\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 191, pp. 1--25.\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-2013/191\hfil Asymptotic behavior]
{Asymptotic behavior of non-autonomous stochastic parabolic
equations with nonlinear Laplacian principal part}

\author[B. Wang, B. Guo \hfil EJDE-2013/191\hfilneg]
{Bixiang Wang, Boling Guo} 

\address{Bixiang Wang \newline
Department of Mathematics, New Mexico Institute of Mining and
Technology \\ Socorro, NM 87801, USA }
\email{bwang@nmt.edu}

\address{Boling Guo \newline
Institute of Applied Physics and Computational Mathematics,
P.O. Box 8009, Beijing 100088, China}
\email{gbl@iapcm.ac.cn}

\thanks{Submitted May 20, 2013. Published August 30, 2013.}
\thanks{Boling Guo was supported by grant DMS 0401708 from the NSF}
\subjclass[2000]{35B40, 35B41, 37L30}
\keywords{Pullback attractor; periodic random attractor; $p$-Laplace equation;
\hfill\break\indent upper semicontinuity}

\begin{abstract}
 We prove the existence and uniqueness of random attractors
 for the $p$-Laplace equation driven simultaneously by
 non-autonomous deterministic and stochastic forcing.
 The nonlinearity of the equation is allowed to have a polynomial
 growth rate of any order which may be greater than $p$.
 We further establish the upper semicontinuity of random attractors
 as the intensity of noise approaches zero.
 In addition, we show the pathwise periodicity of random attractors
 when all non-autonomous deterministic forcing terms are time periodic.
\end{abstract}

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

\section{Introduction}

In this article, we investigate the existence and upper semicontinuity of
random attractors for a class of degenerate parabolic equations
driven simultaneously by non-autonomous deterministic and stochastic
forcing. More precisely, we consider the $p$-Laplace equation
defined in a bounded domain $ {\mathcal{O}} \subseteq \mathbb{R}^n$:
 \begin{equation} \label{intr1}
 {\frac {\partial u}{\partial t}}
 - \operatorname{div} (|\nabla u |^{p-2} \nabla u )
 =f_1(t,x,u) + f_2(t,x, u) +g(t,x) +\alpha u \circ {\frac {dW}{dt}}
 \end{equation}
with homogeneous Dirichlet boundary condition,
where $g \in L^2_{\rm loc}(\mathbb{R}, L^2(\mathcal{O}))$,  $p>2$, $ \alpha >0$, and
$W$ is a two-sided real-valued Wiener process on a probability space.
The symbol $\circ$ means that the stochastic equation is understood
in the sense of Stratonovich integration.
The functions $f_1, f_2: \mathbb{R} \times {\mathcal{O}} \times \mathbb{R}$
$\to \mathbb{R}$ are continuous and satisfy some growth conditions
which will be specified later. In particular, we will assume
that $f_2$ is Lipschitz continuous
in its third argument and $f_1$ has the property:
\begin{equation} \label{intr2}
f_1 (t, x, s) s \le - \lambda
 |s|^q + \psi_1(t, x), \quad \text{for all } t, s \in \mathbb{R}
 \text{ and } x \in {\mathcal{O}},
\end{equation}
where $\lambda>0$, $q>1$ and
$\psi_1 \in L^1_{\rm loc} (\mathbb{R}, L^1( {\mathcal{O}}) )$.

The random attractors of the stochastic $p$-Laplace equation
\eqref{intr1} were studied in \cite{gess1, gess2, gess3} recently, where the
authors successfully established the existence of such attractors
under the condition $q \le p$ for the exponents $p$ and $q$
in \eqref{intr1} and \eqref{intr2}.
In these papers, an abstract framework
of the variational approach was introduced and
equation \eqref{intr1} fits these frameworks only when
 $q \le p$. To the best of our knowledge, the existence of random
 attractors for $q>p$ is still open and the first goal of the present
 paper is to solve this problem.
 In other words, we will show
 the stochastic equation \eqref{intr1} has a random attractor
 in $L^2(\mathcal{O})$
 for any $p>2$ and $q>1$.
In addition, if $f_1$, $f_2$ and $g$ are periodic in their first
 argument, then the corresponding random attractor is pathwise
 periodic. The second goal of this paper is to establish the
 upper semicontinuity of random attractors when the intensity
 of noise approaches zero. Such continuity of attractors
is new for the $p$-Laplace equation even when $q \le p$.

If $f_1$, $f_2$ and $g$ are independent of $t \in \mathbb{R}$, then
 \eqref{intr1} reduces to an autonomous stochastic equation.
 The attractors of autonomous stochastic systems have been
 extensively investigated in
\cite{arn1, bat1, bat2, beyn1, car1, car2,
 car3, car4, car5, chu2, chu3, cra1, cra2,
 fla1, gar1, gar2, gar3, gess1, huang1, kloe1, lv1, schm1, shen1,
 wan1, wan2}
and the references therein.
For non-autonomous random attractors, we refer the
reader to \cite{adi1, car6, dua3, gess2, gess3, kloe2, wan5,
 wan6, wan7, wan8}
 for details.


 This paper is organized as follows.
In the next section, we
review an existence result on random attractors
for non-autonomous stochastic equations.
In Section 3, we establish the well-posedness
of equation \eqref{intr1} in $L^2(\mathcal{O})$ and
define a continuous cocycle for the solution operators.
In Section 4, we derive uniform
estimates of solutions which are necessary for
the existence of random absorbing sets. The asymptotic
compactness of solutions for a non-autonomous deterministic
equation is also contained in this section.
Sections 5 and 6 are devoted to the existence and upper
semicontinuity of random attractors for \eqref{intr1}, respectively.


 In the sequel, we use $\| \cdot \|$ and $(\cdot, \cdot)$ for
the norm and inner product
of $L^2(\mathcal{O})$, respectively.
The norm of a general Banach space $X$
is written as $\|\cdot\|_{X}$,
For convenience, we will use
$c$ or $c_i$ ($i=1, 2, \ldots$) to denote a positive
number whose value is not of significance.

\section{Notation}

In this section, we assume $(X, d)$ is a complete
separable metric space
and $(\Omega, \mathcal{F}, P, \{\theta\}_{t \in \mathbb{R}})$ is a metric dynamical system.
 Denote by $\mathcal{D}$ a collection of families of nonempty subsets of $X$.
We recall the following definitions from \cite{wan5}.

\begin{definition} \label{ds1} \rm
A mapping $\Phi$: $ \mathbb{R}^+ \times \mathbb{R} \times \Omega \times X
\to X$ is called a continuous cocycle on $X$
over $(\Omega, \mathcal{F}, P, \{\theta\}_{t \in \mathbb{R}})$
if for all  $\tau\in \mathbb{R}$, $\omega \in \Omega $
 and $t, s \in \mathbb{R}^+$,
\begin{itemize}
\item [(i)] $\Phi (\cdot, \tau, \cdot, \cdot): \mathbb{R} ^+ \times \Omega \times X
\to X$ is
 $(\mathcal{B} (\mathbb{R}^+) \times \mathcal{F} \times \mathcal{B} (X), \
\mathcal{B}(X))$-measurable;

\item[(ii)] $\Phi(0, \tau, \omega, \cdot) $ is the identity on $X$;

\item[(iii)] $\Phi(t+ s, \tau, \omega, \cdot) =
 \Phi(t, \tau +s, \theta_s \omega, \cdot)
 \circ \Phi(s, \tau, \omega, \cdot)$;

\item[(iv)] $\Phi(t, \tau, \omega, \cdot): X \to X$ is continuous.
 \end{itemize}
 A continuous cocycle $\Phi$ is called
 $T$-periodic if
 $\Phi(t, \tau +T, \omega, \cdot)
= \Phi(t, \tau, \omega, \cdot )$
 for all $t \in \mathbb{R}^+$,
 $\tau \in \mathbb{R}$ and $\omega \in \Omega$.
\end{definition}


\begin{definition}\label{asycomp} \rm
A cocycle $\Phi$ is said to be $\mathcal{D}$-pullback asymptotically
compact in $X$ if for all $\tau \in \mathbb{R}$ and
$\omega \in \Omega$, the sequence
$\{\Phi(t_n, \tau -t_n, \theta_{ -t_n} \omega,
x_n)\}_{n=1}^\infty$ has a convergent subsequence in
$X $
 whenever
 $t_n \to \infty$, and $ x_n\in B(\tau -t_n,
 \theta_{ -t_n} \omega )$ with
$\{B(\tau, \omega): \tau \in \mathbb{R}, \ \omega \in \Omega
\} \in \mathcal{D}$.
\end{definition}


\begin{definition}\label{defatt} \rm
 ${\mathcal{A}} = \{{\mathcal{A}} (\tau, \omega): \tau \in \mathbb{R},
 \omega \in \Omega \} \in {\mathcal{D}}$ is
 called a $\mathcal{D}$-pullback attractor of  $\Phi$ if
\begin{itemize}
\item [(i)] $ {\mathcal{A}}$ is measurable
 and
 $ {\mathcal{A}}(\tau, \omega)$ is compact for all $\tau \in \mathbb{R}$
and $\omega \in \Omega$.

\item[(ii)] $ {\mathcal{A}}$ is invariant, that is,
for every $\tau \in \mathbb{R}$ and
 $\omega \in \Omega$,
$$ \Phi(t, \tau, \omega, {\mathcal{A}}(\tau, \omega) )
= {\mathcal{A}} ( \tau +t, \theta_{t} \omega
), \ \ \forall \ t \ge 0.
$$

\item[(iii)] For every
 $B = \{B(\tau, \omega): \tau \in \mathbb{R}, \omega \in \Omega\}
 \in {\mathcal{D}}$ and for every $\tau \in \mathbb{R}$ and
 $\omega \in \Omega$,
$$ \lim_{t \to \infty} d (\Phi(t, \tau -t,
 \theta_{-t}\omega, B(\tau -t,
 \theta_{-t}\omega) ) , {\mathcal{A}} (\tau, \omega ))=0.
$$
 \end{itemize}
 A pullback attractor $ {\mathcal{A}}$ is said to be $T$-periodic if
 $
 {\mathcal{A}} (\tau +T, \omega) = {\mathcal{A}}(\tau, \omega )$
 for all $ \tau \in \mathbb{R}$ and
 $\omega \in \Omega$.
\end{definition}

 We borrow  the following result from \cite{wan5}.
 Similar results can be found in
\cite{bat1, car6, cra2, fla1, gess2, schm1}.

\begin{proposition} \label{att}
 Let $ {\mathcal{D}}$ be an inclusion-closed
 collection of  families of nonempty subsets of
$X$, and $\Phi$ be a continuous cocycle on $X$
over
$(\Omega, \mathcal{F}, P, \{\theta\}_{t \in \mathbb{R}})$.
If $\Phi$ is $ {\mathcal{D}} $-pullback asymptotically
compact in $X$ and $\Phi$ has a closed measurable
 $ {\mathcal{D}}$-pullback absorbing set
 $K$ in $ {\mathcal{D}}$, then
$\Phi$ has a unique $ {\mathcal{D}}$-pullback
attractor $ {\mathcal{A}} $ which is characterized by,
for each $\tau \in \mathbb{R}$ and
$\omega \in \Omega$,
\begin{align*}
{\mathcal{A}} (\tau, \omega)
&=\Omega(K, \tau, \omega)
=\cup_{B \in {\mathcal{D}}} \Omega(B, \tau, \omega)\\
&=\{\psi(0, \tau, \omega): \psi \text{ is a } {\mathcal{D}} \text{-}
 \text{complete solution of } \Phi \} \\
& =\{\xi( \tau, \omega): \xi \text{ is a } {\mathcal{D}} \text{-}
 \text{complete quasi-solution of } \Phi \} .
\end{align*}
If, in addition, both $\Phi$ and $K$ are $T$-periodic, then so
is the attractor $ {\mathcal{A}}$.
 \end{proposition}

Note that the $\mathcal{F}$-measurability of the
 attractor $ {\mathcal{A}}$ was proved in
\cite{wan7} which is an improvement of the measurability of $ {\mathcal{A}}$
 with respect to  the $P$-completion of $\mathcal{F}$ in \cite{wan5}.

\section{Cocycles associated with degenerate equations}

Let ${\mathcal{O}}$ be a bounded domain in $\mathbb{R}^n$.
Consider the non-autonomous stochastic equation
defined in ${\mathcal{O}}$ for $t > \tau$ with $\tau \in \mathbb{R}$,
\begin{equation}  \label{pde1}
 {\frac {\partial u}{\partial t}}
 - \operatorname{div} \left (|\nabla u |^{p-2} \nabla u \right )
 =f_1(t,x,u) + f_2(t,x, u) +g(t,x) +\alpha u \circ {\frac {dW}{dt}},
 \end{equation}
 with boundary condition
 \begin{equation}  \label{pde2}
 u(t,x) = 0, \quad  x \in \partial {\mathcal{O}}  \text{ and }  t >\tau ,
 \end{equation}
 and initial condition
 \begin{equation} \label{pde3}
 u( \tau, x ) = u_\tau (x), \quad x\in {\mathcal{O}},
 \end{equation}
where $p>2$ and $ \alpha >0$ are constants,
$g \in L^2_{\rm loc}(\mathbb{R}, L^2(\mathcal{O}))$,
$W$ is a two-sided real-valued Wiener process on a probability space.
Throughout this paper, we assume that
$f_1: \mathbb{R} \times {\mathcal{O}} \times \mathbb{R}$
$\to \mathbb{R}$ is continuous and satisfies, for all
$t, s \in \mathbb{R}$ and $x \in {\mathcal{O}}$,
\begin{gather} \label{f1}
f_1 (t, x, s) s \le - \lambda  |s|^q + \psi_1(t, x),\\
\label{f2}
|f_1(t, x, s) | \le \psi_2 (t,x) |s|^{q-1} + \psi_3 (t, x), \\
\label{f3}
{\frac {\partial f_1}{\partial s}} (t, x, s) \le \psi_4 (t,x),
\end{gather}
where $\lambda>0$ and $ q>1$ are constants,
$\psi_1 \in L^1_{\rm loc} (\mathbb{R}, L^1( {\mathcal{O}}) )$,
$\psi_3 \in L^{q_1}_{\rm loc} (\mathbb{R}, L^{q_1} ({\mathcal{O}}))$, and
$\psi_2, \psi_4 \in L^\infty_{\rm loc} (\mathbb{R}, L^\infty ({\mathcal{O}}))$.
In this paper, we always use $p_1$ and $q_1$
for the conjugate exponents
of $p$ and $q$, respectively; that is,
${\frac 1{p_1}} + {\frac 1p} =1$ and
 ${\frac 1{q_1}} + {\frac 1q} =1$.
Let $f_2:  \mathbb{R} \times {\mathcal{O}} \times \mathbb{R}$
$\to \mathbb{R}$ be continuous and satisfy,
 for all $t, s_1, s_2 \in \mathbb{R}$ and $x \in {\mathcal{O}}$,
\begin{equation} \label{f4}
f_2(t,x,0)=0, \quad
|f_2(t,x, s_1) - f_2(t,x, s_2)|
\le \psi_5 (t,x) |s_1 -s_2 |,
\end{equation}
where $\psi_5 \in L^\infty_{\rm loc} (\mathbb{R} , L^\infty ({\mathcal{O}}))$.

In the sequel, we will use the probability
space $(\Omega, \mathcal{F}, P)$ where
$\Omega = \{ \omega \in C(\mathbb{R}, \mathbb{R} ): \ \omega(0) = 0 \}$, $\mathcal{F}$ is
the Borel $\sigma$-algebra induced by the
compact-open topology of $\Omega$, and $P$
is the corresponding Wiener
measure on $(\Omega, \mathcal{F})$.
Let $\{\theta_t\}_{t \in \mathbb{R}}$ be the standard group acting on
 $(\Omega, \mathcal{F}, P)$
 given by
\begin{equation} \label{shift1}
 \theta_t \omega (\cdot) = \omega (\cdot +t) - \omega (t),
 \quad \omega \in \Omega, \; t \in \mathbb{R} .
\end{equation}
In later sections, we will study the dynamics of
 problem \eqref{pde1}-\eqref{pde3} over the parametric
dynamical system
 $(\Omega, \mathcal{F}, P, \{\theta_t\}_{t\in \mathbb{R}})$.
 To that end, we first convert the stochastic equation
 \eqref{pde1} into a non-autonomous deterministic one by using
the random variable $z$ given by:
\begin{equation} \label{zran}
z ( \omega)= - \int^0_{-\infty} e^{ \tau} \omega (\tau) d \tau,
\quad \omega \in \Omega.
\end{equation}
Then $z$ satisfies
\begin{equation} \label{z1}
dz(\theta_t \omega ) + z(\theta_t \omega) dt = d W.
\end{equation}
Moreover, from \cite{arn1},  there exists a $\theta_t$-invariant
set $\widetilde{\Omega}\subseteq \Omega$  with $P(\widetilde{\Omega})=1$
 such that for every $\omega \in \widetilde{\Omega}$,
 $z(\theta_t \omega)$ is  continuous in $t$
and
\begin{equation} \label{z2}
\lim_{t \to \pm \infty} {\frac { |z(\theta_t \omega)|}{|t|}}
= 0 \quad \text{and} \quad
\lim_{t \to \pm \infty} {\frac 1t} \int_0^t z(\theta_t \omega) dt
= E(z) = 0.
\end{equation}
From now on, we only consider the space
$\widetilde{\Omega}$ rather than $\Omega$, and hence
write $\widetilde{\Omega}$ as $\Omega$
for convenience.

Throughout the rest of this article, we set  $V= W^{1,p}_0 ({\mathcal{O}})$.
 Given $u_\tau \in L^2(\mathcal{O})$, a continuous $L^2(\mathcal{O})$-valued process
 $ \{u(t, \tau, \omega, u_\tau)\}_{t \ge \tau}$
 is called a solution of problem \eqref{pde1}-\eqref{pde3}
if  $u(\cdot, \tau, \omega, u_\tau)
 \in L^p_{\rm loc}([\tau, \infty), V) \cap
 L^q_{\rm loc} ([\tau, \infty), L^q(\mathcal{O}))$
 for almost all
 $\omega \in \Omega$, and if for all $\xi \in
 V \cap L^q(\mathcal{O})$,
\begin{align*}
 &(u(t,\tau, \omega, u_\tau), \xi)
 + \int_\tau^t
 \int_{\mathcal{O}} |\nabla u(s, \tau, \omega, u_\tau)|^{p-2}
\nabla u(s, \tau, \omega, u_\tau)  \cdot \nabla \xi\,dx ds
\\
& =(u_\tau, \xi) +
 \int_\tau^t\int_{\mathcal{O}} f_1(s,x, u(s,\tau, \omega, u_\tau)) \xi \,dx\,ds\\
&\quad + \int_\tau^t\int_{\mathcal{O}} f_2(s,x, u(s,\tau, \omega, u_\tau)) \xi \,dx\,ds
\\
&\quad  + \int_\tau^t\int_{\mathcal{O}} g (s,x) \xi \,dx\,ds
 + \alpha \int_\tau^t (u(s, \tau, \omega, u_\tau), \xi) \circ dW ,
\end{align*}
 for all $t \ge \tau$ and for almost all $\omega \in \Omega$.
 Let $u$ be a solution of  \eqref{pde1}-\eqref{pde3} and set
 \begin{equation}  \label{uv}
 v(t, \tau, \omega, v_\tau)
 = e^{-\alpha z(\theta_t \omega)}
 u(t, \tau, \omega, u_\tau)
 \quad \text{with }
 v_\tau = e^{-\alpha z(\theta_\tau \omega)} u_\tau.
 \end{equation}
 By \eqref{pde1} and \eqref{uv} we find
 \begin{equation}  \label{v1}
\begin{aligned}
&{\frac {dv}{dt}}
 -e^{\alpha (p-2) z(\theta_t \omega)} \operatorname{div} (|\nabla v|^{p-2}
\nabla v) \\
&=\alpha z(\theta_t \omega) v
 + e^{-\alpha z(\theta_t \omega)} f_1(t,x, e^{\alpha z(\theta_t \omega)} v)
\\
&\quad + e^{- \alpha z(\theta_t \omega)}f_2(t,x, e^{\alpha z(\theta_t \omega)} v)
 +e^{- \alpha z(\theta_t \omega)} g(t,x),
\end{aligned}
\end{equation}
 with boundary condition
 \begin{equation}  \label{v2}
 v(t,x) = 0, \quad  x \in \partial {\mathcal{O}}  \text{ and }  t >\tau ,
 \end{equation}
 and initial condition
 \begin{equation} \label{v3}
 v( \tau, x ) = v_\tau (x), \quad x\in {\mathcal{O}}.
 \end{equation}

 In the sequel, we will establish the well-posedness
 of problem \eqref{v1}-\eqref{v3}.
 For that purpose, we need to introduce the definition of weak solutions
 for the equation.

 \begin{definition}  \label{defv} \rm
 Given $\tau \in \mathbb{R}$, $\omega \in \Omega$  and
 $v_\tau \in L^2(\mathcal{O})$, a continuous function
 $v(\cdot, \tau, \omega, v_\tau)$: $[\tau, \infty) \to L^2(\mathcal{O})$
 is called a solution of  \eqref{v1}-\eqref{v3}
 if $ v(\tau, \tau, \omega, v_\tau) =v_\tau$ and
\begin{gather*}
v  \in L^p_{\rm loc}([\tau, \infty), V) \cap
 L^q_{\rm loc} ([\tau, \infty), L^q(\mathcal{O})), \\
 {\frac {dv}{dt}} \in L^{p_1}_{\rm loc}([\tau, \infty), V^*)
 + L^{q_1}_{\rm loc} ( [\tau, \infty), L^{q_1} ({\mathcal{O}} ) ),
\end{gather*}
and $v$ satisfies, for every  $\xi \in V \cap L^q(\mathcal{O}) $,
\begin{align*}
&{\frac {d}{dt}} (v, \xi)
 -e^{\alpha (p-2) z(\theta_t \omega)}
 \int_{\mathcal{O}} |\nabla v|^{p-2}\nabla v\cdot \nabla \xi\,dx\\
&=\alpha z(\theta_t \omega) (v,\xi)
 + e^{-\alpha z(\theta_t \omega)}
 \int_{\mathcal{O}} f_1 (t,x, e^{\alpha z(\theta_t \omega)} v) \xi\,dx\\
&\quad + e^{- \alpha z(\theta_t \omega)}
 \int_{\mathcal{O}} f_2(t,x, e^{\alpha z(\theta_t \omega)} v) \xi\,dx
 +e^{- \alpha z(\theta_t \omega)} ( g(t,\cdot), \xi)
\end{align*}
in the sense of distribution on $[\tau, \infty)$.
\end{definition}

 We will employ the Galerkin method to prove the
 existence of solutions for the deterministic equation \eqref{v1}.
For simplicity, we define an operator
 $A$: $V \to V^*$ by
\begin{equation} \label{Av}
 (A(v_1), v_2 )_{(V^*, V)}
 = \int_{\mathcal{O}} | \nabla v_1|^{p-2} \nabla v_1 \cdot \nabla v_2 dx,
 \quad \text{for all }  v_1, v_2 \in V,
\end{equation}
where $ (\cdot, \cdot )_{(V^*, V)}$ is the duality pairing
of $V^*$ and $V$.
It follows from \cite{show1} that $A$ is hemicontinuous and monotone.
 Let $\{ e_j\}_{j=1}^\infty \subseteq V\cap L^q(\mathcal{O})$
 be an orthonormal basis of $ L^2(\mathcal{O})$ such that
 span$\{ e_j: j \in \mathbb{N}\}$ is dense
 in $V\cap L^q(\mathcal{O})$.
Given $n \in \mathbb{N}$, let $X_n$ be the space spanned by
 $\{e_j: j=1, \ldots, n\}$ and $P_n$:
 $L^2(\mathcal{O}) \to X_n$ be the projection given by
 $$
 P_n v = \sum_{j=1}^n (v, e_j) e_j,
 \quad \forall  v \in L^2(\mathcal{O}).
$$
 Note that $P_n$ can be extended to
$V^*$ and $ (L^q(\mathcal{O}))^*$ by
 $$
P_n \phi = \sum_{j=1}^n (\phi (e_j) ) e_j,
 \quad  \text{for }
 \phi \in V^*  \text{ or }  \phi \in (L^q(\mathcal{O}))^*.
 $$
Consider the following  system for $v_n \in X_n$
 defined for $t > \tau$:
 \begin{equation}  \label{ode1}
\begin{aligned}
{\frac {dv_n}{dt}}
 + e^{\alpha (p-2) z(\theta_t \omega)} P_n A (v_n)
&=\alpha z(\theta_t \omega) v_n
 + e^{-\alpha z(\theta_t \omega)}
P_n f_1(t,\cdot, e^{\alpha z(\theta_t \omega)} v_n)\\
&\quad  + e^{- \alpha z(\theta_t \omega)} P_n f_2(t, \cdot,
e^{\alpha z(\theta_t \omega)} v_n)
 +e^{- \alpha z(\theta_t \omega)} P_n g,
\end{aligned}
\end{equation}
 with initial condition
 \begin{equation} \label{ode2}
 v_n ( \tau ) = P_n v_\tau .
 \end{equation}
 Under assumptions \eqref{f1}-\eqref{f4}, we find that
 for every $\tau \in \mathbb{R}$, $\omega \in \Omega$ and
 $v_\tau \in L^2(\mathcal{O})$, problem \eqref{ode1}-\eqref{ode2}
 has a unique maximal solution
 $v_n(\cdot, \tau, \omega) \in C^1 ([\tau, \tau +T), X_n)$
 for some $T>0$. The uniform estimates given below
 imply $T =\infty$, and hence the solution is defined for all
 $t \ge \tau$. In addition,
 $v_n(t, \tau, \omega, v_\tau)$ is measurable
 with respect to $\omega \in \Omega$.
 By examining the limiting behavior of $v_n$, we prove the
 following existence and uniqueness of solutions for
 \eqref{v1}-\eqref{v3}.

 \begin{lemma} \label{exiv}
 Suppose \eqref{f1}-\eqref{f4} hold. Then for every
 $\tau \in \mathbb{R}$, $\omega \in \Omega$ and $v_\tau \in L^2(\mathcal{O})$,
 problem \eqref{v1}-\eqref{v3} has a unique solution
 $v(t, \tau, \omega, v_\tau)$ in the sense of Definition \ref{defv}.
 This solution  is ($\mathcal{F}, \mathcal{B} (L^2(\mathcal{O})))$-measurable
 in $\omega$ and continuous in
 initial data $v_\tau$ in $L^2(\mathcal{O})$. Moreover, the solution $v$ satisfies the
 energy equation:
 \begin{equation} \label{enlem1}
\begin{aligned}
&{\frac d{dt}} \| v(t, \tau, \omega, v_\tau) \|^2
 +2 e^{\alpha (p-2) z(\theta_t \omega)}
 \int_{\mathcal{O}} |\nabla v|^{p} dx\\
& = 2\alpha z(\theta_t \omega) \| v \|^2
 + 2 e^{-\alpha z(\theta_t \omega)}
 \int_{\mathcal{O}} f_1(t,x, e^{\alpha z(\theta_t \omega)} v) v\,dx
\\
&\quad + 2 e^{- \alpha z(\theta_t \omega)}
 \int_{\mathcal{O}}
 f_2(t,x, e^{\alpha z(\theta_t \omega)} v) v dx
 + 2 e^{- \alpha z(\theta_t \omega)} \int_{\mathcal{O}} g(t,x) v dx
\end{aligned}
 \end{equation}
 for almost all $t \ge \tau$.
 \end{lemma}

 \begin{proof}
 The proof consists of several steps.
We first derive uniform estimates
 on the solution $v_n$ of \eqref{ode1}-\eqref{ode2}.

\subsection*{Step 1: Uniform estimates}
 By \eqref{ode1} we obtain
\begin{equation} \label{a1}
\begin{aligned}
&{\frac 12} {\frac d{dt}} \| v_n \|^2
 + e^{\alpha (p-2) z(\theta_t \omega)} \int_{\mathcal{O}} |\nabla v_n |^p dx\\
&=\alpha z(\theta_t \omega) \| v_n \|^2
 + e^{-\alpha z(\theta_t \omega)}
 \int_{\mathcal{O}} f_1(t, x, e^{\alpha z(\theta_t \omega)} v_n) v_n dx\\
&\quad  + e^{- \alpha z(\theta_t \omega)}
 \int_{\mathcal{O}} f_2(t, x, e^{\alpha z(\theta_t \omega)} v_n) v_n dx
 +e^{- \alpha z(\theta_t \omega)} (g(t), v_n).
\end{aligned}
\end{equation}
 By \eqref{f1} we obtain
 \begin{equation} \label{a2}
\begin{aligned}
&e^{-\alpha z(\theta_t \omega)}
 \int_{\mathcal{O}} f_1(t, x, e^{\alpha z(\theta_t \omega)} v_n) v_n dx\\
&\leq -\lambda e^{\alpha (q-2) z(\theta_t \omega) }
 \int_{\mathcal{O}} |v_n |^q dx
 + e^{-2\alpha z(\theta_t \omega) } \int_{\mathcal{O}} \psi_1 (t,x) dx.
\end{aligned}
 \end{equation}
 By \eqref{f4} we have
 \begin{equation} \label{a3}
 e^{- \alpha z(\theta_t \omega)}
 |\int_{\mathcal{O}} f_2(t, x, e^{\alpha z(\theta_t \omega)} v_n) v_n dx|
 \le \| \psi_5 (t) \|_{L^\infty ({\mathcal{O}})} \| v_n \|^2.
 \end{equation}
It follows from \eqref{a1}-\eqref{a3} that
 \begin{equation} \label{a4}
\begin{aligned}
& {\frac d{dt}} \| v_n \|^2
+ 2e^{\alpha (p-2) z(\theta_t \omega)} \int_{\mathcal{O}} |\nabla v_n |^p dx
+  2 \lambda e^{\alpha (q-2) z(\theta_t \omega) }
 \int_{\mathcal{O}} |v_n |^q dx \\
&\leq
\left (1+ 2\alpha z(\theta_t \omega)
+ 2 \| \psi_5 (t)\|_{L^\infty ({\mathcal{O}})} \right ) \| v_n \|^2
+ 2  e^{-2\alpha z(\theta_t \omega) }
\| \psi_1 (t)\|_{L^1({\mathcal{O}})}\\
&\quad + e^{-2\alpha z(\theta_t \omega)} \| g(t) \|^2.
\end{aligned}
\end{equation}
Note that $z(\theta_t \omega)$ is continuous in $t$ for fixed
$\omega$. Therefore, for every $\tau \in \mathbb{R}$,
$\omega \in \Omega$ and $T>0$, we obtain from \eqref{a4} that
\begin{equation} \label{a5}
\{v_n\}_{n=1}^\infty
 \text{ is bounded in }
 L^\infty (\tau, \tau +T; L^2(\mathcal{O}) ) \cap L^q(\tau, \tau +T; L^q(\mathcal{O}))
 \cap L^p(\tau, \tau +T; V).
 \end{equation}
By \eqref{Av} and \eqref{a5} we obtain
 \begin{equation} \label{a6}
 \{A(v_n)\}_{n=1}^\infty \text{ is bounded in }  L^{p_1}(\tau, \tau +T; V^*),
 \end{equation}
where $ {\frac 1{p_1}} + {\frac 1p} =1$. By \eqref{f2} we have
\begin{align*}
&\int_\tau^{\tau +T}
 \int_{\mathcal{O}} |f_1(t,x, e^{\alpha z(\theta_t \omega) }v_n) |^{q_1} \,dx\,dt\\
&\leq c_1  \int_\tau^{\tau +T}
 \int_{\mathcal{O}} |v_n |^q \,dx\,dt
 + \int_\tau^{\tau +T} \int_{\mathcal{O}} | \psi_3 (t,x) |^{q_1} \,dx\,dt,
\end{align*}
with $ {\frac 1{q_1}} + {\frac 1q} =1 $,
which along with \eqref{a5}
 implies
 \begin{equation} \label{a7}
 \{ f_1(t,x, e^{\alpha z(\theta_t \omega) }v_ n)\}_{n=1}^\infty
  \text{ is bounded in }  L^{q_1} (\tau, \tau +T; L^{q_1} ({\mathcal{O}})).
 \end{equation}
 By \eqref{a6} and \eqref{a7}, we infer from \eqref{ode1} that
 \begin{equation} \label{a9}
 \{ {\frac {dv_n}{dt}}\}_{n=1}^\infty
 \text{ is bounded in }
 L^{p_1}(\tau, \tau +T; V^*)  +
 L^{q_1} (\tau, \tau +T; L^{q_1} ({\mathcal{O}})).
 \end{equation}
 Next, we prove the existence of solutions for
\eqref{v1}-\eqref{v3}  based on \eqref{a5}-\eqref{a9}.

\subsection*{Step 2: Existence of solutions}
 It follows from \eqref{a5}-\eqref{a7} that  there exist
 $\widetilde{v}\in L^2(\mathcal{O})$,
 $v \in L^\infty (\tau, \tau +T; L^2(\mathcal{O}))
 \cap L^p(\tau, \tau +T; V) \cap L^q(\tau, \tau +T; L^q(\mathcal{O})) $,
 $\chi_1  \in L^{q_1} (\tau, \tau +T; L^{q_1}({\mathcal{O}}) ) $,
 $\chi_2  \in L^{p_1} (\tau, \tau +T; V^* ) $
 such that, up to a subsequence,
\begin{gather} \label{a10}
 v_n \to v \quad \text{weak-star in } L^\infty (\tau, \tau +T; L^2(\mathcal{O}) ),\\
\label{a11}
 v_n \to v  \quad\text{weakly in }  L^p (\tau, \tau +T; V )
  \text{ and }  L^q(\tau, \tau +T; L^q(\mathcal{O})), \\
\label{a12}
 f_1(t, x, e^{\alpha z(\theta_t \omega) } v_n) \to \chi_1  \quad \text{weakly in }
 L^{q_1} (\tau, \tau +T; L^{q_1} ({\mathcal{O}}) ), \\
\label{a13}
 A( v_n) \to \chi_2  \quad \text{weakly in }
 L^{p_1} (\tau, \tau +T; V^* ), \\
\label{a13a1}
 {\frac {dv_n}{dt}}  \to {\frac {dv}{dt}}  \quad \text{weakly in }
 L^{p_1}(\tau, \tau +T; V^*)  +
 L^{q_1} (\tau, \tau +T; L^{q_1} ({\mathcal{O}})),\\
\label{a13a2}
 v_n(\tau +T,\tau, \omega)  \to {\widetilde{v}}\quad
  \text{weakly in }  L^2(\mathcal{O}).
 \end{gather}
Let $\sigma =\min \{ p_1, q_1\}$. By \eqref{a9} we see that
 $ \{{\frac {dv_n}{dt}} \}$ is bounded
 in $L^\sigma (\tau, \tau +T; (V\cap L^q(\mathcal{O}))^* )$.
Note that $V \hookrightarrow L^2(\mathcal{O})  \hookrightarrow (V\cap L^q(\mathcal{O}))^*$
and the embedding  $V \hookrightarrow L^2(\mathcal{O}) $ is compact.
Therefore, it follows from \cite{lion1} that,  up to a subsequence,
 \begin{equation} \label{a14}
 v_n \to v  \text{ strongly in } L^2(\tau, \tau +T; L^2(\mathcal{O}) ).
 \end{equation}
By \eqref{f4} and \eqref{a14} one can verify that for every
 $j \in \mathbb{N}$ and $\phi \in C_0^\infty (\tau, \tau +T)$,
\begin{equation} \label{a15}
\begin{aligned}
&\lim_{n \to \infty}  \int_\tau^{\tau +T}
 e^{-\alpha z(\theta_t \omega) }
 (P_n f_2(t, \cdot, e^{\alpha z(\theta_t \omega)} v_n ), \phi e_j) dt\\
&= \int_\tau^{\tau +T} e^{-\alpha z(\theta_t \omega) }
 ( f_2(t, \cdot, e^{\alpha z(\theta_t \omega)} v) , \phi e_j) dt.
\end{aligned}
\end{equation}
Letting $ n \to \infty$ in \eqref{ode1},
by \eqref{a10}-\eqref{a13} and \eqref{a15}, we obtain,
for $j \in \mathbb{N}$ and $\phi \in C_0^\infty (\tau, \tau +T)$,
\begin{equation} \label{a16}
\begin{aligned}
&-\int_\tau^{\tau +T} (v, e_j) \phi^\prime dt
 + \int_\tau^{\tau +T}  e^{ \alpha (p-2) z(\theta_t \omega) }
 (\chi_2, \phi e_j)_{(V^*, V)} dt \\
& =\alpha \int_\tau^{\tau +T}
 z(\theta_t \omega ) (v, \phi e_j) dt
 + \int_\tau^{\tau +T} e^{-\alpha z(\theta_t \omega) }
 (\chi_1, \phi e_j)_{(L^{q_1}, L^q )} dt \\
&\quad  + \int_\tau^{\tau +T}
 e^{ - \alpha z(\theta_t \omega) }
 ( f_2(t, \cdot, e^{\alpha z(\theta_t \omega)} v) , \phi e_j) dt
+\int_\tau^{\tau +T} e^{ - \alpha z(\theta_t \omega) }
 (g, \phi e_j) dt.
\end{aligned}
\end{equation}
Since span$\{e_j, j\in \mathbb{N}\}$ is dense  in $V\cap L^q(\mathcal{O})$, we find that
\eqref{a16} is still valid when  $e_j$ is replaced by any element
in $V\cap L^q(\mathcal{O})$. Therefore,  for every $\xi \in V\cap L^q(\mathcal{O})$,
 in the sense of distributions, we have
\begin{equation} \label{a17}
\begin{aligned}
& {\frac {d}{dt}} (v, \xi)
 + e^{\alpha (p-2) z(\theta_t \omega)} (\chi_2, \xi)_{(V^*, V)}\\
&=\alpha z(\theta_t \omega) (v , \xi)
 + e^{-\alpha z(\theta_t \omega)} (\chi_1, \xi)_{(L^{q_1}, L^q)}\\
&\quad + e^{- \alpha z(\theta_t \omega)}
 (f_2(t, \cdot, e^{\alpha z(\theta_t \omega)} v), \xi)
 +e^{- \alpha z(\theta_t \omega)} (g, \xi),
\end{aligned}
\end{equation}
Note that \eqref{a17} implies
\begin{equation} \label{a18}
\begin{aligned}
{\frac {dv}{dt}}
&= - e^{\alpha (p-2) z(\theta_t \omega)} \chi_2
 + \alpha z(\theta_t \omega) v
 + e^{-\alpha z(\theta_t \omega)} \chi_1 \\
&\quad + e^{- \alpha z(\theta_t \omega)}
 f_2(t, \cdot, e^{\alpha z(\theta_t \omega)} v)
 +e^{- \alpha z(\theta_t \omega)} g,
\end{aligned}
\end{equation}
in $L^{p_1}(\tau, \tau +T; V^*)
+ L^{q_1} (\tau, \tau +T; L^{q_1} ({\mathcal{O}}) )$.
Since $v \in L^p(\tau, \tau +T; V)
\cap L^q(\tau, \tau +T; L^q(\mathcal{O}))$
and ${\frac {dv}{dt}} \in L^{p_1}(\tau, \tau +T; V^*)
+ L^{q_1} (\tau, \tau +T; L^{q_1} ({\mathcal{O}}) )$,
as in \cite{lion1}, we find that
$v \in C ([\tau, \tau +T], L^2(\mathcal{O}))$ and
\begin{equation} \label{a20}
{\frac 12} {\frac d{dt}} \| v \|^2
= ( {\frac {dv}{dt}}, v)_{( V^*+ L^{q_1}, V\cap L^q )}
 \quad \text{for almost all } t \in (\tau, \tau +T).
\end{equation}
We now prove $v(\tau) =v_\tau$ and $v(\tau +T) = { \widetilde{v}}$.
Let $\phi \in C^1([\tau, \tau +T])$ and $\xi \in V\cap L^q(\mathcal{O})$.
Multiplying \eqref{ode1} by $\phi \xi$
and then taking the limit as before, we obtain from \eqref{ode2}
and \eqref{a13a2} that
\begin{equation} \label{a21}
\begin{aligned}
&({\widetilde{v}}, \xi ) \phi(\tau +T)  -(v_\tau, \xi) \phi (\tau)\\
& = \int_\tau^{\tau +T} (v, \xi) \phi^\prime dt
- \int_\tau^{\tau +T} e^{ \alpha (p-2) z(\theta_t \omega) }
 (\chi_2, \phi \xi)_{(V^*, V)} dt \\
&\quad  + \alpha \int_\tau^{\tau +T}
 z(\theta_t \omega ) (v, \phi \xi ) dt
 + \int_\tau^{\tau +T}
 e^{-\alpha z(\theta_t \omega) }
 (\chi_1, \phi \xi )_{(L^{q_1}, L^q )} dt \\
&\quad + \int_\tau^{\tau +T} e^{ - \alpha z(\theta_t \omega) }
 ( f_2(t, \cdot, e^{\alpha z(\theta_t \omega)} v) , \phi \xi) dt
 +  \int_\tau^{\tau +T} e^{ - \alpha z(\theta_t \omega) }
 (g, \phi \xi) dt.
\end{aligned}
 \end{equation}
On the other hand, by \eqref{a17} we find that the right-hand side
of \eqref{a21} is given by
 $(v(\tau +T), \xi) \phi(\tau +T)
 -(v(\tau), \xi) \phi (\tau)$. Therefore we obtain
 $$
 (v(\tau +T), \xi) \phi(\tau +T)
 -(v(\tau), \xi) \phi (\tau)
= ({\widetilde{v}}, \xi ) \phi(\tau +T)
 -(v_\tau, \xi) \phi (\tau).
 $$
Choose $\psi \in C^1([\tau, \tau +T])$
such that $\psi (\tau)=1$ and $\psi(\tau +T) =0$.
First letting $\phi =\psi$ and then letting $\phi = 1-\psi$,
from the above, we obtain
 \begin{equation} \label{a24}
 v(\tau) = v_\tau
 \quad \text{and} \quad
 v(\tau +T) = \widetilde{v}.
\end{equation}
By \eqref{a13a2} and \eqref{a24} we obtain
\begin{equation} \label{a25}
 v_n(\tau +T, \tau, \omega)
 \to v(\tau +T)\quad \text{weakly in }  L^2(\mathcal{O}),
\end{equation}
which implies that
\begin{equation}\label{a26}
\liminf_{n \to \infty}\| v_n (\tau +T, \tau, \omega) \| \ge \| v(\tau +T) \|.
\end{equation}
Next, we prove $\chi_1 = f_1(t, \cdot, v)$.
 By \eqref{a14} we see that, up to a subsequence,
 \begin{equation}\label{a30}
 v_n \to v  \quad \text{for almost all }
 (t,x) \in (\tau, \tau +T) \times {\mathcal{O}}.
\end{equation}
 By \eqref{a30} and the continuity of $f_1$, we obtain
\begin{equation} \label{a31}
 f_1(t,x, e^{\alpha z(\theta_t \omega) }v_n)
 \to f_1(t,x, e^{\alpha z(\theta_t \omega) }v)
\end{equation}
for almost all $ (t,x) \in (\tau, \tau +T) \times {\mathcal{O}}$.
By \eqref{a7} and \eqref{a31}, from
 \cite{lion1} it follows that
\begin{equation} \label{a31a1}
 f_1(t, x, e^{\alpha z(\theta_t \omega) } v_n)
 \to f_1(t, x, e^{\alpha z(\theta_t \omega) } v)
 \quad \text{weakly in } L^{q_1}(\tau, \tau +T; L^{q_1} ({\mathcal{O}}) ).
\end{equation}
By \eqref{a12} and \eqref{a31a1} we have
\begin{equation} \label{a32}
 \chi_1 =f_1(t,x, e^{\alpha z(\theta_t \omega) }v).
\end{equation}
We now show that  $\chi_2 = A(v)$.
By \eqref{f1},
 $$
 e^{- 2\alpha z(\theta_t \omega) } \psi_1 (t,x)
 -e^{- \alpha z(\theta_t \omega) }
 f_1(t,x, e^{ \alpha z(\theta_t \omega) } v_n)v_n \ge 0.
 $$
 Therefore,  from Fatou's lemma it follows that
\begin{align*}
 &\liminf_{n \to \infty} \int_\tau^{\tau +T} \int_{\mathcal{O}}
 \left (
 e^{- 2 \alpha z(\theta_t \omega) }
 \psi_1 (t,x) - e^{- \alpha z(\theta_t \omega) }
 f_1(t,x, e^{ \alpha z(\theta_t \omega) } v_n)v_n \right ) \,dx\,dt
 \\
 & \geq \int_\tau^{\tau +T} \int_{\mathcal{O}}
 \liminf_{n \to \infty}
 \left ( e^{- 2 \alpha z(\theta_t \omega) }
 \psi_1 (t,x) - e^{- \alpha z(\theta_t \omega) }
 f_1(t,x, e^{ \alpha z(\theta_t \omega) } v_n)v_n \right ) \,dx\,dt
\end{align*}
which along with \eqref{a30}-\eqref{a31} shows that
\begin{equation} \label{a35}
\begin{aligned}
&\limsup_{n \to \infty} \int_\tau^{\tau +T} \int_{\mathcal{O}}
 e^{- \alpha z(\theta_t \omega) }
 f_1(t,x, e^{ \alpha z(\theta_t \omega) } v_n)v_n\\
&\leq  \int_\tau^{\tau +T} \int_{\mathcal{O}}
 e^{- \alpha z(\theta_t \omega) }
 f_1(t,x, e^{ \alpha z(\theta_t \omega) } v)v\,.
\end{aligned}
\end{equation}
By \eqref{ode1}-\eqref{ode2} we find that
\begin{equation} \label{a36}
\begin{aligned}
&\int_\tau^{\tau +T}
 e^{\alpha (p-2) z(\theta_t \omega)} (A(v_n), v_n)_{(V^*, V)} dt\\
&= {\frac 12} \| v_n (\tau) \|^2 -{\frac 12} \| v_n (\tau +T) \|^2
 + \alpha \int_\tau^{\tau +T}
 z(\theta_t \omega) \| v_n \|^2 dt\\
&\quad + \int_\tau^{\tau +T} \int_{\mathcal{O}}
 e^{-\alpha z(\theta_t \omega)}
 f_1(t,x, e^{\alpha z(\theta_t \omega)} v_n) v_n \,dx\,dt\\
&\quad + \int_\tau^{\tau +T} \int_{\mathcal{O}}
 e^{- \alpha z(\theta_t \omega)} f_2(t, x, e^{\alpha z(\theta_t \omega)} v_n)v_n
 \,dx\,dt \\
&\quad +\int_\tau^{\tau +T} \int_{\mathcal{O}}
 e^{- \alpha z(\theta_t \omega)} g(t,x) v_n \,dx\,dt.
\end{aligned}
\end{equation}
Letting $n \to \infty$ in \eqref{a36}, by \eqref{a14}, \eqref{a26}
and \eqref{a35} we obtain
\begin{equation} \label{a37}
\begin{aligned}
&\limsup_{n \to \infty}  \int_\tau^{\tau +T}
 e^{\alpha (p-2) z(\theta_t \omega)} (A(v_n), v_n)_{(V^*, V)} dt\\
&\leq {\frac 12} \| v (\tau) \|^2 -{\frac 12} \| v (\tau +T) \|^2
 + \alpha \int_\tau^{\tau +T}
 z(\theta_t \omega) \| v \|^2 dt\\
&\quad + \int_\tau^{\tau +T} \int_{\mathcal{O}}
 e^{-\alpha z(\theta_t \omega)}
 f_1(t,x, e^{\alpha z(\theta_t \omega)} v) v \,dx\,dt \\
&\quad + \int_\tau^{\tau +T} \int_{\mathcal{O}}
 e^{- \alpha z(\theta_t \omega)} f_2(t, x, e^{\alpha z(\theta_t \omega)} v)v
 \,dx\,dt\\
&\quad +\int_\tau^{\tau +T} \int_{\mathcal{O}}
 e^{- \alpha z(\theta_t \omega)} g(t,x) v \,dx\,dt.
\end{aligned}
\end{equation}
On the other hand, by \eqref{a18}, \eqref{a20} and
\eqref{a32} we see that the right-hand side of \eqref{a37}
is given by $ \int_\tau^{\tau +T}
 e^{\alpha (p-2) z(\theta_t \omega)} (\chi_2, v )_{(V^*, V)} dt $.
 Therefore we obtain
 \begin{equation} \label{a40}
 \limsup_{n \to \infty}
 \int_\tau^{\tau +T}
 e^{\alpha (p-2) z(\theta_t \omega)} (A(v_n), v_n)_{(V^*, V)} dt
 \le
 \int_\tau^{\tau +T}
 e^{\alpha (p-2) z(\theta_t \omega)} (\chi_2, v)_{(V^*, V)} dt.
 \end{equation}
Since $A: L^p(\tau, \tau +T; V) \to L^{p_1} (\tau, \tau +T; V^*)$
is hemicontinuous and monotone, by \eqref{a11}, \eqref{a13}
 and \eqref{a40} we infer that
 \begin{equation}\label{a41}
 \chi_2 = A(v).
 \end{equation}
It follows from \eqref{a17}, \eqref{a24}, \eqref{a32}
 and \eqref{a41} that
 $v$ is a solution of problem \eqref{v1}-\eqref{v3}
 in the sense of Definition \ref{defv}.
 By \eqref{a18}, \eqref{a20}, \eqref{a32}
 and \eqref{a41} we find that $v$ satisfies the energy equation
 \eqref{enlem1}.

 \subsection*{Step 3: Uniqueness of solutions}
 Suppose $v_1$ and $v_2$ are two solutions of \eqref{v1}-\eqref{v2}
 with initial data $v_{1,\tau}$ and $v_{2, \tau}$, respectively.
 Then $ {\widetilde{v}} = v_1 -v_2$ satisfies
\begin{align*}
& {\frac {d{\widetilde{v}}}{dt}}
 + e^{\alpha (p-2) z(\theta_t \omega) }
 (A(v_1) -A(v_2) )
\\
& =\alpha z(\theta_t \omega) {\widetilde{v}}
 + e^{ - \alpha z(\theta_t \omega) }
 \left ( f_1(t,x, e^{ \alpha z(\theta_t \omega) }v_1)
 -f_1(t,x, e^{ \alpha z(\theta_t \omega) }v_2)
 \right )
\\
&\quad + e^{ - \alpha z(\theta_t \omega) }
\left (f_2(t,x, e^{ \alpha z(\theta_t \omega) } v_1)
-f_2(t,x, e^{ \alpha z(\theta_t \omega) }v_2)
\right ).
\end{align*}
By \eqref{f3}-\eqref{f4} and the monotonicity
of $A$, we obtain for $t \in [\tau, \tau +T]$,
$$
 {\frac 12} {\frac {d}{dt}}
 \| {\widetilde{v}} \|^2
 \le \alpha z(\theta_t \omega) \| {\widetilde{v}} \|^2
 + e^{ - \alpha z(\theta_t \omega) }
 \int_{\mathcal{O}} \psi_4 (t,x) | {\widetilde{v}} |^2 dx
 + \int_{\mathcal{O}} \psi_5 (t,x) | {\widetilde{v}} |^2 dx
 \le c_1 \| {\widetilde{v}} \|^2,
$$
where $c_1$ is a positive constant depending on $\tau,T$ and $\omega$.
Therefore, for all $t \in [\tau, \tau +T]$, we have
$$
 \| v_1(t, \tau, \omega, v_{1,\tau}) - v_2(t, \tau, \omega, v_{2,\tau}) \|^2
 \le e^{c_1 (t-\tau)} \| v_{1,\tau} - v_{2, \tau} \|^2,
$$
which implies the uniqueness and continuous dependence of
solutions on initial data in $L^2(\mathcal{O})$.

By \eqref{a13a2},  \eqref{a24} and the uniqueness of solutions, we infer that
for every $\omega \in \Omega$, the whole sequence $v_n(\tau +T, \tau, \omega)
 \to v(\tau +T, \tau, \omega)$ weakly in $L^2(\mathcal{O})$.
By a similar argument, one can verify that
 $v_n(t, \tau, \omega)  \to v(t, \tau, \omega)$ weakly in $L^2(\mathcal{O})$
for any $t \ge \tau$ and $\omega \in \Omega$.
Then the measurability of $v (t, \tau, \omega)$
follows from that of $v_n(t, \tau, \omega)$.
This completes the proof.
\end{proof}

From the proof of Lemma \ref{exiv}, we see that
the solution operator of problem \eqref{v1}-\eqref{v3}
is compact in $L^2(\mathcal{O})$ as stated below.


\begin{lemma} \label{wcont}
If \eqref{f1}-\eqref{f4} hold, then for every $\omega \in \Omega$
and $t, \tau \in \mathbb{R}$ with $t>\tau$,
the solution operator $v(t, \tau, \omega, \cdot)$:
$L^2(\mathcal{O}) \to L^2(\mathcal{O})$ of problem \eqref{v1}-\eqref{v3}
is compact; that is, for every bounded set $B $ in $L^2(\mathcal{O})$,
 the image $v(t, \tau, \omega, B)$
 is precompact in $L^2(\mathcal{O})$.
\end{lemma}

\begin{proof}
We argue as in \cite{gess3}.
 Suppose $\{v_{0,n} \}_{n=1}^\infty$ is an arbitrary sequence
 in $B$. Choose a positive number
 $T$ such that
 $t \in [\tau, \tau +T]$.
 As in \eqref{a14}, one can show that
 there exists $v \in L^2((\tau, \tau +T), L^2(\mathcal{O}))$
 such that, up to a subsequence,
 $$
 v(\cdot, \tau, \omega, v_{0,n} ) \to v
 \quad \text{in } \ L^2( (\tau, \tau+T), L^2(\mathcal{O}) ) .
 $$
Therefore, there exists a subsequence
 (which is still denoted by $ v(\cdot, \tau, \omega, v_{0,n} )$)
 such that
 \begin{equation} \label{pwcont_1}
 v(s, \tau, \omega, v_{0,n} ) \to v (s)
 \quad \text{in }  L^2(\mathcal{O})
  \text{ for almost all } s \in (\tau, \tau +T) .
 \end{equation}
Since $t>\tau$, we may choose $s_0 \in (\tau, t)$ such that \eqref{pwcont_1}
is fulfilled at $s =s_0$.
It follows from the continuity of solutions in initial data
that
$$
v(t, \tau, \omega, v_{0,n} )
= v(t,s_0, \omega, v(s_0, \tau, \omega, v_{0,n} )
\to v(t,s_0 , \omega, v(s_0)),
$$
and hence $\{v(t, \tau, \omega, v_{0,n} )\}$
has a convergent subsequence in $L^2(\mathcal{O})$.
\end{proof}

We now define a continuous cocycle for
the stochastic equation \eqref{pde1}.
Let $u(t, \tau, \omega, u_\tau) = e^{\alpha z(\theta_t \omega) }
v(t, \tau, \omega, v_\tau) $
with $u_\tau = e^{\alpha z(\theta_\tau \omega) }v_\tau $, where
$v$ is the solution of problem \eqref{v1}-\eqref{v3}.
 It follows from Lemma \ref{exiv} that
 $u$ is a solution of problem \eqref{pde1}-\eqref{pde3}
 which is continuous
in $t \in [\tau, \infty)$ as well as in $u_\tau \in L^2(\mathcal{O})$.
Let $\Phi: \mathbb{R}^+ \times \mathbb{R} \times \Omega \times L^2(\mathcal{O})$
$\to L^2(\mathcal{O})$ be a mapping given by,
for every $t \in \mathbb{R}^+$, $\tau \in \mathbb{R}$, $\omega \in \Omega$ and $u_\tau \in L^2(\mathcal{O})$,
 \begin{equation} \label{pcycle}
 \Phi (t, \tau, \omega, u_\tau) = u (t+\tau, \tau, \theta_{-\tau} \omega, u_\tau)
= e^{\alpha z(\theta_t \omega)} v(t+\tau, \tau, \theta_{ -\tau} \omega, v_\tau),
\end{equation}
where $v_\tau = e^{-\alpha z(\omega)} u_\tau $.

Then $\Phi$ is a continuous cocycle on $L^2(\mathcal{O})$ over $(\Omega, \mathcal{F},
P, \{\theta_t \}_{t\in \mathbb{R}})$.
We will prove the existence of $\mathcal{D}$-pullback attractors for $\Phi$
 for an appropriate collection of families of subsets of $L^2(\mathcal{O})$.
 To define such a collection, we first recall the Poincare inequality:
\begin{equation} \label{b1}
\int_{\mathcal{O}} |\nabla v (x) |^p dx
\ge \beta \int_{\mathcal{O}} |v (x)|^p dx,
\quad \text{for all } v \in W^{1,p}_0 ({\mathcal{O}}) ,
\end{equation}
where $\beta$ is a positive constant depending only on ${\mathcal{O}}$ and $p$.
Note that for any $p \ge 2$,
\begin{equation} \label{b2}
 \int_{\mathcal{O}} |v(x)|^2 dx
 \le {\frac {p-2}p} |{\mathcal{O}}|
 + {\frac 2p} \int_{\mathcal{O}} |v(x) |^p dx,
\end{equation}
where $|{\mathcal{O}}|$ stands for the Lebesgue measure of ${\mathcal{O}}$.
By \eqref{b1} and \eqref{b2} we have
\begin{equation} \label{b3}
\int_{\mathcal{O}} | \nabla v(x) |^p dx
\geq  {\frac 12} \beta p \| v \|^2
 -{\frac 12} \beta (p-2) |{\mathcal{O}}|,
 \quad \text{ for all } \ v \in W^{1,p}_0 ({\mathcal{O}}) .
 \end{equation}
Given $r \in \mathbb{R}$ and $\omega \in \Omega$, denote by
\begin{equation} \label{hro}
 h(r, \omega)
 = {\frac 12} \beta p e^{\alpha (p-2) z(\theta_r \omega)}
 -2 \alpha z(\theta_r \omega).
\end{equation}
 By the ergodic theory, we find that
\begin{equation} \label{ergo1}
 \lim_{s \to -\infty} {\frac 1s} \int_0^s h(r, \omega) dr
 = E \big({\frac 12} \beta p e^{\alpha (p-2) z( \omega)}
 -2 \alpha z( \omega) \big)
 = {\frac 12} \beta p E \big( e^{\alpha (p-2) z( \omega)} \big)>0.
\end{equation}
We now fix a number $\delta$ such  that
\begin{equation} \label{delta1}
 0 < \delta < {\frac 12} \beta p E \big( e^{\alpha (p-2) z( \omega)} \big).
 \end{equation}
It follows from \eqref{ergo1} and \eqref{delta1} that
for every $\varepsilon \in \left (0, \ {\frac 14} \beta p
 E \left ( e^{\alpha (p-2) z( \omega)} \right ) - {\frac 12} \delta
 \right )$, there exists $s_0=s_0 (\omega, \varepsilon)<0$
 such that
 for all $s \le s_0$,
\begin{equation} \label{b10a}
 \int_0^s h(r, \omega) dr
 < (\delta + \varepsilon) s < \delta s.
 \end{equation}
 Let
 $D =\{ D(\tau, \omega): \tau \in \mathbb{R}, \omega \in \Omega \}$ be a family of
bounded nonempty subsets of $L^2(\mathcal{O}) $ with the property:
 for every $\tau \in \mathbb{R}$ and $\omega \in \Omega$,
 \begin{equation} \label{Du1}
 \lim_{s \to - \infty} e^{ \int_0^s h(r, \omega) dr
 -2 \alpha z(\theta_s \omega)} \| D( \tau + s, \theta_s \omega ) \|^2 =0,
 \end{equation}
where $h$ is given by \eqref{hro} and
$\| S \|= \sup_{u \in S} \| u\|_{L^2(\mathcal{O}) }$ for a subset $S$ of $L^2(\mathcal{O})$.
Hereafter, we use $\mathcal{D}$
to denote the collection of all families satisfying \eqref{Du1}:
\begin{equation} \label{Du}
\mathcal{D} = \{
 D =\{ D(\tau, \omega): \tau \in \mathbb{R}, \omega \in \Omega \}: \
 D  \text{ satisfies }  \eqref{Du1} \} .
\end{equation}
From now on, we assume
$\psi_5 \in L^\infty (\mathbb{R}, L^\infty ({\mathcal{O}}) )$ and
\begin{equation}\label{gpsi1}
\int_{-\infty}^0 e^{\delta s}
\left (\| g(s+\tau) \|^2 + \| \psi_1 (s+\tau) \|_{L^1({\mathcal{O}})}
\right ) ds < \infty, \quad \forall \tau \in \mathbb{R},
\end{equation}
where $\delta$ is the fixed number satisfying \eqref{delta1}.

\section{Uniform estimates of solutions}

We will derive uniform estimates of solutions for problem \eqref{v1}-\eqref{v3}
in this section. These estimates will be used to prove  the existence of
pullback absorbing sets and the asymptotic compactness of the stochastic
equation \eqref{pde1}.

\begin{lemma} \label{unif1}
 Suppose \eqref{f1}-\eqref{f4} and \eqref{gpsi1} hold.
Then for every $\sigma \in \mathbb{R}$,
 $\tau \in \mathbb{R}$, $\omega \in \Omega$ and $D=\{D(\tau, \omega)
: \tau \in \mathbb{R}, \omega \in \Omega\} \in \mathcal{D}$,
 there exists $T=T(\tau, \omega, D, \sigma)>0$ such that
 for all $t \ge T$, the solution
 $v$ of problem \eqref{v1}-\eqref{v3} satisfies
\begin{align*}
&\| v(\sigma, \tau -t, \theta_{-\tau} \omega, v_{\tau -t}) \|^2\\
&\le  c \int_{ - \infty}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
e^{ \alpha (p-2) z(\theta_s \omega)} ds
\\
&\quad + c \int_{ -\infty}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
e^{ -2 \alpha z(\theta_s \omega)} ds
 + c \int_{ -\infty}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
 e^{ {\frac {4\alpha (p-1)} {2-p}} z(\theta_s \omega)} ds
\\
&\quad  + 2 \int_{ -\infty}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
 e^{- 2 \alpha z(\theta_s \omega)} \| \psi_1 (s+\tau) \|_{L^1({\mathcal{O}})}
 ds\\
&\quad + \int_{ -\infty}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
\| g(s+\tau) \|^2 ds ,
\end{align*}
where $ e^{\alpha z(\theta_{-t} \omega)} v_{\tau -t}
\in D(\tau -t, \theta_{ -t} \omega)$
and  $c$ is a positive constant independent of $\tau$, $\omega$, $D$
 and $\alpha$.
\end{lemma}

\begin{proof}
We start with the energy identity \eqref{enlem1}.
By \eqref{f1}, the nonlinearity $f_1$ satisfies
\begin{equation} \label{b5}
\begin{aligned}
&2e^{-\alpha z(\theta_t \omega)}
\int_{\mathcal{O}} f_1 (t,x, e^{\alpha z(\theta_t \omega)}v) v dx\\
&\le
-2 \lambda e^{\alpha (q-2) z(\theta_t \omega)} \int_{\mathcal{O}} |v|^q dx
+2 \int_{\mathcal{O}}
e^{- 2 \alpha z(\theta_t \omega)} \psi_1 (t,x) dx.
\end{aligned}
\end{equation}
By the Young inequality and \eqref{f4} we obtain
\begin{equation}\label{b6}
\begin{aligned}
&2e^{-\alpha z(\theta_t \omega)}|\int_{\mathcal{O}} f_2(t,x,
 e^{ \alpha z(\theta_t \omega)} v) v dx |\\
&\leq 2 \| \psi_5 (t) \|_{L^\infty ({\mathcal{O}})} \| v \|^2 \\
&\leq c_1 e^{- 2\alpha z(\theta_t \omega)}
 + {\frac 12} \beta e^{\alpha (p-2) z(\theta_t \omega)} \int_{\mathcal{O}} |v|^p dx,
\end{aligned}
\end{equation}
where $c_1$ is a positive number independent of $\tau, \omega$
and $\alpha$.
Similarly, by the Young inequality, we have
\begin{equation} \label{b7}
\begin{aligned}
&2e^{-\alpha z(\theta_t \omega)}|\int_{\mathcal{O}} g(t,x) v dx|\\
&\le \| g(t) \|^2 + e^{- 2\alpha z(\theta_t \omega)} \| v \|^2\\
&\le \| g(t) \|^2+ c_2 e^{ {\frac {4\alpha (p-1)} {2-p}} z(\theta_t \omega)}
 + {\frac 12} \beta e^{\alpha (p-2) z(\theta_t \omega)} \int_{\mathcal{O}} |v|^p dx.
\end{aligned}
\end{equation}
By \eqref{b1} and \eqref{b3} we obtain
\begin{equation} \label{b7a1}
\begin{aligned}
2 e^{ \alpha (p-2) z(\theta_t \omega)} \int_{\mathcal{O}} | \nabla v|^pdx
&\ge \beta e^{ \alpha (p-2) z(\theta_t \omega)} \int_{\mathcal{O}} |v|^p dx
+ {\frac 12} \beta p e^{ \alpha (p-2) z(\theta_t \omega)} \| v \|^2\\
&\quad -{\frac 12} \beta (p-2) |{\mathcal{O}}| e^{ \alpha (p-2) z(\theta_t \omega)}.
\end{aligned}
\end{equation}
It follows from \eqref{enlem1} and \eqref{b5}-\eqref{b7a1}
that
\begin{equation} \label{b7a2}
\begin{aligned}
{\frac d{dt}} \| v \|^2 + h(t, \omega) \| v \|^2
&\le c_3 e^{ \alpha (p-2) z(\theta_t \omega)}
 +c_3 e^{ -2 \alpha z(\theta_t \omega)}
 + c_3 e^{ {\frac {4\alpha (p-1)} {2-p}} z(\theta_t \omega)}\\
&\quad +2 e^{- 2 \alpha z(\theta_t \omega)} \| \psi_1 (t) \|_{L^1({\mathcal{O}})}
 + \| g(t) \|^2,
\end{aligned}
\end{equation}
where $h(t, \omega)$ is given by \eqref{hro}.
Multiplying \eqref{b7a2} by $e^{\int_0^t h(r, \omega) dr}$
and then solving the inequality, we obtain
for every $\tau \in \mathbb{R}$, $t\in \mathbb{R}^+$,
$\sigma \ge \tau -t$ and $\omega \in \Omega$,
\begin{equation} \label{b7a3}
\begin{aligned}
&\| v(\sigma, \tau -t, \omega, v_{\tau -t}) \|^2\\
&\le e^{\int_\sigma^{\tau -t} h(r, \omega) dr} \| v_{\tau -t} \|^2
 + c_3 \int_{\tau -t}^\sigma e^{\int_\sigma^s h(r,\omega) dr}
e^{ \alpha (p-2) z(\theta_s \omega)} ds
\\
&\quad + c_3 \int_{\tau -t}^\sigma e^{\int_\sigma^s h(r,\omega) dr}
e^{ -2 \alpha z(\theta_s \omega)} ds
+ c_3 \int_{\tau -t}^\sigma e^{\int_\sigma^s h(r,\omega) dr}
 e^{ {\frac {4\alpha (p-1)} {2-p}} z(\theta_s \omega)} ds
\\
&\quad + 2 \int_{\tau -t}^\sigma e^{\int_\sigma^s h(r,\omega) dr}
 e^{- 2 \alpha z(\theta_s \omega)} \| \psi_1 (s) \|_{L^1({\mathcal{O}})}  ds
\\
&\quad + \int_{\tau -t}^\sigma e^{\int_\sigma^s h(r,\omega) dr}
\| g(s) \|^2 ds.
\end{aligned}
\end{equation}
Replacing $\omega$ by $\theta_{-\tau} \omega$ in \eqref{b7a3} we obtain
\begin{equation} \label{b9}
\begin{aligned}
&\| v(\sigma, \tau -t, \theta_{-\tau} \omega, v_{\tau -t}) \|^2\\
&\le e^{\int_\sigma^{\tau -t} h(r, \theta_{-\tau}\omega) dr} \| v_{\tau -t} \|^2
 + c_3 \int_{\tau -t}^\sigma e^{\int_\sigma^s h(r, \theta_{-\tau}\omega) dr}
e^{ \alpha (p-2) z(\theta_{s-\tau} \omega)} ds
\\
&\quad + c_3 \int_{\tau -t}^\sigma e^{\int_\sigma^s
 h(r, \theta_{-\tau}\omega) dr}
e^{ -2 \alpha z(\theta_{s-\tau} \omega)} ds
+ c_3 \int_{\tau -t}^\sigma e^{\int_\sigma^s h(r, \theta_{-\tau}\omega) dr}
 e^{ {\frac {4\alpha (p-1)} {2-p}} z(\theta_{s-\tau} \omega)} ds
\\
&\quad + 2 \int_{\tau -t}^\sigma e^{\int_\sigma^s h(r, \theta_{-\tau} \omega) dr}
 e^{- 2 \alpha z(\theta_{s-\tau} \omega)} \| \psi_1 (s) \|_{L^1({\mathcal{O}})}
 ds\\
&\quad + \int_{\tau -t}^\sigma e^{\int_\sigma^s h(r, \theta_{-\tau}\omega) dr}
\| g(s) \|^2 ds\\
& \le e^{\int_{\sigma-\tau} ^{ -t} h(r, \omega) dr} \| v_{\tau -t} \|^2
 + c_3 \int_{ -t}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
e^{ \alpha (p-2) z(\theta_s \omega)} ds
\\
&\quad + c_3 \int_{ -t}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
e^{ -2 \alpha z(\theta_s \omega)} ds
 + c_3 \int_{ -t}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
 e^{ {\frac {4\alpha (p-1)} {2-p}} z(\theta_s \omega)} ds
\\
&\quad  + 2  \int_{ -t}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
 e^{- 2 \alpha z(\theta_s \omega)} \| \psi_1 (s+\tau) \|_{L^1({\mathcal{O}})}
 ds\\
 &\quad + \int_{ -t}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
\| g(s+\tau) \|^2 ds .
\end{aligned}
\end{equation}
Next, we estimate every term on the right-hand side of
\eqref{b9}.
First, by \eqref{b10a} we find that for all $s \le s_0$,
\begin{align*}
\int_{ -\infty }^{s_0}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
\| g(s+\tau) \|^2 ds
&= e^{\int_{\sigma-\tau} ^0 h(r, \omega) dr}
 \int^{s_0}_{-\infty}
 e^{\int_0 ^s h(r, \omega) dr}
\| g(s+\tau) \|^2 ds
\\
&\le e^{\int_{\sigma-\tau} ^0 h(r, \omega) dr}
 \int^{s_0}_{-\infty} e^{\delta s }
\| g(s+\tau) \|^2 ds\\
&\le e^{\int_{\sigma-\tau} ^0 h(r, \omega) dr}
 \int^0_{-\infty}
 e^{\delta s }
\| g(s+\tau) \|^2 ds < \infty,
\end{align*}
where we have used \eqref{gpsi1} for the last integral.
Therefore,  for $t >0$, we obtain
 \begin{equation} \label{b20}
 \int_{ -t}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
\| g(s+\tau) \|^2 ds
 \leq  \int_{ -\infty}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
\| g(s+\tau) \|^2 ds < \infty.
 \end{equation}
Note that \eqref{z2} implies that for $\varepsilon>0$, there
exists $s_1 \le s_0$ such that for all $s \le s_1$,
\begin{equation} \label{b22}
 -2 \alpha z(\theta_s \omega) < -\varepsilon s.
\end{equation}
By \eqref{b10a}, \eqref{b22} and \eqref{gpsi1} we obtain
\begin{align*}
&\int_{ -\infty }^{s_1}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
 e^{- 2 \alpha z(\theta_s \omega)} \| \psi_1 (s+\tau) \|_{L^1({\mathcal{O}})} ds\\
& = e^{\int_{\sigma-\tau} ^0 h(r, \omega) dr}
 \int_{ -\infty }^{s_1}
 e^{\int_0 ^s h(r, \omega) dr}
 e^{- 2 \alpha z(\theta_s \omega)} \| \psi_1 (s+\tau) \|_{L^1({\mathcal{O}})} ds\\
& \le e^{\int_{\sigma-\tau} ^0 h(r, \omega) dr}
 \int_{ -\infty }^{s_1}
 e^{\delta s }
 \| \psi_1 (s+\tau) \|_{L^1({\mathcal{O}})}
 ds <\infty,
\end{align*}
 and hence
\begin{equation} \label{b23}
\begin{aligned}
&\int_{ - t }^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
 e^{- 2 \alpha z(\theta_s \omega)} \| \psi_1 (s+\tau) \|_{L^1({\mathcal{O}})} ds\\
&\le \int_{ -\infty }^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
 e^{- 2 \alpha z(\theta_s \omega)} \| \psi_1 (s+\tau) \|_{L^1({\mathcal{O}})}
 ds<\infty.
\end{aligned}
\end{equation}
 By a similar argument, one can verify that the following integrals
 are also convergent:
\begin{gather} \label{b24}
 \int_{ - \infty}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
e^{ \alpha (p-2) z(\theta_s \omega)} ds < \infty,\\
\label{b25}
 \int_{ -\infty}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
e^{ -2 \alpha z(\theta_s \omega)} ds < \infty,\\
\label{b26}
 \int_{ -\infty}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
 e^{ {\frac {4\alpha (p-1)} {2-p}} z(\theta_s \omega)} ds
 <\infty.
\end{gather}
For the first term on the right-hand side of \eqref{b9}, since
$e^{\alpha z(\theta_{-t} \omega)}v_{\tau -t}
\in D(\tau -t, \theta_{-t} \omega) \in {\mathcal{D}}$,
 we have
 $$
e^{\int_0 ^{ -t} h(r, \omega) dr} \| v_{\tau -t} \|^2
\le
e^{\int_0^{-t} h(r,\omega) dr - 2\alpha z(\theta_{-t} \omega) }
\| D(\tau -t, \theta_{-t} \omega) \|^2 \to 0,
$$
 as $ t \to \infty$. Therefore, there exists
$T=T(\tau, \omega, D, \sigma)>0$ such that for all $t \ge T$,
\begin{equation} \label{b27}
\begin{aligned}
e^{\int_{\sigma-\tau} ^{ -t} h(r, \omega) dr} \| v_{\tau -t} \|^2
&= e^{\int^0 _{ \sigma -\tau } h(r, \omega) dr}
 e^{\int_0 ^{ -t} h(r, \omega) dr} \| v_{\tau -t} \|^2\\
&\le  \int_{ - \infty}^{\sigma -\tau}
 e^{\int_{\sigma-\tau} ^s h(r, \omega) dr}
e^{ \alpha (p-2) z(\theta_s \omega)} ds .
\end{aligned}
\end{equation}
Thus, the lemma follows from
 \eqref{b9}-\eqref{b27} immediately.
\end{proof}



As a consequence of Lemma \ref{unif1}, we have the following estimates.

\begin{lemma}\label{unif2}
 Suppose \eqref{f1}-\eqref{f4} and \eqref{gpsi1} hold.
Then for every $\tau \in \mathbb{R}$, $\omega \in \Omega$ and
$D=\{D(\tau, \omega): \tau \in \mathbb{R}, \omega \in \Omega\} \in \mathcal{D}$,
there exists $T=T(\tau, \omega, D)>0$ such that for all $t \ge T$, the solution
$v$ of problem \eqref{v1}-\eqref{v3}
 with $ e^{\alpha z(\theta_{-t} \omega)}
 v_{\tau -t} \in D(\tau -t, \theta_{ -t} \omega)$ satisfies,
for all $t \ge T$,
\begin{equation} \label{unif2_1}
\| v(\tau, \tau -t, \theta_{-\tau} \omega, v_{\tau -t}) \|^2
\le R(\tau, \omega, \alpha),
\end{equation}
where
 \begin{equation} \label{unif2_2}
\begin{aligned}
&R(\tau, \omega, \alpha)\\
&= c \int_{ - \infty}^0 e^{\int_0 ^s h(r, \omega) dr}
e^{ \alpha (p-2) z(\theta_s \omega)} ds
\\
&\quad + c \int_{ -\infty}^0 e^{\int_0 ^s h(r, \omega) dr}
e^{ -2 \alpha z(\theta_s \omega)} ds
 + c \int_{ -\infty}^0 e^{\int_0 ^s h(r, \omega) dr}
 e^{ {\frac {4\alpha (p-1)} {2-p}} z(\theta_s \omega)} ds
\\
&\quad + 2 \int_{ -\infty}^0 e^{\int_0 ^s h(r, \omega) dr}
 e^{- 2 \alpha z(\theta_s \omega)} \| \psi_1 (s+\tau) \|_{L^1({\mathcal{O}})} ds
\\
&\quad + \int_{ -\infty}^0 e^{\int_0 ^s h(r, \omega) dr}
\| g(s+\tau) \|^2 ds ,
\end{aligned}
\end{equation}
with  $c$ being a positive constant independent of $\tau$, $\omega$, $D$
 and $\alpha$.
 Moreover, we have
 \begin{equation}\label{unif2_3}
 \lim_{t \to \infty} e^{\int_0^{-t} h(r, \omega) dr} R(\tau- t, \theta_{-t}
\omega, \alpha) =0.
 \end{equation}
\end{lemma}

\begin{proof}
Note that \eqref{unif2_1} follows from Lemma \ref{unif1}
by setting $\sigma =\tau$. We only need to show \eqref{unif2_3}.
By \eqref{unif2_2} we obtain
\begin{align*}
&R(\tau- t, \theta_{-t} \omega, \alpha)\\
&= c \int_{ - \infty}^0 e^{\int_0 ^s h(r, \theta_{-t} \omega) dr}
e^{ \alpha (p-2) z(\theta_{s-t} \omega)} ds
\\
&\quad + c \int_{ -\infty}^0 e^{\int_0 ^s h(r, \theta_{-t}\omega) dr}
e^{ -2 \alpha z(\theta_{s-t} \omega)} ds
 + c \int_{ -\infty}^0
 e^{\int_0 ^s h(r, \theta_{-t}\omega) dr}
 e^{ {\frac {4\alpha (p-1)} {2-p}} z(\theta_{s-t} \omega)} ds
\\
&\quad + 2 \int_{ -\infty}^0
 e^{\int_0 ^s h(r, \theta_{-t}\omega) dr}
 e^{- 2 \alpha z(\theta_{s-t} \omega)} \| \psi_1 (s+\tau-t) \|_{L^1({\mathcal{O}})} ds
\\
&\quad + \int_{ -\infty}^0 e^{\int_0 ^s h(r, \theta_{-t}\omega) dr}
\| g(s+\tau-t) \|^2 ds
\\
&=  c \int_{ - \infty}^{-t}
 e^{\int_{-t} ^s h(r, \omega) dr}
e^{ \alpha (p-2) z(\theta_s \omega)} ds
\\
&\quad + c \int_{ -\infty}^{-t} e^{\int_{-t} ^s h(r, \omega) dr}
e^{ -2 \alpha z(\theta_s \omega)} ds
 + c \int_{ -\infty}^{-t}
 e^{\int_{-t} ^s h(r, \omega) dr}
 e^{ {\frac {4\alpha (p-1)} {2-p}} z(\theta_s \omega)} ds
\\
&\quad  + 2 \int_{ -\infty}^{-t} e^{\int_{-t} ^s h(r, \omega) dr}
 e^{- 2 \alpha z(\theta_s \omega)} \| \psi_1 (s+\tau) \|_{L^1({\mathcal{O}})} ds\\
&\quad + \int_{ -\infty}^{-t} e^{\int_{-t} ^s h(r, \omega) dr}
\| g(s+\tau) \|^2 ds .
\end{align*}
Therefore,
\begin{equation} \label{b40}
\begin{aligned}
&e^{\int_0^{-t} h(r, \omega) dr} R(\tau- t, \theta_{-t} \omega, \alpha)\\
&= c \int_{ - \infty}^{-t} e^{\int_0 ^s h(r, \omega) dr}
e^{ \alpha (p-2) z(\theta_s \omega)} ds
+ c \int_{ -\infty}^{-t} e^{\int_0 ^s h(r, \omega) dr}
e^{ -2 \alpha z(\theta_s \omega)} ds\\
&\quad + c \int_{ -\infty}^{-t} e^{\int_0 ^s h(r, \omega) dr}
 e^{ {\frac {4\alpha (p-1)} {2-p}} z(\theta_s \omega)} ds\\
&\quad + 2 \int_{ -\infty}^{-t}  e^{\int_0 ^s h(r, \omega) dr}
 e^{- 2 \alpha z(\theta_s \omega)} \| \psi_1 (s+\tau) \|_{L^1({\mathcal{O}})} ds\\
&\quad  + \int_{ -\infty}^{-t} e^{\int_0 ^s h(r, \omega) dr}
\| g(s+\tau) \|^2 ds .
\end{aligned}
\end{equation}
By the convergence of the integral
$$
 \int_{ - \infty}^0 e^{\int_0 ^s h(r, \omega) dr}
e^{ \alpha (p-2) z(\theta_s \omega)} ds<\infty
$$
we obtain
\begin{equation} \label{b42}
 \lim_{t \to \infty} \int_{ - \infty}^{-t} e^{\int_0 ^s h(r, \omega) dr}
e^{ \alpha (p-2) z(\theta_s \omega)} ds =0.
\end{equation}
Similarly, we find that all remaining terms on the right-hand side
of \eqref{b40} converge to zero as $t \to \infty$.
Therefore, from \eqref{b40}-\eqref{b42}, the desired
limit \eqref{unif2_3} follows.
\end{proof}

The asymptotic compactness of solutions of equation \eqref{v1}
is stated as follows.

\begin{lemma}\label{asyv}
Suppose \eqref{f1}-\eqref{f4} and \eqref{gpsi1} hold.
Then for every $\tau \in \mathbb{R}$, $\omega \in \Omega$ and $D=\{D(\tau, \omega)
: \tau \in \mathbb{R}, \omega \in \Omega\} \in \mathcal{D}$,
 the sequence  $v(\tau, \tau -t_n, \theta_{-\tau} \omega, v_{0,n} ) $ has a
 convergent subsequence in $L^2(\mathcal{O}) $ provided  $t_n \to \infty$ and
 $ e^{\alpha z(\theta_{-t_n} \omega )} v_{0,n} \in D(\tau -t_n,
\theta_{ -t_n} \omega )$.
 \end{lemma}

 \begin{proof}
Thanks to $t_n \to \infty$ and
 $ e^{\alpha z(\theta_{-t_n} \omega )}
 v_{0,n} \in D(\tau -t_n, \theta_{ -t_n} \omega )$,
it follows from Lemma \ref{unif1} that  there
 exist $c= c(\tau, \omega)>0$
 and $N =N(\tau, \omega, D)>0$ such that for all $n \ge N$,
\begin{equation} \label{pasyv_1}
 \| v(\tau -1, \tau -t_n, \theta_{-\tau} \omega, v_{0,n} ) \|
 \le c (\tau, \omega).
\end{equation}
Note that
 \begin{equation} \label{pasyv_2}
 v(\tau , \tau -t_n, \theta_{-\tau} \omega, v_{0,n} )
 = v(\tau, \tau - 1, \theta_{-\tau} \omega,
 v(\tau -1, \tau -t_n, \theta_{-\tau} \omega, v_{0,n} ) ).
 \end{equation}
Then the precompactness of
 $ \{v(\tau , \tau -t_n, \theta_{-\tau} \omega, v_{0,n} ) \}$
 follows from \eqref{pasyv_1}-\eqref{pasyv_2}
 and Lemma \ref{wcont} immediately.
 \end{proof}

\section{ Existence of Random Attractors}

 In this section, we prove the existence of $\mathcal{D}$-pullback attractor
 for the stochastic problem \eqref{pde1}-\eqref{pde3}.
 We first construct a $\mathcal{D}$-pullback absorbing set for the
 corresponding cocycle based on the uniform estimates derived
 in the previous section.

 \begin{lemma} \label{abs1}
 Suppose \eqref{f1}-\eqref{f4}
 and \eqref{gpsi1} hold. Then the continuous cocycle $\Phi$
 associated with problem \eqref{pde1}-\eqref{pde3} has a closed
 measurable $\mathcal{D}$-pullback absorbing set $K_\alpha=\{ K_\alpha(\tau, \omega):
 \tau \in \mathbb{R}, \omega \in \Omega \} \in \mathcal{D}$ which is given by
\begin{equation} \label{asb1_1}
 K_\alpha(\tau, \omega) = \{ u \in L^2(\mathcal{O}): \| u\|^2 \le e^{2\alpha z (\omega)}
 R(\tau, \omega, \alpha) \},
\end{equation}
where $R(\tau, \omega, \alpha)$ is the number given by
 \eqref{unif2_2}
 \end{lemma}

\begin{proof}
 Replacing $\omega$ by $\theta_{-\tau} \omega$ in \eqref{uv} we obtain
\begin{equation} \label{abp2}
 u(\tau, \tau -t, \theta_{-\tau} \omega, u_{\tau -t} )
 = e^{\alpha z(\omega)} v(\tau, \tau -t, \theta_{-\tau} \omega, v_{\tau -t} )
 \quad \text{with }
 u_{\tau -t} = e^{\alpha z(\theta_{-t } \omega)} v_{\tau -t}.
\end{equation}
 Therefore, for every
 $u_{\tau -t} \in D(\tau -t, \theta_{-t} \omega)$, we have
 $ e^{\alpha z(\theta_{-t} \omega)} v_{\tau -t} \in
 D(\tau -t, \theta_{-t} \omega)$. This along with \eqref{unif2_1}
 shows that there is $T=T(\tau, \omega, D)>0$ such that for all
 $t \ge T$, the solution $v$ of problem \eqref{v1}-\eqref{v3} satisfies
\begin{equation} \label{abp3}
 \| v(\tau, \tau -t, \theta_{-\tau} \omega, v_{\tau-t}) \|^2
 \le R(\tau, \omega, \alpha),
\end{equation}
where $R(\tau, \omega, \alpha)$ is given by \eqref{unif2_2}.
 By \eqref{abp2} and \eqref{abp3} we obtain, for all $t \ge T$,
\begin{equation} \label{abp5}
 \| u(\tau, \tau -t, \theta_{-\tau} \omega, u_{\tau-t}) \|^2
 \le e^{2\alpha z(\omega) } R(\tau, \omega, \alpha).
\end{equation}
 By \eqref{pcycle} and \eqref{abp5} we find that, for all $t \ge T$,
 \begin{equation} \label{abp6}
 \Phi (t, \tau -t, \theta_{-t} \omega, D(\tau -t, \theta_{-t} \omega ) )
 = u(\tau, \tau -t, \theta_{-\tau} \omega, D(\tau -t, \theta_{-t} \omega ) )
 \subseteq K_\alpha(\tau, \omega),
 \end{equation}
where $K_\alpha(\tau, \omega)$ is given by \eqref{asb1_1}.
Note that
\begin{equation} \label{abp7}
 e^{\int_0^s h(r,\omega) dr -2\alpha z(\theta_s \omega)}
 \| K_\alpha(\tau +s, \theta_s \omega ) \|^2
 = e^{\int_0^s h(r,\omega) dr )}
 R(\tau +s, \theta_s \omega, \alpha).
\end{equation}
It follows from \eqref{unif2_3} and \eqref{abp7} that
$$
\lim_{s \to -\infty}
 e^{\int_0^s h(r,\omega) dr -2\alpha z(\theta_s \omega)}
 \| K_\alpha(\tau +s, \theta_s \omega ) \|^2 =0,
 $$
 and hence by \eqref{Du} we have
 $K_\alpha =\{ K_\alpha(\tau, \omega): \tau \in \mathbb{R}, \omega \in \Omega \}
\in \mathcal{D}$.
 By \eqref{unif2_2}, it is evident that
 $R(\tau, \omega, \alpha)$ is measurable in $\omega \in \Omega$ and so is
 the set-valued function $K_\alpha(\tau, \omega)$.
 This along with \eqref{abp6} concludes the proof.
\end{proof}

 Next, we prove the $\mathcal{D}$-pullback
 asymptotic compactness of solutions of
 problem \eqref{pde1}-\eqref{pde3}.


\begin{lemma}\label{asyu}
Suppose \eqref{f1}-\eqref{f4} and \eqref{gpsi1} hold.
Then for every $\tau \in \mathbb{R}$, $\omega \in \Omega$ and
$D=\{D(\tau, \omega): \tau \in \mathbb{R}, \omega \in \Omega\} \in \mathcal{D}$,
 the sequence
$\Phi(t_n, \tau -t_n, \theta_{-t_n } \omega, u_{0,n} ) $
is precompact in $L^2(\mathcal{O}) $ provided
$t_n \to \infty$ and $u_{0,n} \in D(\tau -t_n, \theta_{ -t_n} \omega )$.
\end{lemma}

\begin{proof}
 Let $v_{0,n} = e^{-\alpha z(\theta_{-t_n} \omega )} u_{0,n}$
 for $n \in \mathbb{N}$.  Since $u_{0,n} \in D(\tau -t_n, \theta_{ -t_n} \omega )$,
 we have $e^{\alpha z(\theta_{-t_n} \omega )} v_{0,n}
 \in D(\tau -t_n, \theta_{ -t_n} \omega )$.
 By Lemma \ref{asyv} we find that
 the sequence
 $v(\tau, \tau -t_n, \theta_{-\tau} \omega, v_{0,n} ) $ has a
 convergent subsequence in $L^2(\mathcal{O})$, and so is the
sequence $u(\tau, \tau -t_n, \theta_{-\tau} \omega, u_{0,n} ) $
 by \eqref{abp2}.
Therefore, by \eqref{pcycle}, the sequence
 $\Phi(t_n, \tau -t_n, \theta_{-t_n } \omega, u_{0,n} ) $
 is precompact in $L^2(\mathcal{O})$ as desired.
\end{proof}

We are now in a position to present the existence of
 $\mathcal{D}$-pullback attractors for
 \eqref{pde1}-\eqref{pde3}.

 \begin{theorem} \label{eatt1}
 Suppose \eqref{f1}-\eqref{f4} and \eqref{gpsi1} hold.
 Then the cocycle $\Phi$ for
 problem \eqref{pde1}-\eqref{pde3}
 has a unique $\mathcal{D}$-pullback attractor
$\mathcal{A}_\alpha =\{\mathcal{A}_\alpha(\tau, \omega): \tau \in \mathbb{R}, \ \omega \in \Omega \} \in \mathcal{D}$
in $L^2(\mathcal{O})$. In addition, for each $\tau \in \mathbb{R}$ and
$\omega \in \Omega$,
\begin{align*}
\mathcal{A}_\alpha (\tau, \omega)
&=\Omega(K, \tau, \omega)
=\cup_{B \in \mathcal{D}} \Omega(B, \tau, \omega) \\
&=\{\psi(0, \tau, \omega): \psi \text{ is a } \mathcal{D} {\rm -}
 \text{complete orbit of } \Phi\}\\
&=\{\xi(\tau, \omega): \xi \text{ is a } \mathcal{D} {\rm -}
 \text{complete quasi-solution of } \Phi\} ,
\end{align*}
where $K$ is a $\mathcal{D}$-pullback absorbing set of $\Phi$.
\end{theorem}

\begin{proof}
This result follows directly from Lemmas \ref{abs1},
\ref{asyu} and Proposition \ref{att}.
\end{proof}

For the periodicity of the $\mathcal{D}$-pullback attractor $\mathcal{A}_\alpha$, we have
the following result.

 \begin{theorem} \label{eatt2}
 Let \eqref{f1}--\eqref{f4} hold.
 Suppose that $f_1(t,x,s)$ and $f_2(t,x,s)$  are $T$-periodic in
$t \in \mathbb{R} $ for each fixed $x \in {\mathcal{O}}$ and $s \in \mathbb{R}$.
If $g \in L^2_{\rm loc}(\mathbb{R}, L^2(\mathcal{O}))$
 and $\psi_1 \in L^1_{\rm loc} (\mathbb{R}, L^1({\mathcal{O}}) )$  are also $T$-periodic, then
 problem \eqref{pde1}-\eqref{pde3}  has a unique
 $\mathcal{D}$-pullback attractor $\mathcal{A}_\alpha$  in $L^2(\mathcal{O})$ such that
 $\mathcal{A}_\alpha(\tau +T, \omega) =\mathcal{A}_\alpha(\tau, \omega)$ for all
 $\tau \in \mathbb{R}$ and $\omega \in \Omega$.
\end{theorem}

\begin{proof}
Since $g \in L^2_{\rm loc}(\mathbb{R}, L^2(\mathcal{O}))$
 and $\psi_1 \in L^1_{\rm loc} (\mathbb{R}, L^1({\mathcal{O}}) )$
 are both $T$-periodic, one may check that
 condition \eqref{gpsi1} is fulfilled in this case.
 Therefore, by Theorem \ref{eatt1}, problem
 \eqref{pde1}-\eqref{pde3} has a unique
 $\mathcal{D}$-pullback attractor $\mathcal{A}_\alpha$ in $L^2(\mathcal{O})$.
 By the $T$-periodicity of $f_1$, $f_2$ and $g$,
 we find that the cocycle $\Phi$ is also $T$-periodic;
 more precisely,  for any $u_0 \in L^2(\mathcal{O})$,
 $t \in \mathbb{R}^+$, $\tau \in \mathbb{R}$ and $\omega \in \Omega$,
\begin{align*}
\Phi (t, \tau +T, \omega, u_0 )
&= u(t+ \tau +T, \tau +T, \theta_{ -\tau -T} \omega, u_0 )\\
&=u(t +\tau, \tau, \theta_{ -\tau} \omega, u_0 )\\
&= \Phi (t, \tau, \omega, u_0 ).
\end{align*}
On the other hand, by the $T$-periodicity of $g$ and $\psi_1$, we obtain
from \eqref{unif2_2} that
$R(\tau +T, \omega, \alpha)=R(\tau, \omega, \alpha)$
 for all $\tau \in \mathbb{R}$ and $\omega \in \Omega$,
and hence, by \eqref{asb1_1}, the $\mathcal{D}$-pullback absorbing
set $K_\alpha$ is $T$-periodic.
Thus, the periodicity of $\mathcal{A}_\alpha$ follows from Proposition \ref{att}
immediately.
 \end{proof}

\section{Upper semicontinuity of attractors}

In this section, we establish the upper semicontinuity
 of random attractors of problem \eqref{pde1}-\eqref{pde3}
 as $\alpha \to 0$. Hereafter, we assume $\delta$ is a fixed
 number such that
 \begin{equation} \label{delta2}
 0<\delta < {\frac 12} \beta p.
 \end{equation}
Note that $\lim_{\alpha \to 0} E(e^{\pm \alpha (p-2) |z(\omega) |}) =1$
and $\lim_{\alpha \to 0} E(\alpha |z(\omega)|) =0$.
By \eqref{delta2} we find that there exists
 $\alpha_0 \in (0, 1)$ such that
 for all $\alpha \in [0, \alpha_0]$,
 \begin{equation} \label{delta3}
 \delta < {\frac 12} \beta p E(e^{\alpha (p-2) z(\omega)}),\quad
 \delta < {\frac 12} \beta p E(e^{-\alpha (p-2) |z(\omega)|})
 - 2 \alpha E(|z(\omega)|) .
 \end{equation}
 Therefore, condition \eqref{delta1} is fulfilled
 for any $\alpha \in [0, \alpha_0]$.
Throughout this section, we always assume $\alpha \in [0, \alpha_0]$
and write the the cocycle associated with  \eqref{pde1}-\eqref{pde3}
 as $\Phi_\alpha$.
 By \eqref{f1}-\eqref{f4} and \eqref{delta3}, it follows from
 Theorem \ref{eatt1} that for every $\alpha \in (0,\alpha_0]$,
 $\Phi_\alpha$ has a unique $\mathcal{D}$-pullback attractor
 $\mathcal{A}_\alpha \in \mathcal{D}$ in $L^2(\mathcal{O})$.
Moreover,  the family $K_\alpha = \{K_\alpha
 (\tau, \omega): \tau \in \mathbb{R}, \omega \in \Omega
 \}$ given by \eqref{asb1_1} is a $\mathcal{D}$-pullback absorbing set
of $ \Phi_\alpha$.
Given $r \in \mathbb{R}$ and $\omega \in \Omega$, denote by
\begin{equation} \label{up1}
{\widetilde{h}} (r, \omega)
= {\frac 12} \beta p e^{-\alpha_0 (p-2) |z(\theta_r \omega)|}
- 2\alpha_0 | z(\theta_r \omega)|.
\end{equation}
As in \eqref{unif2_2}, by \eqref{delta3} one can verify the following integrals
are well defined:
\begin{equation} \label{up2}
\begin{aligned}
L(\tau, \omega)
&= c \int_{ - \infty}^0  e^{\int_0 ^s {\widetilde{h}}(r, \omega) dr}
e^{ \alpha (p-2) | z(\theta_s \omega) |} ds
\\
&\quad + c \int_{ -\infty}^0
 e^{\int_0 ^s {\widetilde{h}}(r, \omega) dr}
e^{ 2 \alpha |z(\theta_s \omega) |} ds\\
&\quad + c \int_{ -\infty}^0 e^{\int_0 ^s {\widetilde{h}} (r, \omega) dr}
 e^{ {\frac {4\alpha (p-1)} {p-2}} |z(\theta_s \omega) |} ds
\\
&\quad + 2 \int_{ -\infty}^0 e^{\int_0 ^s {\widetilde{h}} (r, \omega) dr}
 e^{ 2 \alpha |z(\theta_s \omega) |} \| \psi_1 (s+\tau) \|_{L^1({\mathcal{O}})}ds\\
&\quad + \int_{ -\infty}^0  e^{\int_0 ^s {\widetilde{h}} (r, \omega) dr}
\| g(s+\tau) \|^2 ds ,
\end{aligned}
\end{equation}
where  $c$ is the same constant as  in \eqref{unif2_2}.
By \eqref{hro} we find that
 $h(r, \omega) \ge {\widetilde{h}} (r, \omega)$
 for all $r\in \mathbb{R}$, $\omega \in \Omega$
 and $\alpha \in [0, \alpha_0]$.
Therefore, by \eqref{unif2_2} we have
 $R(\tau, \omega, \alpha) \le L(\tau, \omega)$
 for all $\alpha \in [0, \alpha_0]$, which along with
 \eqref{asb1_1} implies that
 for all $\tau \in \mathbb{R}$ and $\omega \in \Omega$,
 \begin{equation} \label{up1b}
 \cup_{0< \alpha \le \alpha_0} \mathcal{A}_\alpha (\tau, \omega)
 \subseteq
 \cup_{0< \alpha \le \alpha_0} K_\alpha (\tau, \omega)
 \subseteq K(\tau, \omega),
 \end{equation}
 where $ K(\tau, \omega)
 =\{ u \in L^2(\mathcal{O}) : \ \| u \|^2 \le e^{2\alpha_0 |z(\omega)|} L(\tau, \omega) \}$.

 We now consider \eqref{pde1} with $\alpha =0$. In this case we have
 \begin{equation} \label{dere1}
 {\frac {\partial u}{\partial t}}
 - \operatorname{div} \left (|\nabla u |^{p-2} \nabla u \right )
 =f_1(t,x,u)  + f_2(t,x, u)
 +g(t,x) , \quad t>\tau,\; x \in {\mathcal{O}},
 \end{equation}
 which is supplemented with
 \begin{equation}  \label{dere2}
 u|_{\partial {\mathcal{O}}} = 0,\quad   u( \tau, \cdot ) = u_\tau .
 \end{equation}
 We will use $\Phi_0$ to denote  the cocycle generated by
\eqref{dere1}-\eqref{dere2}  in $L^2(\mathcal{O})$.
 Let ${\mathcal{D}}_0$ be the collection given by
 $$
 {\mathcal{D}}_0
 =\{ D= \{
 D(\tau) \subseteq L^2(\mathcal{O}) : \tau \in \mathbb{R} \}:
 \lim_{ s \to - \infty}
 e^{{\frac 12} \beta p s} \| D (\tau + s ) \|^2 =0 ,
 \ \forall  \tau \in \mathbb{R}
 \}.
 $$
It is evident that the results in the previous sections also
 hold true for $\alpha =0$.
 Particularly,  $\Phi_0$ has a unique $ {\mathcal{D}}_0$-pullback
 attractor $ {\mathcal{A}}_0 =\{ {\mathcal{A}}_0 (\tau), \tau \in \mathbb{R}\}
 \in {\mathcal{D}}_0 $ in $L^2(\mathcal{O})$ and has a
 $ {\mathcal{D}}_0$-pullback absorbing set
 $K_0 = \{ K_0(\tau): \tau \in \mathbb{R} \}$ with $K_0(\tau)$
given by
 \begin{equation}  \label{up2b}
 K_0 (\tau) =\{ u \in L^2(\mathcal{O}): \ \| u \|^2 \le R_0(\tau) \},
 \end{equation}
where $R_0(\tau)$ is defined by
 \begin{equation} \label{up3}
 R_0(\tau)
 = {\frac {6c}{\beta p}} +
 \int^0_{-\infty} e^{{\frac 12} \beta p s}
 (\| g(s+\tau) \|^2 + 2 \| \psi_1 (s+\tau) \|_{L^1({\mathcal{O}})} ) ds ,
 \end{equation}
 and $c$ is as in \eqref{unif2_2}.
 By \eqref{unif2_2}, \eqref{asb1_1} and
 \eqref{up2b}-\eqref{up3} we obtain
 for all $\tau \in \mathbb{R}$ and $\omega \in \Omega$,
 \begin{equation} \label{up4}
 \limsup_{\alpha \to 0}
 \| K_\alpha (\tau, \omega) \|
 \le \| K_0 (\tau ) \|.
 \end{equation}

Next, we establish the convergence of solutions
of problem \eqref{pde1}-\eqref{pde3} as $\alpha \to 0$,
for which the following condition is needed:
there exists $\psi_6 \in L^\infty_{\rm loc} (\mathbb{R}, L^\infty ({\mathcal{O}}))$
 such that  for all $t, s \in \mathbb{R}$ and $x \in {\mathcal{O}}$,
\begin{equation} \label{f5}
 |{\frac {\partial f_1}{\partial s}} (t, x,s) |
\le \psi_6 (t,x) \left ( 1+ |s|^{q-2} \right ).
\end{equation}


\begin{lemma} \label{sep1}
Let \eqref{f1}-\eqref{f4} and \eqref{f5} hold.
If $u_\alpha$ and $u$ are the solutions of \eqref{pde1}-\eqref{pde3} and
\eqref{dere1}-\eqref{dere2}
with initial data $u_{\alpha, \tau}$ and $u_\tau$, respectively,
then for every $\tau \in \mathbb{R}$, $\omega \in \Omega$,
$T>0$ and $\varepsilon \in [0, 1]$,
there exists  $\alpha_1 = \alpha_1(\tau, \omega, T, \varepsilon)>0$
 such that for all $\alpha \le \alpha_1$ and
 $t \in [\tau, \tau +T]$,
 \begin{equation} \label{sep1_1}
\begin{aligned}
&\| u_\alpha (t, \tau, \omega, u_{\alpha, \tau}) - u(t,\tau, u_\tau) \|^2\\
&\le c_1 e^{c_2 (t -\tau)} \| u_{\alpha, \tau} - u_\tau \|^2 \\
&\quad + c_1 \varepsilon e^{c_2 (t -\tau)}
 \Big( 1 +\|u_\tau\|^2 + \|u_{\alpha, \tau} \|^2
+ \int_{\tau}^t ( \| \psi_1 (s) \|_{L^1({\mathcal{O}})} + \| g(s ) \|^2 )
 ds \Big),
\end{aligned}
 \end{equation}
where $c_1$ and $c_2$ are positive constants
 independent of $\varepsilon$ and $\alpha$.
\end{lemma}

\begin{proof}
Let $v_\alpha$ be the solution of  \eqref{v1}-\eqref{v3}
and $\xi = v_\alpha -u$.
By \eqref{v1} and \eqref{dere1} we obtain
 \begin{equation} \label{psep1_1}
\begin{aligned}
&{\frac 12} {\frac d{dt}} \| \xi \|^2
 + \int_{\mathcal{O}} \left ( e^{\alpha (p-2) z(\theta_t \omega)}
 |\nabla v_\alpha|^{p-2} \nabla v _\alpha
 - |\nabla u |^{p-2} \nabla u
 \right ) \nabla \xi\,dx \\
& =\alpha z(\theta_t \omega) \| \xi \|^2
 +\alpha z(\theta_t \omega) (u, \xi)\\
&\quad +\int_{\mathcal{O}}  \big( e^{-\alpha z(\theta_t \omega) }
f_1 \big(t, x, e^{\alpha z(\theta_t \omega)} v_\alpha
 \big) -f_1(t, x,u) \big) \xi\,dx
\\
&\quad + \int_{\mathcal{O}}  \big( e^{-\alpha z(\theta_t \omega) }
 f_2 (t, x, e^{\alpha z(\theta_t \omega)} v_\alpha) 
-f_2(t, x,u) \big) \xi \,dx\\
&\quad + \big(e^{-\alpha z(\theta_t \omega) } -1 \big)
 \int_{\mathcal{O}} g(t, x ) \xi\,dx.
\end{aligned}
\end{equation}
For the second term on the left-hand side of \eqref{psep1_1} we have
\begin{equation} \label{psep1_2}
\begin{aligned}
&\int_{\mathcal{O}} \left ( e^{\alpha (p-2) z(\theta_t \omega)}
 |\nabla v_\alpha|^{p-2} \nabla v _\alpha
 - |\nabla u |^{p-2} \nabla u
 \right ) \nabla \xi\,dx \\
& = \int_{\mathcal{O}} e^{\alpha (p-2) z(\theta_t \omega)}
 \left ( |\nabla v_\alpha|^{p-2} \nabla v _\alpha
 - |\nabla u |^{p-2} \nabla u
 \right ) \nabla \xi\,dx\\
&\quad + \int_{\mathcal{O}} \big( e^{\alpha (p-2)
 z(\theta_t \omega)} -1 \big)
 |\nabla u|^{p-2} \nabla u  \cdot \nabla \xi\,dx.
\end{aligned}
\end{equation}
From \cite{dibe1}, we know that there is a positive number $\gamma$ such that
\begin{equation} \label{psep1_3}
\left ( |\nabla v_\alpha|^{p-2} \nabla v _\alpha
 - |\nabla u |^{p-2} \nabla u
 \right ) \cdot (\nabla v_\alpha -\nabla u )
 \ge \gamma |\nabla v_\alpha -\nabla u |^p.
 \end{equation}
On the other hand, since
$ z(\theta_t \omega)$ is continuous in $t \in \mathbb{R}$,
by Young's inequality we find that for every $\tau \in \mathbb{R}$,
$\omega \in \Omega$, $T>0$ and $\varepsilon \in [0, 1)$,
there exists  $\alpha_2 = \alpha_2(\tau, \omega, T, \varepsilon)>0$
such that for all $\alpha \in [0, \alpha_2]$ and
$t \in [\tau, \tau +T]$,
 \begin{equation} \label{psep1_4}
\begin{aligned}
&|\int_{\mathcal{O}} \big( e^{\alpha (p-2)
 z(\theta_t \omega)} -1 \big)
 |\nabla u|^{p-2} \nabla u \cdot \nabla \xi\,dx|\\
&\leq {\frac 12} \gamma  e^{\alpha (p-2)
 z(\theta_t \omega)} \int _{\mathcal{O}} |\nabla \xi |^p dx
 + \varepsilon \int_{\mathcal{O}} |\nabla u |^p dx.
\end{aligned}
\end{equation}
 It follows from \eqref{psep1_2}-\eqref{psep1_4} that
 for all $\alpha \in [0, \alpha_2]$ and $t \in [\tau,
 \tau +T]$,
\begin{equation} \label{psep1_5}
\begin{aligned}
&\int_{\mathcal{O}} \left ( e^{\alpha (p-2) z(\theta_t \omega)}
 |\nabla v_\alpha|^{p-2} \nabla v _\alpha
 - |\nabla u |^{p-2} \nabla u
 \right ) \nabla \xi\,dx\\
&\ge {\frac 12} \gamma e^{\alpha (p-2)
 z(\theta_t \omega)} \int _{\mathcal{O}} |\nabla \xi |^p dx
 - \varepsilon \int_{\mathcal{O}} |\nabla u |^p dx.
\end{aligned}
\end{equation}
 For the nonlinearity $f_1$ in \eqref{psep1_1}, by
 \eqref{f2}-\eqref{f3} and \eqref{f5}, we have
\begin{align*}
& \int_{\mathcal{O}} \left ( e^{-\alpha z(\theta_t \omega) } f_1 
\big(t, x, e^{\alpha z(\theta_t \omega)} v_\alpha
 \big) -f_1(t, x,u) \right ) \xi\,dx
 \\
& =\int_{\mathcal{O}}
 e^{-\alpha z(\theta_t \omega) } \big( f_1 (t, x,
 e^{\alpha z(\theta_t \omega)} v_\alpha )
 - f_1 (t, x, e^{\alpha z(\theta_t \omega)} u )
 \big)\xi\,dx
\\
&\quad + \int_{\mathcal{O}} \big(
 e^{-\alpha z(\theta_t \omega) } -1 \big)
 f _1 (t, x, e^{\alpha z(\theta_t \omega) } u )
 \xi\,dx\\
&\quad + \int_{\mathcal{O}} \big( f_1 (t, x, e^{\alpha z(\theta_t \omega)} u )
 - f_1 (t, x, u ) \big) \xi\,dx
 \\
 & = \int_{\mathcal{O}} \xi^2 {\frac {\partial f_1} {\partial s}} (t,x,s) dx
+ \big( e^{-\alpha z(\theta_t \omega)} -1 \big)
 \int_{\mathcal{O}} f _1 (t, x, e^{\alpha z(\theta_t \omega)} u ) \xi\,dx
\\
&\quad  + \big( e^{\alpha z(\theta_t \omega)} -1 \big) \int_{\mathcal{O}}
 \xi u {\frac {\partial f_1} {\partial s}} (t, x,s) dx
\\
& \le \int_{\mathcal{O}} \xi^2 \psi_4 (t,x) dx
 + c | e^{-\alpha z(\theta_t \omega)} -1 |
 \int_{\mathcal{O}} \big(
 e^{\alpha (q-1) z(\theta_t \omega) } |u|^{q-1} |\xi|
 +\psi_3(t,x) | \xi| \big) dx
\\
&\quad  + c | e^{\alpha z(\theta_t \omega) } -1 | \int_{\mathcal{O}}
 \psi_6(t,x) \big(
 e^{\alpha (q-2 ) z(\theta_t \omega)}
 |u|^{q-1} |\xi| + |u|^{q-1} |\xi|
 + |u| |\xi| \big)dx,
\end{align*}
from which we find that there is
 $\alpha_3 = \alpha_3(\tau, \omega, T, \varepsilon) \le \alpha_2$
 such that for all $\alpha \in [0, \alpha_3]$ and $t \in [\tau,
 \tau +T]$,
 \begin{equation} \label{psep1_9}
\begin{aligned}
&\int_{\mathcal{O}}  \left ( e^{-\alpha z(\theta_t \omega) } f_1
 \big(t, x, e^{\alpha z(\theta_t \omega)} v_\alpha
 \big) -f_1(t, x,u) \right ) \xi\,dx\\
&\leq c \| \xi \|^2 + c \varepsilon +
 c \varepsilon \int_{\mathcal{O}} ( |u|^q + |v_\alpha|^q ) dx.
\end{aligned}
 \end{equation}
Similarly, by \eqref{f4}, one can verify that
there exists $\alpha_4 = \alpha_4(\tau, \omega, T, \varepsilon) \le \alpha_3$
such that for all $\alpha \in [0, \alpha_4]$ and
$t \in [\tau, \tau +T]$,
\begin{gather} \label{psep1_10}
\begin{aligned}
& \int_{\mathcal{O}} \left ( e^{-\alpha z(\theta_t \omega) } f_2
\big(t, x, e^{\alpha z(\theta_t \omega)} v_\alpha
 \big) -f_2(t, x,u) \right ) \xi\,dx\\
&\leq c \| \xi \|^2 + c \varepsilon +
 c \varepsilon \int_{\mathcal{O}} |u|^q dx,
\end{aligned}\\
 \label{psep1_11}
 \big(e^{-\alpha z(\theta_t \omega) } -1 \big) \int_{\mathcal{O}} g(t, x ) \xi\,dx
 \le \varepsilon \| \xi \|^2
 + \varepsilon \| g(t)\|^2.
\end{gather}
By \eqref{psep1_1}, \eqref{psep1_5}
and \eqref{psep1_9}-\eqref{psep1_11} we obtain
 for all $\alpha \in [0, \alpha_4]$ and $t \in [\tau, \tau +T]$,
\begin{equation}  \label{psep1_20}
 {\frac d{dt}} \| \xi \|^2
 \le c_1 \| \xi \|^2  + c_2 \varepsilon \left ( 1+ \| \nabla u \|^p_p +
 \|u \|^q_q + \|v_\alpha \|^q_q + \| g(t) \|^2\right ).
\end{equation}
 It follows from \eqref{psep1_20} that
 for all $\alpha \in [0, \alpha_4]$ and $t \in [\tau, \tau +T]$,
 \begin{equation} \label{psep1_21}
 \begin{aligned}
\| \xi (t) \|^2
& \le e^{c_1 (t -\tau)} \| \xi (\tau )\|^2
 + c_2 \varepsilon e^{c_1 (t -\tau)}\\
&\quad\times  \int_\tau^t \Big(1+
 \| v_\alpha (s, \tau, \omega, v_{\alpha, \tau})\|^q_q
 + \| u \|^q_q + \| \nabla u \|^p_p + \| g(s)\|^2 \Big)ds.
\end{aligned}
 \end{equation}
 By \eqref{f1} and \eqref{f4} we obtain from \eqref{enlem1}
 that, for all $\alpha \in [0, 1]$,
\begin{align*}
&{\frac d{dt}} \| v_\alpha (t) \|^2
 + 2 e^{\alpha (p-2) z(\theta_t \omega)} \| \nabla v_\alpha (t) \|^p_p
 + 2\lambda e^{\alpha (q-2) z(\theta_t \omega)} \| v_\alpha (t) \|^q_q
\\
& \le c_3 \| v_\alpha \|^2 + c_4 (\| \psi_1 (t) \|_{L^1({\mathcal{O}})}
 +\| g(t) \|^2 ).
\end{align*}
 Solving this inequality, we obtain that
 for all $\alpha \in [0, 1]$ and
 $t \in [\tau, \tau +T]$,
\begin{equation}\label{psep1_30}
\begin{aligned}
&\| v_\alpha (t, \tau , \omega, v_{\alpha, \tau } ) \|^2
 + 2 \int_\tau^t e^{c_3 (t-s)}
 e^{\alpha (p-2) z(\theta_s \omega)} \| \nabla v_\alpha (s) \|^p_p ds\\
&+ 2\lambda \int_\tau^t e^{c_3 (t-s)}
 e^{\alpha (q-2) z(\theta_s \omega)} \| v_\alpha (s) \|^q_q ds
\\
& \le e^{c_3 (t-\tau)} \|v_{\alpha, \tau} \|^2
 +c_4 \int_\tau^t e^{c_3 (t-s)} (\| \psi_1 (s) \|_{L^1({\mathcal{O}})}
 +\| g(s) \|^2 ) ds.
\end{aligned}
\end{equation}
 By \eqref{psep1_30} we have, for all $\alpha \in [0, 1]$ and
 $t \in [\tau, \tau +T]$,
 \begin{equation} \label{psep1_31}
\begin{aligned}
&\| v_\alpha (t, \tau , \omega, v_{\alpha, \tau } ) \|^2
+ \int_\tau^t
 \left ( \| \nabla v_\alpha (s,\tau, \omega, v_{\alpha, \tau} ) \|^p_p
 + \| v_\alpha (s, \tau, \omega, v_{\alpha, \tau}) \|^q_q \right ) ds
\\
&\le c_5 e^{c_3 (t-\tau)} \|v_{\alpha, \tau} \|^2
 +c_5 e^{c_3 (t-\tau)} \int_\tau^t (\| \psi_1 (s) \|_{L^1({\mathcal{O}})}
 +\| g(s) \|^2 ) ds.
\end{aligned}
\end{equation}
Note that \eqref{psep1_31} is also valid for $\alpha =0$, and hence we have,
 for all $t \in [\tau, \tau +T]$,
\begin{equation} \label{psep1_32}
\begin{aligned}
& \int_\tau^t \left ( \| \nabla u (s,\tau, u_{ \tau} ) \|^p_p
 + \| u (s, \tau, u_{ \tau}) \|^q_q \right ) ds\\
& \le c_5 e^{c_3 (t-\tau)} \|u_{ \tau} \|^2
 +c_5 e^{c_3 (t-\tau)} \int_\tau^t (\| \psi_1 (s) \|_{L^1({\mathcal{O}})}
 +\| g(s) \|^2 ) ds.
 \end{aligned}
\end{equation}
By \eqref{psep1_21} and \eqref{psep1_31}-\eqref{psep1_32}
we obtain  for all $\alpha \in [0, \alpha_4]$ and $t \in [\tau, \tau +T]$,
\begin{equation} \label{psep1_40}
\begin{aligned}
&\| v_\alpha (t, \tau, \omega, v_{\alpha, \tau}) - u(t,\tau, \omega) \|^2\\
&\le e^{c_1 (t -\tau)} \| v_{\alpha, \tau} - u_\tau \|^2\\
&\quad + c_6 \varepsilon e^{c_7 (t -\tau)}
\Big( 1 + \|u_\tau\|^2 + \|v_{\alpha, \tau} \|^2 +
 \int_{\tau}^t ( \| \psi_1 (s) \|_{L^1({\mathcal{O}})} + \| g(s ) \|^2 ) ds \Big).
\end{aligned}
\end{equation}
Note that
\begin{equation} \label{psep1_43}
 \|u_\alpha (t, \tau, \omega, u_{\alpha, \tau})
 - v_\alpha (t, \tau, \omega, v_{\alpha, \tau}) \|
 = | e^{\alpha z(\theta_t \omega) } -1 |
 \|v_\alpha (t, \tau, \omega, v_{\alpha, \tau}) \| .
\end{equation}
 Then \eqref{psep1_31} and
 \eqref{psep1_40}-\eqref{psep1_43}
 conclude the proof.
 \end{proof}

\begin{lemma}\label{sep3}
 Suppose \eqref{f1}-\eqref{f4}, \eqref{gpsi1}  and \eqref{f5} hold.
 Let $\tau \in \mathbb{R}$ and $\omega \in \Omega$ be fixed.
 If $\alpha_n \to 0 $ and $u_n \in {\mathcal{A}}_{\alpha_n}
 (\tau, \omega)$, then
 the sequence $\{ u_n \}_{n=1}^\infty$
 has a convergent subsequence in $L^2(\mathcal{O})$.
\end{lemma}

\begin{proof}
By $u_n \in {\mathcal{A}}_{\alpha_n} (\tau, \omega)$
 and the invariance of $ {\mathcal{A}}_{\alpha_n}$
 we find that
 there exists ${\widetilde{u}}_{n} \in {\mathcal{A}}_{\alpha_n}
 (\tau -1, \theta_{-1} \omega)$
such that
 \begin{equation} \label{psep3_1}
 u_n = u_{\alpha_n} (\tau, \tau -1, \theta_{-\tau} \omega,
 {\widetilde{u}}_{n} ).
 \end{equation}
Let $v_{\alpha_n}$ be the solution
 of \eqref{v1} with $\alpha$ replaced by
 $\alpha_n$. Then we have
 \begin{equation} \label{psep3_3}
 u_{\alpha_n} (\tau, \tau -1, \theta_{-\tau} \omega,
 {\widetilde{u}}_{n} )
 = e^{\alpha_n z(\omega)} v_{\alpha_n}
 (\tau, \tau -1, \theta_{-\tau} \omega, {\widetilde{v}}_{n} )
 \quad  \text{with }  {\widetilde{v}}_{n}
 = e^{-\alpha_n z(\theta_{-1} \omega)} {\widetilde{u}}_{n}.
 \end{equation}
Since $\alpha_n \to 0$,
 by \eqref{up1b}, there exist
 $c=c(\tau, \omega)>0$
and $N=N (\tau, \omega) \ge 1$ such that
 $ \| {\widetilde{u}}_{n} \| \le c$
for all $n \ge N$, which along with \eqref{psep3_3}
shows that  $\{{\widetilde{v}}_{n}\}_{n=1}^\infty$
 is bounded in $L^2(\mathcal{O})$.
Therefore, as in \eqref{pwcont_1}, one can prove that there exists
${\bar{v}} \in L^2((\tau-1, \tau), L^2(\mathcal{O}) )$ such that,
up to a subsequence,
\begin{equation} \label{psep3_7}
 v_{\alpha_n} (s, \tau -1, \theta_{-\tau} \omega,
 {\widetilde{v}}_{n} )
 \to {\bar{v}} (s) \quad
 \text{in $L^2(\mathcal{O})$ for almost all }
  s \in (\tau -1, \tau).
 \end{equation}
 By \eqref{uv}  and \eqref{psep3_7} we obtain
 \begin{equation}\label{psep3_8}
 u_{\alpha_n} (s, \tau -1, \theta_{-\tau} \omega,
 {\widetilde{u}}_{n} )
 \to  {\bar{v}} (s) \quad
 \text{in $L^2(\mathcal{O})$ for almost all }
  s \in (\tau -1, \tau).
 \end{equation}
 Since $\alpha_n \to 0$,  by \eqref{psep3_8} and Lemma \ref{sep1}
 we obtain,  for almost all  $ s \in (\tau -1, \tau)$,
 \begin{equation}\label{psep3_9}
 u_{\alpha_n} \left (\tau , s, \theta_{-\tau} \omega,
 u_{\alpha_n} (s, \tau -1, \theta_{-\tau} \omega,
 {\widetilde{u}}_{n} )
 \right )
 \to u(\tau, s, \theta_{-\tau} \omega,
 {\bar{v}} (s) )
 \quad \text{in }  L^2(\mathcal{O})
 \end{equation}
 where $u$ is the solution of \eqref{dere1}.
 By the cocycle property, we find that
 the left-hand side of \eqref{psep3_9}
 is the same as
 $ u_{\alpha_n} (\tau, \tau -1, \theta_{-\tau} \omega,
 {\widetilde{u}}_{n} ) $.
 Thus, by \eqref{psep3_1} and \eqref{psep3_9}
 we obtain, for almost all
 $ s \in (\tau -1, \tau)$,
 $$
 u_n = u_{\alpha_n} (\tau, \tau -1, \theta_{-\tau} \omega,
 {\widetilde{u}}_{n} )
 \to u(\tau, s, \theta_{-\tau} \omega,
 {\bar{v}} (s) )
 \quad \text{in }  L^2(\mathcal{O}),
 $$
 which completes the proof.
\end{proof}

We are now ready to present the upper semicontinuity
of random attractors as the intensity of noise approaches zero.

\begin{theorem}\label{sep8}
 Suppose \eqref{f1}-\eqref{f4}, \eqref{gpsi1} and \eqref{f5}  hold.
Then for every $\tau \in \mathbb{R}$ and $\omega \in \Omega$,
\begin{equation} \label{sep8_1}
\lim_{\alpha \to 0} \operatorname{dist}_{ L^2(\mathcal{O})}
( \mathcal{A}_\alpha (\tau, \omega), {\mathcal{A}}_0 (\tau) ) =0.
\end{equation}
\end{theorem}

\begin{proof}
Given a sequence $\alpha_n \to 0$ and $u_{0,n} \to u_0$ in $L^2(\mathcal{O})$,
it follows from  Lemma \ref{sep1} that,  for every $t \in \mathbb{R}^+$,
$\tau \in \mathbb{R}$ and $\omega \in \Omega$,
\begin{equation} \label{psep8_1}
\Phi_{\alpha_n} (t, \tau, \omega, u_{0,n}) \to \Phi_0 (t, \tau, u_0)
\quad \text{in }  L^2(\mathcal{O}).
\end{equation}
By \eqref{up4}, \eqref{psep8_1} and Lemma \ref{sep3},
we find that all conditions of Theorem 3.2 in \cite{wan6} are satisfied,
and thus \eqref{sep8_1} follows immediately.
\end{proof}

\begin{remark} \label{rmk6.4} \rm
In this paper, we discuss the random attractors
of parabolic equation of type \eqref{pde1}
driven by linear multiplicative noise and
non-autonomous deterministic forcing $g$.
It is interesting to consider the case when
$g$ is replaced by an additive white noise.
The attractors for such a system have been
investigated recently in \cite{gess2}.
Since the main objective of this paper
is to deal with the nonlinearity $f_1$
with polynomial growth of any order,
we do not consider the additive noise here
for the sake of simplicity,
and leave this case for future investigation.
\end{remark}


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\end{document}