v(r, s-t, \theta _{-s}\omega , \psi )\Vert ^{2}dr\nonumber \\&\qquad +2|z(\theta _{\sigma }\omega )|^{2}\Vert h\Vert ^2+2\int _{s-t}^{s+\sigma }e^{m_f (r-s-\sigma )}|z(\theta _{r-s}\omega )|^{2}\Vert \Delta h\Vert ^{2}dr\nonumber \\&\quad \le ce^{-m_ft}\Vert \psi \Vert _{C(H_0^2)}^{2}+c{\widetilde{\chi }}(s,\omega ) +2|z(\theta _{\sigma }\omega )|^{2}\Vert h\Vert ^2+2\Vert \Delta h\Vert ^{2}e^{2\varrho +4} \int _{-t}^{0}e^{m_fr}|z(\theta _{r}\omega )|^{2}dr\nonumber \\&\quad \le ce^{-m_ft}\left( \Vert \phi \Vert _{C(H_0^2)}^{2}+|z(\theta _{-t}\omega )|^{2}\right) +c{\widetilde{\chi }}(s,\omega )+c|z(\theta _{\sigma }\omega )|^{2}. \end{aligned}$$

(4.11)

(i) For all $\phi \in U(s-t, \theta _{-t}\omega )$ with $U\in {\mathfrak {U}}$, we see from (3.2) and (3.13) that there is a ${\widetilde{T}}={\widetilde{T}}(s, \omega , U)\ge 2\varrho +4$ such that for all $t\ge {\widetilde{T}}$

$$\begin{aligned} ce^{-m_ft}\left( \Vert \phi \Vert _{C(H_0^2)}^{2}+|z(\theta _{-t}\omega )|^{2}\right) \le ce^{-m_ft}(\Vert U(s-t, \theta _{-t}\omega )\Vert _{C(H_0^2)}^{2}+t^{2})\le {\widetilde{\chi }}(s,\omega ). \end{aligned}$$

By letting $s=\tau$ in (4.11), we obtain (4.2) as desired.

(ii) For all $\phi \in D(s-t, \theta _{-t}\omega )$ with $s\le \tau$ and $D\in {\mathfrak {D}}$, it follows from (3.2) and (3.12) that there is a $T=T(\tau , \omega , D)\ge 2\varrho +4$ such that for all $t\ge T$

$$\begin{aligned} ce^{-m_ft}\left( \sup _{s\le \tau }\Vert \phi \Vert _{C(H_0^2)}^{2}+|z(\theta _{-t}\omega )|^{2}\right)&\le \ ce^{-m_ft}\left( \sup _{s\le \tau }\Vert D(s-t, \theta _{-t}\omega )\Vert _{C(H_0^2)}^{2}+|z(\theta _{-t}\omega )|^{2}\right) \nonumber \\&\le \chi (\tau , \omega ), \end{aligned}$$

(4.12)

where $\chi (\cdot , \cdot )$ is given by (4.5). By taking the supremum over $s\in (-\infty , \tau ]$ in (4.11), we obtain (4.4) holds. In addition, by Lemma 3.1, (G) implies that g is backward tempered:

$$\begin{aligned} \sup _{s\le \tau }\int _{-\infty }^{0}e^{m_f r}\Vert g(r+s)\Vert ^{2}dr<+\infty , \ \forall \tau \in {\mathbb {R}}, \end{aligned}$$

which together with (3.1) implies $\chi (\tau , \omega )<+\infty$. The proof is complete. $\square$

Lemma 4.2

For each $(D, \tau , \omega )\in {\mathfrak {D}}\times {\mathbb {R}}\times \Omega$, we have for all $t\ge T\,(T=T(D, \tau , \omega )$ is given as in (4.4)) and $\phi \in D(s-t, \theta _{-t}\omega )$ with $s\le \tau$,

$$\begin{aligned}&\sup _{s\le \tau }\sup _{\sigma \in [-\varrho -2, 0]}\Vert \Delta u(s+\sigma , s-t, \theta _{-s}\omega , \phi )\Vert ^{2}\le c\sup _{\sigma \in [-2\varrho -4, 0]}|z(\theta _{\sigma }\omega )|^{2}+c\chi ^{5}(\tau , \omega ), \end{aligned}$$

(4.13)

$$\begin{aligned}&\sup _{s\le \tau }\int _{s-\varrho }^{s}\Vert \Delta ^{2}u(r,s-t, \theta _{-s}\omega , \phi )\Vert ^{2}dr\le c|z(\theta _{\sigma }\omega )|^{2}+c\sup _{\sigma \in [-2\varrho -4, 0]}|z(\theta _{\sigma }\omega )|^{2}+c\chi ^{5}(\tau , \omega ) . \end{aligned}$$

(4.14)

Proof

Multiplying (3.4) by $\Delta ^{2}v(r, s-t, \theta _{-s}\omega , \psi )$ yields

$$\begin{aligned} \frac{d}{dr}\Vert \Delta v(r)\Vert ^{2}+\Vert \Delta ^{2}v\Vert ^{2}\le \,&c(1+\Vert \Delta v\Vert ^{2}+\Vert \nabla v\Vert ^{4}_{4}+\Vert v\Vert ^{6}_{6}+|z(\theta _{r-s}\omega )|^{6}) \\&\ +c(\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}+\Vert g(r)\Vert ^{2}). \end{aligned}$$

By (4.1), let $j=1, p=4, l=q=r=2, \gamma =\frac{1}{4}$,

$$\begin{aligned} \Vert \nabla v\Vert ^{4}_{4}\le c\Vert v\Vert ^{3}\Vert \Delta v\Vert \le c\Vert \Delta v\Vert ^{2}+c\Vert v\Vert ^{6}. \end{aligned}$$

Again, let $j=0, p=6, l=q=r=2, \gamma =\frac{1}{6}$,

$$\begin{aligned} \Vert v\Vert ^{6}_{6}\le c\Vert v\Vert ^{5}\Vert \Delta v\Vert \le c\Vert \Delta v\Vert ^{2}+c\Vert v\Vert ^{10}. \end{aligned}$$

Hence, by the interpolation inequality, we have

$$\begin{aligned} \frac{d}{dr}\Vert \Delta v(r)\Vert ^{2}+\Vert \Delta ^{2}v\Vert ^{2}\le \,&c(1+\Vert \Delta v\Vert ^{2}+\Vert v\Vert ^{10}+|z(\theta _{r-s}\omega )|^{6} \nonumber \\&\ +c(\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}+\Vert g(r)\Vert ^{2}). \end{aligned}$$

(4.15)

Integrating (4.15) on $[\zeta , s+\sigma ]$ with $\zeta \in [s+\sigma -1, s+\sigma ]$ and $\sigma \in [-\varrho -2, 0]$, we have

$$\begin{aligned} \Vert \Delta v(s+\sigma )\Vert ^{2}&\le \Vert \Delta v(\zeta )\Vert ^{2}+c\int _{\zeta }^{s}(1+\Vert \Delta v\Vert ^{2}+\Vert v\Vert ^{10}+|z(\theta _{r-s}\omega )|^{6})dr \\&\quad +c\int _{\zeta }^{s}(\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}+\Vert g(r)\Vert ^{2})dr. \end{aligned}$$

Integrating it again w.r.t. $\zeta \in [s+\sigma -1, s+\sigma ]$, we obtain

$$\begin{aligned} \Vert \Delta v(s+\sigma )\Vert ^{2}\le \,&c\int _{s-\varrho -3}^{s}(1+\Vert \Delta v\Vert ^{2}+\Vert v\Vert ^{10}+|z(\theta _{r-s}\omega )|^{6} \nonumber \\&\ +c\int _{s-\varrho -3}^{s}(\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}+\Vert g(r)\Vert ^{2})dr. \end{aligned}$$

(4.16)

It follows from the proof of Lemma 4.1 that

$$\begin{aligned}&\sup _{s\le \tau }\sup _{\sigma \in [-2\varrho -4, 0]}\Vert v(s+\sigma , s-t, \theta _{-s}\omega , \psi )\Vert ^{2}\nonumber \\&\quad +\sup _{s\le \tau }\int _{s-\varrho -3}^{s}\Vert \Delta v(r, s-t, \theta _{-s}\omega , \psi )\Vert ^{2}dr\le c\chi (\tau , \omega ). \end{aligned}$$

(4.17)

Since $\chi (\tau , \omega )\ge 1$, it follows

$$\begin{aligned} \sup _{s\le \tau }\int _{s-\varrho -3}^{s}\Vert v(r)\Vert ^{10}dr\le (\varrho +3)\sup _{s\le \tau }\sup _{\sigma \in [-2\varrho -4, 0]}\Vert v(s+\sigma )\Vert ^{10} \le c\chi ^{5}(\tau , \omega ). \end{aligned}$$

(4.18)

And

$$\begin{aligned} \sup _{s\le \tau }\int _{s-\varrho -3}^{s}\Vert \Delta v(r, s-t, \theta _{-s}\omega , \psi )\Vert ^{2}dr \le c\chi ^5(\tau , \omega ). \end{aligned}$$

By (F2) and (4.17), we obtain

$$\begin{aligned}&\sup _{s\le \tau }\int _{s-\varrho -3}^{s}\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}dr \nonumber \\&\quad \le \ e^{m_f(\varrho +3)}\sup _{s\le \tau }\int _{s-\varrho -3}^{s}e^{m_f(r-s)}\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}dr \nonumber \\&\quad \le \ c \sup _{s\le \tau }\int _{s-\varrho -3}^{s}e^{m_f(r-s)}\Vert v(r)\Vert ^{2}dr+\sup _{s\le \tau }\int _{s-\varrho -3}^{s}e^{m_f(r-s)}|z(\theta _{r-s}\omega )|^{2}dr \nonumber \\&\quad \le \ c\sup _{s\le \tau } \sup _{\sigma \in [-\varrho -3,0]}\Vert v(s+\sigma )\Vert ^{2}+ \int _{-\varrho -3}^{0}e^{m_fr}|z(\theta _{r}\omega )|^{2}dr \le c\chi ^{5}(\tau , \omega ). \end{aligned}$$

(4.19)

It is easy to see

$$\begin{aligned} \sup _{s\le \tau }\int _{s-\varrho -3}^{s}(|z(\theta _{r-s}\omega )|^{6}+\Vert g(r)\Vert ^{2})dr\le c \chi ^5(\tau , \omega ). \end{aligned}$$

(4.20)

Substituting (4.18)–(4.20) into (4.16), by $v(s+\sigma , s-t, \theta _{-s}\omega , \psi )={u(s+\sigma , s-t, \theta _{-s}\omega , \phi )}-hz(\theta _{\sigma }\omega )$ we obtain (4.13) as desired. On the other hand, integrating (4.15) on $[s-\varrho , s]$ yields

$$\begin{aligned} \int _{s-\varrho }^{s}\Vert \Delta ^{2}v(r)\Vert ^{2}dr&\le \Vert \Delta v(s-\varrho )\Vert ^{2}+c\int _{s-\varrho }^{s}(1+\Vert \Delta v\Vert ^{2}+\Vert v\Vert ^{10}+|z(\theta _{r-s}\omega )|^{6})dr \nonumber \\&\quad +c\int _{s-\varrho }^{s}(\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}+\Vert g(r)\Vert ^{2})dr. \end{aligned}$$

(4.21)

Taking the supremum w.r.t. $s\le \tau$ and using (4.17) and (4.13), by $v(s+\sigma , s-t, \theta _{-s}\omega , \psi )=u(s+\sigma , s-t, \theta _{-s}\omega , \psi )-hz(\theta _{\sigma }\omega )$ again, we obtain (4.14) as desired. $\square$

Next, let $P_{k}: L^2({\mathcal {O}})\mapsto H_{k}=\text {span}\{e_{1}, e_{2}, \ldots , e_{k}\}$ be the orthogonal projection, where $\{e_{k}\}_{k\in {\mathbb {N}}}\subset H_{0}^{2}({\mathcal {O}})$ is the family of eigenfunctions for $-\Delta$ with the corresponding eigenvalues: $0<\lambda _{1}\le \lambda _{2}\le \cdots \le \lambda _{k}\rightarrow +\infty$ as $k\rightarrow +\infty$, and $\{e_{k}\}_{k\in {\mathbb {N}}}$ forms an orthonormal basis of $L^{2}({\mathcal {O}})$. Then each $v\in H_{0}^{2}({\mathcal {O}})$ has the orthogonal decomposition:

$$\begin{aligned} v=P_{k}v\oplus (I-P_{k})v=:v_{k, 1}+v_{k, 2}, \ \forall k\in {\mathbb {N}} \end{aligned}$$

with the following inequalities:

$$\begin{aligned} \Vert \Delta v_{k, 1}\Vert \le \lambda _{k}\Vert v_{k, 1}\Vert \text { and } \Vert \Delta v_{k, 2}\Vert \ge \lambda _{k+1}\Vert v_{k, 2}\Vert , \ \forall k\in {\mathbb {N}}. \end{aligned}$$

(4.22)

Lemma 4.3

Let $k\in {\mathbb {N}}$, $\varepsilon >0$, $(D, \tau , \omega )\in {\mathfrak {D}}\times {\mathbb {R}}\times \Omega$. There are $\delta =\delta (k,\varepsilon , \tau , \omega )>0$ and $T=T(D, \tau , \omega )>0$ such that for all $\sigma _{1}, \sigma _{2}\in [-\varrho , 0]$ with $|\sigma _{1}-\sigma _{2}|<\delta$ and $t\ge T\,(T$ is given as in (4.4)),

$$\begin{aligned} \sup _{s\le \tau }\Vert P_{k}v(s+\sigma _{1}, s-t, \theta _{-s}\omega , \psi )-P_{k}v(s+\sigma _{2}, s-t, \theta _{-s}\omega , \psi )\Vert _{H_0^2}<\varepsilon . \end{aligned}$$

(4.23)

Proof

We deduce from (4.18)–(4.21) that $\sup _{s\le \tau }\int _{s-\varrho }^{s}\Vert \Delta ^{2}v(r)\Vert ^{2}dr\le c\chi ^5 (\tau , \omega )$. Then, by the same method as in (4.15) we have

$$\begin{aligned}&\sup _{s\le \tau }\int _{s-\varrho }^{s}\Vert \frac{\partial }{\partial r}v(r, s-t, \theta _{-s}\omega , \psi )\Vert ^{2}dr\\&\quad \le c\sup _{s\le \tau }\int _{s-\varrho }^{s}(\Vert \Delta ^{2}v(r)\Vert ^{2}+\Vert \Delta v(r)\Vert ^{2}+\Vert v(r)\Vert ^{10}+|z(\theta _{r-s}\omega )|^{2}+\Vert g(r)\Vert ^{2})dr \\&\qquad + c\sup _{s\le \tau }\int _{s-\varrho }^{s}\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}dr \le c\chi ^5 (\tau , \omega ). \end{aligned}$$

Given $\sigma _{1}, \sigma _{2}\in [-\varrho , 0]$ with $\sigma _{1}\le \sigma _{2}$. By the first inequality in (4.22), we have for all $t\ge T$,

$$\begin{aligned}&\sup _{s\le \tau }\Vert \Delta v_{k, 1}(s+\sigma _{1}, s-t, \theta _{-s}\omega , \psi )-\Delta v_{k, 1}(s+\sigma _{2}, s-t, \theta _{-s}\omega , \psi )\Vert \\&\quad \le \ \lambda _{k}\sup _{s\le \tau }\Vert v_{k, 1}(s+\sigma _{1})-v_{k, 1}(s+\sigma _{2})\Vert \le \lambda _{k}\sup _{s\le \tau }\int _{s+\sigma _{1}}^{s+\sigma _{2}}\left\| \frac{\partial }{\partial r}v_{k, 1}(r)\right\| dr \\&\quad \le \ \lambda _{k}\sup _{s\le \tau }\left( \int _{s-\varrho }^{s}\left\| \frac{\partial }{\partial r}v(r)\right\| ^{2}dr\right) ^{\frac{1}{2}}|\sigma _{1}-\sigma _{2}|^{\frac{1}{2}} \le \ c\lambda _{k}\chi ^{\frac{5}{2}} (\tau , \omega )|\sigma _{1}-\sigma _{2}|^{\frac{1}{2}}. \end{aligned}$$

Then, by the continuity of $z(\theta _{\cdot }\omega )$ there exists a $\delta >0$ such that for all $\sigma _{1}, \sigma _{2}\in [-\varrho , 0]$ with $|\sigma _{1}-\sigma _{2}|<\delta$,

$$\begin{aligned} \sup _{s\le \tau }\Vert \Delta v_{k, 1}(s+\sigma _{1}, s-t, \theta _{-s}\omega , \psi )-\Delta v_{k, 1}(s+\sigma _{2}, s-t, \theta _{-s}\omega , \psi )\Vert <\varepsilon , \end{aligned}$$

which implies (4.23) holds. The proof is complete. $\square$

Lemma 4.4

Let $\varepsilon >0$ and $(D, \tau , \omega )\in {\mathfrak {D}}\times {\mathbb {R}}\times \Omega$. Then there exists a $k_0=k_0(\varepsilon ,\tau , \omega )\in {\mathbb {N}}$ such that for all $k\ge k_0$ and $t\ge T\,(T=T(D, \tau , \omega )$ is given as in (4.4)),

$$\begin{aligned} \sup _{s\le \tau }\sup _{\sigma \in [-\varrho , 0]}\Vert v_{k, 2}(s+\sigma , s-t, \theta _{-s}\omega , \psi )\Vert _{H_0^2}^{2}<\varepsilon . \end{aligned}$$

(4.24)

Proof

Taking the inner product of (3.4) with $\Delta ^{2}v_{k, 2}(r, s-t, \theta _{-s}\omega , \psi )$ in $L^{2}({\mathcal {O}})$, we have

$$\begin{aligned} \frac{d}{dr}\Vert \Delta v_{k, 2}(r)\Vert ^{2}+\Vert \Delta ^{2}v_{k, 2}\Vert ^{2}&\le c(1+\Vert \Delta v\Vert ^{2}+\Vert \nabla v\Vert ^{4}_{4}+\Vert v\Vert ^{6}_{6}+|z(\theta _{r-s}\omega )|^{6}\nonumber \\&\quad +\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}+\Vert g(r)\Vert ^{2}) \nonumber \\&\le c(1+\Vert \Delta v\Vert ^{2}+\Vert \Delta v\Vert ^{4}+\Vert \Delta v\Vert ^{6}+|z(\theta _{r-s}\omega )|^{6}\nonumber \\&\quad +\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}+\Vert g(r)\Vert ^{2}). \end{aligned}$$

(4.25)

By the second inequality in (4.22), we have $\Vert \Delta ^{2}v_{k, 2}\Vert ^{2}\ge \lambda _{k+1}^2\Vert \Delta v_{k, 2}\Vert ^{2}$ and thus (4.25) can be rewritten as

$$\begin{aligned} \frac{d}{dr}(e^{\lambda _{k+1}^2r}\Vert \Delta v_{k, 2}(r)\Vert ^{2})&\le ce^{\lambda _{k+1}^2r}(1+\Vert \Delta v\Vert ^{6}+|z(\theta _{r-s}\omega )|^{6}\nonumber \\&\quad +\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}+\Vert g(r)\Vert ^{2}). \end{aligned}$$

(4.26)

Integrating (4.26) on $[\zeta , s+\sigma ]$ with $\zeta \in [s+\sigma -1, s+\sigma ]$ and $\sigma \in [-\varrho , 0]$, we have

$$\begin{aligned} e^{\lambda _{k+1}^2(s+\sigma )}\Vert \Delta v_{k, 2}(s+\sigma , s-t, \theta _{-s}\omega , \psi )\Vert ^{2}&\le e^{\lambda _{k+1}^2 \zeta }\Vert \Delta v_{k, 2}(\zeta , s-t, \theta _{-s}\omega , \psi )\Vert ^{2} \\&\le c\int _{\zeta }^{s+\sigma }e^{\lambda _{k}r}(1+\Vert \Delta v(r)\Vert ^{6}+|z(\theta _{r-s}\omega )|^{6}\\&\quad +\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}+\Vert g(r)\Vert ^{2})dr. \end{aligned}$$

Integrating the result again w.r.t. $\zeta \in [s+\sigma -1, s+\sigma ]$ yields

$$\begin{aligned}&e^{\lambda _{k+1}^2(s+\sigma )}\Vert \Delta v_{k, 2}(s+\sigma , s-t, \theta _{-s}\omega , \psi )\Vert ^{2} \\&\quad \le c\int _{s+\sigma -1}^{s+\sigma }e^{\lambda _{k+1}^2r}(1+\Vert \Delta v(r)\Vert ^{6}+|z(\theta _{r-s}\omega )|^{6}\\&\qquad +\Vert f(r, v_{r}+hz(\theta _{r-s+\cdot }\omega ))\Vert ^{2}+\Vert g(r)\Vert ^{2})dr \\&\quad = c\int _{s-1}^{s}e^{\lambda _{k+1}^2(r+\sigma )}(1+\Vert \Delta v(r+\sigma )\Vert ^{6}+|z(\theta _{r+\sigma -s}\omega )|^{6})dr \\&\qquad +c\int _{s-1}^{s}e^{\lambda _{k+1}^2(r+\sigma )}(\Vert f(r+\sigma , v_{r+\sigma }+hz(\theta _{r-s+\sigma +\cdot }\omega ))\Vert ^{2}+\Vert g(r+\sigma )\Vert ^{2})dr. \end{aligned}$$

Multiplying the two sides by $e^{-\lambda _{k+1}^2(s+\sigma )}$ with $s\le \tau$ and $\sigma \in [-\varrho , 0]$ yields

$$\begin{aligned}&\Vert \Delta v_{k, 2}(r, s-t, \theta _{-s}\omega , \psi )\Vert ^{2}\le c\int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}dr\nonumber \\&\ +c\int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}\Vert \Delta v(r+\sigma )\Vert ^{6}dr+c\int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}\Vert f(r+\sigma , v_{r+\sigma }+hz(\theta _{r-s+\sigma +\cdot }\omega ))\Vert ^{2}dr \nonumber \\&\ +c\int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}|z(\theta _{r+\sigma -s}\omega )|^{6}dr+c\int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}\Vert g(r+\sigma )\Vert ^{2}dr. \end{aligned}$$

(4.27)

Consider the maximums of all terms on the right-hand side of (4.27) w.r.t. $s\in (-\infty , \tau ]$ and $\sigma \in [-\varrho , 0]$. Then

$$\begin{aligned}&\sup _{s\le \tau }\sup _{\sigma \in [-\varrho , 0]}c\int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}dr\le c \lambda _{k+1}^{-2}\rightarrow 0 \ \text {as} \ k\rightarrow \infty {,} \end{aligned}$$

as $k\rightarrow \infty$. By (4.13), for all $t\ge T$,

$$\begin{aligned}&\sup _{s\le \tau }\sup _{\sigma \in [-\varrho , 0]}c\int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}\Vert \Delta v(r+\sigma ,s-t, \theta _{-s}\omega , \psi )\Vert ^{6}dr\\&\quad \le c\sup _{s\le \tau }\sup _{r\in [s-\varrho -1, s]}\Vert \Delta v(r,s-t, \theta _{-s}\omega , \psi )\Vert ^{6}\int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}dr \\&\quad \le c \left( \sup _{\sigma \in [-2\varrho -4, 0]}|z(\theta _{\sigma }\omega )|^{2}+\chi ^{5}(\tau , \omega )\right) ^{3}\lambda _{k+1}^{-2}\rightarrow 0 \ \ \text {as} \ k\rightarrow \infty . \end{aligned}$$

For the delay forcing term, we deduce from (F1), (F3) and (4.4) that

$$\begin{aligned}&\sup _{s\le \tau }\sup _{\sigma \in [-\varrho , 0]}c\int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}\Vert f(r+\sigma , v_{r+\sigma }+hz(\theta _{r-s+\sigma +\cdot }\omega ))\Vert ^{2}dr\\&\quad \le c \sup _{s\le \tau }\sup _{\sigma \in [-\varrho , 0]} \int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}L_{f}^{2}(r+\sigma )\Vert u_{r+\sigma }\Vert _{C(L^2)}^{2}dr \\&\quad \le \left( c\sup _{\sigma \in [-2\varrho -4, 0]}|z(\theta _{\sigma }\omega )|^{2}+c\chi (\tau , \omega )\right) \sup _{s\le \tau }\sup _{\sigma \in [-\varrho , 0]}\int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}L_{f}^{2}(r+\sigma )dr. \end{aligned}$$

Since $L_{f}(\cdot )$ is backward limiting as in (3.7) and $\lambda _{k+1}^2\rightarrow \infty$ as $k\rightarrow \infty$, it follows that

$$\begin{aligned}&\sup _{s\le \tau }\sup _{\sigma \in [-\varrho , 0]}\int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}L_{f}^{2}(r+\sigma )dr \\&\quad =\sup _{s\le \tau }\sup _{\sigma \in [-\varrho , 0]}\int _{(s+\sigma )-1}^{s+\sigma }e^{\lambda _{k+1}^2(r-(s+\sigma ))}L_{f}^{2}(r)dr=\sup _{s\le \tau }\int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}L_{f}^{2}(r)dr \\&\quad \le \sup _{s\le \tau }\int _{-\infty }^{s}e^{\lambda _{k+1}^2(r-s)}L_{f}^{2}(r)dr\rightarrow 0 \text { as } k\rightarrow \infty . \end{aligned}$$

Hence we have

$$\begin{aligned}&\sup _{s\le \tau }\sup _{\sigma \in [-\varrho , 0]}c\int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}\Vert f(r+\sigma , v_{r+\sigma }+hz(\theta _{r-s+\sigma +\cdot }\omega ))\Vert ^{2}dr\rightarrow 0 \text { as } k\rightarrow \infty . \end{aligned}$$

(4.28)

For the third term, by the continuity of $z(\theta _{\cdot }\omega )$ again, it is easy to know that

$$\begin{aligned} \sup _{s\le \tau }\sup _{\sigma \in [-\varrho , 0]}\int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}|z(\theta _{r+\sigma -s}\omega )|^{6}dr&= \sup _{\sigma \in [-\varrho , 0]}\int _{-1}^{0}e^{\lambda _{k+1}^2r}|z(\theta _{r+\sigma }\omega )|^{6}dr\\&\le \sup _{\sigma \in [-\varrho -1, 0]}|z(\theta _{\sigma }\omega )|^{6}\int _{-1}^{0}e^{\lambda _{k+1}^2r}dr\\&\le \lambda _{k+1}^{-2}\sup _{\sigma \in [-\varrho -1, 0]}|z(\theta _{\sigma }\omega )|^{6}\rightarrow 0 \ \text {as} \ k\rightarrow +\infty . \end{aligned}$$

Since g is backward limitable as in (G), it follows that

$$\begin{aligned} \sup _{s\le \tau }\sup _{\sigma \in [-\varrho , 0]}\int _{s-1}^{s}e^{\lambda _{k+1}^2(r-s)}\Vert g(r+\sigma )\Vert ^{2}dr&= \sup _{s\le \tau }\sup _{\sigma \in [-\varrho , 0]}\int _{(s+\sigma )-1}^{s+\sigma }e^{\lambda _{k+1}^2(r-(s+\sigma ))}\Vert g(r)\Vert ^{2}dr\\&=\sup _{s\le \tau }\int _{s-1}^{s} e^{\lambda _{k+1}^2 (r-s)}\Vert g(r)\Vert ^{2}dr\\&\le \sup _{s\le \tau }\int _{-\infty }^{s} e^{\lambda _{k+1}^2 (r-s)}\Vert g(r)\Vert ^{2}dr\rightarrow 0. \end{aligned}$$

We substitute all limits into (4.27) to obtain (4.24) as desired. $\square$

5 Backward Attractors and Backward Stability of Pullback Attractors

Lemma 5.1

(i) The cocycle $\Phi$ in (3.11) has a random ${\mathfrak {U}}$-pullback absorbing set $\widetilde{{\mathcal {K}}}\in {\mathfrak {U}}$ given by

$$\begin{aligned} \widetilde{{\mathcal {K}}}(\tau , \omega )=\left\{ \varphi \in C(H_0^2): \Vert \varphi \Vert _{C(H_0^2)}^{2}\le c\sup _{\sigma \in [-2\varrho -4, 0]}|z(\theta _{\sigma }\omega )|^{2}+ c {\widetilde{\chi }}^5(\tau , \omega ))\right\} , \end{aligned}$$

(5.1)

for all $\tau \in {\mathbb {R}}, \omega \in \Omega$.

(ii) The cocycle $\Phi$ has an increasing ${\mathfrak {D}}$-backward absorbing set ${\mathcal {K}}\in {\mathfrak {D}}$ defined by

$$\begin{aligned} {\mathcal {K}}(\tau , \omega )=\left\{ \varphi \in C(H_0^2): \Vert \varphi \Vert _{C(H_0^2)}^{2}\le c\sup _{\sigma \in [-2\varrho -4, 0]}|z(\theta _{\sigma }\omega )|^{2}+c\chi ^5(\tau , \omega ))\right\} , \end{aligned}$$

(5.2)

for all $\tau \in {\mathbb {R}}, \omega \in \Omega$, where $\chi (\tau , \omega )\,($is given as in (4.5)) is the backward supremum of ${\widetilde{\chi }}(\tau , \omega )\,($is given as in (4.3)).

Proof

It is easy to see that $\omega \rightarrow \int _{-\infty }^{0}e^{m_f r}|z(\theta _{r}\omega )|^{6}dr$ is ${\mathfrak {F}}$-measurable. Hence, by (4.3),

$$\begin{aligned} {\widetilde{\chi }}(\tau , \omega )=1+\int _{-\infty }^{0}e^{m_f r}|z(\theta _{r}\omega )|^{6}dr+\int _{-\infty }^{0}e^{m_f r}\Vert g(r+\tau )\Vert ^{2}dr \end{aligned}$$

is ${\mathfrak {F}}$-measurable in $\omega$ for each $\tau \in {\mathbb {R}}$. So, $\widetilde{{\mathcal {K}}}$ is a random set in $C(H_0^2)$. Since $\chi (\tau , \omega ):=\sup _{s\le \tau }{\widetilde{\chi }}(s, \omega )$ obviously increases as $\tau$ increases, it follows that $\tau \mapsto {\mathcal {K}}(\tau , \omega )$ is increasing according to the inclusion relationship.

By (4.13) in Lemma 4.2, for $D\in {\mathfrak {D}}$, there is a $T>0$ such that for all $t\ge T$,

$$\begin{aligned} \cup _{s\le \tau }\Psi (t,s-t, \theta _{-t}\omega )D(s-t, \theta _{-t}\omega )\subset {\mathcal {K}}(\tau , \omega ), \end{aligned}$$

which means that ${\mathcal {K}}$ is ${\mathfrak {D}}$-backward absorbing. By the same method as in (4.13), we see from (i) of Lemma 4.1 that for each $U\in {\mathfrak {U}}$, there is a ${\widetilde{T}}>0$ such that for all $t\ge {\widetilde{T}}$,

$$\begin{aligned} \Psi (t,\tau -t, \theta _{-t}\omega )U(\tau -t, \theta _{-t}\omega )\subset \widetilde{{\mathcal {K}}}(\tau , \omega ), \end{aligned}$$

which means that $\widetilde{{\mathcal {K}}}$ is ${\mathfrak {U}}$-pullback absorbing in $C(H_0^2)$.

It suffices to prove $\widetilde{{\mathcal {K}}}\in {\mathfrak {U}}$ and ${\mathcal {K}}\in {\mathfrak {D}}$. Let $\gamma$ be an arbitrary positive number. Since $s\rightarrow \chi (s,\omega )$ is increasing, it follows that for all $t\ge 0$,

$$\begin{aligned} e^{-\gamma t}\sup _{s\le \tau }\Vert {\mathcal {K}}(s-t, \theta _{-t}\omega )\Vert _{C(H_0^2)}^{2}&= c e^{-\gamma t}\left( \sup _{\sigma \in [-2\varrho -4, 0]}|z(\theta _{\sigma }\omega )|^{2}+\sup _{s\le \tau }\chi ^5(s-t, \theta _{-t}\omega )\right) \\&= ce^{-\gamma t}\left( \sup _{\sigma \in [-2\varrho -4, 0]}|z(\theta _{\sigma }\omega )|^{2}+\chi ^5(\tau -t, \theta _{-t}\omega )\right) \\&=ce^{-\gamma t}\sup _{\sigma \in [-2\varrho -4, 0]}|z(\theta _{\sigma }\omega )|^{2}\\&\quad +ce^{-\gamma t}\left( 1+ \int _{-\infty }^{0}e^{m_f r}|z(\theta _{r-t}\omega )|^{6}dr \right. \\&\left. \quad +\sup _{s\le \tau -t}\int _{-\infty }^{0}e^{m_f r}\Vert g(r+s)\Vert ^{2}dr\right) ^5. \end{aligned}$$

By Lemma 3.1, g is backward tempered with the rate $m_f$, that is,

$$\begin{aligned} \sup _{s\le \tau }\int _{-\infty }^{0}e^{m_f r}\Vert g(r+s)\Vert ^{2}dr<\infty \text { and so } e^{-\gamma t}\left( \sup _{s\le \tau }\int _{-\infty }^{0}e^{m_f r}\Vert g(r+s)\Vert ^{2}dr\right) ^5\rightarrow 0 \end{aligned}$$

as $t\rightarrow +\infty$. By (3.1), $z(\theta _{s}\omega )/s\rightarrow 0$ as $s\rightarrow -\infty$ and thus there is a random variable $Z(\omega )>0$ such that

$$\begin{aligned} |z(\theta _{r-t}\omega )|\le (t-r)+Z(\omega ),\ \forall r\le 0, \ t\ge 0, \end{aligned}$$

which further implies

$$\begin{aligned} e^{-\gamma t}\left( \int _{-\infty }^{0}e^{m_f r}|z(\theta _{r-t}\omega )|^{6}dr\right) ^5&\le e^{-\gamma t}\left( \int _{-\infty }^{0}e^{m_f r}(t-r+Z(\omega ))^{6}dr\right) ^5\\&\le ct^{30}e^{-\gamma t}+cZ^{30}(\omega )e^{-\gamma t}\\&\quad +ce^{-\gamma t}\left( \int _{-\infty }^{0}e^{m_f r}r^6dr\right) ^5\rightarrow 0 \end{aligned}$$

as $t\rightarrow +\infty$, in view of $\int _{-\infty }^{0}e^{m_f r}r^6dr<+\infty$. By the continuity of $z(\theta _{\cdot }\omega )$ we have

$$\begin{aligned} ce^{-\gamma t}\sup _{\sigma \in [-2\varrho -4, 0]}|z(\theta _{\sigma }\omega )|^{2}\rightarrow 0 \end{aligned}$$

as $t\rightarrow +\infty$. Thereby, we obtain

$$\begin{aligned}&e^{-\gamma t}\Vert \widetilde{{\mathcal {K}}}(\tau -t, \theta _{-t}\omega )\Vert _{C(H_0^2)}^{2}\le e^{-\gamma t}\sup _{s\le \tau }\Vert {\mathcal {K}}(s-t, \theta _{-t}\omega )\Vert _{C(H_0^2)}^{2}\rightarrow 0 \end{aligned}$$

as $t\rightarrow +\infty$, which means $\widetilde{{\mathcal {K}}}\in {\mathfrak {U}}$ and ${\mathcal {K}}\in {\mathfrak {D}}$. $\square$

Lemma 5.2

The cocycle $\Phi$ is ${\mathfrak {D}}$-backward asymptotically compact in $C([-\varrho ,0], H_0^2({\mathcal {O}}))$.

Proof

Let $t_n\rightarrow +\infty$, $s_n\le \tau$ and $\phi _{n}\in D(s_n-t_n,\theta _{-t_{n}}\omega )$ with $D\in {\mathfrak {D}}$ and $\tau \in {\mathbb {R}}$. We show in two steps that the sequence $\{\Phi (t_{n}, s_{n}-t_{n}, \theta _{-t_{n}}\omega )\phi _{n}\}_{n\in {\mathbb {N}}}$ is pre-compact in $C(H_0^2)$.

Step 1. Prove that the sequence $\{(\Phi (t_{n}, s_{n}-t_{n}, \theta _{-t_{n}}\omega )\phi _{n})(\cdot )\}_{n\in {\mathbb {N}}}$ is equi-continuous from $[-\varrho , 0]$ to $H_0^2({\mathcal {O}})$.

Denote by $T=T(D,\tau , \omega )$ the same entry-time in all above lemmas. Then there is $N\in {\mathbb {N}}$ such that $t_{n}\ge T$ for all $n\ge N$. Given $\varepsilon >0$, it follows from Lemma 4.4 that there is $k_0\in {\mathbb {N}}$ such that

$$\begin{aligned} \sup _{\sigma \in [-\varrho , 0]}\Vert (I-P_{k_{0}})v(s_{n}+\sigma , s_{n}-t_{n}, \theta _{-s_{n}}\omega , \psi _{n})\Vert _{H_0^2}<\varepsilon , \ \forall n\ge N. \end{aligned}$$

(5.3)

By Lemma 4.3, there is $\delta >0$ such that for all $\sigma _1,\sigma _2\in [-\varrho ,0]$ with $|\sigma _1-\sigma _2|<\delta$,

$$\begin{aligned} \Vert P_{k_{0}}v(s_{n}+\sigma _{1}, s_{n}-t_{n}, \theta _{-s_{n}}\omega , \psi _{n})-P_{k_{0}}v(s_{n}+\sigma _{2}, s_{n}-t_{n}, \theta _{-s_{n}}\omega , \psi _{n})\Vert _{H_0^2} <\varepsilon , \ \forall n\ge N, \end{aligned}$$

which together with (5.3) implies that for all $n\ge N$,

$$\begin{aligned}&\Vert (\Phi (t_{n}, s_{n}-t_{n}, \theta _{-t_{n}}\omega )\phi _{n})(\sigma _{1})-(\Phi (t_{n}, s_{n}-t_{n}, \theta _{-t_{n}}\omega )\phi _{n})(\sigma _{2})\Vert _{H_0^2} \\&\quad =\ \Vert u(s_{n}+\sigma _{1}, s_{n}-t_{n}, \theta _{-s_{n}}\omega , \phi _{n})-u(s_{n}+\sigma _{2}, s_{n}-t_{n}, \theta _{-s_{n}}\omega , \phi _{n})\Vert _{H_0^2} \\&\quad \le \ \Vert v(s_{n}+\sigma _{1}, s_{n}-t_{n}, \theta _{-s_{n}}\omega , \psi _{n})-v(s_{n}+\sigma _{2}, s_{n}-t_{n}, \theta _{-s_{n}}\omega , \psi _{n})\Vert _{H_0^2}\\&\qquad +\Vert h\Vert _{H_0^2}|z(\theta _{\sigma _{1}}\omega )-z(\theta _{\sigma _{2}}\omega )| \\&\quad \le \ \Vert P_{k_{0}}v(s_{n}+\sigma _{1}, s_{n}-t_{n}, \theta _{-s_{n}}\omega , \psi _{n})-P_{k_{0}}v(s_{n}+\sigma _{2}, s_{n}-t_{n}, \theta _{-s_{n}}\omega , \psi _{n})\Vert _{H_0^2} \\&\qquad \ +2\sup _{\sigma \in [-\varrho , 0]}\Vert (I-P_{k_{0}})v(s_{n}+\sigma , s_{n}-t_{n}, \theta _{-s_{n}}\omega , \psi _{n})\Vert _{H_0^2}\\&\qquad +\Vert h\Vert _{H_0^2}|z(\theta _{\sigma _{1}}\omega )-z(\theta _{\sigma _{2}}\omega )| < c\varepsilon , \end{aligned}$$

where we use the continuity of $z(\theta _{\cdot }\omega )$. Hence, $\{\Phi (t_{n}, s_{n}-t_{n}, \theta _{-t_{n}}\omega )\phi _{n}: n\ge N\}$ is equi-continuous from $[-\varrho , 0]$ to X. On the other hand, the finite set $\{\Phi (t_{n}, s_{n}-t_{n}, \theta _{-t_{n}}\omega )\phi _{n}: n<N\}$ of continuous functions is obviously equi-continuous from $[-\varrho , 0]$ to X. Then, the whole sequence $\{\Phi (t_{n}, s_{n}-t_{n}, \theta _{-t_{n}}\omega )\phi _{n}\}_{n\in {\mathbb {N}}}$ is equi-continuous.

Step 2. We prove that for each $\sigma \in [-\varrho , 0]$, the following sequence

$$\begin{aligned} (\Phi (t_{n}, s_{n}-t_{n}, \theta _{-t_{n}}\omega )\phi _{n})(\sigma )=u(s_{n}+\sigma , s_{n}-t_{n}, \theta _{-s_{n}}\omega , \phi _{n}) \end{aligned}$$

is pre-compact in $H_0^2({\mathcal {O}})$.

Indeed, by (4.13), $\{P_{k_{0}}v(s_{n}+\sigma , s_{n}-t_{n}, \theta _{-s_{n}}\omega , \psi _{n})\}_{n\in {\mathbb {N}}}$ is bounded in $H_0^2({\mathcal {O}})$ and thus pre-compact in the $k_{0}$-dimensional subspace $H_{k_{0}}=\text {span}\{e_1,\ldots , e_{k_0}\}$. Then, there is an index subsequence $n_{i}$ of n such that $\{P_{k_{0}}v(s_{n_{i}}+\sigma ; s_{n_{i}}-t_{n_{i}}, \theta _{-s_{n_{i}}}\omega , \psi _{n_{i}})\}_{i\in {\mathbb {N}}}$ is a Cauchy sequence in $H_{k_{0}}$. If i and j are large enough, then we see from (5.3) that

$$\begin{aligned}&\Vert {u(s_{n_{i}}+\sigma , s_{n_{i}}-t_{n_{i}}, \theta _{-s_{n_{i}}}\omega , \phi _{n_{i}})}-u(s_{n_{j}}+\sigma , s_{n_{j}}-t_{n_{j}}, \theta _{-s_{n_{j}}}\omega , \phi _{n_{j}})\Vert _{H_0^2} \\&\quad =\ \Vert v(s_{n_{i}}+\sigma , s_{n_{i}}-t_{n_{i}}, \theta _{-s_{n_{i}}}\omega , \psi _{n_{i}})-v(s_{n_{j}}+\sigma , s_{n_{j}}-t_{n_{j}}, \theta _{-s_{n_{j}}}\omega , \psi _{n_{j}})\Vert _{H_0^2} \\&\quad \le \ \Vert P_{k_{0}}v(s_{n_{i}}+\sigma , s_{n_{i}}-t_{n_{i}}, \theta _{-s_{n_{i}}}\omega , \psi _{n_{i}})-P_{k_{0}}v(s_{n_{j}}+\sigma , s_{n_{j}}-t_{n_{j}}, \theta _{-s_{n_{j}}}\omega , \psi _{n_{j}})\Vert _{H_0^2} \\&\quad \ +\Vert (I-P_{k_{0}})v(s_{n_{i}}+\sigma , s_{n_{i}}-t_{n_{i}}, \theta _{-s_{n_{i}}}\omega , \psi _{n_{i}})\Vert _{H_0^2} \\&\quad \ +\Vert (I-P_{k_{0}})v(s_{n_{j}}+\sigma , s_{n_{j}}-t_{n_{j}}, \theta _{-s_{n_{j}}}\omega , \psi _{n_{j}})\Vert _{H_0^2} < 3\varepsilon . \end{aligned}$$

Hence, $\{u(s_{n_{i}}+\sigma , s_{n_{i}}-t_{n_{i}}, \theta _{-s_{n_{i}}}\omega , \phi _{n_{i}})\}_{i\in {\mathbb {N}}}$ is a Cauchy sequence and thus convergent in $H_0^2({\mathcal {O}})$.

Then, it follows from Step 1 and Step 2 that all conditions of the Ascoli–Arzelà theorem are satisfied. Hence, we obtain $\Phi$ is ${\mathfrak {D}}$-backward asymptotically compact in $C(H_0^2)$. The proof is concluded. $\square$

Theorem 5.3

Assume (F1)–(F3), (G) and let $\Phi$ be the cocycle generated from the delayed stochastic modified Swift–Hohenberg equations. Then

• (a) $\Phi$ has a unique ${\mathfrak {D}}$-backward attractor ${\mathcal {B}}\in {\mathfrak {D}}$ and a unique ${\mathfrak {D}}$-pullback attractor ${\mathcal {A}}\in {\mathfrak {D}}$ such that

  $$\begin{aligned} {\mathcal {B}}(\tau , \omega )=\overline{\bigcup _{s\le \tau }{\mathcal {A}}(s, \omega )}\ \text { and } {\mathcal {A}}(\tau , \omega )=\bigcap _{T>0}\overline{\bigcup _{t\ge T}\Phi (t, \tau -t, \theta _{-t}\omega ){\mathcal {K}}(\tau -t, \theta _{-t}\omega )}, \end{aligned}$$

  (5.4)

  where the closures are taken under the norm of $C([-\varrho ,0], H_0^2({\mathcal {O}}))$;

• (b) $\Phi$ has a random ${\mathfrak {U}}$-pullback attractor ${\mathcal {A}}_{{\mathfrak {U}}}\in {\mathfrak {U}}$ such that ${\mathcal {A}}_{{\mathfrak {U}}}={\mathcal {A}}$ in $C(H_0^2)$, in particular, ${\mathcal {A}}$ is random and thus ${\mathcal {B}}$ is dividedly random.

• (c) Both ${\mathcal {A}}$ and ${\mathcal {B}}$ are backward stable to the same backward controller given by

  $$\begin{aligned} {\mathcal {A}}_{-\infty }(\omega )={\mathcal {B}}_{-\infty }(\omega )=\lim _{\tau \rightarrow -\infty }{\mathcal {B}}(\tau , \omega ), \end{aligned}$$

  (5.5)

  where the last limit of sets is taken under the (symmetric) Hausdorff distance, more precisely, ${\mathcal {A}}_{-\infty }(\omega )$ is the minimal nonempty compact set such that

  $$\begin{aligned} \lim _{\tau \rightarrow -\infty } \textrm{dist}_{C(H_0^2)}({\mathcal {A}}(\tau , \omega ), {\mathcal {A}} _{-\infty }(\omega ))=0. \end{aligned}$$

Proof

• (a) By Lemma 5.1 (ii), $\Phi$ has an increasing ${\mathfrak {D}}$-backward absorbing set ${\mathcal {K}}\in {\mathfrak {D}}$. By Lemma 5.2, $\Phi$ is ${\mathfrak {D}}$-backward asymptotically compact in $C(H_0^2)$. Then all conclusions follow from the abstract Theorems  2.9 and 2.6.

• (b) By the similar method as in Lemma 5.2, $\Phi$ is ${\mathfrak {U}}$-backward asymptotically compact in $C(H_0^2)$. Then the abstract result [25, Theorem 2.23] implies that $\Psi$ has a unique ${\mathfrak {U}}$-pullback attractor ${\mathcal {A}}_{{\mathfrak {U}}}\in {\mathfrak {U}}$. By Lemma 5.1 (i), the ${\mathfrak {U}}$-pullback absorbing set $\widetilde{{\mathcal {K}}}$ is random and so is ${\mathcal {A}}_{{\mathfrak {U}}}$. Since ${\mathcal {K}}(\tau , \omega )\supset \widetilde{{\mathcal {K}}}(\tau , \omega )$ and ${\mathcal {K}}\in {\mathfrak {D}}\subset {\mathfrak {U}}$, it follows that the ${\mathfrak {D}}$-pullback absorbing set ${\mathcal {K}}$ is ${\mathfrak {U}}$-pullback absorbing too. Then its omega-limit ${\mathcal {A}}$ (is given as in (5.4)) is a ${\mathfrak {U}}$-pullback attractor too. By the uniqueness of ${\mathfrak {U}}$-pullback attractors, we know ${\mathcal {A}}={\mathcal {A}}_{{\mathfrak {U}}}$. Then ${\mathcal {A}}$ is random too and thus, from (5.4), ${\mathcal {B}}$ is dividedly random.

• (c) By the abstract Theorem  2.6, all assertions in (c) follow from the existence of the backward attractor ${\mathcal {B}}$ is given as in (a). In particular, by (2.10),

  $$\begin{aligned} \text {dist}_{C(H_0^2)}^{\text {sym}}({\mathcal {B}}(\tau , \omega ), {\mathcal {B}}_{-\infty }(\omega ))\rightarrow 0 \text { as }\tau \rightarrow -\infty , \end{aligned}$$

  which means ${\mathcal {B}}(\tau , \omega )\rightarrow {\mathcal {B}}_{-\infty }(\omega )$ (as $\tau \rightarrow -\infty$) in the space forming from all nonempty closed subsets of $C(H_0^2)$ w.r.t. the Hausdorff distance. $\square$

To close this article, we provide two important examples for the delay force f. Such examples includes the wider range than those in [2, 3, 10, 27] since we have not restricted the upper bound of $c_f$ or $L_f$ in (F2)–(F3).

Example 5.4

(Variable delay) Let $F: {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ be a measurable function such that

$$\begin{aligned} F(t, 0)=0, \ |F(t, y)-F(t, z)|\le L_{F}(t)|y-z|, \ \ \forall t, y, z\in {\mathbb {R}}, \end{aligned}$$

(5.6)

where $L_{F}(\cdot )$ is positive, continuous and increasing such that $L:=\lim _{t\rightarrow \infty }L_{F}(t)<\infty$. Let

$$\begin{aligned} f(t, u_{t})(x):=F(t, u(t-\rho (t), x)), \ \forall t\in {\mathbb {R}},\ x\in {\mathcal {O}}, \end{aligned}$$

where $\rho (\cdot )$ is a positive function such that $\rho (\cdot )\in C^{1}({\mathbb {R}})$ and

$$\begin{aligned} \varrho :=\sup _{t\in {\mathbb {R}}}\rho (t)< +\infty , \ \ \ \rho _{*}:=\sup _{t\in {\mathbb {R}}}\rho '(t)<1. \end{aligned}$$

It is easy to show that f is Lipschtz continuous (i.e. (3.6) holds). The Lipschtz bound $L_{F}(\cdot )$ is increasing and thus strongly backward tempered. By Lemma 3.2, $L_{f}(\cdot )$ is backward limiting and thus (F3) holds true. Let $m_f>0$ be arbitrary. Then for all $m\in [0,m_f]$,

$$\begin{aligned} \int _{\tau }^{t}e^{mr}\Vert f(r, u_{r})-f(r, v_{r})\Vert ^{2}dr&=\int _{\tau }^{t}e^{mr}\Vert F(r, u(r-\rho (r)))-F(r, v(r-\rho (r)))\Vert ^{2}dr \\&\le \ L^{2}\int _{\tau }^{t}e^{mr}\Vert u(r-\rho (r))-v(r-\rho (r))\Vert ^{2}dr\\&\le \frac{L^{2}e^{m_f\varrho }}{1-\rho _{*}}\int _{\tau -\varrho }^{t}e^{mr}\Vert u(r)-v(r)\Vert ^{2}dr. \end{aligned}$$

Hence (F2) holds with $c_f=Le^{m_f\varrho /2}(1-\rho _{*})^{-1/2}$.

Example 5.5

(Distributed delay) Suppose F is given as in Example 5.4 and $\varrho >0$ is arbitrary. Consider

$$\begin{aligned} f(t, u_{t})(x):=\int _{-\varrho }^{0}F(t+r, u(t+r, x))dr, \ \forall t\in {\mathbb {R}},\ x\in {\mathcal {O}}. \end{aligned}$$

We need to verify (F2)–(F3). Since $L_{F}(\cdot )$ is increasing, it follows from (5.6) that

$$\begin{aligned} \Vert f(t, u_{t})-f(t, v_{t})\Vert ^{2}&=\int _{{\mathcal {O}}}\left| \int _{-\varrho }^{0}(F(t+r, u(t+r, x))-F(t+r, v(t+r, x)))dr\right| ^{2}dx \\&\le \int _{{\mathcal {O}}}\left( \int _{-\varrho }^{0}L_{F}(t+r)|u(t+r, x)-v(t+r, x)|dr\right) ^{2}dx \\&\le \varrho L_{F}^{2}(t+0)\int _{{\mathcal {O}}}\int _{-\varrho }^{0}|u(t+r, x)-v(t+r, x)|^{2}drdx \\&\le \varrho ^{2}L_{F}^{2}(t)\sup _{r\in [-\varrho , 0]}\int _{{\mathcal {O}}}|u(t+r, x)-v(t+r, x)|^{2}dx\\&=\varrho ^{2}L_{F}^{2}(t)\Vert u_{t}-v_{t}\Vert _{C(L^2)}^{2}. \end{aligned}$$

By Lemma 3.2, the Lipschtz bound $L_{f}(\cdot ):=\varrho L_{F}(\cdot )$ is backward limiting and thus (F3) holds true. Let $m_f>0$ be arbitrary. Then for all $m\in [0,m_f]$ and $t\ge \tau$,

$$\begin{aligned}&\int _{\tau }^{t}e^{mr}\Vert f(r, u_{r})-f(r, v_{r})\Vert ^{2}dr \\&\quad = \int _{\tau }^{t}e^{mr}\int _{{\mathcal {O}}}\left| \int _{-\varrho }^{0}[F(r+s, u(r+s, x))-F(r+s, v(r+s, x))]ds\right| ^{2}dxdr \\&\quad \le \int _{\tau }^{t}e^{mr}\int _{{\mathcal {O}}}\left( \int _{-\varrho }^{0}L_{F}(r+s)|u(r+s, x)-v(r+s, x)|ds\right) ^{2}dxdr \\&\quad \le \varrho L^{2}\int _{\tau }^{t}e^{mr}\int _{{\mathcal {O}}}\int _{-\varrho }^{0}|u(r+s, x)-v(r+s, x)|^{2}dsdxdr \\&\quad = \varrho L^{2}\int _{-\varrho }^{0}e^{-ms}\left[ \int _{\tau +s}^{t+s}e^{m r}\int _{{\mathcal {O}}}|u(r, x)-v(r, x)|^{2}dxdr\right] ds \\&\quad \le \varrho ^{2}L^{2}e^{m_f\varrho }\int _{\tau -\varrho }^{t}e^{mr}\Vert u(r)-v(r)\Vert ^{2}dr. \end{aligned}$$

Hence (F2) holds with $c_f=\varrho Le^{m_f\varrho /2}$.