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Random attractors for non-autonomous stochastic wave equations with nonlinear damping and white noise

Abstract

This paper is concerned with the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with additive white noise, for which the nonlinear damping has a critical cubic growth rate. By showing the pullback asymptotic compactness of the stochastic dynamic systems, we prove the existence of a random attractor in \(H_{0}^{1}\times L^{2}\).

Introduction

This paper deals with the existence of random attractors for the following non-autonomous stochastic wave equation with white noise in a bounded domain \(U \subset R^{3}\) with smooth boundary ∂U:

$$\begin{aligned}& u_{tt} + q(u)u_{t} + \alpha u_{t} - \Delta u + f(u, x) = g(x, t) + ah(x)\dot{W}(t), \end{aligned}$$
(1)
$$\begin{aligned}& u(x, t)\arrowvert_{x \in\partial U } = 0,\quad t\geq\tau, \tau\in R, \end{aligned}$$
(2)
$$\begin{aligned}& u(x, \tau) = u_{0\tau}(x),\qquad u_{t}(x, \tau) = u_{1\tau}(x),\quad x\in U, \tau\in R, \end{aligned}$$
(3)

for \((x, t) \in U \times(\tau, +\infty)\) with \(\tau\in R\), where \(h \in H_{0}^{1}(U)\cap H^{2}(U)\) and \(\alpha\geq0\) is the damping coefficient. Here \(u(x, t)\) is a real-valued function on \(U\times[\tau, +\infty)\); \(g(x, \cdot) \in C_{b}(R, H_{0}^{1}(U))\) is a time-dependent driving force; \(C_{b}(R, H_{0}^{1}(U))\) denotes the set of continuous bounded functions from R into \(H_{0}^{1}(U)\); and \(W(t)\) is a two-sided real-valued Wiener processes on the probability space \((\varOmega, \mathcal{F}, {P})\). In addition, the function \(q: R\rightarrow R\) and the nonlinear function f satisfy the following assumptions:

\((H_{1})\):

The function \(q \in\mathcal{C}^{1}\) is not identically equal to zero, and there exist three constants \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\) and \(\alpha_{2}\geq|\alpha_{1}|\) such that

$$\begin{aligned} -\alpha< \alpha_{1} \leq q(s) \leq\alpha_{2} < +\infty,\qquad \bigl\vert q'(s) \bigr\vert \leq\alpha_{3}, \quad\forall s\in R. \end{aligned}$$
(4)
\((H_{2})\):

Let \(f(u,x)=f_{1}(u,x)+f_{2}(u,x)\) and \(F_{i}=\int_{0}^{u}f_{i}(r,x)\,dr\), where \(f_{1}(\cdot,x)\in \mathcal{C}^{2}(R,R)\), \(f_{2}(\cdot,x)\in\mathcal{C}^{1}(R,R)\). Furthermore, \(f_{1}\), \(f_{2}\) meet conditions that there exist constants \(c_{1},c_{2},c_{3},c_{4} >0\) and functions \(\phi_{i}(x)\in L^{1}(U)\), \(i = 1,2\), such that

$$\begin{aligned}& f'_{1,u}(0,x)=0, \qquad f_{1}(u,x)u\geq0,\qquad \bigl\vert f''_{1,u}(u,x) \bigr\vert \leq c_{1}\bigl(1+ \vert u \vert \bigr), \end{aligned}$$
(5)
$$\begin{aligned}& f_{2}(0,x)=0,\qquad \bigl\vert f'_{2,u}(u,x) \bigr\vert \leq c_{2}\bigl(1+ \vert u \vert ^{p} \bigr),\quad 0\leq p\leq2, \end{aligned}$$
(6)
$$\begin{aligned}& c_{3}u^{4}-\phi_{1}(x)\leq F_{i}(u,x)\leq c_{4}uf_{i}(u,x)+\phi _{2}(x), \quad\forall u\in R, x\in U. \end{aligned}$$
(7)

In the deterministic damped wave equation (i.e., \(a=0\)), global attractors have been studied by many authors, such as [13] and the reference therein. In addition, uniform attractors and pullback attractors also attracted many experts’ attention, cf. [48]. If the function g does not depend on time, (1)–(3) is an autonomous stochastic wave equation, and its random attractors have been explored in [913]. For many problems, such as wave propagation through the atmosphere or the ocean, the more realistic models must take the random fluctuation into account. So it is important and interesting to study random attractors. For non-autonomous random dynamical systems, Wang established an efficacious theory about the existence of random attractors [1417]. Particularly, Li [18] studied the asymptotic dynamics for a stochastic damped wave equation with multiplicative noise defined on unbounded domains and proved the existence of random attractors. For the non-autonomous stochastic strongly damped wave equation, the existence of random attractors is proved in [1921]. Lv and Wang [10] also studied the existence of random attractors for the stochastic wave equation and showed the upper semicontinuous dependence of the random attractor on parameters. The authors in [22] studied the asymptotic behavior of a class of non-autonomous nonlocal fractional stochastic parabolic equations driven by multiplicative white noise on the entire space \(R^{n}\).

In this paper, (1)–(3) is a non-autonomous stochastic system where the external term g is time-dependent. We shall transform the stochastic wave equation into a deterministic one with random parameter and random initial data through an Ornstein–Uhlenbeck process \(z(\theta_{t}\omega)\), then prove the existence of a random attractor for the random dynamical system generated by (1)–(3). It is well known that the key step in proving the existence of attractors in both deterministic and random systems is to establish the compactness of the system in some sense. Motivated by [23], we will work out this problem.

The paper is arranged as follows. In Sect. 2, we collect some basic concepts and background material about random attractor for the random dynamical system generated by (1)–(3), then the existence and uniqueness of solutions is established in Sect. 3. In Sect. 4, we consider the concrete bounds of the solution and decompose the solution of (12)–(13) into two parts. In Sect. 5, we establish the asymptotic compactness of the random dynamical system and obtain the existence of the random attractor.

Random dynamical systems

In this section, we collect some basic definitions and known results about general random dynamical systems (see [17, 24, 25] for details).

Let \((\varOmega,\mathcal{F},{P})\) be a probability space, where \(\varOmega=\{\omega\in C(R,R):\omega(0)=0\}\) is endowed with compact-open topology. \(\mathcal{F}\) is the Borel σ-algebra on Ω and P is the corresponding Wiener measure on \(\mathcal {F}\). For any t, let \((\theta_{t})_{t\in R}\) on Ω via

$$\theta_{t}\omega(\cdot)=\omega(\cdot+t)-\omega(t),\quad t\in R, $$

thus \((\varOmega,\mathcal{F},{P},(\theta_{t})_{t\in R})\) is an ergodic metric dynamical system [24]. In the following, X labels as a Banach or Hilbert space with the Borel σ-algebra \(B(X)\).

Definition 2.1

Let \(\{\theta_{t}\}_{t\in R}\) be a family of \((B({R}\times\mathcal{F}),\mathcal{F})\)-measurable mappings, \(\theta_{t}:R\times\varOmega\rightarrow\varOmega\) such that \(\theta_{0}(\cdot)\) is the identity on Ω, \(\theta_{s+t}(\cdot)=\theta_{t}(\cdot)\circ\theta_{s}(\cdot)\) for all \(t,s\in R\) and \(P\theta_{t}=P\) for all \(t\in R\).

Definition 2.2

Let \((\varOmega,\mathcal{F},P,(\theta _{t})_{t\in R})\) be a parametric dynamical system. A mapping \(\varPhi:R^{+}\times R\times\varOmega\times X\rightarrow X\) is called a continuous cocycle on X over R and \((\varOmega,\mathcal{F},P,(\theta_{t})_{t\in R})\) if, for all \(\tau\in R\), \(\omega\in\varOmega\), and \(t,s \in R^{+}\), the following conditions (i)–(iv) are satisfied:

  1. (i)

    \(\varPhi(\cdot,\tau,\cdot,\cdot):R^{+}\times\varOmega\times X\rightarrow X\) is a \((B(R^{+})\times\mathcal{F}\times B(X), B(X))\)-measurable mapping;

  2. (ii)

    \(\varPhi(0,\tau,\omega,\cdot)\) is the identity on X;

  3. (iii)

    \(\varPhi(t+s,\tau,\omega,\cdot)=\varPhi(t,\tau+s,\theta_{s}\omega ,\cdot)\circ\varPhi(s,\tau,\omega,\cdot)\);

  4. (iv)

    \(\varPhi(t,\tau,\omega,\cdot):X\rightarrow X \) is continuous.

Definition 2.3

  1. (1)

    Let \(2^{X}\) be the collection of all subsets of X. A set-valued mapping \((\tau,\omega)\rightarrow D(\tau,\omega):R\times\varOmega\rightarrow2^{X}\) is called measurable with respect to \(\mathcal{F}\) in Ω if \(D(\tau,\omega)\) is a (usually closed) nonempty subset of X and the mapping \(\omega\in \varOmega\rightarrow d(x,D(\tau,\omega))\) is \((\mathcal{F,B(R)})\)-measurable for every fixed \(x\in X\) and \(\tau\in R\), then \(D={D(\tau,\omega):\tau\in R, \omega\in\varOmega}\) is called a random set.

  2. (2)

    Let D be a collection of random sets in a Polish space X. A continuous cocycle Φ is said to be pullback \(\mathcal{D}\)-asymptotically compact \((\mathcal{D}\text{-a.c.})\) in X if, for any \(\tau\in R\), \(\omega\in\varOmega\), \(D\in\mathcal{D}\) and any sequences \(t_{n}\rightarrow+\infty\), \(x_{N}\in D(\tau-t_{n},\theta_{-t_{n}}\omega)\), the sequence \({\varPhi(t_{n},\tau-t_{n},\theta_{-t_{n}}\omega,x_{n})}\) has a convergent subsequence in X.

  3. (3)

    Let \(K={K(\tau,\omega):\tau\in R,\omega\in\varOmega}\in D\). Then K is called a pullback \(\mathcal{D}\)-absorbing set for Φ if, for all \(\tau\in R\), \(\omega\in\varOmega\) and for every \(D\in \mathcal{D}\), there exists \(t_{0}(K,\tau,\omega)> 0\) such that \(\varPhi(t,\tau-t,\theta_{-t}\omega,D(\tau-t,\theta_{-t}\omega))\subset K(\tau,\omega)\) for any \(t\geq t_{0}\).

  4. (4)

    A family \(C={C(\tau,\omega):\tau\in R,\omega\in\varOmega}\in \mathcal{D}\) is said to be pullback \(\mathcal{D}\)-attracting if \(\lim_{t\rightarrow +\infty}d(\varPhi(t,\tau-t,\theta_{-t}\omega,D(\tau-t,\theta_{-t}\omega ),C(\tau,\omega)))=0\) for all \(D\in\mathcal{D}\).

  5. (5)

    A compact set \(A=\{A(\tau,\omega):\tau\in R,\omega\in \varOmega\}\in\mathcal{D}\) is called a pullback \(\mathcal{D}\)-attractor for Φ if A attracts every \(B\in \mathcal{D}\) and A is invariant in the sense that \(\varPhi(t,\tau,\omega,A(\tau,\omega))=A(\tau+ t,\theta_{t}\omega)\) for every \(t\geq0\), \(\tau\in R\), and \(\omega\in\varOmega\).

In addition, if there exists \(T> 0\) such that \(A(\tau+T,\omega)=A(\tau,\omega)\) for any \(\tau\in R\), \(\omega\in \varOmega\), then A is periodic with period T.

Proposition 2.1

LetDbe a neighborhood-closed collection of\((\tau,\omega)\)-parametrized families of nonempty subsets ofXandΦbe a continuous cocycle onXoverRand\((\varOmega,\mathcal{F},P,\{\theta_{t}\}_{t\in R})\). ThenΦhas a pullback D-attractorAinDif and only ifΦis pullback\(\mathcal{D}\)-asymptotically compact inXandΦhas a closed\(\mathcal{F}\)-measurable pullback\(\mathcal{D}\)-absorbing setKinD. The unique pullbackD-attractor\(A={A(\tau,\omega)}\)is given by

$$A(\tau,\omega)=\bigcap_{r\geq0}\overline{\bigcup _{t\geq r}\varPhi(t, \tau-t,\theta_{-t} \omega),K(\tau-t,\theta_{-t}\omega)},\quad\tau\in R,\omega \in \varOmega. $$

Proposition 2.2

LetDbe a neighborhood-closed collection of\((\tau,\omega)\)-parametrized families of nonempty subsets ofX. IfΦis a continuousτ-periodic cocycle with period\(T>0\)onXoverRand\((\varOmega,\mathcal{F},P,\{\theta_{t}\}_{t\in R})\)and ifΦhas a pullback\(\mathcal{D}\)-attractor\(A\in D\), thenAisτ-periodic with periodTif and only ifΦhas a closed\(\mathcal{F}\)-measurable pullback\(\mathcal{D}\)-absorbing set\(K\in D\)with\(K=K(\tau,\omega)\)being periodic inτwith periodTfor each\(\omega\in\varOmega\).

Notation

Set \(E=H_{0}^{1}(U)\times L^{2}(U)\) with its inner product and norm as follows:

$$(z_{1},z_{2})_{E}=\bigl((u_{1},u_{2}) \bigr)+(v_{1},v_{2}), \qquad \Vert z \Vert _{E}=(z,z)_{E}^{\frac{1}{2}} $$

for all \(z_{i}=(u_{i},v_{i})^{T}\), \(i=1,2\), and \(z=(u,v)^{T}\) in E.

$$(u,v)= \int_{U}u(x)v(x)\,dx, \qquad \Vert u \Vert = \Vert u \Vert _{L^{2}}=(u,u)^{\frac{1}{2}} $$

for all \(u,v\in L^{2}(U)\), and

$$\bigl((u,v)\bigr)= \int_{U}\nabla u(x)\nabla v(x)\,dx, \qquad \Vert u \Vert _{1}= \Vert u \Vert _{H_{1}^{0}}=\bigl((u,u) \bigr)^{\frac{1}{2}} $$

for all \(u,v\in H_{0}^{1}(U)\). More generally, denote \(E_{s}=W^{s,2}(U)\cap H_{0}^{1}\times W^{s-1,2}(U)\) for \(s\in R\).

The letters c and \(c_{i}\) (\(i = 1, 2,\dots\)) are generic positive constants which do not depend on ω, τ, t, a.

Existence and uniqueness of solutions

In this section, motivated by [26, 27], we establish the existence and uniqueness of solutions for Eqs. (1)–(3). Let λ be the first eigenvalue of the operator \(A: = -\Delta\) on U with Dirichlet boundary conditions. Note that \(A : H^{1}_{0}(U)\cap H^{2}(U)\rightarrow L^{2}(U)\), so \(D(A)= H^{1}_{0}(U)\cap H^{2}(U)\). In the following, we convert problem (1)–(3) into a random system without noise terms. Identify \(\omega(t)\) with \(W(t)\), i.e., \(\omega(t)=W(t)\), \(t\in R\), and let \(z(\theta_{t}\omega):= -\int_{-\infty}^{0}e^{s}(\theta_{t}\omega)(s)\,ds\) (\(t\in R\)) be a Ornstein–Uhlenbeck stationary process which solves the Itô equation \(dz+z\,dx=dW(t)\).

Let \(\varepsilon=\frac{(\alpha+\alpha_{1})\lambda_{1}}{2(\alpha+\alpha _{2})^{2}+3\lambda_{1}}\). By the transformation

$$\varphi_{1}=u, \qquad\varphi_{2}=u_{t}+ \varepsilon u-ah(x)z(\theta_{t}\omega), $$

Equations (1)–(3) are equivalent to the following determined system with random parameters in E:

$$\begin{aligned}& \frac{d\varphi_{1}}{dt} =\varphi_{2}-\varepsilon \varphi_{1}+ah(x)z(\theta_{t}\omega), \end{aligned}$$
(8)
$$\begin{aligned}& \frac{d\varphi_{2}}{dt} =\Delta \varphi_{1}+\varepsilon(\alpha- \varepsilon)\varphi_{1}+(\varepsilon -\alpha)\varphi_{2} -q(\varphi_{1}) (\varphi_{2}-\varepsilon \varphi_{1}) \\& \phantom{\frac{d\varphi_{2}}{dt} =}{} -\bigl(q(\varphi_{1})+\alpha-\varepsilon-1\bigr)ah(x)z( \theta_{t}\omega) -f(\varphi_{1},x)+g(x,t), \end{aligned}$$
(9)
$$\begin{aligned}& \varphi_{1}(x,t)|_{\partial U}=0, \end{aligned}$$
(10)
$$\begin{aligned}& \varphi_{1}(\tau,\tau,x)=u_{\tau}(x),\qquad \varphi_{2}(\tau,\tau,x)=v_{\tau}(x)=u_{1\tau}+ \varepsilon u_{\tau}(x)-ah(x)z(\theta_{\tau}\omega). \end{aligned}$$
(11)

Then (8)–(11) can also be rewritten as a vector form:

$$\begin{aligned}& \dot{\varphi}+L(\varphi)=G(\varphi,\theta_{t}\omega,t), \end{aligned}$$
(12)
$$\begin{aligned}& \varphi(\tau,\omega)=\varphi_{\tau}(\omega)=\bigl(u_{\tau},u_{1,\tau }+ \varepsilon u_{\tau}-ah(x)z(\theta_{\tau}\omega)\bigr)^{T},\quad \tau\in R,t\geq\tau, \end{aligned}$$
(13)

where

$$\begin{aligned} \varphi= \left ( \textstyle\begin{array}{c} \varphi_{1} \\ \varphi_{2} \end{array}\displaystyle \right ),\qquad L = \left ( \textstyle\begin{array}{c@{\quad}c} \varepsilon I & -I\\ A-\varepsilon(\alpha-\varepsilon)I & (\alpha-\varepsilon)I \end{array}\displaystyle \right ) \end{aligned}$$
(14)

and

$$\begin{aligned}& G(\varphi,\theta_{t}\omega,t) \\& \quad=\left ( \textstyle\begin{array}{c} ah(x)z(\theta_{t}\omega) \\ -q(\varphi_{1})(\varphi_{2}-\varepsilon\varphi_{1})-[q(\varphi _{1})+\alpha-\varepsilon-1]ah(x)z(\theta_{t}\omega)-f(u,x)+g(x,t) \end{array}\displaystyle \right ). \end{aligned}$$
(15)

It is known from [28] that −L is the infinitesimal generator of a \(C^{0}\)-semigroup \(e^{-Lt}\) on E. By assumption \((H_{2})\) and the embedding relation \(H_{0}^{1}(U)\hookrightarrow L^{6}(U)\), the function \(G(\varphi,\theta_{t}\omega,t): E\rightarrow E\) is Lipschitz with respect to φ for t in a bounded interval and \(\omega\in \varOmega\), continuous in \((\varphi,t)\), and measurable in ω w.r.t \(\mathcal{F}\). Thus, by the classical semigroup theory on the local existence and the uniqueness of solutions of evolution differential equations in [25], we have the following theorem.

Theorem 1

Consider the initial value problem (12)(13), if assumptions\((H_{1})\)and\((H_{2})\)hold, then for each\(\tau\in R\), \(\omega\in\varOmega\)and any\(\varphi_{\tau}\in E\), there exists\(T>0\)such that Eqs. (12)(13) have a unique mild function\(\varphi(\cdot)=\varphi(\cdot,\tau,\omega,\varphi_{\tau})\in C([\tau,\tau+T);E)\), where\(\varphi(\tau,\tau,\omega,\varphi_{\tau})=\varphi_{\tau}\)and\(\varphi(t)\)satisfies the integral equation

$$\begin{aligned} \varphi(t,\tau,\omega,\varphi_{\tau}) =&e^{-L(t-\tau)} \varphi_{\tau }(\omega) \\ &{}+ \int_{\tau}^{t} e^{-L(t-r)}G\bigl( \varphi(r,\tau,\omega,\varphi_{\tau}),\theta_{r}\omega,r \bigr)\,dr,\quad \forall t\geq\tau. \end{aligned}$$
(16)

System (12)–(13) generates a continuous random dynamical system over R and \((\varOmega,\mathcal{F},P,(\theta_{t})_{t\in R})\)

$$\begin{aligned} \varPhi:R^{+}\times R\times\varOmega\times E\rightarrow E,\quad (t,\tau, \omega,\varphi_{\tau})\rightarrow \varPhi(t,\tau,\omega, \varphi_{\tau}), \end{aligned}$$
(17)

where

$$\begin{aligned}& \begin{gathered} \varPhi\bigl(t,\tau,\omega,\varphi_{\tau}(\omega)\bigr) \\ \quad=\varphi \bigl(t+\tau,\tau,\theta_{-\tau}\omega,\varphi_{\tau}( \theta_{-\tau}\omega )\bigr) \\ \quad= \left ( \textstyle\begin{array}{c} u(t+\tau,\tau,\theta_{-\tau}\omega,\varphi_{-\tau}(\theta_{-\tau}\omega )) \\ u_{t}(t+\tau,\tau,\theta_{-\tau}\omega,\varphi_{-\tau}(\theta_{-\tau }\omega))+\varepsilon u(t+\tau,\tau,\theta_{-\tau}\omega,\varphi_{-\tau}(\theta_{-\tau}\omega ))-ah(x)z(\theta_{t}\omega) \end{array}\displaystyle \right ),\end{gathered} \\& \begin{gathered} \varPhi(0,\tau,\omega,\varphi_{\tau}\omega) \\ \quad=\varphi _{\tau}(\theta_{-\tau}\omega) \\ \quad=\left ( \textstyle\begin{array}{c}u(\tau,\tau,\theta_{-\tau}\omega,\varphi_{\tau}(\theta _{-\tau}\omega))\\ u_{t}(\tau,\tau,\theta_{-\tau}\omega,\varphi_{\tau}(\theta_{-\tau}\omega )+\varepsilon u(\tau,\tau,\theta_{-\tau}\omega,\varphi_{\tau}(\theta_{-\tau}\omega )-ah(x)z(\omega) \end{array}\displaystyle \right ),\end{gathered} \end{aligned}$$

and

$$\begin{aligned} \varPhi\bigl(t,\tau-t,\theta_{-t}\omega,\varphi_{\tau-t}( \theta_{-t}\omega )\bigr)=\varphi\bigl(\tau,\tau-t, \theta_{-\tau}\omega,\varphi_{\tau-t}(\theta _{-\tau} \omega)\bigr). \end{aligned}$$

So we have \(\varGamma(t,\tau,\omega,Z_{\tau})=R^{-1}_{\varepsilon,\theta_{t}\omega }\varPhi(t,\tau,\omega,\varphi_{\tau})R_{\varepsilon,\theta_{t}\omega}: Z_{\tau}\rightarrow Z(t+\tau,\tau,\theta_{-\tau}\omega,Z_{\tau})\). Next, we use the transformation

$$\psi_{1}=u,\qquad \psi_{2}=u_{t}+\varepsilon u. $$

By using

$$ \psi= \left ( \textstyle\begin{array}{c} \psi_{1} \\ \psi_{2} \end{array}\displaystyle \right ), \qquad\tilde{G}(\psi)= \left ( \textstyle\begin{array}{c}0 \\ g(x,t)-f(u,x)+ah(x)z(\theta_{t}\omega) \end{array}\displaystyle \right ) $$
(18)

and

$$ H(\psi)= \left ( \textstyle\begin{array}{c} \varepsilon\psi_{1}-\psi_{2}\\ A\psi_{1}-\varepsilon(\alpha-\varepsilon)\psi_{1}+(\alpha-\varepsilon )\psi_{2}+q(\psi_{1})(\psi_{2} -\varepsilon\psi_{1}) \end{array}\displaystyle \right ), $$
(19)

Eqs. (1)–(3) can be rewritten as

$$\begin{aligned} \dot{\psi}+H(\psi)=\tilde{G}(\psi),\qquad \psi_{\tau}(\omega)=(u_{\tau},u_{1,\tau}+ \varepsilon u_{\tau})^{T}. \end{aligned}$$
(20)

Thus

$$\begin{aligned} \varPsi(t,\tau,\omega,\psi_{\tau})=T_{\varepsilon}\varGamma(t,\tau ,\omega,Z_{\tau}) T_{-\varepsilon}:\psi_{\tau}\rightarrow \varphi(t+\tau,\tau,\theta_{-\tau}\omega,\psi_{\tau}), \end{aligned}$$
(21)

where

$$\begin{aligned} \varPsi(t,\tau,\omega,\psi_{\tau}) =&\psi(t+\tau,\tau, \theta_{-\tau }\omega,\psi_{\tau}) \\ =&\varphi\bigl(t+\tau,\tau,\theta_{-\tau}\omega,\varphi_{\tau}( \theta_{-\tau }\omega)\bigr)+\bigl(0,ah(x)z(\theta_{t}\omega) \bigr)^{T}. \end{aligned}$$
(22)

Since \(R_{\varepsilon,\theta_{t}\omega}:(a,b)^{T}\rightarrow (a,b+\varepsilon a-ah(x)z(\theta_{t}\omega))^{T}\) is an isomorphism of E, then Φ, Γ, Ψ are equivalent to each other in dynamics.

Therefore, the existence of random attractors in any of these stochastic dynamical systems means that random attractors also exist in other dynamical systems. We will consider the existence of a random attractor for RDSΦ in the following.

Pullback absorbing set

Let \(\varphi=(\varphi_{1},\varphi_{2})^{T}\) be a solution of system (12)–(13). Rewriting system (12)–(13) as

$$\begin{aligned} \dot{\varphi}+Q\varphi=\bar{G}(\varphi,\theta_{t}\omega),\qquad \varphi(- \tau,\omega)=\bigl(u_{0},u_{1}+\varepsilon u_{0}-ah(x)z(\theta_{-\tau}\omega)\bigr)^{T}, \end{aligned}$$
(23)

where

$$Q\varphi= \left ( \textstyle\begin{array}{c} \varepsilon\varphi_{1}-\varphi_{2} \\ A\varphi_{1}-\varepsilon(\alpha-\varepsilon)\varphi_{1}+(\alpha -\varepsilon)\varphi_{2}+q(\varphi_{1}) (\varphi_{2}-\varepsilon\varphi_{1}) \end{array}\displaystyle \right ) $$

and

$$\bar{G}(\varphi,\theta_{t}\omega)= \left ( \textstyle\begin{array}{c} ah(x)z(\theta_{t}\omega) \\ -(q(\varphi_{1})+\alpha-\varepsilon-1)ah(x)z(\theta_{t}\omega)-f(\varphi _{1},x)+g(x,t) \end{array}\displaystyle \right ), $$

we have the following lemmas.

Lemma 1

([26, 27])

For any\(\varphi=(\varphi_{1},\varphi_{2})^{T}\in E\), \((Q\varphi,\varphi)_{E}\geq\frac{\varepsilon}{2}\|\varphi\| _{E}^{2}+\frac{\varepsilon}{4}\|\varphi_{1}\|_{H_{1}^{0}} +\frac{\alpha+\alpha_{1}}{2}\|\varphi_{2}\|^{2}_{L^{2}}\).

Lemma 2

If assumptions\((H_{1})\)\((H_{2})\)hold, then for any\(\tau\in R\), \(\omega\in \varOmega\), there exists a tempered variable\(M_{0}(\omega)\) (independent ofτ) such that, for any set\(B\in D(E)\)and\(\varphi_{\tau-t}(\theta_{-\tau}\omega)\in B(\tau-t,\theta_{-t}\omega)\), there exists\(T=T(\tau,\omega,B)\geq0\)such that, for\(t\geq T\), the solution\(\varphi(\tau,\tau-t, \theta_{-\tau}\omega,\varphi_{\tau-t}(\theta _{-\tau}\omega))\in E\)of (12)(13) satisfies

$$\begin{aligned} \bigl\Vert \varphi\bigl(\tau,\tau-t,\theta_{-\tau}\omega, \varphi_{\tau-t}(\theta _{-\tau}\omega)\bigr) \bigr\Vert _{E}^{2}\leq M_{0}^{2}(\omega),\quad \forall t\geq T(\tau,\omega,B). \end{aligned}$$
(24)

Proof

For any \(\tau\in R\), \(\omega\in\varOmega\), let \(\varphi(r)=\varphi(r,\tau-t,\theta_{\tau-t}\omega,\varphi_{\tau-t}(\theta _{-\tau}\omega))=(\varphi_{1},\varphi_{2})^{T}\in E\) (\(r>\tau-t\)) be a solution of (12)–(13) with

$$\varphi(\tau-t)=\varphi_{\tau-t}(\theta_{-\tau}\omega)= \bigl(u_{\tau -t},u_{1,\tau-t}+\varepsilon u_{\tau-t}-ah(x)z( \theta_{-\tau}\omega)\bigr)^{T}\in E. $$

Taking the inner product \((\cdot,\cdot)_{E}\) of (12) with \(\varphi(r)\), according to Lemma 1, we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert \varphi \Vert ^{2}_{E}+\frac{\varepsilon}{2} \Vert \varphi \Vert ^{2}_{E}+ \frac{\varepsilon}{4} \Vert \varphi_{1} \Vert ^{2}_{H^{0}_{1}}+ \frac{\alpha+\alpha_{1}}{2} \Vert \varphi_{2} \Vert ^{2}_{L^{2}} \leq\bigl(\bar{G}(\varphi,\theta_{r-\tau}\omega ,t),\varphi \bigr)_{E} \end{aligned}$$
(25)

and

$$\begin{aligned}& \bigl(\bar{G}(\varphi,\theta_{r-\tau}\omega,t),\varphi \bigr)_{E} \\& \quad=\bigl(\bigl(ah(x)z(\theta_{r-\tau}\omega), \varphi_{1}\bigr)\bigr)-\bigl(\bigl(\alpha -\varepsilon-1 +q( \varphi_{1})\bigr)ah(x)z(\theta_{r-\tau}\omega), \varphi_{2}\bigr) \\& \qquad{}+\bigl(g(x,r),\varphi_{2}\bigr)-\bigl(f(u,x), \varphi_{2}\bigr). \end{aligned}$$
(26)

By some simple computations, we obtain

$$\begin{aligned}& \bigl(\bigl(ah(x)z(\theta_{r-\tau}\omega),\varphi_{1} \bigr)\bigr)\leq \frac{1}{\sigma} \bigl\vert az(\theta_{r-\tau} \omega) \bigr\vert ^{2} \bigl\Vert h(x) \bigr\Vert ^{2}_{1}+ \frac{\sigma}{4} \Vert \varphi_{1} \Vert ^{2}_{1}, \end{aligned}$$
(27)
$$\begin{aligned}& \begin{gathered}[b]\bigl(-q(\varphi_{1})ah(x)z(\theta_{r-\tau}), \varphi_{2}\bigr) \\ \quad=\delta \Vert \varphi_{2} \Vert _{L^{2}}^{2}+ C_{\delta}(\alpha+\alpha_{2})^{2} \vert a \vert ^{2} \bigl\Vert h(x) \bigr\Vert ^{2}_{L^{2}} \bigl\vert z(\theta _{r-\tau}\omega) \bigr\vert ^{2},\end{gathered} \end{aligned}$$
(28)

and

$$\begin{aligned}& \begin{gathered}[b] \bigl(-(\alpha-\varepsilon-1)ah(x)z(\theta_{r-t}\omega), \varphi_{2}\bigr) \\ \quad\leq\delta \Vert \varphi_{2} \Vert _{L^{2}}^{2}+C_{\delta}( \alpha-\varepsilon)^{2}a^{2} \bigl\Vert h(x) \bigr\Vert ^{2} \bigl\vert z(\theta_{r-\tau}\omega) \bigr\vert ^{2},\end{gathered} \end{aligned}$$
(29)
$$\begin{aligned}& \bigl(g(x,r),\varphi_{2}\bigr)\leq \frac{1}{\alpha} \Vert g \Vert ^{2}+\frac{\alpha}{2} \Vert \varphi_{2} \Vert ^{2}, \end{aligned}$$
(30)

where \(\|g\|^{2}=\sup_{r\in R}\|g(\cdot,r)\|^{2}<\infty\), \(\varphi_{2}=u_{t}+\varepsilon u-ah(x)z(\theta_{t}\omega)\).

By (5)–(7), we have

$$\begin{aligned} \bigl(f(u,x),\varphi_{2}\bigr) =& \bigl(f(u,x),u_{t}+ \varepsilon u-ah(x)z(\theta_{r-\tau}\omega)\bigr) \\ =& \frac{d}{dt} \int_{U}F\bigl(u(r,x),x\bigr)\,dx+\varepsilon \bigl(f(u,x),u \bigr) \\ &{}-\bigl(f(u,x),ah(x)z(\theta_{r-\tau}\omega)\bigr). \end{aligned}$$
(31)

From assumption \((H_{2})\), it is clear that

$$\begin{aligned} \bigl(f(u,x),u\bigr)= \int_{U}f(u,x)u\,dx\geq\frac{1}{c_{1}}\biggl( \int_{U}F\bigl(u(r,x),x\bigr)\, dx- \int_{U}\phi_{2}\,dx\biggr). \end{aligned}$$
(32)

With \(u^{4}\leq\frac{1}{c_{2}}F(u,x)+\phi_{1}\) and \(|f(u,x)|\leq c_{3}(1+u^{4})\), we get

$$\begin{aligned}& \bigl(f(u,x),ah(x)z(\theta_{r-\tau}\omega)\bigr) \\& \quad \leq c_{4} \vert a \vert \bigl\Vert h(x) \bigr\Vert \bigl\vert z(\theta_{r-\tau}\omega) \bigr\vert +c_{5} \vert a \vert \biggl( \int_{U} \vert u \vert ^{4}\,dx \biggr)^{\frac{3}{4}} \bigl\Vert h(x) \bigr\Vert _{L^{4}} \bigl\vert z(\theta _{r-\tau}\omega) \bigr\vert \\& \quad\leq c_{4} \vert a \vert \Vert h \Vert \bigl\vert z(\theta_{r-\tau}\omega) \bigr\vert +\frac{\varepsilon}{2c_{2}}\bar {F}(u,x) \\& \qquad{}+c_{6} \int_{U}\phi_{1}\,dx+c_{7}a^{4} \bigl\Vert h(x) \bigr\Vert ^{4}_{1} \bigl\vert z( \theta _{r-\tau}\omega) \bigr\vert ^{4}, \end{aligned}$$
(33)

here \(\bar{F}(u,x)=\int_{U}F(u,x)\,dx\). By taking (26)–(33) into (25), we have

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\bigl[ \Vert \varphi \Vert ^{2}+2\bar{F}(u,x)\bigr]+\frac{1}{2} \Vert \varphi \Vert _{E}^{2} +\frac{\varepsilon}{4} \Vert \varphi_{1} \Vert _{H_{0}^{1}}^{2}+ \frac{\alpha +\alpha_{2}}{2} \Vert \varphi_{2} \Vert ^{2}_{L^{2}} \\& \qquad{}+\frac{\varepsilon}{c_{1}}\biggl[\bar{F}(u,x)-2 \int_{U}\phi _{2}(x)\,dx\biggr]+c_{4} \vert a \vert \bigl\Vert h(x) \bigr\Vert \bigl\vert z( \theta_{r-\tau}\omega) \bigr\vert +\frac{\varepsilon}{2c_{2}}\bar{F}(u,x) \\& \qquad{}+c_{6} \int _{U}\phi_{1}(x)\,dx+c_{7}a^{4} \bigl\Vert h(x) \bigr\Vert ^{4}_{1} \bigl\vert z( \theta_{r-\tau}\omega ) \bigr\vert ^{4} \\& \quad\leq \frac{1}{\varepsilon}a^{2} \bigl\vert z( \theta_{r-\tau}\omega) \bigr\vert ^{2} \bigl\Vert h(x) \bigr\Vert ^{2}_{1}+\frac{\varepsilon}{4} \Vert u \Vert ^{2}_{1} +\frac{1}{\alpha}\bigl[ \Vert g \Vert ^{2}+a^{2} \bigl\vert z(\theta_{r-\tau} \omega) \bigr\vert ^{2} \bigl\Vert h(x) \bigr\Vert ^{2}\bigr] \\& \quad\quad{}+\frac{\alpha}{2} \Vert \varphi_{2} \Vert ^{2}+\delta \Vert \varphi_{2} \Vert ^{2}_{L^{2}} +C_{\delta}(2\alpha+ \alpha_{2})^{2}a^{2} \bigl\Vert h(x) \bigr\Vert ^{2}_{L^{2}} \bigl\vert z(\theta _{r-\tau} \omega) \bigr\vert ^{2}. \end{aligned}$$
(34)

Let

$$\begin{aligned}[b] \beta(\theta_{r-\tau}\omega)&=-\frac{2\varepsilon}{c_{1}} \int_{U}\phi _{2}(x)\,dx+c_{4} \vert a \vert \bigl\Vert h(x) \bigr\Vert \bigl\vert z( \theta_{r-\tau}\omega) \bigr\vert \\ &\quad+ c_{6} \int_{U}\phi_{1}(x)\,dx+c_{7}a^{4} \bigl\Vert h(x) \bigr\Vert ^{4}_{1} \bigl\vert z( \theta_{r-\tau }\omega) \bigr\vert ^{4} \\ &\quad+\frac{1}{\alpha}\bigl[ \Vert g \Vert ^{2}+a^{2} \bigl\vert z(\theta_{r-\tau}\omega) \bigr\vert ^{2} \bigl\Vert h(x) \bigr\Vert ^{2}\bigr] \\ &\quad+ C_{\delta}(2\alpha+\alpha_{2})^{2}a^{2} \bigl\Vert h(x) \bigr\Vert ^{2}_{L^{2}} \bigl\vert z( \theta _{r-\tau}\omega) \bigr\vert ^{2} \\ &=c_{8}+c_{9}a^{4} \bigl\vert z( \theta_{r-\tau}\omega) \bigr\vert ^{4}.\end{aligned} $$
(35)

By choosing δ small enough, we get

$$\begin{aligned} \frac{d}{dt}y(r)+\rho y(r)\leq\beta(\theta_{r-\tau},\omega),\quad \forall r\geq\tau-t, \end{aligned}$$
(36)

where \(y(r)=\|\varphi(r)\|^{2}_{E}+2\bar{F}(u,x)\) and \(\rho= \min \{\frac{\varepsilon+2}{4}, \frac{\alpha+\alpha_{2}+2}{2}, \frac{2\varepsilon}{c_{1}} \}\). By Gronwall’s inequality to Eq. (36), we have

$$\begin{aligned}& y\bigl(r,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t}( \theta_{-\tau}\omega )\bigr) \\& \quad\leq y\bigl(\tau-t,\tau-t,\theta_{-\tau}\omega, \varphi_{\tau-t}(\theta_{-\tau }\omega)\bigr)e^{-\rho(r+t-\tau)} \\& \quad\quad{}+ \int_{\tau-t}^{r}\beta(\theta_{s-\tau} \omega)e^{-\rho(r-s)}\,ds, \end{aligned}$$
(37)

where

$$\begin{aligned}& y\bigl(\tau-t,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t}( \theta_{-\tau }\omega)\bigr) \\& \quad= \bigl\Vert \varphi_{\tau-t}(\theta_{-\tau}\omega) \bigr\Vert _{E}^{2}+2 \int _{U}F\bigl(u(\tau-t,x),x\bigr)\,dx \\& \quad\leq \bigl\Vert \varphi_{\tau-t}(\theta_{-\tau} \omega) \bigr\Vert ^{2}_{E}+2c_{10}\biggl( \vert U \vert + \Vert u_{\tau -t} \Vert _{1}^{4}+2 \int_{U}\phi_{2}(x)\,dx\biggr), \end{aligned}$$
(38)
$$\begin{aligned}& \int_{\tau-t}^{r}\beta(\theta_{s-\tau})e^{-\sigma(r-s)}\,ds = \frac{c_{8}}{\sigma}+c_{9}a^{4} \int_{\tau-t}^{r} \bigl\vert z( \theta_{s-r}\omega ) \bigr\vert ^{4}e^{-\sigma(r-s)} \end{aligned}$$
(39)

and

$$\begin{aligned}& y\bigl(r,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t}( \theta_{-\tau}\omega )\bigr) \\& \quad\geq \bigl\Vert \varphi\bigl(r,\tau-t,\theta_{-\tau}\omega , \varphi_{\tau-t}(\theta_{-\tau}\omega)\bigr) \bigr\Vert _{E}^{2}-2 \int_{U}\phi _{1}(x)\,dx. \end{aligned}$$
(40)

Thus by (37)–(40), for \(r\geq\tau-t\), we have

$$\begin{aligned}& \bigl\Vert \varphi\bigl(r,\tau-t,\theta_{-\tau}\omega, \varphi_{\tau-t}(\theta_{-\tau }\omega)\bigr) \bigr\Vert ^{2}_{E} \\& \quad\leq y(r)+2 \int_{U}\phi_{1}(x)\,dx \\& \quad\leq \biggl( \bigl\Vert \varphi_{\tau-t}(\theta_{-\tau} \omega) \bigr\Vert ^{2}_{E}+2c_{10}\biggl( \vert U \vert + \Vert u_{\tau-t} \Vert _{1}^{4}+2 \int_{U}\phi_{2}(x)\,dx\biggr) \biggr)e^{-\sigma (r+t-\tau)} \\& \quad\quad{}+2 \int_{U}\phi_{1}(x)\, dx+c_{11}+c_{12}a^{4} \int_{\tau-t}^{r} \bigl\vert z( \theta_{s-r}\omega ) \bigr\vert ^{4}e^{-\sigma(r-s)}\,ds. \end{aligned}$$
(41)

Therefore

$$\begin{aligned}& \bigl\Vert \varphi(r,\tau-t,\theta_{-\tau}\omega), \varphi_{\tau-t}(\theta _{-\tau}\omega) \bigr\Vert ^{2}_{E} \\& \quad\leq \biggl( \bigl\Vert \varphi_{\tau-t}(\theta_{-\tau} \omega) \bigr\Vert ^{2}_{E}+2c_{10}\biggl( \vert U \vert + \Vert u_{\tau-t} \Vert _{1}^{4}+2 \int_{U}\phi_{2}(x)\,dx\biggr) \biggr)e^{-\sigma t} \\& \quad\quad{}+2 \int_{U}\phi_{1}(x)\, dx+c_{11}+c_{12}a^{4} \int_{\tau-t}^{r} \bigl\vert z( \theta_{s-r}\omega ) \bigr\vert ^{4}e^{-\sigma t}\,ds. \end{aligned}$$
(42)

For any set \(B(\tau,\omega)\in B\in D(E)\),

$$\begin{aligned}& \varphi_{\tau-t}(\theta_{-\tau}\omega) \\& \quad=\bigl(u_{\tau-t},u_{1,\tau-t}+\varepsilon u_{\tau-t}-ah(x)z(\theta_{-t}\omega)\bigr)^{T} \\& \quad\in B(\tau-t,\theta_{-t}\omega)\in D(E). \end{aligned}$$
(43)

We have

$$\begin{aligned}& \lim\sup_{t\rightarrow +\infty}\biggl( \bigl\Vert \varphi_{\tau-t}( \theta_{-\tau}\omega) \bigr\Vert _{E}^{2} \\& \quad{}+2c_{10}\biggl( \vert U \vert + \Vert u_{\tau-t} \Vert _{1}^{4}+2 \int_{U}\phi_{2}(x)\, dx\biggr) \biggr)e^{-\rho t}=0. \end{aligned}$$
(44)

Taking

$$M_{0}^{2}(\omega)=2c_{11}+2c_{12}a^{4} \int_{-\infty}^{0} \bigl\vert z(\theta _{s}\omega) \bigr\vert ^{4}e^{\rho s}\,ds< \infty, $$

which is a tempered random variable, \(B_{0}(\omega)=\{\varphi\in E:\|\varphi\|_{E}\leq M_{0}(\omega)\}\) is a close measurable absorbing ball in \(D(E)\), and there exists \(T(\tau,\omega,B)\geq0\) for all \(t\geq T(\tau,\omega,B)\) such that

$$ \varphi\bigl(\tau,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau -t}( \theta_{-\tau}\omega)\bigr)\in B_{0}(\omega). $$
(45)

\(B_{0}(\omega)\) is the random absorbing set for Φ. The proof is completed. □

Decomposition of the equations

In this section, for proving asymptotic compactness of the random dynamical system Φ, we decompose the solution of Eq. (12)–(13) with different initial data into a sum of two parts, one part decays exponentially and another one is bounded in a higher regular space by using the method in [4, 13].

For any \(\tau\in R\) and \(\omega\in\varOmega\), assume that

$$B_{1}(\tau,\omega)=\bigcup_{t\geq T(\tau,\omega,B_{0})} \varphi\bigl(\tau,\tau -t,\theta_{-t}\omega,B_{0}( \theta_{-t}\omega)\bigr)\subseteq B_{0}(\omega). $$

Let \(\varphi(r)=\varphi(r,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau -t}(\theta_{-\tau}\omega))\) be a solution of system (12)–(13), with \(\varphi_{\tau-t}(\theta_{-\tau}\omega)\in B_{1}(\tau-t,\theta_{-t}\omega)\subseteq B_{0}(\theta_{-t}\omega)\). Thus \(\varphi(r)\in B_{0}(\theta_{r-\tau}\omega)\) for all \(r\geq\tau-t\). We decompose \(\varphi(r)\) into

$$\varphi(r)=\varphi_{L}(r)+\varphi_{N}(r),\quad \varphi_{L}(r)=(u_{L},v_{L})^{T}, \varphi_{N}(r)=(u_{N},\varphi_{2N})^{T}, $$

where \(\varphi_{L}(r)\) and \(\varphi_{N}(r)\) satisfy

{φ˙L+QφL+(0f1(uL,x))=0,φL(r,τt,θτω,φτt(θτω))=φL,τt=(uτt,u1,τt+εuτt)T
(46)

and

{φ˙N+QφN+(0f(u,x)f1(u,x))=(ah(x)z(θtω)(q(u)+αε)ah(x)z(θtω)+g(x,r)),[3pt]φN(r,τt,θτω,φτt(θτω))=(0,ah(x)z(θtω))T.
(47)

First, let us estimate the component \(\varphi_{L}\) which decays exponentially.

Lemma 3

Under assumptions\((H_{1})\)\((H_{2})\), for any\(\tau\in R\), \(\omega\in\varOmega\), \(t\geq0\), \(r\geq\tau-t\), and\(\varphi_{\tau-t}(\theta_{-\tau}\omega)\in B_{0}(\theta_{-t}\omega)\), the solution\(\varphi_{L}(r)=\varphi_{L}(r,\tau-t,\varphi_{L,\tau-t})\)of (46) satisfies that

$$\begin{aligned} \bigl\Vert \varphi_{L}(r,\tau-t,\varphi_{L,\tau-t}) \bigr\Vert _{E}^{2}\leq M_{L}^{2}e^{-2\sigma_{1}(t+r-\tau)} \end{aligned}$$
(48)

holds.

Proof

Let \(\varphi_{L}=(\varphi_{L,1},\varphi _{L,2})=(u_{L},v_{L})=(u_{L},u_{L,t}+\varepsilon u)\). Taking the inner product \((\cdot,\cdot)_{E}\) of Eq. (46) with \(\varphi_{L}(r)\), we have

$$ \frac{1}{2}\frac{d}{dt} \bigl\Vert \varphi_{L}(r) \bigr\Vert _{E}^{2}+2(Q\varphi_{L}, \varphi _{L})_{E}=\left(\left ( \textstyle\begin{array}{cc} 0, \\ -f_{1}(u_{L},x) \end{array}\displaystyle \right ),\varphi_{L}\right), $$
(49)

where ((0,f1(uL,x)),φL)=ddtUF(φL1,x)εUf(φL1,x)φL1. By Lemma 1, we see that

$$ 2\bigl(Q(\varphi_{L}),\varphi_{L}\bigr)\geq \varepsilon \Vert \varphi_{L} \Vert _{E}^{2}+ \frac{\varepsilon}{2} \Vert \varphi _{L,1} \Vert _{H^{2}_{0}}^{2}+(\alpha+\alpha_{1}) \Vert \varphi_{L,2} \Vert ^{2}_{L^{2}}. $$
(50)

By assumption \((H_{2})\), we obtain

$$ f(\varphi_{L1},x),\varphi_{L1})=\bigl(f(u,x),u\bigr) \geq\frac{1}{C_{4}} \bar{F}_{1}(u_{L},x)- \varepsilon \int_{U}\phi_{2}(x)\,dx. $$
(51)

Thus, by Eqs. (49)–(51), we have

$$\begin{aligned}& \frac{d}{dt}\bigl[ \Vert \varphi_{L} \Vert ^{2}_{E}+2\bar {F_{1}}(u_{L},x) \bigr]+\varepsilon \Vert \varphi_{L} \Vert ^{2}_{E}+ \frac{\varepsilon}{2} \Vert \varphi_{L,1} \Vert _{H^{2}_{0}}^{2} \\& \qquad{}+(\alpha+\alpha_{1}\|\varphi_{L,2})\| ^{2}_{L^{2}} +\frac{2\varepsilon}{C_{4}}\bar{F_{1}}(u_{L},x) \\& \quad\leq 2\varepsilon \int_{U}\phi_{2}(x)\,dx. \end{aligned}$$
(52)

Hence we can conclude that there exists \(\sigma_{L}=\min\{\frac{2\varepsilon}{C_{4}},\frac{\varepsilon}{2},\frac {\alpha+\alpha_{2}}{2}\}\) such that

$$ \frac{d}{dt}y_{L}+\sigma_{L}y_{L}(r) \leq\beta_{L}, $$
(53)

where \(y_{L}(r)=\|\varphi_{L}(r)\|^{2}_{E}+2\bar{F_{L}}(u_{L},x)\geq 0\), \(\beta_{L}=2\varepsilon\int_{U}\phi_{2}(x)\,dx\). Since \(\varphi_{\tau-t}(\theta_{-\tau}\omega)+ (0, ah(x)z(\theta_{-t}\omega ))^{T}\in B_{0}(\theta_{-t}\omega)\), we have

$$\Vert \varphi_{L,\tau-t} \Vert _{E}\leq M_{0}(\theta_{-t}\omega)+ \vert a \vert \bigl\Vert h(x) \bigr\Vert . \bigl\vert z(\theta_{-t}\omega) \bigr\vert . $$

Notice that \(\varphi_{L,\tau-t}\) is independent of ω, so replacing ω by \(\theta_{-t}\omega\), then

$$\Vert \varphi_{L,\tau-t} \Vert _{E}\leq M_{0}(\omega)+ \vert a \vert \bigl\Vert h(x) \bigr\Vert . \bigl\vert z(\omega) \bigr\vert . $$

Applying Gronwall’s inequality to Eq. (53), we have

$$\begin{aligned} \bigl\Vert \varphi_{L}(r,\tau-t,\varphi_{L,\tau-t}) \bigr\Vert _{E}^{2} \leq& y_{L}(r,\tau -t, \varphi_{L,\tau-t}) \\ \leq& y(\tau-t,\tau-t,\varphi_{L,\tau-t})e^{-\sigma_{L}(t+r-\tau )}+ \frac{\beta_{L}}{\sigma_{L}} \\ \leq& \biggl( \Vert \varphi_{L,\tau-t} \Vert ^{2}_{E}+c_{10} \biggl( \vert U \vert + \Vert u_{\tau-t} \Vert ^{4}_{1}+ \int_{U}\phi_{2}\,dx\biggr)\biggr)e^{-\sigma_{L}(t+r-\tau)} +\frac{\beta_{L}}{\sigma_{L}} \\ =&M_{1}^{2}(\omega). \end{aligned}$$
(54)

Next, we consider Eq. (49). Due to \((f_{1}(u_{L},x),u_{L})\geq0\), \(|f_{1}(u_{L}(r),x)|\leq c_{13}(|u_{L}(r)|^{3}+|u_{L}(r)|)\) and assumption \((H_{2})\), according to Sobolev embedding \(H_{0}^{1}(U)\subset L^{4}(U)\subset L^{2}(U)\), there exists \(M_{2}(\omega)>0\) such that

$$ 0\leq\bar{F_{1}}\bigl(u_{L}(r),x\bigr)\leq c_{14}\bigl( \bigl\Vert u_{L}(r) \bigr\Vert _{L^{4}}^{4}+ \bigl\Vert u_{L}(r) \bigr\Vert ^{2}\bigr)\leq M_{2}(\omega) \bigl\Vert u_{L}(r) \bigr\Vert ^{2}_{1}. $$
(55)

That is, \(\|u_{L}(r)\|^{2}_{1}\geq \frac{1}{M_{2}(\omega)}\bar{F}_{1}(u_{L}(r),x)\). From (49)–(50) and (55), for any \(r\geq\tau-t\), we have

$$\begin{aligned} \frac{d}{dt}\bigl[ \Vert \varphi_{L} \Vert _{E}^{2}+2\bar{F}_{1}(u_{L},x) \bigr]+\frac {\varepsilon}{2} \Vert \varphi_{L} \Vert _{E}^{2}+\frac{\varepsilon}{2M_{2}(\omega )}\bar{F}_{1}(u_{L},x) \leq 0. \end{aligned}$$
(56)

So the inequality

$$\frac{d}{dt}\bigl[ \Vert \varphi_{L} \Vert _{E}^{2}+2\bar{F}_{1}(u_{L},x) \bigr]+2\sigma _{1}(\omega)\bigl[ \Vert \varphi_{L} \Vert ^{2}_{E}+2\bar{F}_{1}(u_{L},x) \bigr]\leq 0 $$

holds, where \(\sigma_{1}=\min\{\frac{\varepsilon}{4},\frac{\varepsilon}{8M_{2}(\omega )}\}>0\). Thus, we get

$$\begin{aligned} \bigl\Vert \varphi_{L}(r,\tau-t,\varphi_{L,\tau-t}) \bigr\Vert ^{2}_{E} \leq&\bigl[ \Vert \varphi _{L,\tau-t} \Vert ^{2}_{E}+2 \bar{F}_{1}(u_{L,\tau-t},x)\bigr]e^{-2\sigma _{1}(r+t-\tau)} \\ \leq&\biggl( \Vert \varphi_{L,\tau-t} \Vert ^{2}_{E}+c_{10} \biggl( \vert U \vert + \Vert u_{\tau-t} \Vert ^{4}_{1} + \int_{U}\phi_{2}\,dx\biggr)\biggr)e^{-2\sigma_{1}(t+r-\tau)} \\ =& M_{L}^{2}e^{-2\sigma_{1}(r+t-\tau)}. \end{aligned}$$
(57)

The proof is completed. □

For the component \(\varphi_{N}\), which is ultimately pullback bounded in a higher regular space, we have the following estimate.

Lemma 4

If assumptions\((H_{1})\)\((H_{2})\)hold, then for any\(\tau\in R\), \(\omega \in\varOmega\), and\(t\geq0\), there exist a positive constant\(\nu\in(0,\min\{\frac{1}{4},\frac{3-p}{4}\})\)and a positive-value random variable\(M_{N}(\omega)>0\)such that the solution\(\varphi_{N}(r)=(\varphi_{1N}(r),\varphi _{2N}(r))^{T}\)of Eq. (47) satisfies the following:

$$\begin{aligned} \bigl\Vert A^{\nu}\varphi_{N}\bigl(r,\tau-t, \theta_{-\tau}\omega,\varphi_{\tau -t}(\theta_{-\tau} \omega)\bigr) \bigr\Vert _{E}^{2}\leq M_{N}^{2}(\omega) \end{aligned}$$
(58)

for\(t\geq0\)and\(\varphi_{\tau-t}(\theta_{-\tau}\omega)\in B_{0}(\theta_{-t}\omega)\).

Proof

Taking the inner product of Eq. (47) in E with

$$A^{2\nu}\varphi_{N}=\bigl(A^{2\nu} \varphi_{1N},A^{2\nu}\varphi_{2N}\bigr)= \bigl(A^{2\nu }u_{N},A^{2\nu}\varphi_{2N} \bigr), $$

we have

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\biggl[A^{\nu} \Vert \varphi_{N} \Vert ^{2}+2 \int _{U}\bigl[f(u,x)-f_{1}(u_{L},x) \bigr]A^{2\nu}u_{N}\,dx\biggr]+\bigl(Q\bigl( \varphi_{N},A^{2\nu}\bigr)\bigr) \\& \qquad{}+\varepsilon \int_{U}\bigl[f(u,x)-f_{1}(u_{L},x) \bigr]A^{2\nu}u_{N}\,dx \\& \qquad {}- \int_{U}\bigl(\bigl[f'_{1,u}(u,x)-f'_{1,u}(u_{L},x) \bigr]u_{t}+f'_{1,u}(u_{L},x)u_{Nt}+f'_{2,u}(u,x)u_{t} \bigr)A^{2\nu}u_{N}\,dx \\& \quad=\bigl(ah(x)z(\theta_{r-\tau}\omega),A^{2\nu }u_{N} \bigr)_{1}-\bigl(f(u,x)-f_{1}(u_{L},x),A^{2\nu} ah(x)z(\theta_{r-\tau}\omega)\bigr) \\& \qquad{} +\bigl(g(x,r)-(\alpha-\varepsilon-1)ah(x)z(\theta_{r-\tau} \omega),A^{2\nu }\varphi_{2N}\bigr) \\& \quad\quad{} +\bigl(-q(\varphi_{1N})ah(x)z(\theta_{r-\tau} \omega), A^{2\nu}\varphi_{2N}\bigr). \end{aligned}$$
(59)

Similar to (50), we get

$$\begin{aligned} \frac{\varepsilon}{2} \bigl\Vert A^{\nu}\varphi_{N} \bigr\Vert _{E}^{2}+\frac{\varepsilon }{4} \bigl\Vert A^{\nu}u_{N} \bigr\Vert ^{2}_{H_{0}^{1}}+ \frac{\alpha+\alpha_{1}}{2} \bigl\Vert A^{\nu }\varphi_{2N} \bigr\Vert _{L^{2}}^{2} \leq\bigl(Q \varphi_{N},A^{2\nu} \varphi_{N}\bigr)_{E}. \end{aligned}$$
(60)

By some computations, we have

$$\begin{aligned}& \bigl(ah(x)z(\theta_{r-\tau}\omega), A^{2\nu}u_{N} \bigr)\leq \frac{2a^{2}}{\varepsilon}z^{2}(\theta_{r-\tau}\omega) \bigl\Vert h(x) \bigr\Vert _{1}^{2}+ \frac{\varepsilon}{8} \bigl\Vert A^{\nu+\frac{1}{2}}u_{N} \bigr\Vert ^{2}, \end{aligned}$$
(61)
$$\begin{aligned}& \bigl(f(u,x)-f_{1}(u_{L},x), A^{2\nu}ah(x)z( \theta_{r-\tau}\omega)\bigr) \\& \quad\leq K_{1}(r,\tau-t,\theta_{-t} \omega)+c_{15}a^{2} \bigl\Vert h(x) \bigr\Vert _{1}^{2} \bigl\vert z(\theta_{r-\tau}\omega) \bigr\vert ^{2}, \end{aligned}$$
(62)

and

$$\begin{aligned}& \bigl(g(x,r)-(\alpha-\varepsilon-1)ah(x) z(\theta_{r-\tau}\omega), A^{2\nu }\varphi_{2N}\bigr) \\& \quad\leq \frac{2}{\alpha}\bigl[ \Vert g \Vert _{1}^{2}+( \alpha-\varepsilon-1)^{2}a^{2} \bigl\Vert h(x) \bigr\Vert ^{2}_{2}z^{2}(\theta_{r-\tau}\omega)\bigr]+\frac{\alpha}{4} \bigl\Vert A^{\nu}\varphi_{2N} \bigr\Vert ^{2}, \end{aligned}$$
(63)

where \(\|g\|^{2}_{1}=\sup_{r\in R}\|g(\cdot,r)\|^{2}_{1}<\infty\). Thus, by taking (60)–(63) into (59), we obtain

$$\begin{gathered}[b] \frac{1}{2}\frac{d}{dt}\biggl( \bigl\Vert A^{\nu}\varphi_{N} \bigr\Vert ^{2}_{E}+2 \int _{U}\bigl[f(u,x)-f_{1}(u_{L},x) \bigr]A^{2\nu}u_{N}\,dx\biggr) +\sigma_{2} \bigl\Vert A^{\nu}\varphi_{N} \bigr\Vert _{E}^{2} \\ \qquad{}+\varepsilon \int_{U}\bigl[f(u,x)-f_{1}(u_{L},x) \cdot A^{2\nu}u_{N}\,dx\bigr] - \int_{U}\bigl(\bigl[f'_{1,u}(u,x) \\ \qquad{} -f'_{1,u}(u_{L},x) \bigr]u_{t}+f'_{1,u}(u_{L},x)u_{N,t}+f'_{2,u}(u,x)u_{t} \bigr)\cdot A^{2\nu}u_{N}\,dx \\ \quad\leq K_{2}(r,r-t,\theta_{-t} \omega)+c_{16}a^{2}z^{2}({ \theta_{r-\tau}\omega}).\end{gathered} $$
(64)

Let \(\sigma_{2}=\min\{\frac{\varepsilon}{4},\frac{2\varepsilon+\alpha}{4}\}\), then by Hölder’s inequality, we have

$$\begin{aligned}& \bigl\vert \bigl(f'_{1,u}(u_{L},x)u_{N,t},A^{2\nu} u_{N}\bigr) \bigr\vert \\& \quad\leq c_{17}\biggl( \int_{U}\bigl(1+u_{L}^{2} \bigr)^{3}\,dx\biggr)^{\frac{1}{3}}\biggl( \int_{U} \bigl\vert A^{2\nu }u_{N} \bigr\vert ^{\frac{6}{1+4\nu}}\,dx\biggr)^{\frac{1+4\nu}{6}} \biggl( \int_{u} \vert u_{N,t} \vert ^{\frac{6}{3-4\nu}}\,dx\biggr)^{\frac{3-4\nu}{6}} \\& \quad\leq c_{18} \bigl\Vert A^{\nu+\frac{1}{2}}u_{N} \bigr\Vert \cdot \bigl\Vert A^{\nu}u_{N,t} \bigr\Vert \end{aligned}$$
(65)

for \(r>\tau-t\). We have

$$\begin{aligned}& \bigl\vert \bigl(f'_{2,u}(u,x)u_{t}, A^{2\nu} u_{N}\bigr) \bigr\vert \\& \quad\leq c_{19} \int_{U} \vert u_{t} \vert \cdot \bigl(1+ \vert u \vert ^{p}\bigr)\cdot \bigl\vert A^{2\nu}u_{N} \bigr\vert \,dx \\& \quad\leq \biggl( \int_{U} \vert u_{t} \vert ^{2} \,dx\biggr)^{\frac{1}{2}}\biggl( \int_{U}\bigl(1+ \vert u \vert ^{p} \bigr)^{\frac {6}{2-4\nu}}\,dx\biggr)^{\frac{2-4\nu}{6}}\biggl( \int_{U} \bigl\vert A^{2\nu}u_{N} \bigr\vert ^{\frac {6}{1+4\nu}}\biggr)^{\frac{1+4\nu}{6}} \\& \quad\leq \Vert u_{t} \Vert _{0}\bigl(1+ \Vert u \Vert _{1}^{p}\bigr) \bigl\Vert A^{\nu+\frac{1}{2}}u_{N} \bigr\Vert \leq K_{3}(r, \tau-t,\theta_{-t}\omega) \bigl\Vert A^{\nu+\frac{1}{2}}u_{N} \bigr\Vert \end{aligned}$$
(66)

and

$$\begin{aligned}& \bigl\vert \bigl(\bigl[f'_{1,u}(u,x)-f'_{1,u}(u_{L},x) \bigr]u_{t},A^{2\nu}u_{N}\bigr) \bigr\vert \\& \quad\leq \int_{U}|u_{t}|\bigl(1+ \vert u_{N} \vert + \vert u_{L} \vert \bigr) \vert u_{N}| \bigl\vert A^{2\nu}u_{N} \bigr\vert \,dx \\& \quad\leq \biggl( \int_{U} \vert u_{t} \vert ^{2} \,dx\biggr)^{\frac{1}{2}}\biggl( \int _{u}\bigl(1+ \vert u_{N} \vert + \vert u_{L} \vert \bigr)^{6}\,dx \biggr)^{\frac{1}{6}} \\& \qquad{}\times\biggl( \int_{U} \vert u_{N} \vert ^{\frac{6}{3-4\nu}}\,dx\biggr)^{\frac{3-4\nu}{6}} \biggl( \int_{U} \bigl\vert A^{2\nu}u_{N} \bigr\vert ^{\frac{6}{1+4\nu}}\,dx\biggr)^{\frac{1+4\nu }{6}} \\& \quad\leq K_{4}(r,\tau-t,\theta_{-t}\omega) \bigl\Vert A^{\nu+\frac{1}{2}}u_{N} \bigr\Vert . \end{aligned}$$
(67)

By putting the above inequalities into (64), we get

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\biggl( \bigl\Vert A^{\nu}\varphi_{N} \bigr\Vert ^{2}_{E}+2 \int _{U}\bigl[f(u,x)-f_{1}(u_{L},x) \bigr]A^{2\nu}u_{N}\,dx\biggr) \\& \qquad{}+\sigma_{1} \biggl[ \bigl\Vert A^{\nu}\varphi_{N} \bigr\Vert ^{2}_{E}+2 \int _{U}\bigl[f(u,x)-f_{1}(u_{L},x) \bigr]A^{2\nu}u_{N}\,dx\biggr] \\& \quad\leq c_{20} \bigl\Vert A^{\nu+\frac{1}{2}}u_{N} \bigr\Vert \bigl\Vert A^{\nu} u_{N,t} \bigr\Vert +K_{2}(r,\tau-t,\theta_{-t}\omega)+ c_{16}a^{2}z^{2}( \theta_{r-\tau}\omega) \\& \quad\quad{}+ \Vert u_{t} \Vert _{0}\bigl(1+ \Vert u \Vert ^{p}_{1} \bigl\Vert A^{\nu+\frac{1}{2}}u_{N} \bigr\Vert \bigr)+K_{4}(r, \tau-t,\theta_{-t}\omega) \bigl\Vert A^{\nu+\frac{1}{2}}u_{N} \bigr\Vert . \end{aligned}$$
(68)

That is,

$$\begin{aligned}& \frac{d}{dt}y_{1}\bigl(r,\tau-t,\theta_{-t} \omega,\varphi_{\tau-t}(\theta _{-t}\omega)\bigr)+ \sigma_{1}y_{1}\bigl(r,\tau-t,\theta_{-t} \omega,\varphi_{\tau -t}(\theta_{-t}\omega)\bigr) \\& \quad\leq K_{5}(r,\tau-t,\theta_{-t} \omega)+c_{21}a^{2}z^{2}( \theta_{r-\tau}\omega) \end{aligned}$$
(69)

and

$$\begin{aligned}& y_{1}(r)= \bigl\Vert A^{\nu}\varphi_{N} \bigr\Vert ^{2}_{E}+2 \int _{U}\bigl[f(u,x)-f_{1}(u_{L},x) \bigr]A^{2\nu}u_{N}, \end{aligned}$$
(70)
$$\begin{aligned}& \beta_{2}(\theta_{r-\tau }\omega)=K_{5}(r, \tau-t,\theta_{-t}\omega)+c_{21}a^{2}z^{2}( \theta _{r-\tau}\omega). \end{aligned}$$
(71)

It follows from (69) that

$$\begin{aligned} \frac{d}{dt}y_{1}(r)+\sigma_{1}y_{1}(r) \leq \beta_{2}(\theta_{r-\tau}\omega),\quad \forall r\geq\tau-t. \end{aligned}$$
(72)

Note that \(y_{1}(\tau-t,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t}\omega)\leq a^{2}\|A^{\nu}h(x)\|^{2}z^{2}(\theta_{-t}\omega)\), then by applying Gronwall’s inequality to (72) on \([\tau-t,r]\) (\(r\geq \tau-t\)), we have

$$\begin{aligned}& y_{1}(r,\tau-t,\theta_{-t}\omega,\varphi_{\tau-t} \omega) \\& \quad\leq y_{1}\bigl(\tau-t,\tau-t,\theta_{-\tau}\omega, \varphi_{\tau-t}(\theta_{-\tau }\omega)\bigr)+ \int_{\tau-t}^{r}\beta_{2}( \theta_{\tau-t}\omega)e^{r-\tau+t} \\& \quad\leq c_{21} \bigl\Vert h(x) \bigr\Vert _{1}^{2}a^{2}z^{2}( \theta_{-t}\omega)e^{r-\tau+t}+ \int _{\tau-t}^{r}\beta_{2}( \theta_{\tau-t}\omega)e^{r-\tau+t} \\& \quad\leq a^{2}M_{8}(\omega)+M_{9}( \tau,\omega) \end{aligned}$$
(73)

for

$$\begin{aligned}& \biggl\vert \biggl[ \int_{U}f(u,x)-f_{1}(u_{L},x) \biggr] \biggr\vert A^{2\nu}u_{N}\,dx \\& \quad= \int_{u}\bigl[f_{2}(u,x)+f_{1}(u,x)-f_{1}(u_{L},x) \bigr]A^{2\nu}u_{N}\,dx, \end{aligned}$$
(74)

where

$$\begin{aligned} \biggl\vert \int_{U}f_{2}(u,x)A^{2\nu}u_{N} \,dx \biggr\vert \leq& c_{31} \int_{U}\bigl(1+ \vert u \vert ^{p} \bigr)A^{2\nu}u_{N}\,dx \\ \leq& c_{31}\biggl( \int_{U}\bigl(1+ \vert u \vert ^{p} \bigr)^{\frac{6}{5-4\nu}}\biggr)^{\frac{5-4\nu}{6}}\biggl( \int _{u} \bigl\vert A^{2\nu}u_{N} \bigr\vert ^{\frac{6}{1+4\nu}}\biggr)^{\frac{1+4\nu}{6}} \\ \leq& K_{8}(r,\tau-t,\theta_{-t}\omega) \bigl\Vert A^{\nu+\frac{1}{2}}u_{N} \bigr\Vert \\ \leq& K^{2}_{8}(r,\tau-t,\theta_{-t} \omega)+\frac{1}{2} \bigl\Vert A^{\nu+\frac {1}{2}}u_{N} \bigr\Vert ^{2} \end{aligned}$$
(75)

and

$$\begin{aligned}& \int_{U} \bigl\vert \bigl[f_{1}(u,x)-f_{1}(u_{L},x) \bigr]A^{2\nu}u_{N} \bigr\vert \,dx \\& \quad\leq c_{32} \int_{U}\bigl(1+ \vert u_{N} \vert ^{2}+ \vert u_{L} \vert ^{2}\bigr) \vert u_{N} \vert \bigl\vert A^{2\nu}u_{N} \bigr\vert \,dx \\& \quad\leq c_{33}\biggl( \int_{U}\bigl(1+ \vert u_{N} \vert ^{2}+ \vert u_{L} \vert ^{2} \bigr)^{3}\biggr)^{\frac{1}{3}}\biggl( \int _{U} \vert u_{N} \vert ^{\frac{6}{3-4\nu}}\biggr)^{\frac{3-4\nu}{6}} \\& \quad\quad{}\times\biggl( \int_{U} \bigl\vert A^{2\nu}u_{N} \bigr\vert ^{\frac{6}{1+4\nu}}\biggr)^{\frac{1+4\nu }{6}} \\& \quad\leq K_{9}^{2}(r,\tau-t,\theta_{-t} \omega)+\frac{1}{8} \bigl\Vert A^{\nu+\frac {1}{2}}u_{N} \bigr\Vert ^{2}. \end{aligned}$$
(76)

It follows from above that

$$y_{1}(r)= \bigl\Vert A^{\nu}\varphi_{N} \bigr\Vert ^{2}+2 \int _{U}\bigl[f(u,x)-f_{1}(u_{L},x) \bigr]A^{2\nu}u_{N}\,dx, $$

where

$$\begin{aligned}& \int_{u}\bigl[f(u,x)-f_{1}(u_{L},x) \bigr]A^{2\nu}u_{N}\,dx \\& \quad\leq K_{8}^{2}(r,\tau-t,\theta_{-t} \omega)+\frac{1}{8} \bigl\Vert A^{\nu+\frac {1}{2}}u_{N} \bigr\Vert ^{2} \\& \quad\quad{}+K_{9}^{2}(r,\tau-t,\theta _{-t}\omega)+\frac{1}{8} \bigl\Vert A^{\nu+\frac{1}{2}}u_{N} \bigr\Vert ^{2} \\& \quad\leq K_{10}^{2}(r,\tau-t,\theta_{-t} \omega)+\frac{1}{4} \bigl\Vert A^{\nu+\frac {1}{2}}u_{N} \bigr\Vert ^{2}. \end{aligned}$$
(77)

So we obtain

$$\begin{aligned} \bigl\Vert A^{\nu}\varphi_{N} \bigr\Vert =& \bigl\Vert A^{\nu+\frac{1}{2}}u_{N} \bigr\Vert ^{2}+ \bigl\Vert A^{\nu }u_{N,t} \bigr\Vert ^{2} \\ \leq& y_{1}\bigl(r,\tau-t,\theta_{-t}\omega, \varphi_{\tau-t}(\theta_{-t}\omega )\bigr)+K_{10}^{2}(r, \tau-t,\theta_{-t}\omega) \\ \leq& \alpha^{2}M_{8}(\omega)+M_{9}+K_{10}^{2}(r, \tau-t,\theta_{-t}\omega)\leq M_{N}^{2}( \omega). \end{aligned}$$
(78)

The proof is completed. □

Lemma 5

For any\(\tau\in R\), \(\omega\in\varOmega\), and\(t>0\), assume that\(B_{\nu}(\tau,\omega)\subseteq B_{1}(\tau,\omega)\subseteq B_{0}(\omega)\)and\(B_{\nu}(\tau,\omega)\in D_{E^{\nu}}\), whereνis as in Lemma 4, then if assumptions\((H_{1})\)\((H_{2})\)hold, then there exist a random variable\(t_{\nu}(\omega)> 0\)and a tempered random variable\(M_{\nu}(\omega)> 0\)such that, for any\(t\geq t_{\nu}(\omega)\), \(\varphi_{\tau-t}(\theta_{-t}\omega)\subset B_{\nu}(\tau-t,\theta_{-t}\omega)\subseteq B_{0}(\theta_{-t}\omega)\cap D(E^{\nu})\), the solutionφof Eqs. (12)(13) satisfies

$$\begin{aligned}& \bigl\Vert \varphi\bigl(\tau,\tau-t,\theta_{-\tau}\omega, \varphi_{\tau-t}(\theta _{-\tau}\omega)\bigr) \bigr\Vert _{E^{\nu}}^{2} \\& \quad = \bigl\Vert A^{\nu}\varphi\bigl(\tau,\tau-t, \theta_{-\tau}\omega,\varphi_{\tau -t}(\theta_{\tau-t} \omega)\bigr) \bigr\Vert _{E}^{2} \leq M_{\nu}^{2}(\omega). \end{aligned}$$
(79)

Proof

Taking the inner product of Eqs. (12)–(13) in E with \(A^{2\nu}\varphi=(A^{2\nu}u,A^{2\nu}v)^{T}\), then for any \(r\geq \tau-t\), we have

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\biggl( \bigl\Vert A^{2\nu}\varphi \bigr\Vert _{E}^{2}+2 \int _{U}f(u,x)A^{2\nu}u \,dx\biggr)+\bigl(Q \varphi,A^{2\nu}\varphi\bigr)_{E} \\& \quad \quad{}+\varepsilon \bigl(f(u,x),A^{2\nu}\varphi\bigr)+ \bigl(f(u,x),aA^{2\nu}h(x)z(\theta_{r-\tau}\omega )\bigr) \\& \quad=\bigl(ah(x)z(\theta_{r-\tau}\omega),A^{2\nu}u\bigr) + \bigl(g(x,r) \\& \quad\quad {}+(\varepsilon-\alpha+1)ah(x)z(\theta_{r-\tau} \omega),A^{2\nu}v\bigr). \end{aligned}$$
(80)

The same to (75), the following inequality holds:

$$\begin{aligned} \biggl\vert \int_{U}f(u,x)A^{2\nu}u\,dx \biggr\vert \leq& c_{3}\int_{U}\bigl\vert \bigl(1+u^{4}\bigr) \bigr\vert \bigl\vert A^{2\nu}u \bigr\vert \,dx \\ \leq& c_{3}\biggl( \int_{U}\bigl(1+u^{4}\bigr)^{\frac{6}{5-4\nu}}\,dx \biggr)^{\frac{5-4\nu}{6}} \biggl( \int_{U} \bigl\vert A^{2\nu}u \bigr\vert ^{\frac{6}{1+4\nu}}\biggr)^{\frac{1+4\nu}{6}} \\ \leq& c_{3}M_{6}(\theta_{r-\tau} \omega)+ \frac{\mu}{4} \bigl\Vert A^{\nu+\frac{1}{2}}u \bigr\Vert ^{2}. \end{aligned}$$
(81)

Similar to (72), by (80) and (81), we get

$$ \frac{d}{dt}y_{2}+\sigma_{1}y_{2} \leq \beta_{3}(\theta_{r-\tau}\omega), $$
(82)

where

$$\begin{aligned}& y_{2}= \bigl\Vert A^{\nu}\varphi(r) \bigr\Vert _{E}^{2}+2 \int _{U}f(u,x)A^{2\nu}\,dx \geq \frac{1}{2} \bigl\Vert A^{\nu}\varphi(r) \bigr\Vert _{E}^{2}-c_{3}M_{6}( \theta_{r-\tau }\omega), \end{aligned}$$
(83)
$$\begin{aligned}& \beta_{3}=K_{11}(r,\tau-t,\theta_{t}\omega )+c_{22}a^{2}z^{2}(\theta_{r-\tau} \omega), \end{aligned}$$
(84)
$$\begin{aligned}& y_{2}(r)\leq y_{2}(\tau-t)e^{-\sigma_{1}(r+t-\tau)}+ \int_{\tau-t}^{r}\beta_{3}(\theta _{\xi-\tau}\omega)e^{-\sigma_{1}(r-\xi)}\,d\xi. \end{aligned}$$
(85)

By applying Gronwall’s inequality to (82) on \([\tau-t,r]\), one has

$$\begin{aligned}& \bigl\Vert A^{\nu}\varphi\bigl(\tau,\tau-t,\theta_{-\tau} \omega,\varphi_{\tau -t}(\theta_{-\tau}\omega)\bigr) \bigr\Vert \\& \quad\leq 2y_{2}\bigl(\tau,\tau-t,\theta_{-\tau}\omega, \varphi_{\tau-t}(\theta_{-\tau }\omega)\bigr)+c_{3}M_{6}( \omega) \\& \quad\leq 2y_{2}(\tau-t)e^{-\sigma_{1}t}+2 \int_{\tau-t}^{r}\beta_{3}( \theta_{\xi -t}\omega)e^{-\sigma_{1}(\tau-\xi)}\,d\xi+4c_{3}M_{6}( \omega) \\& \quad\leq 2y_{2}(\tau-t)e^{-\sigma_{1}t}+2\biggl[ \int_{-\tau }^{0}(K_{11}(r,\tau-t, \theta_{t}\omega)+c_{22}a^{2}z^{2}( \theta_{r-\tau}\omega)\biggr]e^{\sigma_{1}\xi}\,d\xi \\& \quad\quad{} +4c_{3}M_{6}( \omega). \end{aligned}$$
(86)

From (81), (83), (85), and \(\varphi_{\tau-t}(\theta_{-\tau}\omega)\in B_{0}(\theta_{-t}\omega)\cap D(E)\), it is clear that as \(t\rightarrow+\infty\),

$$y_{2}(\tau-t)e^{-\sigma_{1}t}\leq\biggl(\frac{3}{2} \bigl\Vert A^{\nu}\varphi_{\tau -t} \bigr\Vert \biggr)_{E}^{2} +2c_{3}M_{6}( \theta_{-t}\omega)e^{-\sigma_{1}t}\rightarrow0. $$

Taking

$$M_{\nu}^{2}=4c \int_{-\infty}^{0}(K_{11}(r,\tau-t, \theta_{t}\omega )+c_{22}a^{2}z^{2}( \theta_{r-\tau}\omega)]e^{\sigma_{1}\xi}\,d\xi +8c_{3}M_{6}( \omega), $$

then the proof is completed. □

Existence of random attractor

Lemma 6

If assumptions\((H_{1})\)\((H_{2})\)hold, then for any\(\tau\in R\), \(\omega \in \varOmega\), there exist\(T_{\nu}(\omega)\geq0\), a random bounded ball\(\hat{B}_{1}\)ofE, a positive numberρ̂, and a tempered random variable\(\hat{Q}(\omega)\)such that, for any\(t\geq T_{\nu}(\omega)\)and\(\varphi_{\tau-t}(\theta_{-\tau}\omega)\in B_{1}(\tau-t,\theta_{-t}\omega)\), the solutionφof (12)(13) satisfies

$$\begin{aligned} d_{E}\bigl(\varphi\bigl(\tau,\tau-t,\theta_{-\tau} \omega,B_{1}(\tau-t,\theta_{-t}\omega)\bigr),\hat{B}_{1}(\omega)\bigr)\leq \hat{Q}(\theta_{-t} \omega)e^{-\hat{\rho}t}. \end{aligned}$$
(87)

By Lemma 7.6 in [2], Lemma 3, and Lemma 4, one can prove Lemma 6. Since the proof of Lemma 6 is similar to that of Lemma 3.8 in [29], we omit it here. From Lemmas 5 and 6, it is easy to see the existence of a random attractor for the cocycle Φ.

Theorem 2

If assumptions\((H_{1})\)and\((H_{2})\)hold, then the cocycleΦassociated with (12)(13) possesses a\(D(E)\)-pullback random attractor\(A\in D(E)\)such that, for any\(\tau\in R\), \(\omega\in\varOmega\), \(A(\tau,\omega)\subseteq\hat{B_{1}}(\omega)\cap B_{0}(\omega)\), where\(B_{0}(\omega)\)and\(\hat{B}_{1}(\omega)\)are as in (45) and Lemma 6, respectively.

Proof

For any \(\tau\in R \) and \(\omega\in\varOmega\), by Lemma 6 and the compactness of embedding \(E^{\nu}\rightarrow E\), \(\hat{B}_{1}(\omega)\) is a compact measurable \(D(E)\)-pullback attracting ball in E. By Proposition 2.1, the cocycle Φ has a \(D(E)\)-pullback random attractor \(\mathcal{A}\in D(E)\) such that, for any \(\tau\in R\), \(\omega\in\varOmega\), \(\mathcal{A}(\tau,\omega)\subseteq \hat{B}_{1}(\omega)\cap B_{0}(\omega)\). The proof is completed. □

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Acknowledgements

The authors would like to thank the anonymous referees for helpful comments and suggestions that improved the original paper.

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The authors are supported by the National Natural Science Foundation of China (11872264).

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HY completed the proofs of the main theorems, the rest of this paper was accomplished by JZ. All authors read and approved the final manuscript.

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Yao, H., Zhang, J. Random attractors for non-autonomous stochastic wave equations with nonlinear damping and white noise. Adv Differ Equ 2020, 221 (2020). https://doi.org/10.1186/s13662-020-02664-3

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Keywords

  • Random attractors
  • Additive white noise
  • Nonlinear damping
  • Non-autonomous stochastic wave equation