\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{amssymb,mathrsfs}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 111, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/111\hfil Random attractor]
{Attractors for stochastic strongly damped
plate equations with additive noise}

\author[W. Ma,  Q. Ma \hfil EJDE-2013/111\hfilneg]
{Wenjun Ma, Qiaozhen Ma}  % in alphabetical order

\address{Wenjun Ma \newline
College of Mathematics and Statistics,
Northwest Normal University, Lanzhou, Gansu 730070, China}
\email{mawenjun.0222@163.com}

\address{Qiaozhen Ma \newline
College of Mathematics and Statistics,
Northwest Normal University, Lanzhou, Gansu 730070, China}
\email{maqzh@nwnu.edu.cn}


\thanks{Submitted April 12, 2013. Published April 30, 2013.}
\subjclass[2000]{35B41, 60H15}
\keywords{Stochastic strongly damped plate equation; attractor;
\hfill\break\indent random dynamical system}

\begin{abstract}
 We study the asymptotic behavior of stochastic plate equations
 with homogeneous Neumann boundary conditions. We show the existence
 of an attractor for the random dynamical system associated with
 the equation.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks


\section{Introduction}

Let $\Omega$ be a bounded open set of $\mathbb{R}^n$ $(n=5)$ with 
a smooth boundary $\partial\Omega$. We consider the  
stochastic strongly damped plate equation with additive noise,
\begin{equation}
\begin{gathered}
du_{t}+du+(f(u)+\Delta^2u+\Delta^2 u_{t})dt
=gdt+\sum^m_{j=1}h_jdW_j,   \\
u(x,0)= u_0(x),  \quad u_{t}(x,0)= u_1(x),\\
u|_{\partial\Omega}=\frac{\partial u}{\partial n}|_{\partial\Omega}= 0, \quad
t\geqslant0,
\end{gathered} \label{e1.1}
\end{equation}
for $(x,t)\in \Omega\times[0,+\infty)$, where $u_0\in{H}^2_0(\Omega)$
 and $u_1\in L^2(\Omega)$. Here $u=u(x,t)$ is a real valued function
on $\Omega\times[0,+\infty), g\in{H}^2_0(\Omega)$ is a given external force.
The nonlinear term $f$ is a $C^{1}$-function with $f(0)=0$, that satisfies
the following conditions:
\begin{gather}
\liminf_{|s|\to \infty}\frac{f(s)}{s}
> -\lambda^2_1, \quad \forall s\in\mathbb{R},\label{e1.2}\\
|f'(s)|\leqslant C(1+|s|^8),  \quad \forall s\in\mathbb{R},\label{e1.3}\\
f(s+\varpi)=f(s), \quad  \forall s\in\mathbb{R}, \varpi\geqslant0,\label{e1.4}
\end{gather}
where $\lambda_1$ is the first eigenvalue of $\Delta^2$ on
${H}^2_0(\Omega)$ and $C$ is a positive constant.
$h_j\in H^{4}(\Omega)\cap{H}^2_0(\Omega)$ with
$\frac{\partial h_j}{\partial n}=0$ on $\partial\Omega, j=1,\dots,m$, and
$\{W_j\}^m_{j=1}$ are independent
two-sided real-valued Wiener processes on a probability space
$(\Theta,\mathscr{F},\mathbb{P})$, where
$$
\Theta=\{\omega=(\omega_1,\omega_{2},\dots,\omega_{m})
\in C(\mathbb{R},\mathbb{R}^m):\omega(0)=0\}
$$
is endowed with compact open topology, $\mathbb{P}$ is the corresponding
Wiener measure, and $\mathscr{F}$ is the
 $\mathbb{P}$-completion of Borel $\sigma$-algebra on $\Theta$.

 We identify $\omega$ with $(W_1,W_{2},\dots,W_{m})$,
 as $\omega(t)=(W_1(t),W_{2}(t),\dots,W_{m}(t))$ for $t\in\mathbb{R}$.
Define
$$
\theta_{t}\omega(\cdot)=\omega(\cdot+t)-\omega(t),\quad t\in\mathbb{R},
\omega\in\Theta.
$$

A random attractor of a random dynamical system is a measurable and
compact invariant random set attracting all the orbits.
When such an attracting set exists, it is the smallest attracting compact
set and the largest invariant set \cite{c2}. This seems to be a good generalization
of the now classical concept of a global attractor for deterministic
dynamical systems \cite{a1}. The notion of a random attractor is very useful
for many infinite-dimensional random dynamical systems (RDS), see
\cite{c2,c3}.

Many authors have studied the existence of a random attractor for an RDS.
For instance, Crauel and Flandoli in \cite{c3} introduced the notion of
 a random attractor and obtained a general theorem on the existence
of a random attractor for the RDS. Their theorem has been successfully
applied to the stochastic reaction-diffusion equations and the stochastic
Navier-Stokes equations. In \cite{c2} they generalized the notion of a random
attractor for the stochastic dynamical system introduced previously
and considered the stochastic nonlinear wave equations.
The asymptotic behavior of solutions for stochastic wave equation
 has been studied by several authors (see \cite{c1,f1,f2,l2,y2}).
The existence of global attractor for plate equation was studied in
\cite{k1,y4}. And in \cite{y1}, the author have investigated the
existence of uniform attractor about the non-autonomous case.
Recently, Yang and Kloeden in \cite{y3} studied the existence of
a random attractor for a class of stochastic semi-linear degenerate
parabolic equations. But there were no results on the random attractor
for the stochastic strongly damped plate equation with additive noise.
It is therefore necessary to investigate this problem.
In this article, we consider the asymptotic dynamics of the stochastic
plate equation with homogeneous Neumann boundary condition.

This article is organized as follows.
In section 2, we recall some basic concepts and properties for
general random dynamical systems. In section 3, we first provide
some basic settings about \eqref{e1.1} and show that it generates
a random dynamical system in proper function space, and then we prove
the existence of a unique random attractor of the random dynamical system.


\section{Random dynamical systems}

In this section, we recall some basic knowledge about general random
dynamical systems (see \cite{a1} for details).

Let $(X,\|\cdot\|_X)$ be a separable Hilbert space with Borel
$\sigma$-algebra $\mathscr{B}(X)$ and let
$(\Theta,\mathscr{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})$
be a metric dynamical system.


\begin{definition} \label{def2.1} \rm
 Let $(\Theta,\mathscr{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})$
be a metric dynamical system. Suppose that the mapping
$\phi:\mathbb{R}^{+}\times\Omega\times X\to X$
is $(\mathscr{B}(\mathbb{R}^{+})\times\mathscr{F}\times \mathscr{B}(X),
\mathscr{B}(X))$-measurable and satisfies the following two properties:
\begin{itemize}
\item[(1)]  $\phi(0,\omega)x=x$, and
\item[(2)]  $\phi(s,\theta_{t}\omega)\circ\phi(t,\omega)x=\phi(s+t,\omega)x$
\end{itemize}
for all $s,t\in\mathbb{R}^{+}, x\in X$ and $\omega\in\Theta$.
Then $\phi$ is called a random dynamical system $(RDS)$. Moreover,
$\phi$ is called a continuous RDS if  $\phi$ is continuous with respect to
$x$ for $t\geqslant0$ and $\omega\in\Theta$.
\end{definition}

To study the asymptotic behavior of the RDS determined by \eqref{e1.1},
we first need to recall some definitions and properties.

A set-valued mapping $B:\Theta\to2^{X}$ is called a random closed set
if $B(\omega)$ is closed, nonempty, and $\omega\mapsto d(x,B(\omega))$
is measurable for all $x\in X$ for each $\omega\in\Theta$. A random set
$\mathscr{B}:=\{B(\omega)\}_{\omega\in\Theta}$ is said to tempered if
$$
\lim_{t\to\infty}e^{-\eta t}\operatorname{diam}(B(\theta_{-t}\omega))=0
$$
for a.e. $\omega\in\Theta$ and all $\eta>0$, where
$\operatorname{diam}(B):=\sup_{x,y\in B}d(x,y)$.

Let $\mathscr{D}$ be the collection of all tempered random sets in $X$.
 We will only deal with the system $\mathscr{D}$ of tempered random
sets in this article.

\begin{definition} \label{def2.2} \rm
A random set $\mathscr{A}:=\{A(\omega)\}_{\omega\in\Theta}\in X$
is called a $\mathscr{D}$-random attractor for an RDS  $\phi$ if
\begin{itemize}
\item[(1)] $\mathscr{A}$ is a random compact set, i.e. $A(\omega)$
is nonempty and compact for a.e. $\omega\in\Theta$ and
$\omega\mapsto d(x,A(\omega))$
is measurable for every $x\in X$;

\item[(2)] $\mathscr{A}$ is $\phi$-invariant, i.e.
$\phi(t,\omega,A(\omega))=A(\theta_{t}\omega)$, for all
$T\geqslant0$ and a.e. $\omega\in\Theta$;

\item[(3)]  $\mathscr{A}$ attracts all tempered random sets
$B\in \mathscr{D}$ in the sense that
$$
\lim_{t\to\infty}\operatorname{dist}
(\phi(t,\theta_{-t}\omega,B(\theta_{-t}\omega)),A(\omega))=0,\quad
\text{a.e. }\omega\in\Theta.
$$
\end{itemize}
\end{definition}

\begin{theorem} \label{thm2.3}
Let $\phi$ be a continuous random dynamical system over dynamical system
$(\Omega,\mathscr{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})$.
Suppose that there exists a $\mathscr{D}$-random absorbing set
 $\{B(\omega)\}_{\omega\in\Omega}$ which absorbs every tempered random
set $D\in\mathscr{D}$. Then,\\ $\phi$ has a unique $\mathscr{D}$-random
attractor $\mathscr{A}=\{A(\omega)\}_{\omega\in\Omega}$, which is unique
in the class of tempered random sets with
$$
A(\omega)=\cap_{\tau\geqslant0}\overline{\cup_{t\geqslant\tau}
\phi(t,\theta_{-t}\omega,B(\theta_{-t}\omega))}, \quad \omega\in\Omega.
$$
\end{theorem}

\section{Attractor for the strongly damped plate equation}

\subsection{Basic settings}
In this subsection, we give some basic settings about \eqref{e1.1}
and show that it generates a random dynamical system.

Let $A=\Delta^2$, then
$D(A)=\{u\in H^{4}(\Omega)\cap{H}^2_0(\Omega):
\frac{\partial u}{\partial n}|_{\partial\Omega}= 0\}$.
Clearly, $A$ is a self-adjoint, positive linear operator with the
eigenvalues $\{\lambda_i\}_{i\in\mathbb{N}}$:
$$
0=\lambda_0<\lambda_1\leqslant\lambda_{2}\leqslant\dots\leqslant\lambda_i
\leqslant\dots, \lambda_i\to+\infty\quad (i\to+\infty).
$$
Let $E={H}^2_0(\Omega)\times L^2(\Omega)$, which is a separable Hilbert
space endowed with the usual norm
\begin{equation}
\|Y\|_{{H}^2_0\times L^2}=(\|\Delta u\|^2+\|v\|^2)^{1/2}\quad
\text{for }Y=(u,v)^{\top},\label{e3.1}
\end{equation}
where $\|\cdot\|$ denotes the usual norm in $L^2(\Omega)$ and $\top$
stands for the transposition.

For our purpose, it is convenient to convert the problem \eqref{e1.1}
into a deterministic system with a random parameter, and then show
that it generates a random
dynamical system. Consider Ornstein-Uhlenbeck equations
\begin{equation}
dz_j+z_jdt=dW_j(t), \quad j=\{1,2,\dots,m\},\label{e3.2}
\end{equation}
and Ornstein-Uhlenbeck processes
$$
z_j(\theta_{t}\omega_j)=-\int^{0}_{-\infty}e^{s}(\theta_{t}\omega_j)(s)ds, \quad
t\in\mathbb{R}.
$$
From \cite{b1}, it is known that the random variable $|z_j(\omega_j)|$ is tempered,
and there is a $\theta_{t}$-invariant set $\widetilde{\Theta}\subset\Theta$
of full $\mathbb{P}$ measure such that $t\mapsto z_j(\theta_{t}\omega_j)$
is continuous in $t$ for every $\omega\in\Theta$ and $j=1,2,\dots,m$. Put
\begin{equation}
z(\theta_{t}\omega)=z(x,\theta_{t}\omega)
=\sum^m_{j=1}h_jz_j(\theta_{t}\omega_j),\label{e3.3}
\end{equation}
which is a solution to
$$
dz+zdt=\sum^m_{j=1}h_jdW_j.
$$

\begin{lemma}[\cite{s1}] \label{lem3.1}
For any $\epsilon>0$, there exist tempered random variable
$r,r^{(l)}:\Theta\mapsto\mathbb{R}^{+},l=\frac{1}{2},1$, such that for all
 $t\in\mathbb{R},\omega\in\Theta$,
\begin{gather*}
\|z(\theta_{t}\omega)\|\leqslant e^{\epsilon|t|}r(\omega), \quad
e^{-\epsilon|t|}r(\omega)\leqslant r(\theta_{t}\omega)\leqslant
e^{\epsilon|t|}r(\omega), \\
\|A^{(l)}z(\theta_{t}\omega)\|\leqslant e^{\epsilon|t|}r^{(l)}(\omega), \quad
e^{-\epsilon|t|}r^{(l)}(\omega)\leqslant r^{(l)}(\theta_{t}\omega)
\leqslant e^{\epsilon|t|}r^{(l)}(\omega),
\end{gather*}
where $r^{(l)}(\omega)=\sum^m_{j=1}r_j(\omega_j)\|A^{(l)}h_j\|$.
\end{lemma}
It is convenient to reduce \eqref{e1.1} to a evolution equation of
 first order in time
\begin{equation}
\begin{gathered}
\dot{u}=v,\\
\dot{v}=-Av-Au-v-f(u)+g+\sum^m_{j=1}h_j\dot{W_j},\\
u(x,0)=u_0(x),    v(x,0)=u_1(x),  x\in\Omega,\\
\end{gathered} \label{e3.4}
\end{equation}
Let
\begin{gather*}
Y= \begin{pmatrix}
u  \\
v
\end{pmatrix}, \quad
M= \begin{pmatrix}
0 & I \\
-A & -A-I
\end{pmatrix},\\
F(t,\omega,Y)= \begin{pmatrix}
0 \\
-f(u)+g+\sum^m_{j=1}h_j\dot{W_j}
\end{pmatrix}.
\end{gather*}
Then problem \eqref{e3.4} has the  simple matrix form
\begin{equation}
\dot{Y}=MY+F(t,\omega,Y).\label{e3.5}
\end{equation}
Let $\psi_1=u,\psi_{2}=v-z(\theta_{t}\omega)$, then \eqref{e3.4}
can be rewritten as the equivalent system, in $E$,
\begin{equation}
\begin{gathered}
\dot{\psi_1}=\psi_{2}+z(\theta_{t}\omega),\\
\dot{\psi_{2}}=-A\psi_1-A\psi_{2}-\psi_{2}-f(\psi_1)+g-Az(\theta_{t}\omega),\\
\psi_1(x,0)=u_0(x), \quad   \psi_{2}(x,0)=u_1(x)-z(\omega), \quad
 x\in\Omega, \\
\end{gathered} \label{e3.6}
\end{equation}
which has the vector form
\begin{equation}
\dot{\psi}=M\psi+\overline{F}(\theta_{t}\omega,\psi),\label{e3.7}
\end{equation}
where
\begin{equation}
\psi= \begin{pmatrix}
\psi_1  \\
\psi_{2}
\end{pmatrix},
\overline{F}(\theta_{t}\omega,\psi)
= \begin{pmatrix}
z(\theta_{t}\omega) \\
-f(\psi_1)+g-Az(\theta_{t}\omega)
\end{pmatrix}
\label{e3.8}
\end{equation}

We will consider \eqref{e3.5} or \eqref{e3.7} for $\omega\in\widetilde{\Theta}$
and write $\widetilde{\Theta}$ as $\Theta$ from now on.
From \cite{l1}, $M$ is an unbounded closed operator on $E$ with domain $D(M)$,
\begin{equation}
D(M)=\{(u,v)^{\top}:u,v\in{H}^2_0(\Omega),u+v\in D(A)\}.\label{e3.9}
\end{equation}
Moreover, the spectral set of $M$ consists of only following points
\begin{equation}
\mu^{\pm}_i=\frac{-(\lambda_i+1)\pm\sqrt{(\lambda_i+1)^2
-4\lambda_i}}{2}, \quad i=0,1,2,\dots \label{e3.10}
\end{equation}
and $M$ generates a $C^{0}$-semigroup of bounded linear operators
$\{e^{Mt}\}_{t\geqslant0}$ on $E$.

Let $\overline{F}^{\omega}(t,\psi):=\overline{F}(\theta_{t}\omega,\psi)$,
it is easy to verify that
$\overline{F}^{\omega}(\cdot,\cdot):[0,+\infty)\times E\to E$
is continuous in $t$ and globally Lipschitz continuous in $\psi$ for each
$\omega\in\Theta$. By the classical semigroup theory on the existence and
uniqueness of the solutions \cite{t1}, we have the following theorem.

\begin{theorem} \label{thm3.2}
Consider \eqref{e3.7}. For each $\omega\in\Theta$ and each $\psi_0\in E$,
there exists a unique function
$\psi(\cdot,\omega,\psi_0)\in C([0,+\infty);E)$ such that
$\psi(0,\omega,\psi_0)=\psi_0$ and $\psi(t,\omega,\psi_0)$
satisfies the integral equation
\begin{equation}
\psi(t,\omega,\psi_0)=e^{Mt}\psi_0+\int^{t}_0e^{M(t-s)}
\overline{F}(\theta_{s}\omega,\psi(s,\omega,\psi_0))ds,\label{e3.11}
\end{equation}
$\psi(t,\omega,\psi_0)$ is jointly continuous in $t,\psi_0$, and is measurable
in $\omega$. Furthermore, if $\psi_0\in D(M)$, there exists
$\psi(\cdot,\omega,\psi_0)\in C([0,+\infty);D(M))\cap C^{1}([0,+\infty);E)$,
 which satisfies $\eqref{e3.7}$. Hence the solution mapping
\begin{equation}
\overline{S}(t,\omega):\psi_0\mapsto\psi(t,\omega,\psi_0)\label{e3.12}
\end{equation}
generates a random dynamical system.
\end{theorem}

Define a mapping $S(t,\omega)$ by
\begin{equation}
S(t,\omega):Y_0=\psi_0+(0,z(\omega))^{\top}\mapsto Y(t,\omega,Y_0)
=\psi(t,\omega,\psi_0)+(0,z(\theta_{t}\omega))^{\top},\label{e3.13}
\end{equation}
where $Y_0=(u_0,u_1)^{\top}$ and $\psi_0=(u_0,u_1-z(\omega))^{\top}$.
Then $S(t,\omega)$ is a continuous random dynamical system associated
with the problem \eqref{e3.5} or \eqref{e1.1} on $E$.
 $S(t,\omega)$ has the following relation with $\bar{S}(t,\omega)$
\begin{equation}
\overline{S}(t,\omega)=R(\theta_{t}\omega)S(t,\omega)R^{-1}
(\theta_{t}\omega),\label{e3.14}
\end{equation}
where $R(\theta_{t}\omega):(a,b)^{\top}\mapsto(a,b-z(\theta_{t}\omega))^{\top}$
is a homeomorphism of $E$.

We will also use the transformation
$$
\varphi_1=u=\psi_1,  \quad\varphi_{2}=v+\varepsilon u-z(\theta_{t}\omega),
$$
where $\varepsilon$ is a given positive number. Then the \eqref{e3.7}
can be rewritten as
\begin{equation}
\dot{\varphi}=M_{\varepsilon}\varphi+\overline{F}_{\varepsilon}
(\theta_{t}\omega,\varphi),
  \varphi_0(x,0)=(u_0(x),u_1(x)+\varepsilon u_0(x)-z(\omega)),\label{e3.15}
\end{equation}
where
\begin{gather}
\varphi= \begin{pmatrix}
\varphi_1  \\
\varphi_{2}
\end{pmatrix}, \quad
M_{\varepsilon}= \begin{pmatrix}
-\varepsilon I & I \\
\varepsilon(1-\varepsilon)I+\varepsilon A-A & (\varepsilon-1)I-A
\end{pmatrix}, \label{e3.16}
\\
\overline{F}_{\varepsilon}(\theta_{t}\omega,\varphi)
= \begin{pmatrix}
z(\theta_{t}\omega) \\
-f(\varphi_1)+g+\varepsilon z(\theta_{t}\omega)-Az(\theta_{t}\omega)
\end{pmatrix}.
\label{e3.17}
\end{gather}
Then the mapping
\begin{equation}
\bar{S}_{\varepsilon}(t,\omega)=T_{\varepsilon}\overline{S}
(t,\omega)T_{-\varepsilon}:\varphi_0\mapsto\varphi
(t,\omega,\varphi_0),\label{e3.18}
\end{equation}
generates a random dynamical system associated with \eqref{e3.15},
 where $\varphi_0=(u_0,u_1+\varepsilon u_0-z(\omega))^{\top}$,
and $T_{\varepsilon}:(a,b)^{\top}\mapsto(a,b+\varepsilon a)^{\top}$
is a isomorphism of $E$.

Notice that all the above random dynamical systems
$S(t,\omega),\overline{S}(t,\omega),\overline{S}_{\varepsilon}(t,\omega)$
are equivalent. Hence, we only need to consider the random dynamical
 system $\overline{S}(t,\omega)$.

Let $p_0=(\varpi,0)^{\top}=\varpi\eta_0\in E_1$, then $Mp_0=0$.
Thus, by the periodicity of function $f$, the random dynamical system
$\overline{S}(t,\omega)$ is $p_0$-translation invariant in the sense that
\begin{equation}
\psi(t,\omega,\psi_0+p_0)=\psi(t,\omega,\psi_0)+p_0, t\geqslant0,
\omega\in\Theta, \psi_0\in E,\label{e3.19}
\end{equation}
which implies that the average of the first component of
$\psi(t,\omega,\psi_0+p_0)$ will be unbounded in $E_1$
(corresponding to the direction of  $\eta_0$ with respect to 0 eigenvalue),
hence $\overline{S}(t,\omega)$ is unbounded in the direction of
$\eta_0$ in $E_1$, which means that it is impossible to obtain a
bounded attractor for $\overline{S}(t,\omega)$ as usual.
So we need to introduce a random dynamical system $\overline{\Phi}(t,\omega)$
defined on cylinder induced from $\overline{S}(t,\omega)$ according to
 $p_0$-translation invariance of $\bar{S}(t,\omega)$. To this end, we
introduce some space and notation.

For any $u\in L^2(\Omega)$, define the spatial average of $u$ as
\begin{equation}
\overline{u}=\frac{1}{|\Omega|}\int_{\Omega}u(x)dx.\label{e3.20}
\end{equation}
Let
\[
\dot{L}^2(\Omega)=\{u\in L^2(\Omega):\bar{u}=0\}, \quad
\dot{H}^2_0(\Omega)={H}^2_0(\Omega)\cap\dot{L}^2(\Omega), \\
E_{22}=\dot{H}^2_0(\Omega)\times\dot{L}^2(\Omega)
\] %\label{e3.21}
By \eqref{e3.10}, $M$ has two real eigenvalues 0 and -1 with eigenvectors
$\eta_0=(1,0)^{\top}$ and $\eta_{-1}=(1,-1)^{\top}$. Let
\[
E_1=\operatorname{span}\{\eta_0\},  \quad
E_{-1} =\operatorname{span}\{\eta_{-1}\}, \quad
E_{11}=E_1\oplus E_{-1}=\mathbb{R}^2, \quad
E_{2}=E_{-1}\oplus E_{22}. 
\] % \label{e3.22}
Then
\begin{equation}
E=E_{11}\oplus E_{22}=\mathbb{R}^2\oplus E_{22}
=E_1\oplus E_{-1}\oplus E_{22}=E_1\oplus E_{2},\label{e3.23}
\end{equation}
and $E_1$ is positive invariant under $M$.

Let $\mathbb{T}^{1}=E_1/p_0\mathbb{Z}$ and
$\mathbf{E}=\mathbb{T}^{1}\oplus E_{2}
=\mathbb{T}^{1}\oplus E_{-1}\oplus E_{22}
=\mathbb{T}^{1}\times E_{-1}\times E_{22}$.
 For $\Psi_0:=\psi_0(mod p_0)=\Psi_0+p_0\mathbb{Z}\subset E$
denotes the equivalence class of $\Psi_0$, which is an element
of $\mathbf{E}$. And the norm on $\mathbf{E}$ is denoted by
$$
\|\Psi_0\|_{\mathbf{E}}=\inf_{y\in p_0\mathbb{Z}}\|\psi_0+y\|_{E}.
$$
Note that, $\psi(t,\omega,\psi_0+kp_0)=\psi(t,\omega,\psi_0)+kp_0$, for all
$k\in\mathbb{Z}$ for $t\geqslant0, \omega\in\Theta$ and
$ \psi_0\in E$. With this, we define
\begin{equation}
\overline{\Phi}(t,\omega):\Psi_0\mapsto \Psi(t,\omega,\Psi_0)
=\psi(t,\omega,\psi_0)(mod p_0).\label{e3.24}
\end{equation}
It is easy to see that $\overline{\Phi}(t,\omega)$ is a random
dynamical system on $\mathbf{E}$.

Similarly, the random dynamical system $S(t,\omega)$ also induces a
random dynamical system $\Phi(t,\omega)$ on $\mathbf{E}$
defined by
\begin{equation}
\Phi(t,\omega):\mathbf{Y}_0\mapsto\mathbf{Y}(t,\omega,\mathbf{Y}_0)
=\Psi(t,\omega,\Psi_0)+\mathbf{Z}(\theta_{t}\omega)(mod p_0),\label{e3.25}
\end{equation}
where $\mathbf{Y}_0=Y_0(mod p_0),\mathbf{Z}(\theta_{t}\omega)
=(0,Z(\theta_{t}\omega))^{\top}$ and
$\Psi_0=\mathbf{Y}_0-\mathbf{Z}(\omega)(mod p_0)$.

We introduce a new norm which is equivalent to the usual norm
$\|\cdot\|_{{H}^2_0\times L^2}$ on $E$ in \eqref{e3.1}.
For $Y_i=(u_i,v_i)^{\top}\in E_{11}, i=1,2$, let
\begin{equation}
\begin{aligned}
&\langle Y_1,Y_{2}\rangle_{E_{11}}\\
&=\frac{1}{4}\langle u_1,u_{2}\rangle
+\langle \frac{1}{2}u_1+v_1,\frac{1}{2}u_{2}+v_{2}\rangle,
\end{aligned}\label{e3.26}
\end{equation}
where $\langle\cdot,\cdot\rangle$ denotes the inner product on $L^2(\Omega)$,
and for  $Y_i=(u_i,v_i)^{\top}\in E_{22}, i=1,2$, let
\begin{equation}
\langle Y_1,Y_{2}\rangle_{E_{22}}=\mu\langle A^{1/2}u_1,A^{1/2}u_{2}\rangle
+\langle v_1,v_{2}\rangle,\label{e3.27}
\end{equation}
where $A^{1/2}=\Delta$ and $\mu$ is chosen such that
$\mu=1-\varepsilon\in(\frac{1}{2},1)$ in which $\varepsilon\in(0,1)$
is a small positive number. By the generalized Poincar\'{e}
inequality
$$
\|A^{1/2}u\|^2\geqslant\lambda^{1/2}_1\|u\|^2, \quad
\forall u\in\dot{H}^2_0(\Omega),
$$
Expression \eqref{e3.27} is then positive definite.
 A bilinear form on $E$ can be induced from \eqref{e3.26} and \eqref{e3.27},
\begin{equation}
\langle X,Y\rangle_{E}=\langle \overline{X},\overline{Y}\rangle_{E_{11}}
+\langle X-\overline{X},Y-\overline{Y}\rangle_{E_{22}},\label{e3.28}
\end{equation}
where $X=\overline{X}+X-\overline{X}\in E, Y=\overline{Y}+Y-\overline{Y}\in E$,
 with $\overline{X},\overline{Y}\in E_{11}$ and
$X-\overline{X},Y-\overline{Y}\in E_{22}$.
It is easy to obtain the following fact.

\begin{lemma} \label{lem3.3}
Expressions \eqref{e3.26} and \eqref{e3.27} define inner products
on $E_{11}$ and $E_{22}$, respectively. Meanwhile, \eqref{e3.28} defines
an inner products on $E$, and the corresponding norm $\|\cdot\|_{E}$
is equivalent to the usual norm $\|\cdot\|_{{H}^2_0\times L^2}$
in \eqref{e3.1}.
\end{lemma}

Under the inner product
$\langle \cdot,\cdot\rangle_{E}, E_1\bot E_{-1}, E_{11}\bot E_{22}, E_1\bot E_{2}$.
Denote by $P,\overline{Q}$ and $Q$ the projections from $E$
into $E_1,E_{-1}$ and $E_{22}$, respectively:
\[
PY= \begin{pmatrix}
\overline{u}+\overline{v}  \\
0
\end{pmatrix} \in E_1,\quad
\overline{Q}Y= \begin{pmatrix}
-\overline{v}  \\
\overline{v}
\end{pmatrix} \in E_{-1}, \quad
QY= Y-\overline{Y}=\begin{pmatrix}
u-\overline{u}  \\
v-\overline{v}
\end{pmatrix}\in E_{22},
\]% \label{e3.29}
where $Y=(u,v)^{\top}\in E$. Sometimes we write $Qu=u-\overline{u}$
for $u\in L^2(\Omega)$.

\subsection{Random attractor}

First we consider the boundedness of the component $\overline{Q}\psi$
of solution $\psi$ of \eqref{e3.7} in $E_{-1}$. Taking the average of
\eqref{e3.6}, by Green's formula and Neumann boundary condition \eqref{e1.2},
and take the second equation, we have
\begin{equation}
\dot{\overline{\psi}_{2}}=-\overline{\psi}_{2}-\overline{f(\psi_1)}
+\overline{g},
\overline{\psi}_{2}(0)=\overline{u}_1-\overline{z(\omega)},\label{e3.30}
\end{equation}
then
\begin{equation}
\frac{d}{dt}|\overline{\psi}_{2}(t,\omega)|^2
\leqslant-|\overline{\psi}_{2}(0,\omega)|^2+(c_1+|\overline{g}|)^2,
t\geqslant0,\label{e3.31}
\end{equation}
thus,
\[
|\overline{\psi}_{2}(t,\omega)|^2\leqslant|\overline{\psi}_{2}(0,\omega)|^2e^{-t}
+(c_1+|\bar{g}|)^2
=|\overline{u}_1-\overline{z(\omega)}|^2e^{-t}+(c_1+|\overline{g}|)^2, \quad
t\geqslant0.\] %\label{e3.32}
So if $|\overline{u}_1-\overline{z(\omega)}|$ is tempered, then there
exists $\overline{t}_0\geqslant0$ such that
\begin{equation}
|\overline{\psi}_{2}(t,\omega)|\leqslant2(c_1+|\overline{g}|),\quad
 t\geqslant\overline{t}_0.\label{e3.33}
\end{equation}
This show the uniformly boundedness of
$\overline{Q}\psi=(-\overline{\psi}_{2},\overline{\psi}_{2})$
of solution of \eqref{e3.7} in one-dimensional subspace $E_{-1}$
of $\mathbb{R}^2$, which implies that $\overline{Q}\psi$ possesses
a compact absorbing set $\{B_{-1}(\omega)\}$ in $E_{-1}$.

Next we prove that $Q\psi$ of solution $\psi$ of \eqref{e3.7} possesses
a compact attracting set in $E_{22}$.

\begin{lemma} \label{lem3.4}
There exists a small positive constant $0<\sigma<\varepsilon$ such that
\begin{equation}
\langle M_{\varepsilon}QY,QY\rangle_{E}\leqslant-\sigma\|QY\|^2_{E}
-\frac{1}{2}\|A^{1/2}Qv\|^2-\frac{1}{2}\|Qv\|^2 \label{e3.34}
\end{equation}
 for $Y=(u,v)^{\top}\in E$,  and
\begin{align*}
\langle M_{\varepsilon}QY,AQY\rangle_{E}
&\leqslant -\sigma\|A^{1/2}QY\|^2_{E}-\frac{1}{2}\|AQv\|^2
-\frac{1}{2}\|A^{1/2}Qv\|^2 for Y\\
&=(u,v)^{\top}\in D(M)\cap E.
\end{align*} % \label{e3.35}
\end{lemma}

The proof of the above lemma is similar to that of \cite[Lemma 1]{z1}, and
it is omitted.

\begin{lemma} \label{lem3.5}
Assume that \eqref{e1.2}--\eqref{e1.4} and
$g\in{H}^2_0(\Omega)$ hold. Then there exists a random ball
$\{B_0(\omega)\}\in\mathscr{D}$ centered at 0 with random radius
 $\varrho(\omega)>0$ such that for any $\{\widehat{B}(\omega)\}\in\mathscr{D}$,
there is a $T_{\widehat{B}}(\omega,\varrho)>0$ such that for any
$\varphi_0(\theta_{-t}\omega)\in\widehat{B}(\theta_{-t}\omega)$
satisfies for a.e. $\omega\in\Theta$,
\begin{equation}
\|Q\varphi(t,\theta_{-t}\omega,\varphi_0(\theta_{-t}\omega))\|_{E}
\leqslant\varrho(\omega) \quad\forall t\geqslant T_{\widehat{B}}(\omega,\varrho).
\label{e3.36}
\end{equation}
\end{lemma}

\begin{proof}
 By \eqref{e3.15} and $QM_{\varepsilon}=M_{\varepsilon}Q$, we have
\begin{equation}
Q\dot{\varphi}=M_{\varepsilon}Q\varphi+Q\overline{F}_{\varepsilon}
(\theta_{t}\omega,\varphi),\label{e3.37}
\end{equation}
where
\begin{equation}
Q\overline{F}_{\varepsilon}(\theta_{t}\omega,\varphi)
= \begin{pmatrix}
Qz(\theta_{t}\omega) \\
Q[-f(u)+g+\varepsilon z(\theta_{t}\omega)-Az(\theta_{t}\omega)]
\end{pmatrix}.
\label{e3.38}
\end{equation}
Taking the inner product $\langle\cdot,\cdot\rangle_{E}$ of \eqref{e3.37}
with $Q\varphi\in E_{22}$, we note that
\begin{gather*}
\begin{aligned}
\mu\langle A^{1/2}Qz(\theta_{t}\omega),A^{1/2}Q\varphi_1\rangle
&\leqslant\mu\|A^{1/2}z(\theta_{t}\omega)\|\cdot
\|A^{1/2}Q\varphi_1\|\\
&\leqslant\frac{\mu}{2\sigma}\|A^{1/2}z(\theta_{t}\omega)\|^2
+\frac{\sigma\mu}{2}\|A^{1/2}Q\varphi_1\|^2,
\end{aligned}
\\
\langle-(f(u)-\overline{f(u)}),Q\varphi_{2}\rangle
 \leqslant2C(\varphi_1+|\varphi_1|^{9})\cdot\|Q\varphi_{2}\|
 \leqslant(2C(\varphi_1+|\varphi_1|^{9}))^2
+\frac{1}{4}\|Q\varphi_{2}\|^2,
\\
\langle g-\overline{g},Q\varphi_{2}\rangle
\leqslant\|g-\overline{g}\|\cdot\|Q\varphi_{2}\|
\leqslant4\|g\|^2+\frac{1}{4}\|Q\varphi_{2}\|^2,
\\
\begin{aligned}
\langle\varepsilon(z(\theta_{t}\omega)-\overline{z(\theta_{t}\omega)}),
 Q\varphi_{2}\rangle
&\leqslant\|\varepsilon(z(\theta_{t}\omega)-\overline{z(\theta_{t}\omega)})
 \|\cdot\|Q\varphi_{2}\|\\
&\leqslant\frac{\varepsilon^2}{\sigma}\|z(\theta_{t}\omega)\|^2
+\frac{\sigma}{2}\|Q\varphi_{2}\|^2,
\end{aligned}\\
\langle Az(\theta_{t}\omega),Q\varphi_{2}\rangle
\leqslant\|A^{1/2}z  (\theta_{t}\omega)\|\cdot\|A^{1/2}Q\varphi_{2}\|
\leqslant \frac{1}{2}\|A^{1/2}z(\theta_{t}\omega)\|^2
 +\frac{1}{2}\|A^{1/2}Q\varphi_{2}\|^2,
\\
\langle M_{\varepsilon}Q\varphi,Q\varphi\rangle_{E}
\leqslant-\sigma\|Q\varphi\|^2_{E}-\frac{1}{2}\|A^{1/2}Q\varphi_{2}\|^2
-\frac{1}{2}\|Q\varphi_{2}\|^2.
\end{gather*}
From the above inequalities, we have
\begin{equation}
\frac{d}{dt}\|Q\varphi\|^2_{E}+2\sigma\|Q\varphi\|^2_{E}
\leqslant2R_0(\theta_{t}\omega),\label{e3.39}
\end{equation}
where
\begin{equation}
R_0(\theta_{t}\omega)=\frac{\mu+\sigma}{2\sigma}\|A^{1/2}z(\theta_{t}\omega)\|^2
+(2C(|\varphi_1|+|\varphi_1|^{9}))^2+4\|g\|^2+
\frac{\varepsilon^2}{\sigma}\|z(\theta_{t}\omega)\|^2.\label{e3.40}
\end{equation}
Applying the Gronwall lemma, for all $t\geqslant0$, we have
\begin{equation}
\|Q\varphi(t,\omega,\varphi_0(\omega))\|^2_{E}
\leqslant e^{-2\sigma t}\|\varphi_0(\omega)\|^2_{E}
+2\int^{t}_0R_0(\theta_{s}\omega)e^{-2\sigma (t-s)}ds.\label{e3.41}
\end{equation}
By replacing $\omega$ by $\theta_{-t}\omega$, we get from \eqref{e3.41} that,
for all $t\geqslant0$,
\begin{align*}
\|Q\varphi(t,\theta_{-t}\omega,\varphi_0(\theta_{-t}\omega))\|^2_{E}
&\leqslant e^{-2\sigma t}\|\varphi_0(\theta_{-t}\omega)\|^2_{E}
+2\int^{t}_0R_0(\theta_{s-t}\omega)e^{-2\sigma (t-s)}ds\\
&=e^{-2\sigma t}\|\varphi_0(\theta_{-t}\omega)\|^2_{E}
+2\int^{0}_{-t}R_0(\theta_{\tau}\omega)e^{2\sigma\tau}d\tau.
\end{align*}%\label{e3.42}
By Lemma \ref{lem3.1} with $\epsilon=\frac{\sigma}{4}$, we have that
\begin{equation}
\int^{0}_{-t}R_0(\theta_{\tau}\omega)e^{2\sigma\tau}d\tau
\leqslant\int^{0}_{-t}\widetilde{R_0}(\tau,\omega)e^{2\sigma\tau}d\tau
\leqslant \int^{0}_{-\infty}\widetilde{R_0}(\tau,\omega)e^{2\sigma\tau}d\tau
<+\infty,\label{e3.43}
\end{equation}
where
\[
\widetilde{R_0}(\tau,\omega)
=\frac{\mu+\sigma}{2\sigma}(e^{\frac{\sigma}{4}|\tau|}
r^{(1/2)}(\omega))^2+(2C(|\varphi_1|+|\varphi_1|^{9}))^2
+4\|g\|^2+\frac{\varepsilon^2}{\sigma}(e^{\frac{\sigma}{4}|\tau|}r(\omega))^2.
\] %label{e3.44}
Note that $\{\widehat{B}(\omega)\}\in\mathscr{D}$ is tempered,
then for any $\varphi_0(\theta_{-t}\omega)\in\widehat{B}(\theta_{-t}\omega)$,
$$
\lim_{t\to+\infty}e^{-2\sigma t}\|\varphi_0(\theta_{-t}\omega)\|^2_{E}=0.
$$
Hence, there exists a $T_{\widehat{B}}(\omega,\varrho)>0$ such that for
any $\varphi_0(\theta_{-t}\omega)\in\widehat{B}(\theta_{-t}\omega)$
satisfies for a.e. $\omega\in\Theta$,
\begin{equation}
\|Q\varphi(t,\theta_{-t}\omega,\varphi_0(\theta_{-t}\omega))\|_{E}
\leqslant\varrho(\omega) for all t\geqslant T_{\widehat{B}}(\omega,\varrho),
\label{e3.45}
\end{equation}
where
\begin{equation}
\varrho^2(\omega)=2\int^{0}_{-\infty}\widetilde{R_0}
(\tau,\omega)e^{2\sigma\tau}d\tau.\label{e3.46}
\end{equation}
So, the proof is complete.
\end{proof}

We now construct a random compact attracting set for
RDS $\bar{S}_{\varepsilon}(t,\omega)$. For this purpose, we split
the solution $\varphi$ of the system \eqref{e3.7} with the initial
value $\varphi_0=(u_0,v_0+\varepsilon u_0-z(\omega))^{\top}$ into two
 parts
$\varphi=\varphi^{a}+\varphi^{b}=(u^{a},v^{a}
+\varepsilon u^{a})^{\top}+(u^{b},v^{b}
+\varepsilon u^{b}-z(\theta_{t}\omega))^{\top}$, where $\varphi^{a}$ solves
\begin{equation}
\dot{\varphi}^{a}=M_{\varepsilon}\varphi^{a},
\varphi^{a}_0=(u_0,v_0+\varepsilon u_0)^{\top},\label{e3.47}
\end{equation}
and $\varphi^{b}$ solves
\begin{equation}
\dot{\varphi}^{b}=M_{\varepsilon}\varphi^{b}
+\overline{F}_{\varepsilon}(\theta_{t}\omega,\varphi), \quad
\varphi^{b}_0=(0,-z(\omega))^{\top}.\label{e3.48}
\end{equation}

\begin{lemma} \label{lem3.6}
Assume that \eqref{e1.2}--\eqref{e1.4} and
$g\in{H}^2_0(\Omega)$ hold. Then there exists a random variable
 $\varrho_1(\omega)>0$ such that for any
$\{\widehat{B}(\omega)\}\in\mathscr{D}$ and
$\varphi_0(\omega)\in\widehat{B}(\omega)$, there is a
$T_{\widehat{B}}(\omega,\varrho_1)>0$ such that for any $\varphi$
of the system $\eqref{e3.7}$ satisfies for a.e. $\omega\in\Theta$,
\begin{equation}
\|Q\varphi^{a}(t,\theta_{-t}\omega,\varphi^{a}_0(\theta_{-t}\omega))\|_{E}
\leqslant e^{-2\sigma t}\|\varphi^{a}_0(\theta_{-t}\omega)\|_{E}\to0,
\quad\text{as }t\to+\infty,\label{e3.49}
\end{equation}
and
\begin{equation}
\|A^{1/2}Q\varphi^{b}(t,\theta_{-t}\omega,\varphi^{b}_0
(\theta_{-t}\omega))\|_{E}\leqslant\varrho_1(\omega),\quad
\forall t\geqslant T_{\widehat{B}}(\omega,\varrho_1),\label{e3.50}
\end{equation}
where $Q\varphi^{a}$ and $Q\varphi^{b}$ satisfy \eqref{e3.47} and
\eqref{e3.48}.
\end{lemma}

\begin{proof}
By \eqref{e3.47}, we have
\begin{equation}
Q\dot{\varphi}^{a}=M_{\varepsilon}Q\varphi^{a}.\label{e3.51}
\end{equation}
Take the inner product $\langle\cdot,\cdot\rangle_{E}$ of \eqref{e3.51}
with $Q\varphi^{a}$. By Lemma \ref{lem3.5}, we obtain
\begin{equation}
\|Q\varphi^{a}(t,\theta_{-t}\omega,\varphi^{a}_0(\theta_{-t}\omega))\|^2_{E}
\leqslant e^{-2\sigma t}\|\varphi^{a}_0(\theta_{-t}\omega)\|^2_{E}.\label{e3.52}
\end{equation}
Then, the first assertion is valid.

From \eqref{e3.48}, we have
\begin{equation}
Q\dot{\varphi}^{b}=M_{\varepsilon}Q\varphi^{b}+Q\bar{F}_{\varepsilon}
(\theta_{t}\omega,\varphi^{b}).\label{e3.53}
\end{equation}
Take the inner product $\langle\cdot,\cdot\rangle_{E}$ of \eqref{e3.53}
 with $AQ\varphi^{b}$. By Lemma \ref{lem3.4}, we have
\begin{equation}
\langle M_{\varepsilon}Q\varphi^{b},AQ\varphi^{b}\rangle_{E}
\leqslant-\sigma\|A^{1/2}Q\varphi^{b}\|^2_{E}
-\frac{1}{2}\|AQ\varphi^{b}_{2}\|^2
-\frac{1}{2}\|A^{1/2}Q\varphi^{b}_{2}\|^2.\label{e3.54}
\end{equation}
By the Cauchy-Schwarz inequality, we obtain
\begin{gather*}
\begin{aligned}
\mu\langle A^{1/2}Qz(\theta_{t}\omega),A^{1/2}AQ\varphi^{b}_1\rangle
&\leqslant\mu\|AQz(\theta_{t}\omega)\|\cdot \|AQ\varphi^{b}_1\|\\
&\leqslant\frac{\mu}{2\sigma}\|Az(\theta_{t}\omega)\|^2
 +\frac{\sigma\mu}{2}\|AQ\varphi^{b}_1\|^2,
\end{aligned}\\
\langle Qf(\varphi^{b}_1),AQ\varphi^{b}_{2}\rangle
 \leqslant\|A^{1/2}Qf(\varphi^{b}_1)\|\cdot\|A^{1/2}Q\varphi^{b}_{2}\|
\leqslant\|A^{1/2}f(\varphi^{b}_1)\|^2+\frac{1}{4}\|A^{1/2}Q\varphi^{b}_{2}\|^2,
\\
\langle Qg,AQ\varphi^{b}_{2}\rangle
 \leqslant\|A^{1/2}g\|\cdot\|A^{1/2}Q\varphi^{b}_{2}\|\leqslant\|A^{1/2}g\|^2
+\frac{1}{4}\|A^{1/2}Q\varphi^{b}_{2}\|^2,
\\
\begin{aligned}\langle\varepsilon Qz(\theta_{t}\omega),AQ\varphi^{b}_{2}\rangle
&\leqslant\|\varepsilon A^{1/2}Qz(\theta_{t}\omega)\|\cdot\|A^{1/2}
 Q\varphi^{b}_{2}\|\\
&\leqslant\frac{\varepsilon^2}{2\sigma}\|A^{1/2}z(\theta_{t}\omega)\|^2
+\frac{\sigma}{2}\|A^{1/2}Q\varphi^{b}_{2}\|^2,
\end{aligned}\\
\langle QAz(\theta_{t}\omega),AQ\varphi^{b}_{2}\rangle
\leqslant\|QAz(\theta_{t}\omega)\|\cdot\|AQ\varphi^{b}_{2}\|
\leqslant \|Az(\theta_{t}\omega)\|^2+\frac{1}{2}\|QA\varphi^{b}_{2}\|^2.
\end{gather*}
From the above inequalities and \eqref{e3.54}, we have
\begin{equation}
\frac{d}{dt}\|A^{1/2}Q\varphi^{b}\|^2_{E}+2\sigma\|A^{1/2}Q\varphi^{b}\|^2_{E}
\leqslant2R_1(\theta_{t}\omega),\label{e3.55}
\end{equation}
where
\[
R_1(\theta_{t}\omega)
=\frac{\mu+2\sigma}{2\sigma}\|Az(\theta_{t}\omega)\|^2
+\|A^{1/2}f(\varphi^{b}_1)\|^2+\|A^{1/2}g\|^2
+\frac{\varepsilon^2}{2\sigma}\|A^{1/2}z(\theta_{t}\omega)\|^2.
\]%\label{e3.56}
By Gronwall's lemma, for all $t\geqslant0$, 
\begin{equation}
\begin{aligned}
&\|A^{1/2}Q\varphi^{b}(t,\omega,\varphi^{b}_0(\omega))\|^2_{E}\\
&\leqslant e^{-2\sigma t}\|A^{1/2}\varphi^{b}_0(\omega)\|^2_{E}
+2\int^{t}_0R_1(\theta_{s}\omega)e^{-2\sigma (t-s)}ds \\
&=e^{-2\sigma t}\|A^{1/2}z(\omega)\|^2_{E}
 +2\int^{t}_0R_1(\theta_{s}\omega)e^{-2\sigma (t-s)}ds.
\end{aligned}\label{e3.57}
\end{equation}
Replacing $\omega$ by $\theta_{-t}\omega$, in \eqref{e3.57} we obtain that
for all $t\geqslant0$,
\begin{equation}
\begin{aligned}
&\|A^{1/2}Q\varphi^{b}(t,\theta_{-t}\omega,
\varphi^{b}_0(\theta_{-t}\omega))\|^2_{E}\\
&\leqslant e^{-2\sigma t}\|A^{1/2}z(\theta_{-t}\omega)\|^2
 +2\int^{t}_0R_1(\theta_{s-t}\omega)e^{-2\sigma (t-s)}ds\\
&=e^{-2\sigma t}\|A^{1/2}z(\theta_{-t}\omega)\|^2_{E}
+2\int^{0}_{-t}R_1(\theta_{\tau}\omega)e^{2\sigma\tau}d\tau.
\end{aligned}\label{e3.58}
\end{equation}
By Lemma \ref{lem3.1} with $\epsilon=\frac{\sigma}{4}$, we have
\begin{gather*}
\lim_{t\to+\infty}e^{-2\sigma t}\|A^{1/2}z(\theta_{-t}\omega)\|^2
\leqslant\lim_{t\to+\infty}e^{-2\sigma t}(e^{\frac{\sigma}{4}|\tau|}
r^{(1/2)}(\omega))^2=0,\\
\int^{0}_{-t}R_1(\theta_{\tau}\omega)e^{2\sigma\tau}d\tau
\leqslant\int^{0}_{-t}\widetilde{R_1}(\tau,\omega)e^{2\sigma\tau}d\tau
\leqslant \int^{0}_{-\infty}\widetilde{R_1}(\tau,\omega)e^{2\sigma\tau}d\tau
<+\infty,
\end{gather*}
where
$$
\widetilde{R_1}(\tau,\omega)=\frac{\mu+2\sigma}{2\sigma}
(e^{\frac{\sigma}{4}|\tau|}r^{1}(\omega))^2+\|A^{1/2}f(\varphi^{b}_1)\|^2
+\|A^{1/2}g\|^2+\frac{\varepsilon^2}{2\sigma}(e^{\frac{\sigma}{4}|\tau|}
r^{(1/2)}(\omega))^2.
$$
Set
\begin{equation}
\varrho^2_1(\omega)=2\int^{0}_{-\infty}\widetilde{R_1}(\tau,\omega)
e^{2\sigma\tau}d\tau,\label{e3.59}
\end{equation}
Hence, there exists a $T_{\widehat{B}}(\omega,\varrho)>0$ such that for any
$\varphi$ of the system $\eqref{e3.7}$ satisfies for a.e. $\omega\in\Theta$,
\begin{equation}
\|A^{1/2}Q\varphi^{b}(t,\theta_{-t}\omega,\varphi^{b}_0
(\theta_{-t}\omega))\|_{E}\leqslant \varrho_1(\omega), \quad\forall
t\geqslant T_{\widehat{B}}(\omega,\varrho_1),\label{e3.60}
\end{equation}
So, the second assertion is valid.
\end{proof}

Notice that
\begin{align*}
\|A^{1/2}Q\varphi^{b}(t,\theta_{-t}\omega,\varphi^{b}_0(\theta_{-t}\omega))\|_{E}
&=\Big\|\begin{pmatrix}
A^{1/2}Q\varphi^{b}_1 \\
A^{1/2}Q\varphi^{b}_{2}
\end{pmatrix}\Big\|_{E}
\geqslant\widetilde{c_1}
\Big\|\begin{pmatrix}
A^{1/2}Q\varphi^{b}_1 \\
A^{1/2}Q\varphi^{b}_{2}
\end{pmatrix}\Big\|_{{H}^2_0\times L^2}\\
&=\widetilde{c_1}(\|AQ\varphi^{b}_1\|^2+\|A^{1/2}Q\varphi^{b}_{2}\|^2)^{1/2},
\end{align*}%\label{e3.61}
which along with \eqref{e3.36}, yields that for
$\varphi_0(\omega)\in\widehat{B}(\omega)\in\mathscr{D}$,
\begin{equation}
\|Q\varphi^{b}(t,\theta_{-t}\omega,\varphi_0(\theta_{-t}\omega))\|_{H^{4}
\times{H}^2_0}
\leqslant K_0(\varrho_1(\omega)+\varrho(\omega)), \label{e3.62}
\end{equation}
for all $t\geqslant T_{\widehat{B}}(\omega,\varrho_1)
+T_{\widehat{B}}(\omega,\varrho)$
for a constant $K_0>0$.
Let $\{B_1(\omega)\}$ be a closed ball of $E$:
\begin{equation}
B_1(\omega)=\{b(\omega)\in E:\|Qb(\omega)\|_{{H}^{4}\times {H}^2_0}
\leqslant K_0(\varrho_1(\omega)+\varrho(\omega))\}.\label{e3.63}
\end{equation}
By \eqref{e3.49}, \eqref{e3.62}, and
\begin{equation}
Q\varphi(t,\theta_{-t}\omega,\varphi_0(\theta_{-t}\omega))
=Q\varphi^{a}(t,\theta_{-t}\omega,\varphi_0(\theta_{-t}\omega))
+Q\varphi^{b}(t,\theta_{-t}\omega,\varphi_0(\theta_{-t}\omega)),\label{e3.64}
\end{equation}
we have for a.e. $\omega\in\Theta$,
\begin{equation}
d_{E}(\varphi(t,\theta_{-t}\omega,B_0(\theta_{-t}\omega)),B_1(\omega))\to 0
\quad\text{as } t\to+\infty,\label{e3.65}
\end{equation}
this implies that for a.e. $\omega\in\Theta$,
\begin{equation}
d_{E}(T_{-\varepsilon}\varphi(t,\theta_{-t}\omega,B_0(\theta_{-t}\omega)),
T_{-\varepsilon}B_1(\omega))\to0 as t\to+\infty,\label{e3.66}
\end{equation}
where $QT_{-\varepsilon}B_1(\omega)\subset E_{22}$ is bounded in the norm
of $H^{4}(\Omega)\times{H}^2_0(\Omega)$ by \eqref{e3.62} and \eqref{e3.63}.
By the compact embedding of $\widetilde{E}=H^{4}(\Omega)\times{H}^2_0(\Omega)$
into $E$, $\{QT_{-\varepsilon}B_1(\omega)\}$ is compact in $E_{22}$,
which imply that
 $\omega\mapsto\mathbf{B}_0(\omega):=(B_1(\omega)+B_{-1}(\omega))(mod p_0)$
is a tempered random compact  attracting set for $\overline{\Phi}(t,\omega)$.
Thus for Theorem \ref{thm2.3}, we have the following result.

\begin{theorem} \label{thm3.7}
Assume that \eqref{e1.2}--\eqref{e1.4} and $g\in{H}^2_0(\Omega)$ hold.
Then the random dynamical system $\overline{\Phi}(t,\omega)$ defined in
\eqref{e3.5} has a unique random attractor $\{\mathscr{A}_0(\omega)\}$ in
$\mathbf{E}$, where
$$
\mathscr{A}_0(\omega)=\cap_{t>0}\overline{\cup_{\tau\geqslant t}
\Psi(\tau,\theta_{-\tau}\omega,\mathbf{B}_0(\theta_{-\tau}\omega))},
   \omega\in\Omega,
$$
in which $\{\mathbf{B}_0(\omega)\}$ is a tempered random compact
attracting set for $\overline{\Phi}(t,\omega)$.
\end{theorem}

\subsection*{Acknowledgments}
This work was partly supported by grant 11101334 from the  NSFC,
grant 1107RJZA223 from   the NSF of Gansu Province,
and by the Fundamental Research Funds for the Gansu Universities.

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