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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 87, pp. 1--27.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/87\hfil Stochastic p-Laplacian equations]
{Existence and upper semicontinuity of random attractors for stochastic
$p$-Laplacian equations  on unbounded domains}

\author[J. Li, Y. Li, H. Cui \hfil EJDE-2014/87\hfilneg]
{Jia Li, Yangrong Li, Hongyong Cui}

\address{Jia Li \newline
School of Mathematics  and statistics, Southwest University, 
Chongqing 400715, China}
\email{bfreedom@swu.edu.cn}

\address{Yangrong Li \newline
School of Mathematics  and statistics, Southwest University, 
Chongqing 400715, China}
\email{liyr@swu.edu.cn}

\address{Hongyong Cui \newline
School of Mathematics  and statistics, Southwest University, 
Chongqing 400715, China}
\email{chy0321@126.com}

\thanks{Submitted January 19, 2014. Published April 2, 2014.}
\subjclass[2000]{37L30, 35B40, 35B41, 35K55, 35K57, 35Q80}
\keywords{Stochastic p-Laplacian equations; multiplicative noise; 
\hfill\break\indent random dynamical system, random attractor;
upper semicontinuity}

\begin{abstract}
 The existence of a pullback attractor is established for a stochastic
 p-Laplacian equation on $\mathbb{R}^n$. Furthermore, the limiting behavior of
 random attractors of the random dynamical systems as stochastic perturbations
 approach zero is studied and the upper semicontinuity is proved.
\end{abstract}

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


\section{Introduction}\label{1}

 It is known that p-Laplacian equation is always used to
model a variety of physical phenomena. In this paper, we investigate
the asymptotic behavior of
solutions to the following stochastic p-Laplacian equation with
multiplicative noise defined on the entire space $\mathbb{R}^n$:
\begin{equation}
du+(-\operatorname{div}(|\nabla u|^{p-2}\nabla u)+\lambda u)dt
=(f(x,u)+g(x))dt +\varepsilon u\circ dW(t), \label{e1.1}
\end{equation}
where $\varepsilon>0$ is a small positive parameter, $\lambda>0$,
$p\geq 2$ are fixed constants. $g$ is a given function defined on
$\mathbb{R}^n$, $f$ is a smooth nonlinear function satisfying
certain conditions, and $W$ is a two-sided real-valued Wiener
processes on a probability space which will be specified later.

The long-term behavior of random systems is captured by a pullback
random attractor, which was introduced by \cite{Hcr1,Hcr2} as an
extension of the attractors theory of deterministic systems in
\cite{Avb,Jkh,Jcr3,Rse,Tem}. In the case of bounded domains, the
existence of random attractors for stochastic PDEs has been studied
extensively by many authors
(see \cite{Lar,Hcr1,Hcr2,Jia,Yan1,Yan2,Glw,Bix2,Wqz2}) and the reference
therein. Since sobolev embeddings are not compact on unbounded
domains, it is more difficult to discuss the existence of random
attractors for PDEs defined on unbounded domains. Nevertheless, the
existence of such attractors for some stochastic PDEs on unbounded
domains has been proved in \cite{Pwb,Anh,Bwa3,Bwa2, Zha,Bix,Wqz}.

The first aim of this paper is to investigate the existence of
random attractors for the stochastic p-Laplacian equation \eqref{e1.1}
defined on $\mathbb{R}^n$. We mention that the existence of global attractors
for the p-Laplacian equation in the deterministic case has been
discussed by many authors, for examples, in \cite{Mhy,Cai} for
bounded domains and in \cite{Akh1,Akh2} for unbounded domains.
Recently \cite{Bix2} investigate the existence of random attractors
for the p-Laplacian equation with multiplicative noise. However, in
the paper \cite{Bix2}, the p-Laplace equation is defined in a
bounded domain where compactness of Sobolev embeddings is available.
To overcome the difficulty caused by the non-compactness of Sobolev
embedding on $\mathbb{R}^n$, we use the tail-estimate method which is always
used to deal with the problem caused by the unboundedness of domains
(see \cite{Pwb,Bwa3,Bwa2,Zha,Bix}). So far as we know, there were no
results on random attractors for stochastic p-Laplacian equation
with multiplicative noise on unbounded domains.

The second aim of this paper is to examine the limiting behavior of
random attractors when $\varepsilon\to 0$ and prove the
upper semicontinuity of these perturbed random attractors. It is
worth mentioning that such continuity of attractors has been
investigated, for examples, in \cite{Jkh1, Jkh2, Jkh3,Tem} for
deterministic equations, in \cite{Tca1,Tca2,Jcr,Bix2} for stochastic
PDEs in bounded domains and in \cite{Anh,Bix} for stochastic PDEs on
unbounded domains.


The paper is arranged as follows. In the next section, we review the
pullback random attractors theory for random dynamical systems. In
section 3, we define a continuous random dynamical system for the
stochastic p-Laplacian equation on $\mathbb{R}^n$. Section 4 is devoted to
obtaining uniform estimates of solutions as $t\to\infty$.
These estimate are necessary for proving the existence of bounded
absorbing sets and the asymptotic compactness of the solution
operator. 
Finally, we prove the upper semicontinuity of random attractors 
for \eqref{e1.1} in the last section.

We denote by $\|\cdot\|$ and $(\cdot,\cdot)$ the norm and the inner
product in $L^2(\mathbb{R}^n)$ respectively and $\|\cdot\|_p$ to denote the
norm in $L^p(\mathbb{R}^n)$. Otherwise, the norm of a general Banach space
$X$ is written as $\|\cdot\|_X$.

The letters $c$ and $C(\omega)$ are generic positive constants and
positive random variable respectively, which don't depend on
$\varepsilon$ and may change their values from line to line or even
in the same line.

\section{Preliminaries}\label{2}

 In this section, we recall some basic concepts related to
RDS (see \cite{Lar,Ich,Hcr1,Hcr2,Bix} for details).

Let $(X,\|\cdot\|_X)$ be a Banach space with Borel $\sigma$-algebra.

\begin{definition} \rm
 Let $(\Omega,\mathcal{F},\mathrm{P})$ be a
probability space and $\{\theta_t:\Omega\to\Omega,t\in \mathbb{R}\}$
a family of measure preserving transformations such that
$(t,\omega)\mapsto \theta_t\omega$ is measurable, $\theta_0=id$ and
$\theta_{t+s}=\theta_t\theta_s$ for all $s,t\in \mathbb{R}$. The flow
$\theta_t$ together with the corresponding probability space
$(\Omega,\mathcal{F},\mathrm{P},\theta_t)$ is called a measurable
dynamical system.
 \end{definition}

\begin{definition}\rm
A continuous random
dynamical system(RDS) on $X$ over $\theta$ on $(\Omega,\mathcal{F},\mathrm{P})$ is
a measurable map
$$
\varphi: \mathbb{R}^+\times\Omega\times X\mapsto X,\;
 (t,\omega,x)\mapsto\varphi(t,\omega)x
$$
such that P-a.s.
\begin{itemize}
\item[(i)] $\varphi(0,\omega)=id$ on $X$;

\item[(ii)] $\varphi(t+s,\omega)=\varphi(t,\theta_s\omega)\varphi(s,\omega)$
for all $s,t\in \mathbb{R}^+$ (cocycle property);

\item[(iii)] $\varphi(t,\omega):X\mapsto X$ is continuous.
\end{itemize}
\end{definition}

\begin{definition} \rm
 A random compact set
$\{K(\omega)\}_{\omega\in\Omega}$ is a family of compact sets
indexed by $\omega$ such that for every $x\in X$ the mapping
$\omega\mapsto d(x,K(\omega))$ is measurable with respect to
$\mathcal{F}$. A random set $\{K(\omega)\}_{\omega\in\Omega}$  is
said to be bounded if there exist $u_0\in X$ and a random variable
$R(\omega)>0$ such that
$$
K(\omega)\subset \{u\in X, \|u-u_0\|_X\leq R(\omega)\}\quad
\text{for all }\omega\in\Omega.
$$
\end{definition}

\begin{definition} \rm
A random bounded set
$\{B(\omega)\}_{\omega\in\Omega}$ of $X$ is called tempered with
respect to $(\theta_t)_{t\in \mathbb{R}}$ if for P-a.e.$\omega\in\Omega$,
$$
\lim_{t\to\infty}e^{-\beta t}d(B(\theta_{-t}\omega))=0\quad \text{for all }\beta>0,
$$
where $d(B)=\sup_{x\in B}\|x\|_X$.
 \end{definition}

 \begin{definition} \rm
Let $\mathcal{D}$ be a collection of random subsets of X and
$\{K(\omega)\}_{\omega\in\Omega}\in \mathcal{D}$. Then
$\{K(\omega)\}_{\omega\in\Omega}$ is called a random absorbing set
for $\varphi$ in $\mathcal{D}$ if for every $B\in \mathcal{D}$ and
P-a.e. $\omega\in\Omega$, there exists $t_B(\omega)>0$ such that
$$
\varphi(t,\theta_{-t}\omega,B(\theta_{-t}\omega))\subseteq K(\omega)\quad
\text{for all } t\geq t_B(\omega).
$$
\end{definition}

\begin{definition} \rm
 Let $\mathcal{D}$ be a collection of random subsets of X.
Then $\varphi$ is said to be
$\mathcal{D}$-pullback asymptotically compact in $X$ if for
P-a.e. $\omega\in\Omega$,
$\{\varphi(t_n,\theta_{-t_n}\omega,x_n)\}^{\infty}_{n=1}$ has a
convergent subsequence in $X$ whenever $t_n\to\infty$, and
$x_n\in B(\theta_{-t}\omega)$ with
$\{B(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$.
\end{definition}

\begin{definition}\rm
Let $\mathcal{D}$ be a collection of random subsets of $X$. Then a
random set $\{{\mathcal{A}(\omega)}\}_{\omega\in\Omega}$ of $X$ is
called a $\mathcal{D}$-random attractor(or $\mathcal{D}$-pullback
attractor) for $\varphi$ if the following conditions are satisfied,
for P-a.e.$\omega\in\Omega$,
\begin{itemize}
\item[(i)] $\mathcal{A}(\omega)$ is a random compact set;

\item[(ii)] $\mathcal{A}(\omega)$ is invariant, that is,
$\varphi(t,\omega,\mathcal{A}(\omega))$=${\mathcal{A}}(\theta_t\omega)$,
for all $t\geq 0$;

\item[(iii)] $\mathcal{A}(\omega)$ attracts every set in $\mathcal{D}$, that
is, for every $B=\{B(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$,
$$
\lim_{t\to\infty}d(\varphi(t,\theta_{-t}\omega,B(\theta_{-t}\omega)),
{\mathcal{A}}(\omega))=0,
$$
where $d$ is the Hausdorff semi-metric.
\end{itemize}
\end{definition}

 \begin{proposition} \label{existence prop}
Let $\mathcal{D}$ be an inclusion-closed collection of random
subsets of $X$ and $\varphi$ a continuous RDS on $X$ over
$(\Omega,\mathcal{F},\mathrm{P},(\theta_t)_{t\in \mathbb{R}})$. Suppose that
$\{K(\omega)\}_{\omega\in\Omega}$ is a closed random absorbing set
for $\varphi$ in $\mathcal{D}$ and $\varphi$ is
$\mathcal{D}$-pullback asymptotically compact in $X$. Then $\varphi$
has a unique $\mathcal{D}$-random attractor
$\{{\mathcal{A}(\omega)}\}_{\omega\in\Omega}$ which is given by
$$
{\mathcal{A}}(\omega)=\cap_{T\geq 0}\overline{\cup_{t\geq T}
\varphi(t,\theta_{-t}\omega,K(\theta_{-t}\omega))}.
$$
\end{proposition}


\begin{proposition} \label{uppersemicontinuity}
Let $\mathcal{D}$ be an inclusion-closed collection of random
subsets of $X$. Given $\sigma>0$, suppose
$\varphi_\sigma$ is a random dynamical system over a metric system
$(\Omega, {\mathcal{F}}, \mathbb{P}, (\theta_t)_{t\in \mathbb{R}})$ which has a
$\mathcal{D}$-random attractor ${\mathcal{A}}_\sigma$ and $\varphi_0$ is a
deterministic dynamical system defined on $X$ which has a global attractor
 ${\mathcal{A}}_0$. Assume that  the
following conditions be satisfied: (i) For $\mathbb{P}$-a.e.
$\omega \in \Omega$, $t\ge 0$, $\sigma_n
\to 0$, and $x_n$, $x \in X$ with $x_n \to x$, there holds
\begin{equation} \label{2.1}
\lim_{n \to \infty} \varphi_{\sigma_n} (t, \omega,  x_n ) =
\varphi_0(t) x.
\end{equation}

(ii) Every $\varphi_\sigma$ has a random absorbing set
$E_\sigma = \{E_\sigma (\omega)\}_{\omega \in \Omega} \in {\mathcal{D}}$  such
that for some deterministic positive constant $c$ and  for
$\mathbb{P}$-a.e. $\omega \in \Omega$,
\begin{equation} \label{2.2}
\limsup_{\sigma \to  0} \| E_\sigma (\omega) \|_X \le c,
\end{equation}
where $\| E_\sigma (\omega) \|_X =\sup_{x \in  E_\sigma (\omega)} \| x \|_X$.

(iii) There exists $\sigma_0 >0$ such that
for $\mathbb{P}$-a.e. $\omega \in \Omega$,
\begin{equation} \label{2.3}
\cup_{0<\sigma \le \sigma_0} {\mathcal{A}}_\sigma (\omega)
\quad \text{is precompact in }  X.
\end{equation}
Then for $\mathbb{P}$-a.e.
$\omega \in \Omega$,
\begin{equation} \label{2.4}
 \operatorname{dist}({\mathcal{A}}_\sigma (\omega), {\mathcal{A}}_0 )
\to 0, \quad \text{as } \sigma \to  0.
\end{equation}
\end{proposition}

\section{Stochastic p-Laplacian equation with multiplicative noise}\label{3}

 Here we show that there is a continuous random dynamical
system generated by the stochastic p-Laplacian equation defined on
$\mathbb{R}^n$ with multiplicative noise:
\begin{equation}  \label{3.1}
 du+(-\operatorname{div}(|\nabla u|^{p-2}\nabla u)+\lambda u)dt=(f(x,u)+g(x))dt
+\varepsilon u\circ dW(t),
 \end{equation}
for $x\in \mathbb{R}^n$, $t>0$,
with the initial condition
\begin{equation}   \label{3.2}
 u(x,0)=u_0(x),\quad  x\in \mathbb{R}^n.
 \end{equation}
where $\varepsilon>0$, $\lambda>0$, $p\geq 2$ are constants,
$g\in L^2(\mathbb{R}^n)$, $W$ is a two-sided real-valued Wiener processes on a
probability space which will be specified below, and $f$ is a smooth
nonlinear function satisfying the following conditions: For all
$x\in \mathbb{R}^n$ and $s\in \mathbb{R}^n$,
\begin{gather}  \label{3.3}
f(x,s)s\leq-\alpha_1 |s|^p+\psi_1(x), \\
 \label{3.5}
|f(x,s)|\leq \alpha_2|s|^{p-1}+\psi_2(x),
\end{gather}
where $\alpha_1,\alpha_2$ are positive constants,
$\psi_1\in L^1(\mathbb{R}^n)\cap L^{\frac{p}{2}}(\mathbb{R}^n)$,
$\psi_2\in L^2(\mathbb{R}^n)\cap L^q(\mathbb{R}^n)$
with $\frac{1}{p}+\frac{1}{q}=1$.

In the sequel, we consider the probability space
$(\Omega,{\mathcal{F}},\mathrm{P})$ 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 $\mathrm{P}$ the
corresponding Wiener measure on $(\Omega,{\mathcal{F}})$. Define the
time shift by
$$
\theta_t\omega(\cdot)=\omega(\cdot+t)-\omega(t),\quad \omega\in\Omega,\;
t\in \mathbb{R}.
$$
Then $(\Omega,{\mathcal{F}},\mathrm{P},(\theta_t)_{t\in \mathbb{R}})$ is a
metric dynamical system.

We now associate a continuous random dynamical system with the
equation over $(\Omega,{\mathcal{F}},\mathrm{P},(\theta_t)_{t\in \mathbb{R}})$.
To this end, we need to convert the stochastic equation with a random
multiplicative term into a deterministic equation with a random
parameter.

Consider the stationary solutions of the one-dimensional
Ornstein-Uhlenbeck equation:
\begin{equation}  \label{3.6}
dz+zdt=dW(t) .
\end{equation}
 The solution to \eqref{3.6} is given by
 \begin{equation}  \label{3.7}
z(\theta_t\omega)=-\int_{-\infty}^0e^{\tau}(\theta_t\omega)(\tau)d\tau,\
\quad t\in \mathbb{R}^n.
\end{equation}
From \cite{Lar,Pwb,Bwa3,Bwa2}, the random variable $|z(\omega)|$ is tempered,
and there is a $\theta_t$-invariant set $\widetilde{\Omega}\subset\Omega$ of full
$\mathrm{P}$ measure such that $z(\theta_t\omega)$ is continuous in
$t$ for every $\omega\in\widetilde{\Omega}$ and
\begin{gather}  \label{3.8}
\lim_{t\to\pm\infty}\frac{|z(\theta_t\omega)|}{|t|}=0; \\
  \label{3.9}
\lim_{t\to\pm\infty}\frac{1}{t}\int_0^tz(\theta_s\omega)ds=0.
\end{gather}
Following the properties of the O-U process and
\eqref{3.8},\eqref{3.9}, it is easy to show for for all
$0<\varepsilon\leq 1$,
\begin{equation}
\begin{aligned}
0\leq M_\varepsilon(\omega)
&:=\int_{-\infty}^0e^{-2\varepsilon
z(\theta_s \omega)+2\varepsilon\int_s^0z(\theta_\tau
\omega)d\tau+\lambda s}ds\\
&\leq \int_{-\infty}^0e^{2|z(\theta_s \omega)|+2|\int_s^0z(\theta_\tau
\omega)d\tau|+\lambda s}ds<+\infty\label{3.10} ,
\end{aligned}
\end{equation}
and
\begin{equation}  \label{3.11}
0\leq K(\omega):=\max_{-2\leq \tau\leq
0}|z(\theta_\tau(\omega))|<+\infty,
 \end{equation}
which will be used frequently in the following paper.
And it is easy to see that
$M_\varepsilon(\omega)$ and $K(\omega)$ are both tempered.

 To show that problem \eqref{3.1}-\eqref{3.2} generates a random
dynamical system, let
\begin{equation}  \label{3.12}
v(t)=e^{-\varepsilon z(\theta_t\omega)}u(t) ,
 \end{equation}
 where $u$ is a solution
of problem \eqref{3.1}-\eqref{3.2}. Then $v$ satisfies
\begin{equation}  \label{3.13}
\frac{dv}{dt}=e^{\varepsilon(p-2)z(\theta_t\omega)}\operatorname{div}(|\nabla
v|^{p-2}\nabla v)-\lambda v+e^{-\varepsilon
z(\theta_t\omega)}(f(x,u)+g(x))+\varepsilon v z(\theta_t\omega),
\end{equation}
and the initial condition
\begin{equation}  \label{3.14}
v(x,0)=v_0(x)=e^{-\varepsilon z(\omega)}u_0(x),\quad x\in \mathbb{R}^n.
 \end{equation}

Using the standard Galerkin method, one may show that for all
$v_0\in L^2(\mathbb{R}^n)$, problem \eqref{3.13}-\eqref{3.14} has a
unique solution
\[
v(\cdot,\omega,v_0)\in C([0,\infty),L^2(\mathbb{R}^n))\cap
L^2((0,T),W^{1,p}(\mathbb{R}^n)).
\]
 Furthermore, the solution is continuous
with respect to $v_0$ in $L^2(\mathbb{R}^n)$ for all $t\geq0$. Let
\begin{equation}  \label{3.15}
u(t,\omega,u_0)=e^{\varepsilon z(\theta_t \omega)}v(t,\omega,v_0) ,
\end{equation}
where
\begin{equation}  \label{3.16}
v_0=e^{-\varepsilon z(\omega)}u_0 .
\end{equation}
We can associate a random dynamical system $\Phi_\varepsilon$ with
problem \eqref{3.1}-\eqref{3.2} via $u$ for each $\varepsilon>0$,
where $\Phi_\varepsilon:\mathbb{R}^+\times \Omega\times
L^2(\mathbb{R}^n)\mapsto L^2(\mathbb{R}^n)$ is given by
\begin{equation}  \label{3.17}
\Phi_\varepsilon(t,\omega)u_0 =u(t,\omega,u_0),\quad\text{for every }
(t,\omega,u_0)\in \mathbb{R}^+\times \Omega\times L^2(\mathbb{R}^n) .
\end{equation}
Then $\Phi_\varepsilon$ is a continuous random dynamical system over
$(\Omega,\mathcal{F},\mathrm{P},(\theta_t)_{t\in \mathbb{R}})$ in
$L^2(\mathbb{R}^n)$.
 In the sequel, we always assume that $\mathcal{D}$ is the collection of all
tempered random subsets of $L^2(\mathbb{R}^n)$.

In the following, we will first prove that $\Phi_\varepsilon$ has a
unique $\mathcal{D}$-pullback random attractor
$\{\mathcal{A}_\varepsilon(\omega)\}_{\omega\in\Omega}$. When
$\varepsilon=0$, problem \eqref{3.1}-\eqref{3.2} defines a
continuous deterministic dynamical system $\Phi$ in $L^2(\mathbb{R}^n)$. We
use $\mathcal{A}_0$ to denote the global attractor for the
deterministic dynamical system. At last, we will establish the
relationship of
$\{\mathcal{A}_\varepsilon(\omega)\}_{\omega\in\Omega}$ and
$\mathcal{A}_0$ when $\varepsilon\to 0$.


\section{Uniform estimates of solutions}\label{4}

  In this section, we derive uniform estimates on the
solution of the stochastic p-Laplacian equation on $\mathbb{R}^n$ when
$t\to\infty$ with the purpose of proving the existence of a
bounded random absorbing set and the asymptotic compactness of the
random dynamical system associated with the equation. In particular,
we will show that the tails of the solutions, i.e.,solutions
evaluated at large values of $|x|$, are uniformly small when time is
sufficiently large.

\begin{lemma}\label{estimate1}
Let $0<\varepsilon\leq 1, g\in L^2(\mathbb{R}^n)$ and \eqref{3.3}-\eqref{3.5}
hold. Then for every
$B=\{B(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$ and
P-a.e.$\omega\in\Omega$, there is $T(B,\omega)>0$, independent of
$\varepsilon$, such that for all $u_0(\theta_{-t}\omega)\in
B(\theta_{-t}\omega)$ and $t\geq T(B,\omega)$,
\begin{equation}  \label{4.1}
\|u(t,\theta_{-t}\omega,u_0(\theta_{-t}\omega))\|^2\leq
\rho_1^2(\omega):=1+ce^{2\varepsilon z(\omega)}M_\varepsilon(\omega).
\end{equation}
Furthermore $\rho_1(\omega)$ is a tempered function.
\end{lemma}

\begin{proof}
Multiplying \eqref{3.13} by $v$ and then integrating over $\mathbb{R}^n$,
 we find that
\begin{equation}
\begin{aligned}
\frac{1}{2}\frac{d}{dt}\|v\|^2
&=-e^{\varepsilon(p-2)z(\theta_t\omega)}\int_{\mathbb{R}^n}|\nabla v|^pdx
-\lambda\|v\|^2+e^{-\varepsilon z(\theta_t\omega)}\int_{\mathbb{R}^n}f(x,u)v
dx \\
&\quad + e^{-\varepsilon z(\theta_t\omega)}\int_{\mathbb{R}^n}g(x)v dx
+\varepsilon z(\theta_t\omega)\|v\|^2
\end{aligned}\label{4.2} .
\end{equation}

For the nonlinear term, by \eqref{3.3}, we have
\begin{equation}
\begin{aligned}
e^{-\varepsilon z(\theta_t\omega)}\int_{\mathbb{R}^n} f(x,u)v
dx&= e^{-2\varepsilon z(\theta_t\omega)}\int_{\mathbb{R}^n} f(x,u)u dx \\
&\leq -\alpha_1e^{-2\varepsilon z(\theta_t\omega)}\int_{\mathbb{R}^n} |u|^p
dx+e^{-2\varepsilon z(\theta_t\omega)}\int_{\mathbb{R}^n} |\psi_1(x)|dx \\
&= -\alpha_1e^{-2\varepsilon
z(\theta_t\omega)}\|u\|^p_p+e^{-2\varepsilon
z(\theta_t\omega)}\|\psi_1\|_{L_1},
\end{aligned} \label{4.3}
\end{equation}
And
\begin{equation} \label{4.4}
\begin{aligned}
e^{-\varepsilon z(\theta_t\omega)}\int_{\mathbb{R}^n}g(x)v dx\leq
e^{-\varepsilon
z(\theta_t\omega)}\|g\|\cdot\|v\|\leq\frac{1}{2\lambda}
e^{-2\varepsilon z(\theta_t\omega)}\|g\|^2+
\frac{\lambda}{2}\|v\|^2.
\end{aligned}
\end{equation}
 Then it
from \eqref{4.2}-\eqref{4.4} it follows that
\begin{equation}
\begin{aligned}
\frac{d}{dt}\|v\|^2
&\leq -2e^{\varepsilon(p-2)z(\theta_t\omega)}\int_{\mathbb{R}^n}|\nabla
v|^pdx+(2\varepsilon
z(\theta_t\omega)-\lambda)\|v\|^2 \\
&\quad -2\alpha_1e^{-2\varepsilon
z(\theta_t\omega)}\|u\|^p_p+(\frac{1}{\lambda}\|g\|^2
 +2\|\psi_1\|_{L_1})e^{-2\varepsilon
z(\theta_t\omega)} .
\end{aligned} \label{4.5}
\end{equation}
Thus,
\begin{equation}  \label{4.6}
\frac{d}{dt}\|v\|^2
\leq (2\varepsilon z(\theta_t\omega)-\lambda)\|v\|^2
+ce^{-2\varepsilon z(\theta_t\omega)}.
\end{equation}
 By the Gronwall Lemma,
\begin{equation}  \label{4.7}
\begin{aligned}
&\|v(t,\omega,v_0(\omega)\|^2\\
&\leq\|v_0(\omega)\|^2e^{2\varepsilon\int_0^tz(\theta_s\omega)ds-\lambda
t}+c\int_0^te^{-2\varepsilon z(\theta_s\omega)
+2\varepsilon\int_s^tz(\theta_\tau\omega)d\tau-\lambda(t-s)}ds .
\end{aligned}
\end{equation}


Replace $\omega$ by $\theta_{-t}\omega$ in \eqref{4.7}, we have
\begin{equation}
\begin{aligned}
\|v(t,\theta_{-t}\omega,v_0(\theta_{-t}\omega)\|^2
&\leq \|v_0(\theta_{-t}\omega)\|^2e^{2\varepsilon\int_0^tz(\theta_{s-t}\omega)ds
-\lambda t}\\
&\quad +c\int_0^te^{-2\varepsilon
z(\theta_{s-t}\omega)+2\varepsilon\int_s^tz(\theta_{\tau-t}\omega)d\tau
 -\lambda(t-s)}ds
\\
&= \|v_0(\theta_{-t}\omega)\|^2e^{2\varepsilon\int_{-t}^0z(\theta_{s}\omega)d
s-\lambda t}\\
&\quad +c\int_{-t}^0e^{-2\varepsilon z(\theta_{s}\omega)
 +2\varepsilon\int_s^0z(\theta_{\tau}\omega)d\tau+\lambda s}ds \\
&\leq \|v_0(\theta_{-t}\omega)\|^2e^{2\varepsilon
 \int_{-t}^0z(\theta_{s}\omega)ds-\lambda t}+cM_\varepsilon(\omega),
\end{aligned}  \label{4.8}
\end{equation}
where $M_\varepsilon(\omega)$ is defined in \eqref{3.10}.

It follows from \eqref{4.8} and \eqref{3.12} that
\begin{equation}
\begin{aligned}
&\|u(t,\theta_{-t}\omega,u_0(\theta_{-t}\omega)\|^2\\
&= \|e^{\varepsilon
z(\omega)}v(t,\theta_{-t}\omega,v_0(\theta_{-t}\omega)\|^2
\\
&= e^{2\varepsilon
z(\omega)}\|v(t,\theta_{-t}\omega,v_0(\theta_{-t}\omega)\|^2
\\
&\leq e^{2\varepsilon z(\omega)}(
\|v_0(\theta_{-t}\omega)\|^2e^{2\varepsilon\int_{-t}^0z(\theta_{s}\omega)ds-\lambda
t}+cM_\varepsilon(\omega)) \\
&= e^{2\varepsilon z(\omega)}(e^{-2\varepsilon
z(\theta_{-t}\omega)}
\|u_0(\theta_{-t}\omega)\|^2e^{2\varepsilon\int_{-t}^0z(\theta_{s}\omega)ds
 -\lambda t}+cM_\varepsilon(\omega)) .
\end{aligned}  \label{4.9}
\end{equation}
Since $B=\{B(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$ and
 $u_0(\theta_{-t}\omega)\in
B(\theta_{-t}\omega)$, due to \eqref{3.8}\eqref{3.9}, there exists
$T(B,\omega)>0$, independent of $\varepsilon$, such that for all
$t\geq T(B,\omega)$
\begin{equation}
\begin{aligned}
&\|u_0(\theta_{-t}\omega)\|^2e^{2\varepsilon
z(\omega)-2\varepsilon
z(\theta_{-t}\omega)+2\varepsilon\int_{-t}^0z(\theta_{s}\omega)ds-\lambda
t} \\
&\leq \|u_0(\theta_{-t}\omega)\|^2e^{2|z(\omega)|+2|z(\theta_{-t}\omega)|+2|\int_{-t}^0z(\theta_{s}\omega)ds|-\lambda
t} \\
&\leq  \|u_0(\theta_{-t}\omega)\|^2 e^{-\frac{\lambda}{2}t}\leq 1 ,
\end{aligned} \label{4.10}
\end{equation}
which along with \eqref{4.9} implies that for all $t\geq T(B,\omega)$
\begin{equation}  \label{4.11}
\|u(t,\theta_{-t}\omega,u_0(\theta_{-t}\omega))\|^2\leq
\rho_1^2(\omega):=1+ce^{2\varepsilon z(\omega)}M_\varepsilon(\omega) .
\end{equation}
It is easy to prove that $\rho_1(\omega)$ is a tempered function.
\end{proof}

\begin{lemma}\label{estimate2}
Let $0<\varepsilon\leq 1, g\in L^2(\mathbb{R}^n)$ and \eqref{3.3}-\eqref{3.5}
hold. Then for every
$B=\{B(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$ and
P-a.e. $\omega\in\Omega$, there is $T(B,\omega)>0$, independent of
$\varepsilon$, such that for all $u_0(\theta_{-t}\omega)\in
B(\theta_{-t}\omega)$ and $t\geq T(B,\omega)$
\begin{gather}  \label{4.12}
\int_t^{t+1}\|\nabla
u(s,\theta_{-t-1}\omega,u_0(\theta_{-t-1}\omega))\|_p^p\leq
1+ce^{2p\varepsilon K(\omega)}M_\varepsilon(\omega) ,
\\
\label{4.13}
\int_{t-1}^{t+1}\|
u(s,\theta_{-t-1}\omega,u_0(\theta_{-t-1}\omega))\|_p^p\leq
1+ce^{6\varepsilon K(\omega)}M_\varepsilon(\omega) .
\end{gather}
\end{lemma}

\begin{proof}
From \eqref{4.5}, we have
\begin{equation}
\begin{aligned}
\frac{d}{dt}\|v\|^2
&\leq -2e^{\varepsilon(p-2)z(\theta_t\omega)}\|\nabla
v\|_p^pdx+(2\varepsilon
z(\theta_t\omega)-\lambda)\|v\|^2 \\
&\quad -2\alpha_1e^{-2\varepsilon
z(\theta_t\omega)}\|u\|^p_p+ce^{-2\varepsilon z(\theta_t\omega)} .
\end{aligned}\label{4.14}
\end{equation}
Using Gronwall Lemma, for all $t\geq T\geq 0$, we have
\begin{equation}
\begin{aligned}
&\|v(t,\omega,v_0(\omega))\|^2\\
&\leq  e^{2\varepsilon
\int_T^tz(\theta_s\omega)ds-\lambda(t-T)}\|v(T,\omega,v_0(\omega))\|^2\\
&\quad +c\int_T^te^{-2\varepsilon
z(\theta_s\omega)+2\varepsilon\int_s^tz(\theta_\tau\omega)d\tau+\lambda(s-t)}
\\
&\quad -2\alpha_1\int_T^te^{-2\varepsilon
z(\theta_s\omega)+2\varepsilon\int_s^tz(\theta_\tau\omega)d\tau+\lambda(s-t)}\|u(s,\omega,u_0(\omega))\|_p^pds
\\
&\quad - 2\int_T^te^{\varepsilon(p-2)
z(\theta_s\omega)+2\varepsilon\int_s^tz(\theta_\tau\omega)d\tau+\lambda(s-t)}\|\nabla
v(s,\omega,v_0(\omega))\|_p^pds .
\end{aligned} \label{4.15}
\end{equation}
Replace $\omega$ by $\theta_{-t}\omega$ and $t$ by $T$ in
\eqref{4.7}, we have
\begin{equation}
\begin{aligned}
\|v(T,\theta_{-t}\omega,v_0(\theta_{-t}\omega)\|^2
&\leq \|v_0(\theta_{-t}\omega)\|^2e^{2\varepsilon\int_0^Tz(\theta_{s-t}\omega)ds
-\lambda T}\\
&\quad +c\int_0^Te^{-2\varepsilon
z(\theta_{s-t}\omega)+2\varepsilon\int_s^Tz(\theta_{\tau-t}\omega)d\tau
-\lambda(T-s)}ds  .
\end{aligned} \label{4.16}
\end{equation}
Multiplying the two sides of \eqref{4.16} by
$e^{2\varepsilon\int_T^tz(\theta_{s-t}\omega)ds-\lambda(t-T)}$, then
for all $t\geq T$,
\begin{equation}
\begin{aligned}
& e^{2\varepsilon\int_T^tz(\theta_{s-t}\omega)ds-\lambda(t-T)}\|v(T,\theta_{-t}\omega,v_0(\theta_{-t}\omega)\|^2
 \\
&\leq \|v_0(\theta_{-t}\omega)\|^2e^{2\varepsilon\int_0^tz(\theta_{s-t}\omega)ds-\lambda
t}
+c\int_0^Te^{-2\varepsilon
z(\theta_{s-t}\omega)+2\varepsilon\int_s^tz(\theta_{\tau-t}\omega)d\tau
-\lambda(t-s)}ds.
\end{aligned}  \label{4.17}
\end{equation}
Thus, replace $\omega$ by $\theta_{-t}\omega$ in \eqref{4.15} and
together with \eqref{4.17}, it follows that
\begin{equation}
\begin{aligned}
& 2\alpha_1\int_T^te^{-2\varepsilon
z(\theta_{s-t}\omega)+2\varepsilon\int_s^tz(\theta_{\tau-t}\omega)d\tau+\lambda(s-t)}\|u(s,\theta_{-t}\omega,u_0(\theta_{-t}\omega))\|_p^pds
\\
& + 2\int_T^te^{\varepsilon(p-2)
z(\theta_{s-t}\omega)+2\varepsilon\int_s^tz(\theta_{\tau-t}\omega)d\tau+\lambda(s-t)}\|\nabla
v(s,\theta_{-t}\omega,v_0(\theta_{-t}\omega))\|_p^pds \\
&\leq \|v_0(\theta_{-t}\omega)\|^2e^{2\varepsilon\int_0^tz(\theta_{s-t}\omega)ds-\lambda
t}+c\int_0^te^{-2\varepsilon
z(\theta_{s-t}\omega)+2\varepsilon\int_s^tz(\theta_{\tau-t}\omega)d\tau-\lambda(t-s)}ds \\
&=\|v_0(\theta_{-t}\omega)\|^2e^{2\varepsilon\int_{-t}^0z(\theta_{s}\omega)ds-\lambda
t}+c\int_{-t}^0e^{-2\varepsilon
z(\theta_{s}\omega)+2\varepsilon\int_s^0z(\theta_{\tau}\omega)d\tau+\lambda
s}ds .
\end{aligned}  \label{4.18}
\end{equation}
Replace $t$ by $t+1$ and $T$ by $t$ in \eqref{4.18}, we have
\begin{align*}
& 2\int_t^{t+1}e^{\varepsilon(p-2)
z(\theta_{s-t-1}\omega)+2\varepsilon\int_s^{t+1}z(\theta_{\tau-t-1}\omega)d\tau
+\lambda(s-t-1)}\\
&\times \|\nabla v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pds \\
&\leq \|v_0(\theta_{-t-1}\omega)\|^2e^{2\varepsilon\int_{-t-1}^0
z(\theta_{s}\omega)ds-\lambda (t+1)}+c\int_{-t-1}^0e^{-2\varepsilon
z(\theta_{s}\omega)+2\varepsilon\int_s^0z(\theta_{\tau}\omega)d\tau+\lambda s}ds .
\end{align*} % \label{4.19}
Using \eqref{3.11}, we have
\begin{align*}
&\int_t^{t+1}e^{\varepsilon(p-2)
z(\theta_{s-t-1}\omega)+2\varepsilon\int_s^{t+1}z(\theta_{\tau-t-1}\omega)d\tau
+\lambda(s-t-1)}\\
&\times \|\nabla v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pds \\
&\geq \int_t^{t+1}e^{-\varepsilon(p-2) K(\omega)-2\varepsilon
K(\omega)-\lambda}\|\nabla
v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pds \\
&= e^{-p\varepsilon K(\omega)-\lambda}\int_t^{t+1}\|\nabla
v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pds .
\end{align*} %\label{4.20}
Thus
\begin{equation}
\begin{aligned}
& \int_t^{t+1}\|\nabla
v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pds
\\
&\leq \frac{1}{2}\|v_0(\theta_{-t-1}\omega)\|^2e^{p\varepsilon
K(\omega)+2\varepsilon\int_{-t-1}^0z(\theta_{s}\omega)ds-\lambda
t}\\
&\quad +ce^{p\varepsilon
K(\omega)+\lambda}\int_{-t-1}^0e^{-2\varepsilon
z(\theta_{s}\omega)+2\varepsilon\int_s^0z(\theta_{\tau}\omega)d\tau+\lambda
s}ds\\
&\leq \frac{1}{2}\|v_0(\theta_{-t-1}\omega)\|^2e^{p\varepsilon
K(\omega)+2\varepsilon\int_{-t-1}^0z(\theta_{s}\omega)ds-\lambda
t}+ce^{p\varepsilon K(\omega)+\lambda}M_\varepsilon(\omega) .
\end{aligned} \label{4.21}
\end{equation}
It follows from \eqref{4.21} that
\begin{equation}
\begin{aligned}
& \int_t^{t+1}\|\nabla
u(s,\theta_{-t-1}\omega,u_0(\theta_{-t-1}\omega))\|_p^pds
\\
&=\int_t^{t+1}e^{p\varepsilon z(\theta_{s-t-1}\omega)}\|\nabla
v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pds
\\
&\leq  e^{p\varepsilon K(\omega)}\int_t^{t+1}\|\nabla
v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pds
\\
&\leq \frac{1}{2}\|v_0(\theta_{-t-1}\omega)\|^2e^{2p\varepsilon
K(\omega)+2\varepsilon\int_{-t-1}^0z(\theta_{s}\omega)ds-\lambda
t}+ce^{2p\varepsilon
K(\omega)+\lambda}M_\varepsilon(\omega)\\
&\leq c\|u_0(\theta_{-t-1}\omega)\|^2e^{-2\varepsilon
z(\theta_{-t-1}\omega)+2p\varepsilon
K(\omega)+2\varepsilon\int_{-t-1}^0z(\theta_{s}\omega)ds-\lambda
t}\\
&\quad +ce^{2p\varepsilon K(\omega)+\lambda}M_\varepsilon(\omega)  .
 \end{aligned} \label{4.22}
\end{equation}
On the other hand, since
\begin{equation}
\begin{aligned}
&\int_{t-1}^{t+1}e^{-2\varepsilon
z(\theta_{s-t-1}\omega)+2\varepsilon\int_s^{t+1}z(\theta_{\tau-t-1}\omega)d\tau
+\lambda(s-t-1)}\|u(s,\theta_{-t-1}\omega,u_0(\theta_{-t-1}\omega))\|_p^pds
\\
&\geq e^{-6\varepsilon
K(\omega)-2\lambda}\int_{t-1}^{t+1}\|u(s,\theta_{-t-1}\omega,
u_0(\theta_{-t-1}\omega))\|_p^pds .
\end{aligned} \label{4.23}
\end{equation}
Replace $t$ by $t+1$ and $T$ by $t-1$ in \eqref{4.18} and using
\eqref{4.23}, we obtain
\begin{equation}
\begin{aligned}
&\int_{t-1}^{t+1}\|u(s,\theta_{-t-1}\omega,u_0(\theta_{-t-1}\omega))\|_p^pds\\
&\leq  c\|v_0(\theta_{-t-1}\omega)\|^2e^{6\varepsilon
K(\omega)+2\varepsilon\int_{-t-1}^0z(\theta_{s}\omega)ds-\lambda
t+\lambda}\\
&\quad +ce^{6\varepsilon
K(\omega)+2\lambda}\int_{-t-1}^0e^{-2\varepsilon
z(\theta_{s}\omega)+2\varepsilon\int_s^0z(\theta_{\tau}\omega)d\tau+\lambda
s}ds\\&\leq  c\|v_0(\theta_{-t-1}\omega)\|^2e^{6\varepsilon
K(\omega)+2\varepsilon\int_{-t-1}^0z(\theta_{s}\omega)ds-\lambda
t+\lambda}+ce^{6\varepsilon
K(\omega)+2\lambda}M_\varepsilon(\omega)\\
&\leq  c\|u_0(\theta_{-t-1}\omega)\|^2e^{-2\varepsilon
z(\theta_{-t-1}\omega)+6\varepsilon
K(\omega)+2\varepsilon\int_{-t-1}^0z(\theta_{s}\omega)ds-\lambda
t+\lambda}\\
&\quad +ce^{6\varepsilon K(\omega)+2\lambda}M_\varepsilon(\omega) .
\end{aligned} \label{4.24}
\end{equation}
Since $B=\{B(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$ and
 $u_0(\theta_{-t}(\omega))\in
B(\theta_{-t}(\omega))$, similar to \eqref{4.10}, there exists
$T(B,\omega)>0$, independent of $\varepsilon$, such that for all
$t\geq T(B,\omega)$
\begin{gather} \label{4.25}
c\|u_0(\theta_{-t-1}\omega)\|^2e^{-2\varepsilon
z(\theta_{-t-1}\omega)+2p\varepsilon
K(\omega)+2\varepsilon\int_{-t-1}^0z(\theta_{s}\omega)ds-\lambda
t}\leq 1,
\\
 \label{4.26}
c\|u_0(\theta_{-t-1}\omega)\|^2e^{-2\varepsilon
z(\theta_{-t-1}\omega)+6\varepsilon
K(\omega)+2\varepsilon\int_{-t-1}^0z(\theta_{s}\omega)ds-\lambda
t+\lambda}\leq 1.
\end{gather}
From \eqref{4.22}, \eqref{4.24} and using \eqref{4.25}, \eqref{4.26},
we obtain that for all $t\geq T(B,\omega)$,
\begin{gather}  \label{4.28}
\int_t^{t+1}\|\nabla
u(s,\theta_{-t-1}\omega,u_0(\theta_{-t-1}\omega))\|_p^p\leq
1+ce^{2p\varepsilon K(\omega)}M_\varepsilon(\omega) ,
\\
\label{4.29}
\int_{t-1}^{t+1}\|u(s,\theta_{-t-1}\omega,u_0(\theta_{-t-1}\omega))\|_p^p\leq
1+ce^{6\varepsilon K(\omega)}M_\varepsilon(\omega).
\end{gather}
\end{proof}


\begin{lemma}\label{estimate3}
Let $0<\varepsilon\leq 1$, $g\in L^2(\mathbb{R}^n)$ and \eqref{3.3}-\eqref{3.5}
hold. Then for every
$B=\{B(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$ and
P-a.e. $\omega\in\Omega$, there is $T(B,\omega)>0$, independent of
$\varepsilon$, such that for all $u_0(\theta_{-t}\omega)\in
B(\theta_{-t}\omega)$ and $t\geq T(B,\omega),\forall \tau\in[t,t+1]$
\begin{equation}  \label{4.30}
\|v(\tau,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^p\leq
ce^{p\varepsilon K(\omega)}(\varepsilon K(\omega)+1)(e^{6\varepsilon
K(\omega)}M_\varepsilon(\omega)+1) .
\end{equation}
\end{lemma}

\begin{proof}
Multiplying \eqref{3.13} with $|v|^{p-2}v$ and then integrating over
$\mathbb{R}^n$, it yields that
\begin{equation}
\begin{aligned}
\frac{1}{p}\frac{d}{dt}\|v\|_p^p &= e^{\varepsilon
(p-2)z(\theta_t\omega)}(\operatorname{div}(|\nabla v|^{p-2}\nabla
v),|v|^{p-2}v)-\lambda \|v\|_p^p\\
&\quad +e^{-\varepsilon z(\theta_t\omega)}(f(x,u),|v|^{p-2}v)
+ e^{-\varepsilon z(\theta_t\omega)}(g(x),|v|^{p-2}v)\\
&\quad +\varepsilon z(\theta_t\omega)\|v\|_p^p .
\end{aligned}\label{4.31}
\end{equation}
We now estimate every term of \eqref{4.31}. First by our assumption
$p\geq 2$, we have
\begin{equation}
\begin{aligned}
& e^{\varepsilon (p-2)z(\theta_t\omega)}(\operatorname{div}(|\nabla
v|^{p-2}\nabla v),|v|^{p-2}v)\\&= e^{\varepsilon
(p-2)z(\theta_t\omega)}\int_{\mathbb{R}^n}\sum_{i=1}^n\frac{\partial}{\partial
x_i}(|\nabla v|^{p-2}\frac{\partial v}{\partial x_i})|v|^{p-2}vdx
\\
&=-e^{\varepsilon
(p-2)z(\theta_t\omega)}\sum_{i=1}^n[\int_{\mathbb{R}^n}(|\nabla
v|^{p-2}\frac{\partial v}{\partial x_i})(p-2)|v|^{p-2}\frac{\partial
v}{\partial x_i}dx   \\
&\quad +\int_{\mathbb{R}^n}(|\nabla v|^{p-2}\frac{\partial v}{\partial
x_i})|v|^{p-2}\frac{\partial v}{\partial
x_i}dx]\\
&=-e^{\varepsilon (p-2)z(\theta_t\omega)}(p-1)\int_{\mathbb{R}^n}|\nabla
v|^p|v|^{p-2}dx
\leq 0    .
\end{aligned}\label{4.32}
\end{equation}
To estimate the nonlinear term, from \eqref{3.3}, we have
\begin{equation}
\begin{aligned}
f(x,u)v=e^{-\varepsilon
z(\theta_t\omega)}f(x,u)u
&\leq -\alpha_1e^{-\varepsilon
z(\theta_t\omega)}|u|^p+e^{-\varepsilon
z(\theta_t\omega)}\psi_1(x)\\
&=-\alpha_1e^{(p-1)\varepsilon
z(\theta_t\omega)}|v|^p+e^{-\varepsilon
z(\theta_t\omega)}\psi_1(x) .
\end{aligned} \label{4.33}
\end{equation}
From which it follows by Young's inequality that
\begin{equation}
\begin{aligned}
&e^{-\varepsilon
z(\theta_t\omega)}(f(x,u),|v|^{p-2}v)\\
&= e^{-\varepsilon
z(\theta_t\omega)}\int_{\mathbb{R}^n}f(x,u)|v|^{p-2}v \\
&\leq -\alpha_1e^{(p-2)\varepsilon
z(\theta_t\omega)}\|v\|_{2p-2}^{2p-2}+e^{-2\varepsilon
z(\theta_t\omega)}\int_{\mathbb{R}^n}\psi_1(x)|v| ^{p-2}dx \\
&\leq -\alpha_1e^{(p-2)\varepsilon
z(\theta_t\omega)}\|v\|_{2p-2}^{2p-2}+ce^{-p\varepsilon
z(\theta_t\omega)}\|\psi_1\|_{\frac{p}{2}}^{\frac{p}{2}}
+\frac{(p-1)\lambda}{p}\|v\|_p^p .
\end{aligned}  \label{4.34}
\end{equation}
On the other hand, the fourth term on the right-hand side of
\eqref{4.31} is bounded by
\begin{equation}  \label{4.35}
|e^{-\varepsilon
z(\theta_t\omega)}(g(x),|v|^{p-2}v)|\leq\frac{\alpha_1}{2}e^{(p-2)\varepsilon
z(\theta_t\omega)}\|v\|_{2p-2}^{2p-2}+ce^{-p\varepsilon
z(\theta_t\omega)}\|g\|^2 .
\end{equation}
Then it follows from
\eqref{4.31}\eqref{4.32}, \eqref{4.34},\eqref{4.35} that
\begin{equation}  \label{4.36}
\frac{d}{dt}\|v\|_p^p \leq(p\varepsilon
z(\theta_t\omega)-\lambda)\|v\|_p^p+ce^{-p\varepsilon
z(\theta_t\omega)} .
\end{equation}
Integrating \eqref{4.36} from $s(t-1\leq s\leq t)$ to
$\tau(t\leq\tau\leq t+1)$, we obtain
\begin{equation}
 \begin{aligned}
& \|v(\tau,\omega,v_0(\omega))\|_p^p\\
&\leq  \|v(s,\omega,v_0(\omega))\|_p^p
 +\int_s^\tau|p\varepsilon z(\theta_{s'}\omega)-\lambda|
 \|v(s',\omega,v_0(\omega))\|_p^pds'\\
 &\quad +c\int_s^\tau e^{-p\varepsilon
 z(\theta_{s'}\omega)}ds'  .
\end{aligned} \label{4.37}
\end{equation}
Replace $\omega$ by $\theta_{-t-1}\omega$ in \eqref{4.37}, we have
\begin{equation}
\begin{aligned}
&\|v(\tau,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^p\\
&\leq  \|v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^p
+c\int_{t-1}^{t+1} e^{-p\varepsilon
 z(\theta_{s'-t-1}\omega)}ds'\\
&\quad + \int_{t-1}^{t+1}|p\varepsilon z(\theta_{s'-t-1}\omega)-\lambda|
 \|v(s',\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pds'.
\end{aligned} \label{4.38}
\end{equation}
Integrating \eqref{4.38} with respect to $s$ from $t-1$ to $t$,
we obtain that for all $\tau\in[t,t+1]$,
\begin{align}
 & \|v(\tau,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^p \nonumber\\
&\leq \int_{t-1}^t\|v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pds
+c\int_{-2}^{0} e^{-p\varepsilon  z(\theta_s\omega)}ds\nonumber \\
&\quad +c(\varepsilon K(\omega)+1)\int_{t-1}^{t+1}
 \|v(s',\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pds' \nonumber\\
&\leq \int_{t-1}^{t+1}\|v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pds+c
e^{p\varepsilon  K(\omega)}\\
&\quad +c(\varepsilon K(\omega)+1)\int_{t-1}^{t+1}
 \|v(s',\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pds' \nonumber\\
&\leq \int_{t-1}^{t+1}e^{-p\varepsilon
 z(\theta_{s-t-1}\omega)}\|u(s,\theta_{-t-1}\omega,u_0(\theta_{-t-1}\omega))\|_p^pds
 +c e^{p\varepsilon  K(\omega)} \nonumber\\
&\quad +c(\varepsilon K(\omega)+1)\int_{t-1}^{t+1} e^{-p\varepsilon
 z(\theta_{s'-t-1}\omega)} \|u(s',\theta_{-t-1}\omega,
 u_0(\theta_{-t-1}\omega))\|_p^pds' \nonumber\\
&\leq c(\varepsilon K(\omega)+1)e^{p\varepsilon
 K(\omega)}\int_{t-1}^{t+1}\|u(s',\theta_{-t-1}\omega,
 u_0(\theta_{-t-1}\omega))\|_p^pds'+c e^{p\varepsilon  K(\omega)}  . \label{4.39}
\end{align}   
Let $T(B,\omega)$ be the positive constant in Lemma \ref{estimate2}
and $t\geq T(B,\omega)$, together with \eqref{4.13} and
\eqref{4.39}, we have for $t\leq\tau\leq t+1$
\begin{equation}  \label{4.40}
\|v(\tau,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^p\leq
ce^{p\varepsilon K(\omega)}(\varepsilon K(\omega)+1)(e^{6\varepsilon
K(\omega)}M_\varepsilon(\omega)+1) .
\end{equation}
\end{proof}


\begin{lemma}\label{estimate 4}
Let $0<\varepsilon\leq 1, g\in L^2(\mathbb{R}^n)$ and \eqref{3.3}-\eqref{3.5}
hold. Then for every
$B=\{B(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$ and
P-a.e.$\omega\in\Omega$, there is $T(B,\omega)>0$, independent of
$\varepsilon$, such that for all $u_0(\theta_{-t}\omega)\in
B(\theta_{-t}\omega)$ and $t\geq T(B,\omega)$
\begin{equation}  \label{4.41}
\int_t^{t+1}\|u(s,\theta_{-t-1}\omega,u_0(\theta_{-t-1}\omega))\|_{2p-2}^{2p-2}ds\leq
ce^{c\varepsilon K(\omega)}(\varepsilon K(\omega)+1)(e^{c\varepsilon
K(\omega)}M_\varepsilon(\omega)+1) .
\end{equation}
\end{lemma}
\begin{proof}
Using \eqref{4.31} together with  \eqref{4.32} \eqref{4.34}and
\eqref{4.35}, we have
\begin{equation}  \label{4.42}
\frac{d}{dt}\|v\|_p^p
\leq-ce^{\varepsilon(p-2)z(\theta_t\omega)}\|v\|_{2p-2}^{2p-2}+(p\varepsilon
z(\theta_t\omega)-\lambda)\|v\|_p^p+ce^{-p\varepsilon
z(\theta_t\omega)} .
\end{equation}
Using Gronwall's Lemma, for all $t\geq T\geq 0$
\begin{equation}
\begin{aligned}
&\|v(t,\omega,v_0(\omega))\|_p^p\\
&\leq \|v(T,\omega,v_0(\omega))\|_p^pe^{p\varepsilon
\int_T^tz(\theta_s\omega)ds-\lambda(t-T)}\\
&\quad +c\int_T^te^{-p\varepsilon z(\theta_s\omega)+p\varepsilon
\int_s^tz(\theta_{\tau}\omega)d\tau-\lambda(t-s)}ds\\
&\quad -c\int_T^te^{\varepsilon (p-2)z(\theta_s\omega)+p\varepsilon
\int_s^tz(\theta_{\tau}\omega)d\tau
-\lambda(t-s)}\|v(s,\omega,v_0(\omega))\|_{2p-2}^{2p-2}ds.
\end{aligned}  \label{4.43}
\end{equation}
Replace $\omega$ by $\theta_{-t}\omega$ in \eqref{4.43}. It follows
that
\begin{equation}
\begin{aligned}
& c\int_T^te^{\varepsilon (p-2)z(\theta_{s-t}\omega)+p\varepsilon
\int_s^tz(\theta_{\tau-t}\omega)d\tau-\lambda(t-s)}\|v(s,\theta_{-t}\omega,v_0(\theta_{-t}\omega))\|_{2p-2}^{2p-2}ds\\
&\leq \|v(T,\theta_{-t}\omega,v_0(\theta_{-t}\omega))\|_p^pe^{p\varepsilon
\int_T^tz(\theta_{s-t}\omega)ds-\lambda(t-T)}  \\
&\quad + c\int_T^te^{-p\varepsilon
z(\theta_{s-t}\omega)+p\varepsilon
\int_s^tz(\theta_{\tau-t}\omega)d\tau-\lambda(t-s)}ds .
\end{aligned}  \label{4.44}
\end{equation}
Replacing $t$ by $t+1$ and $T$ by $t$, we have
\begin{equation}
\begin{aligned}
&c\int_t^{t+1}e^{\varepsilon
(p-2)z(\theta_{s-t-1}\omega)+p\varepsilon
\int_s^{t+1}z(\theta_{\tau-t-1}\omega)d\tau-\lambda(t+1-s)}\\
&\times \|v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_{2p-2}^{2p-2}ds\\
&\leq \|v(t,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pe^{p\varepsilon
\int_t^{t+1}z(\theta_{s-t-1}\omega)ds-\lambda}  \\
&\quad + c\int_t^{t+1}e^{-p\varepsilon
z(\theta_{s-t-1}\omega)+p\varepsilon
\int_s^{t+1}z(\theta_{\tau-t-1}\omega)d\tau-\lambda(t+1-s)}ds \\
&\leq \|v(t,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pe^{p\varepsilon
K(\omega)-\lambda}+ce^{2p\varepsilon K(\omega)} .
\end{aligned}  \label{4.45}
\end{equation}
Note that
\begin{equation}
\begin{aligned}
&\int_t^{t+1}e^{\varepsilon
(p-2)z(\theta_{s-t-1}\omega)+p\varepsilon
\int_s^{t+1}z(\theta_{\tau-t-1}\omega)d\tau-\lambda(t+1-s)}\\
&\times \|v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_{2p-2}^{2p-2}ds\\
&\geq e^{-2p\varepsilon K(\omega)+2\varepsilon
K(\omega)-\lambda}\int_t^{t+1}\|v(s,\theta_{-t-1}\omega,v_0
(\theta_{-t-1}\omega))\|_{2p-2}^{2p-2}ds.
\end{aligned} \label{4.46}
\end{equation}
Thus together with \eqref{4.45}-\eqref{4.46}, we obtain
\begin{equation}
\begin{aligned}
&\int_t^{t+1}\|u(s,\theta_{-t-1}\omega,u_0(\theta_{-t-1}\omega))\|_{2p-2}^{2p-2}ds
\\
&=\int_t^{t+1}e^{\varepsilon
z(\theta_{s-t-1}\omega)(2p-2)}\|v(s,\theta_{-t-1}\omega,
 v_0(\theta_{-t-1}\omega))\|_{2p-2}^{2p-2}ds
\\
&\leq e^{(2p-2)\varepsilon K(\omega)}
\int_t^{t+1}\|v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_{2p-2}^{2p-2}ds
\\
&\leq  ce^{5p\varepsilon K(\omega)-4\varepsilon
K(\omega)}\|v(t,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^p+ce^{6p\varepsilon
K(\omega)-4\varepsilon K(\omega)+\lambda } .
\end{aligned}  \label{4.47}
\end{equation}
Let $T(B,\omega)$ be the positive constant in Lemma \ref{estimate3}.
Then for $t\geq T(B,\omega)$, from \eqref{4.30}, we have
\begin{equation}
\begin{aligned}
&\quad \int_t^{t+1}\|u(s,\theta_{-t-1}\omega,
u_0(\theta_{-t-1}\omega))\|_{2p-2}^{2p-2}ds \\
&\leq ce^{6p\varepsilon K(\omega)-4\varepsilon
K(\omega)}(\varepsilon K(\omega)+1)(e^{6\varepsilon
K(\omega)}M_\varepsilon(\omega)+1)+ce^{6p\varepsilon
K(\omega)-4\varepsilon K(\omega)+\lambda } \\
&= ce^{c\varepsilon K(\omega)}(\varepsilon
K(\omega)+1)(e^{c\varepsilon K(\omega)}M_\varepsilon(\omega)+1).
\end{aligned} \label{4.48}
\end{equation}
\end{proof}

\begin{lemma}\label{estimate5}
Let $0<\varepsilon\leq 1, g\in L^2(\mathbb{R}^n)$ and \eqref{3.3}-\eqref{3.5}
hold. Then for every
$B=\{B(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$ and
P-a.e. $\omega\in\Omega$, there is $T(B,\omega)>0$, independent of
$\varepsilon$, such that for all $u_0(\theta_{-t}\omega)\in
B(\theta_{-t}\omega)$ and $t\geq T(B,\omega)$
\begin{equation}  \label{4.49}
\|\nabla u(t,\theta_{-t}\omega,u_0(\theta_{-t}\omega))\|_p^p\leq
ce^{c\varepsilon K(\omega)}(\varepsilon
K(\omega)+1)(M_\varepsilon(\omega)+1) .
\end{equation}
\end{lemma}

\begin{proof}
Take the inner product of \eqref{3.13} with $v_t$ in $L^2(\mathbb{R}^n)$,
we obtain
\begin{equation} \label{4.50}
\begin{aligned}
\|v_t\|^2&=-e^{\varepsilon(p-2)z(\theta_{t}\omega)}\frac{1}{p}\frac{d}{dt}\|\nabla
v\|_p^p-\lambda(v,v_t)+e^{-\varepsilon
z(\theta_{t}\omega)}(f(x,u),v_t)\\
&\quad  +e^{-\varepsilon
z(\theta_{t}\omega)}(g(x),v_t)+\varepsilon z(\theta_t\omega)(v,v_t) .
\end{aligned}
\end{equation}
By \eqref{3.5}, the Cauchy-Schwartz inequality and the Young
inequality, we find that
\begin{gather}
\begin{aligned}
&|e^{-\varepsilon z(\theta_{t}\omega)}(f(x,u),v_t)|\\
&\leq e^{-2\varepsilon z(\theta_{t}\omega)}\|f(x,u)\|^2+\frac{1}{4}\|v_t\|^2 \\
&\leq 2\alpha_2^2e^{-2\varepsilon
z(\theta_t\omega)}\|u\|_{2p-2}^{2p-2}+2e^{-2\varepsilon
z(\theta_t\omega)}\|\psi_2\|^2+\frac{1}{4}\|v_t\|^2 ,
\end{aligned} \label{4.51}
\\
 \label{4.52}
|e^{-\varepsilon z(\theta_{t}\omega)}(g(x),v_t)|\leq
e^{-2\varepsilon z(\theta_{t}\omega)}\|g\|^2+\frac{1}{4}\|v_t\|^2 ,
\\
 \label{4.53}
|(\varepsilon z(\theta_t\omega)-\lambda)(v,v_t)|\leq |\varepsilon
z(\theta_t\omega)-\lambda|^2\|v\|^2+\frac{1}{4}\|v_t\|^2 .
\end{gather}
It follows from \eqref{4.50}-\eqref{4.53} that
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\|\nabla v\|_p^p\\
&\leq 2p\alpha_2^2e^{-p\varepsilon
z(\theta_t\omega)}\|u\|_{2p-2}^{2p-2}+2pe^{-p\varepsilon
z(\theta_t\omega)}\|\psi_2\|^2 \\
&\quad +p e^{-p\varepsilon
z(\theta_{t}\omega)}\|g\|^2+pe^{-\varepsilon
(p-2)z(\theta_{t}\omega)}|\varepsilon
z(\theta_t\omega)-\lambda|^2\|v\|^2 \\
&= ce^{-p\varepsilon
z(\theta_t\omega)}\|u\|_{2p-2}^{2p-2}+ce^{-p\varepsilon
z(\theta_t\omega)}+ce^{\varepsilon
(2-p)z(\theta_{t}\omega)}|\varepsilon
z(\theta_t\omega)-\lambda|^2\|v\|^2 .
\end{aligned}  \label{4.54}
\end{equation}
Let $T_0(B,\omega)$ be the positive constant in Lemma \ref{estimate2},
take $t\geq T_0(B,\omega)$ and $s\in(t,t+1)$. Integrate
\eqref{4.54} over $(s,t+1)$ to get
\begin{equation}
\begin{aligned}
&\|\nabla v(t+1,\omega,v_0(\omega))\|_p^p\\
&\leq \|\nabla
v(s,\omega,v_0(\omega))\|_p^p+c\int_s^{t+1}e^{-p\varepsilon
z(\theta_{\tau}\omega)}\|u(\tau,\omega,u_0(\omega))\|_{2p-2}^{2p-2}d\tau
\\
&\quad +c\int_s^{t+1}e^{-p\varepsilon
z(\theta_{\tau}\omega)}d\tau+c\int_s^{t+1}e^{\varepsilon
(2-p)z(\theta_{\tau}\omega)}|\varepsilon
z(\theta_{\tau}\omega)-\lambda|^2\|v(\tau,\omega,v_0(\omega))\|^2d\tau  .
\end{aligned} \label{4.55}
\end{equation}
Integrating  with respect to $s$ over $(t,t+1)$, it
follows that
\begin{equation}
\begin{aligned}
&\|\nabla
v(t+1,\omega,v_0(\omega))\|_p^p\\
&\leq \int_t^{t+1}\|\nabla v(s,\omega,v_0(\omega))\|_p^pds
 +c\int_t^{t+1}e^{-p\varepsilon
z(\theta_{\tau}\omega)}\|u(\tau,\omega,u_0(\omega))\|_{2p-2}^{2p-2}d\tau
\\
&\quad +c\int_t^{t+1}e^{-p\varepsilon z(\theta_{\tau}\omega)}d\tau
+c\int_t^{t+1}e^{\varepsilon (2-p)z(\theta_{\tau}\omega)}|\varepsilon
z(\theta_{\tau}\omega)-\lambda|^2\|v(\tau,\omega,v_0(\omega))\|^2d\tau .
\end{aligned} \label{4.56}
\end{equation}
Replace $\omega$ by $\theta_{-t-1}\omega$ in the above inequality,
 we have
\begin{equation}
\begin{aligned}
&\|\nabla
v(t+1,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^p\\
&\leq \int_t^{t+1}\|\nabla
v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pds\\
&\quad +c\int_t^{t+1}e^{-p\varepsilon
z(\theta_{\tau-t-1}\omega)}\|u(\tau,\theta_{-t-1}\omega,u_0(\theta_{-t-1}\omega))
\|_{2p-2}^{2p-2}d\tau
\\
&\quad +c\int_t^{t+1}e^{-p\varepsilon
z(\theta_{\tau-t-1}\omega)}d\tau\\
&\quad +c\int_t^{t+1}e^{\varepsilon
(2-p)z(\theta_{\tau-t-1}\omega)}|\varepsilon
z(\theta_{\tau-t-1}\omega)-\lambda|^2\|v(\tau,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|^2d\tau\\
&\leq \int_t^{t+1}\|\nabla
v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pds\\
&\quad +ce^{p\varepsilon K(\omega)}+ce^{p\varepsilon
K(\omega)}\int_t^{t+1}\|u(\tau,\theta_{-t-1}\omega,u_0(\theta_{-t-1}\omega))\|_{2p-2}^{2p-2}d\tau\\
&\quad +c(K^2(\omega)+1)\int_t^{t+1}e^{\varepsilon
(2-p)z(\theta_{\tau-t-1}\omega)}\|v(\tau,\theta_{-t-1}\omega,
v_0(\theta_{-t-1}\omega))\|^2d\tau ,
\end{aligned} \label{4.57}
\end{equation}
where we used $0<\varepsilon\leq 1$.

We now estimate each part on the right. Firstly, by the Lemma
\ref{estimate2}, for $t\geq T_0(B,\omega)$,
\begin{equation}
\begin{aligned}
& \int_t^{t+1}\|\nabla
v(s,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|_p^pds\\
&=\int_t^{t+1}e^{-p\varepsilon z(\theta_{s-t-1}\omega)}\|\nabla
u(s,\theta_{-t-1}\omega,u_0(\theta_{-t-1}\omega))\|_p^pds\\
&\leq e^{p\varepsilon K(\omega)}\int_t^{t+1}\|\nabla
u(s,\theta_{-t-1}\omega,u_0(\theta_{-t-1}\omega))\|_p^pds\\
&\leq  e^{p\varepsilon K(\omega)}(1+ce^{2p\varepsilon
K(\omega)}M_{\varepsilon}(\omega)) .
\end{aligned} \label{4.58}
\end{equation}
To estimate the last part, firstly, replace $t$ by $\tau+t+1$ and
$\omega$ by $\theta_{-t-1}\omega$ in \eqref{4.7}, we obtain
\begin{equation}
\begin{aligned}
& \|v(\tau+t+1,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega)\|^2\\
&\leq \|v_0(\theta_{-t-1}\omega)\|^2e^{2\varepsilon
\int_0^{\tau+t+1}z(\theta_{s-t-1}\omega)ds-\lambda (\tau+t+1)}\\
&\quad +c\int_0^{\tau+t+1}e^{-2\varepsilon z(\theta_{s-t-1}\omega)
 +2\varepsilon\int_s^{\tau+t+1}z(\theta_{s'-t-1}\omega)ds'-\lambda(\tau+t+1-s)}ds  .
\end{aligned} \label{4.59}
\end{equation}
Thus
\begin{equation}
\begin{aligned}
 &\int_t^{t+1}e^{\varepsilon
(2-p)z(\theta_{\tau-t-1}\omega)}\|v(\tau,\theta_{-t-1}\omega,
v_0(\theta_{-t-1}\omega))\|^2d\tau \\
&\leq \int_{-1}^0e^{\varepsilon
(2-p)z(\theta_{\tau}\omega)}\|v(\tau+t+1,\theta_{-t-1}\omega,
 v_0(\theta_{-t-1}\omega))\|^2d\tau \\
&\leq \int_{-1}^0\|v_0(\theta_{-t-1}\omega)\|^2
e^{2\varepsilon\int_0^{\tau+t+1}z(\theta_{s-t-1}\omega)ds-\lambda
(\tau+t+1)+\varepsilon
(2-p)z(\theta_{\tau}\omega)}d\tau  \\
& +   c\int_{-1}^0\int_0^{\tau+t+1}e^{\varepsilon
(2-p)z(\theta_{\tau}\omega)-2\varepsilon
z(\theta_{s-t-1}\omega)
 +2\varepsilon\int_s^{\tau+t+1}z(\theta_{s'-t-1}\omega)ds'
-\lambda(\tau+t+1-s)}dsd\tau  .
\end{aligned} \label{4.60}
\end{equation}
Since
\begin{equation}
\begin{aligned}
&\int_{-1}^0\|v_0(\theta_{-t-1}\omega)\|^2e^{2\varepsilon\int_0^{\tau+t+1}z(\theta_{s-t-1}\omega)ds-\lambda
(\tau+t+1)+\varepsilon (2-p)z(\theta_{\tau}\omega)}d\tau\\
&\leq \|u_0(\theta_{-t-1}\omega)\|^2e^{-2\varepsilon
z(\theta_{-t-1}\omega)-\lambda(t+1)+2\varepsilon\int_{-t-1}^0z(\theta_{s}\omega)ds} \\
&\quad\times \int_{-1}^0e^{2\varepsilon\int_0^{\tau}z(\theta_{s}\omega)ds-\lambda
\tau+\varepsilon
(2-p)z(\theta_{\tau}\omega)}d\tau \label{4.61} .
\end{aligned}
\end{equation}
For $B=\{B(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$ and
 $u_0(\theta_{-t}(\omega))\in B(\theta_{-t}(\omega))$, similar to \eqref{4.10},
there exists $T_1(B,\omega)>0$, independent of $\varepsilon$, such that for all
$t\geq T_1(B,\omega)$,
\begin{equation}
\begin{aligned}
& \|u_0(\theta_{-t-1}\omega)\|^2e^{-2\varepsilon
z(\theta_{-t-1}\omega)-\lambda(t+1)+2\varepsilon
\int_{-t-1}^0z(\theta_{s}\omega)ds}\\
&\times \int_{-1}^0e^{2\varepsilon\int_0^{\tau}z(\theta_{s}\omega)ds-\lambda
\tau+\varepsilon (2-p)z(\theta_{\tau}\omega)}d\tau\\
&\leq \|u_0(\theta_{-t-1}\omega)\|^2e^{-\frac{\lambda}{2}(t+1)}
\leq 1  .
\end{aligned} \label{4.62}
\end{equation}
For the second part of \eqref{4.60},
\begin{equation}
\begin{aligned}
& c\int_{-1}^0\int_0^{\tau+t+1}e^{\varepsilon
(2-p)z(\theta_{\tau}\omega)-2\varepsilon
z(\theta_{s-t-1}\omega)+2\varepsilon\int_s^{\tau+t+1}z(\theta_{s'-t-1}\omega)ds'
-\lambda(\tau+t+1-s)}dsd\tau\\
&= c\int_{-1}^0\int_{-t-1}^{\tau}e^{\varepsilon
(2-p)z(\theta_{\tau}\omega)-2\varepsilon
z(\theta_{s}\omega)+2\varepsilon\int_s^{\tau}z(\theta_{s'}\omega)ds'
-\lambda(\tau-s)}dsd\tau\\
&\leq c\int_{-1}^0e^{2\varepsilon\int_0^{\tau}z(\theta_{s'}\omega)ds'
-\lambda\tau+\varepsilon
(2-p)z(\theta_{\tau}\omega)}M_{\varepsilon}(\omega) d\tau\\
&\leq ce^{p\varepsilon K(\omega)}M_{\varepsilon}(\omega) .
\end{aligned} \label{4.63}
\end{equation}
Thus \eqref{4.60}-\eqref{4.63} imply that for all $t\geq T_1(B,\omega)$
\begin{equation}
\begin{aligned}
&\int_t^{t+1}e^{\varepsilon
(2-p)z(\theta_{\tau-t-1}\omega)}\|v(\tau,\theta_{-t-1}\omega,v_0(\theta_{-t-1}\omega))\|^2d\tau\\
&\leq 1+ce^{p\varepsilon K(\omega)}M_{\varepsilon}(\omega) .
\end{aligned} \label{4.64}
\end{equation}
Let $T(B,\omega)=max\{T_0(B,\omega),T_1(B,\omega)\}$, it follows
from
\eqref{3.11}, \eqref{3.12}, \eqref{4.41}, \eqref{4.57}-\eqref{4.58}, \eqref{4.64}
that for all $t\geq T(B,\omega)$,
\begin{equation}
\begin{aligned}
&\|\nabla u(t+1,\theta_{-t-1},u_0(\theta_{-t-1}\omega))\|_p^p\\
&= e^{p\varepsilon z(\omega)}\|\nabla
v(t+1,\theta_{-t-1},v_0(\theta_{-t-1}\omega))\|_p^p\\
&\leq e^{p\varepsilon z(\omega)}[ce^{p\varepsilon
K(\omega)}(1+e^{2p\varepsilon
K(\omega)}M_{\varepsilon}(\omega))+ce^{p\varepsilon K(\omega)}\\
&\quad +ce^{c\varepsilon K(\omega)}(\varepsilon
 K(\omega)+1)(1+e^{c\varepsilon
 K(\omega)}M_{\varepsilon}(\omega))
+ c(K^2(\omega)+1)(1+e^{p\varepsilon
 K(\omega)}M_{\varepsilon}(\omega))]\\
&\leq ce^{c\varepsilon K(\omega)}(1+\varepsilon
K(\omega))(1+M_{\varepsilon}(\omega)) .
\end{aligned} \label{4.65}
\end{equation}
\end{proof}

\begin{lemma}\label{estimate6}
Let $0<\varepsilon\leq 1, g\in L^2(\mathbb{R}^n)$ and \eqref{3.3}-\eqref{3.5}
hold. Then for every
$\eta>0,B=\{B(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$ and
P-a.e. $\omega\in\Omega$, there is $T(B,\omega,\eta)>0$ and
$R(\omega,\eta)>0$, independent of $\varepsilon$, such that for all
$u_0(\theta_{-t}\omega)\in B(\theta_{-t}\omega)$ and
$t\geq T(B,\omega,\eta)$
\begin{equation}  \label{4.66}
\int_{|x|\geq R}|u(t,\theta_{-t}\omega,u_0(\theta_{-t}\omega))|^2dx\leq\eta .
\end{equation}
\end{lemma}

\begin{proof}
Choose a smooth function $\xi$, such that $0\leq\xi(s)\leq 1$ for
$s\in \mathbb{R}^+$, and
$$
\xi(s)= \begin{cases}
  0, & 0\leq s\leq 1,\\
  1, & s\geq 2.
       \end{cases}
$$
Then, there exists a constant $M$, such that $|\xi'(s)|\leq M$ for
$s\in \mathbb{R}^+$. Taking the inner product of \eqref{3.13} with
$\xi(\frac{|x|^2}{r^2})v$ in $L^2(\mathbb{R}^n)$, we obtain
\begin{equation}
\begin{aligned}
& \frac{d}{dt}\int_{\mathbb{R}^n} \xi(\frac{|x|^2}{r^2})|v|^2dx\\
&=2e^{\varepsilon(p-2)z(\theta_t\omega)}\int_{\mathbb{R}^n}(\operatorname{div}|\nabla
v|^{p-2}\nabla v)\xi(\frac{|x|^2}{r^2})vdx+2(\varepsilon
z(\theta_t\omega)-\lambda)\int_{\mathbb{R}^n}\xi(\frac{|x|^2}{r^2})|v|^2dx\\
&\quad +2e^{-\varepsilon z(\theta_t\omega)}\int_{\mathbb{R}^n}
f(x,u)\xi(\frac{|x|^2}{r^2})vdx+2e^{-\varepsilon
z(\theta_t\omega)}\int_{\mathbb{R}^n}
g(x)\xi(\frac{|x|^2}{r^2})vdx .
\end{aligned} \label{4.68}
\end{equation}
We now estimate the terms in \eqref{4.68} as follows. First we have
\begin{equation}
\begin{aligned}
& 2e^{\varepsilon(p-2)z(\theta_t\omega)}
 \int_{\mathbb{R}^n}(\operatorname{div}|\nabla
v|^{p-2}\nabla v)\xi(\frac{|x|^2}{r^2})vdx\\
 &= -2e^{\varepsilon(p-2)z(\theta_t\omega)}\int_{\mathbb{R}^n}\sum_{i=1}^n|\nabla
v|^{p-2}\frac{\partial v}{\partial x_i}
\xi(\frac{|x|^2}{r^2})\frac{\partial v}{\partial
x_i}dx\\
&\quad -2e^{\varepsilon(p-2)z(\theta_t\omega)}\int_{\mathbb{R}^n}\sum_{i=1}^n|\nabla
v|^{p-2}\frac{\partial v}{\partial x_i}
\xi'(\frac{|x|^2}{r^2})\frac{2x_i}{r^2} vdx\\
&= -2e^{\varepsilon(p-2)z(\theta_t\omega)}\int_{\mathbb{R}^n}|\nabla v|^{p}
\xi(\frac{|x|^2}{r^2})dx\\
&\quad -2e^{\varepsilon(p-2)z(\theta_t\omega)}\int_{r\leq |x|\leq
\sqrt{2}r}v|\nabla v|^{p-2}
\xi'(\frac{|x|^2}{r^2})\frac{2}{r^2}x\cdot\nabla vdx\\
&\leq -2e^{\varepsilon(p-2)z(\theta_t\omega)}\int_{\mathbb{R}^n}|\nabla
v|^{p}
\xi(\frac{|x|^2}{r^2})dx\\
&\quad +\frac{4\sqrt{2}M}{r}e^{\varepsilon(p-2)z(\theta_t\omega)}\int_{r\leq
|x|\leq \sqrt{2}r}|v||\nabla v|^{p-1} dx\\
&\leq -2e^{\varepsilon(p-2)z(\theta_t\omega)}\int_{\mathbb{R}^n}|\nabla
v|^{p}
\xi(\frac{|x|^2}{r^2})dx+\frac{4\sqrt{2}M}{r}e^{\varepsilon(p-2)z(\theta_t\omega)}(\|v\|^p_p+\|\nabla
v\|^p_p)  .
\end{aligned} \label{4.69}
\end{equation}
For the third term of \eqref{4.68}, using \eqref{3.3}, we
have
\begin{equation}
\begin{aligned}
&2e^{-\varepsilon z(\theta_t\omega)}\int_{\mathbb{R}^n}
f(x,u)\xi(\frac{|x|^2}{r^2})vdx\\
&= 2e^{-2\varepsilon z(\theta_t\omega)}\int_{\mathbb{R}^n}
f(x,u)u\xi(\frac{|x|^2}{r^2})dx\\
&\leq  -2e^{-2\varepsilon z(\theta_t\omega)}\alpha_1\int_{\mathbb{R}^n}
|u|^p\xi(\frac{|x|^2}{r^2})dx+2e^{-2\varepsilon
z(\theta_t\omega)}\int_{\mathbb{R}^n}
|\psi_1(x)|\xi(\frac{|x|^2}{r^2})dx  .
\end{aligned} \label{4.70}
\end{equation}
For the last term of \eqref{4.68}, we have
\begin{equation}
\begin{aligned}
&2e^{-\varepsilon z(\theta_t\omega)}\int_{\mathbb{R}^n}
g(x)\xi(\frac{|x|^2}{r^2})vdx\\
& \leq \lambda \int_{\mathbb{R}^n}
\xi(\frac{|x|^2}{r^2})|v|^2dx+\frac{1}{\lambda} e^{-2\varepsilon
z(\theta_t\omega)}\int_{\mathbb{R}^n}
g^2(x)\xi(\frac{|x|^2}{r^2})dx .
\end{aligned} \label{4.71}
\end{equation}
From \eqref{4.68}-\eqref{4.71}, it follows that
\begin{equation}
\begin{aligned}
& \frac{d}{dt}\int_{\mathbb{R}^n}
\xi(\frac{|x|^2}{r^2})|v|^2dx\\
&\leq (2\varepsilon z(\theta_t\omega)-\lambda)\int_{\mathbb{R}^n}
\xi(\frac{|x|^2}{r^2})|v|^2dx+\frac{c}{r}e^{\varepsilon(p-2)z(\theta_t\omega)}(\|\nabla
v\|^p_p+\|v\|_p^p)\\
&\quad +2e^{-2\varepsilon z(\theta_t\omega)}\int_{\mathbb{R}^n}
|\psi_1(x)|\xi(\frac{|x|^2}{r^2})dx+\frac{1}{\lambda}
e^{-2\varepsilon z(\theta_t\omega)}\int_{\mathbb{R}^n}
g^2(x)\xi(\frac{|x|^2}{r^2})dx  .
\end{aligned} \label{4.72}
\end{equation}
By the Gronwall Lemma, for any $t\geq 0$,
\begin{equation}
\begin{aligned}
& \int_{\mathbb{R}^n}
\xi(\frac{|x|^2}{r^2})|v(t,\omega,v_0(\omega))|^2dx\\
&\leq
e^{2\varepsilon\int_0^tz(\theta_\tau\omega)d\tau-\lambda t}
\int_{\mathbb{R}^n} \xi(\frac{|x|^2}{r^2})|v_0(\omega)|^2dx\\
&\quad
+\frac{c}{r}\int_0^te^{\varepsilon(p-2)z(\theta_s\omega)+2\varepsilon\int_{s}^tz(\theta_\tau\omega)d\tau-\lambda(t-s)}(\|\nabla
v\|^p_p+\|v\|_p^p)ds\\
&\quad +2\int_0^te^{-2\varepsilon
z(\theta_s\omega)+2\varepsilon\int_{s}^tz(\theta_\tau\omega)d\tau-\lambda(t-s)}
\int_{\mathbb{R}^n} \xi(\frac{|x|^2}{r^2})|\psi_1(x)|\,dx\,ds\\
&\quad +\frac{1}{\lambda}\int_0^te^{-2\varepsilon
z(\theta_s\omega)+2\varepsilon\int_{s}^tz(\theta_\tau\omega)d\tau-\lambda(t-s)}
\int_{\mathbb{R}^n} \xi(\frac{|x|^2}{r^2})g^2(x)\,dx\,ds .
\end{aligned} \label{4.73}
\end{equation}
By replacing $\omega$ by $\theta_{-t}\omega$ , it
follows that
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}^n}
\xi(\frac{|x|^2}{r^2})|v(t,\theta_{-t}\omega,v_0(\theta_{-t}\omega))|^2dx\\
&\leq e^{2\varepsilon\int_0^tz(\theta_{\tau-t}\omega)d\tau-\lambda t}
\int_{\mathbb{R}^n} \xi(\frac{|x|^2}{r^2})|v_0(\theta_{-t}\omega)|^2dx\\
&\quad +\frac{c}{r}\int_0^te^{\varepsilon(p-2)z(\theta_{s-t}\omega)
+2\varepsilon\int_{s}^tz(\theta_{\tau-t}\omega)d\tau-\lambda(t-s)}(\|\nabla
v(s,\theta_{-t}\omega,v_0(\theta_{-t}\omega))\|^p_p\\
&\quad +\|v(s,\theta_{-t}\omega,v_0(\theta_{-t}\omega))\|_p^p)ds\\
&\quad +\int_0^te^{-2\varepsilon
z(\theta_{s-t}\omega)+2\varepsilon\int_{s}^tz(\theta_{\tau-t}\omega)d\tau
-\lambda(t-s)}
\int_{\mathbb{R}^n}
\xi(\frac{|x|^2}{r^2})(2|\psi_1(x)|+\frac{1}{\lambda}g^2(x))\,dx\,ds.
\end{aligned}  \label{4.74}
\end{equation}
We then estimate each term on the right-hand side of \eqref{4.74}.
Firstly, Since $B\in {\mathcal{D}}$ and $u_0(\theta_{-t}\omega)\in
B(\theta_{-t}\omega)$, it follows from \eqref{3.8}, \eqref{3.9} that
there exists $T_1(B,\omega,\eta)>0$, independent of $\varepsilon$,
such that for all $t\geq T_1(B,\omega,\eta)$
\begin{equation}
\begin{aligned}
&e^{2\varepsilon\int_0^tz(\theta_{\tau-t}\omega)d\tau-\lambda t}
\int_{\mathbb{R}^n}
\xi(\frac{|x|^2}{r^2})|v_0(\theta_{-t}\omega)|^2dx\\
&\leq e^{2\varepsilon\int_0^tz(\theta_{\tau-t}\omega)d\tau-\lambda
t}\|v_0(\theta_{-t}\omega)\|^2\\
&= e^{2\varepsilon\int_{-t}^0z(\theta_{\tau}\omega)d\tau-\lambda
t-2\varepsilon z(\theta_{-t}\omega)}\|u_0(\theta_{-t}\omega)\|^2\\
&\leq e^{-\frac{\lambda}{2}t}\|u_0(\theta_{-t}\omega)\|^2
<\frac{\eta}{3} .
\end{aligned} \label{4.75}
\end{equation}
To estimate the second term on the right-hand side of \eqref{4.74},
from \eqref{4.18} with $T$ replaced by $0$, we have
\begin{align*}
&\int_0^te^{-2\varepsilon
z(\theta_{s-t}\omega)+2\varepsilon\int_s^tz(\theta_{\tau-t}\omega)d\tau+\lambda(s-t)}\|
u(s,\theta_{-t}\omega,u_0(\theta_{-t}\omega))\|_p^p\\
&+\int_0^te^{\varepsilon(p-2)
z(\theta_{s-t}\omega)+2\varepsilon\int_s^tz(\theta_{\tau-t}\omega)d\tau+\lambda(s-t)}\|\nabla
v(s,\theta_{-t}\omega,v_0(\theta_{-t}\omega))\|_p^pds
\\
&=\int_0^te^{\varepsilon(p-2)
z(\theta_{s-t}\omega)+2\varepsilon\int_s^tz(\theta_{\tau-t}\omega)d\tau+\lambda(s-t)}(\|
v(s,\theta_{-t}\omega,v_0(\theta_{-t}\omega))\|_p^p\\&\quad +\|\nabla
v(s,\theta_{-t}\omega,v_0(\theta_{-t}\omega))\|_p^p)ds \\
&\leq c\|v_0(\theta_{-t}\omega)\|^2e^{2\varepsilon
 \int_{-t}^0z(\theta_{s}\omega)ds-\lambda t}
+c\int_{-t}^0e^{-2\varepsilon
z(\theta_{s}\omega)+2\varepsilon\int_s^0z(\theta_{\tau}\omega)d\tau+\lambda s}ds  .
\end{align*}
Thus
\begin{equation}
\begin{aligned}
&\frac{c}{r}\int_0^te^{\varepsilon(p-2)z(\theta_{s-t}\omega)
+2\varepsilon\int_{s}^tz(\theta_{\tau-t}\omega)d\tau-\lambda(t-s)}(\|\nabla
v(s,\theta_{-t}\omega,v_0(\theta_{-t}\omega))\|^p_p\\
& +\|v(s,\theta_{-t}\omega,v_0(\theta_{-t}\omega))\|_p^p)ds\\
&\leq \frac{c}{r}[\|u_0(\theta_{-t}\omega)\|^2e^{-2\varepsilon
z(\theta_{-t}\omega)+2\varepsilon\int_{-t}^0z(\theta_{\tau}\omega)d\tau-\lambda
t}+M_{\varepsilon}(\omega)].
\end{aligned} \label{4.77}
\end{equation}
Since $B\in\mathcal{D},u_0(\theta_{-t}\omega)\in
B(\theta_{-t}\omega)$, together with \eqref{3.8}-\eqref{3.10}, it
follows that there exists $T_2(B,\omega,\eta)>0$ and
$R_1(\eta,\omega)>0$, both independent of $\varepsilon$, such that
for $t\geq T_2(B,\omega,\eta)$ and $r\geq R_1(\eta,\omega)$,
\begin{equation}
\begin{aligned}
&\frac{c}{r}\int_0^te^{\varepsilon(p-2)z(\theta_{s-t}\omega)
+2\varepsilon\int_{s}^tz(\theta_{\tau-t}\omega)d\tau-\lambda(t-s)}\\
&\times (\|\nabla v(s,\theta_{-t}\omega,v_0(\theta_{-t}\omega))\|^p_p
+\|v(s,\theta_{-t}\omega,v_0(\theta_{-t}\omega))\|_p^p)ds
< {\frac{\eta}{3}} .
\end{aligned}\label{4.78}
\end{equation}
For the last term of \eqref{4.74}, since $g\in L^2(\mathbb{R}^n)$ and
$\psi_1\in L^1(\mathbb{R}^n)$, there exists $R_2(\omega,\eta)>0$, independent
of $\varepsilon$, such that
\begin{equation}
\begin{aligned}
&  \int_0^te^{-2\varepsilon
z(\theta_{s-t}\omega)+2\varepsilon\int_{s}^tz(\theta_{\tau-t}\omega)
d\tau-\lambda(t-s)} \int_{\mathbb{R}^n}
\xi(\frac{|x|^2}{r^2})(2|\psi_1(x)|+\frac{1}{\lambda}g^2(x))\,dx\,ds\\
&\leq \int_0^te^{-2\varepsilon
z(\theta_{s-t}\omega)+2\varepsilon\int_{s}^tz(\theta_{\tau-t}\omega)d\tau-\lambda(t-s)}ds
(2\int_{|x|\geq r}
|\psi_1(x)|dx+\frac{1}{\lambda}\int_{|x|\geq r}g^2(x)dx)\\
&\leq  M_\varepsilon(\omega)(2\int_{|x|\geq r}
|\psi_1(x)|dx+\frac{1}{\lambda}\int_{|x|\geq r}g^2(x))dx
<\frac{\eta}{3}.
\end{aligned} \label{4.79}
\end{equation}
Let
\[
T(B,\omega,\eta)=\max\{T_1,T_2\},\quad
R(\omega,\eta)=\max\{R_1,R_2\} .
\]
Note that $T(B,\omega,\eta),R(\omega,\eta)$ are both independent of
$\varepsilon$. And for $t\geq T(B,\omega,\eta)$ and $r\geq
R(\omega,\eta)$, it follows from
\eqref{4.74}, \eqref{4.75}, \eqref{4.78} and \eqref{4.79} that
\begin{equation} \label{4.82}
\int_{|x|>r}|v(t,\theta_{-t\omega},v_0(\theta_{-t}\omega))|^2dx
=\int_{\mathbb{R}^n} \xi(\frac{|x|^2}{r^2})|v(t,\theta_{-t}
\omega,v_0(\theta_{-t}\omega))|^2dx< \eta ,
\end{equation}
which along with \eqref{3.12} implies the Lemma.
\end{proof}

\begin{lemma}\label{estimate7}
Let $0<\varepsilon\leq 1$, $g\in L^2(\mathbb{R}^n)$ and \eqref{3.3}-\eqref{3.5}
hold. Then the random dynamical system generated by \eqref{3.1} is
$\mathcal{D}$-pullback asymptotically compact in $L^2(\mathbb{R}^n)$, that
is, for P-a.e. $\omega\in\Omega$, the sequence
$\{\Phi(t_n,\theta_{-t_n}\omega,u_{0,n}(\theta_{-t_n}\omega))\}$ has
a convergent subsequence in $L^2(\mathbb{R}^n)$ provided
$t_n\to\infty, \{B(\omega)\}\in\mathcal{D}$ and
$u_{0,n}(\theta_{-t_n}\omega)\in B(\theta_{-t_n}\omega)$.
\end{lemma}

\begin{proof}
Let $t_n\to\infty,\{B(\omega)\}\in\mathcal{D}$ and
$u_{0,n}(\theta_{-t_n}\omega)\in B(\theta_{-t_n}\omega)$. Then by
Lemma \ref{estimate1}, for P-a.e. $\omega\in\Omega$, we have that
\begin{equation} \label{4.83}
\{\Phi(t_n,\theta_{-t_n}\omega,u_{0,n}(\theta_{-t_n}\omega))\}^{\infty}_{n=1}\
\text{ is bounded in } L^2(\mathbb{R}^n).
\end{equation}
Hence, there is $\zeta\in L^2(\mathbb{R}^n)$ such that, up to a subsequence
which is still denoted by
$\{\Phi(t_n,\theta_{-t_n}\omega,u_{0,n}(\theta_{-t_n}\omega))\}$,
such that
\begin{equation} \label{4.84}
\Phi(t_n,\theta_{-t_n}\omega,u_{0,n}(\theta_{-t_n}\omega))\to
\zeta\quad \text{weakly in }\ L^2(\mathbb{R}^n).
\end{equation}
Next, we prove the weak convergence of \eqref{4.84} is actually
strong convergence.

Given $\eta>0$, by Lemma \ref{estimate6}, there is
$T_1(B,\omega,\eta)>0$ and $R_1(\eta,\omega)>0$, independent of
$\varepsilon$, such that for all $t\geq T_1$
\begin{equation} \label{4.85}
 \int_{|x|\geq R_1}|\Phi(t,\theta_{-t}\omega,u_{0}(\theta_{-t}\omega))|^2\leq\eta.
\end{equation}
Since $t_n\to\infty$, there is $N_1$ such that $t_n\geq T_1$
for every $n\geq N_1$. Hence, it follows from \eqref{4.85} that for
all $n\geq N_1$
\begin{equation} \label{4.86}
\int_{|x|\geq R_1}|\Phi(t_n,\theta_{-t_n}\omega,u_{0,n}(\theta_{-t_n}\omega))
|^2\leq\eta.
\end{equation}
Note that $\zeta\in L^2(\mathbb{R}^n)$, therefore there exists $R_2(\eta)>0$
such that
\begin{equation} \label{4.87}
\int_{|x|\geq R_2}|\zeta(x)|^2\leq\eta.
\end{equation}
Let $R_3=\max \{R_1,R_2\}$, denote $Q_{R_3}=\{x\in \mathbb{R}^n, |x|\leq
R_3\}$, from Lemma \ref{estimate5}, there is $T_2=T_2(B,\omega)>0$,
such that for all $t\geq T_2$,
\begin{equation} \label{4.88}
\|\Phi(t,\theta_{-t}\omega,u_{0}(\theta_{-t}\omega))\|^p_{W^{1,p}
(Q_{R_3})}
\leq ce^{c\varepsilon K(\omega)}(1+\varepsilon
K(\omega))(1+M_\varepsilon(\omega)) .
\end{equation}
Then, by the compactness of the embedding
$W^{1,p}(Q_{R_3})\hookrightarrow L^2(Q_{R_3})$,  from
\eqref{4.88}  it follows that, up to a subsequence,
\begin{equation} \label{4.89}
\Phi(t_n,\theta_{-t_n}\omega,u_{0,n}(\theta_{-t_n}\omega))\to
\zeta\quad \text{strongly in } L^2(Q_{R_3}),
\end{equation}
 which implies that
for the given $\eta>0$, there exists $N_2>0$ such that for all
$n\geq N_2$,
\begin{equation} \label{4.90}
\|\Phi(t_n,\theta_{-t_n}\omega,u_{0,n}(\theta_{-t_n}\omega))
-\zeta\|^2_{L^2(Q_{R_3})}\leq\eta.
\end{equation}
Let $N=\max \{N_1,N_2\}$. By
\eqref{4.86} \eqref{4.87} \eqref{4.90}, we find that for all $n\geq
N$,
\begin{equation}
\begin{aligned}
&\|\Phi(t_n,\theta_{-t_n}\omega,u_{0,n}(\theta_{-t_n}\omega))
 -\zeta\|^2_{L^2(\mathbb{R}^n)}\\
&= \int_{|x|\leq
R_3}|\Phi(t_n,\theta_{-t_n}\omega,u_{0,n}(\theta_{-t_n}\omega))-\zeta|^2dx\\
&\quad +\int_{|x|\geq R_3}
|\Phi(t_n,\theta_{-t_n}\omega,u_{0,n}(\theta_{-t_n}\omega))-\zeta|^2dx\\
&\leq  \eta+2\int_{|x|\geq R_3}
|\Phi(t_n,\theta_{-t_n}\omega,u_{0,n}(\theta_{-t_n}\omega))|^2dx+2\int_{|x|\geq
R_3} |\zeta(x)|^2dx\\
&\leq  5\eta ,
\end{aligned}\label{4.91}
\end{equation}
which shows that
\begin{equation} \label{4.92}
\Phi(t_n,\theta_{-t_n}\omega,u_{0,n}(\theta_{-t_n}\omega))\to
\zeta\quad \text{strongly in } L^2(\mathbb{R}^n)
\end{equation}
 as desired.
\end{proof}

It is now sufficient to show the existence of a $\mathcal{D}$-random
attractor for $\Phi$ in $L^2(\mathbb{R}^n)$.


\begin{theorem}\label{existence}
Let $0<\varepsilon\leq 1, g\in L^2(\mathbb{R}^n)$ and \eqref{3.3}-\eqref{3.5}
hold. Then the random dynamical system $\Phi$ generated by
\eqref{3.1} has a unique $\mathcal{D}$-random attractor in
$L^2(\mathbb{R}^n)$.
\end{theorem}

\begin{proof}
Notice that $\Phi$ has a closed random absorbing set
$\{K(\omega)\}_{\omega\in\Omega}$ in $\mathcal{D}$ by Lemma
\ref{estimate1}, and is $\mathcal{D}$-pullback asymptotically
compact in $L^2(\mathbb{R}^n)$ by Lemma \ref{estimate7}. Hence the existence
of a unique $\mathcal{D}$-random attractor for $\Phi$ follows from
Proposition \ref{existence prop} immediately.
\end{proof}


\section{Upper semicontinuity of random attractors for p-Laplacian
equation on $\mathbb{R}^n$}\label{5}

In this section, we prove the upper semicontinuity of random
attractors for the p-Laplacian equation defined on $\mathbb{R}^n$ when the
stochastic perturbations approach zero. We may further assume that
the nonlinear function $f$ satisfies, for all $x\in \mathbb{R}^n$
 and $s\in \mathbb{R}$
\begin{gather}  \label{3.4}
\frac{\partial f}{\partial s}(x,s)\leq \beta , \\
 \label{5.0}
|\frac{\partial f}{\partial s}(x,s)|\leq \alpha_3|s|^{p-2}+\psi_3(x) ,
\end{gather}
where $\alpha_3,\beta$ are positive constants,
$\psi_3\in L^\infty(\mathbb{R}^n)$ if $p=2$ and
$\psi_3\in L^{\frac{p}{p-2}}(\mathbb{R}^n)$ if
$p>2$.

To indicate dependence of solutions on $\varepsilon$, in this
section, we write the solution of problem \eqref{3.1}-\eqref{3.2} as
$u^\varepsilon$, and the corresponding cocycle as
$\Phi_\varepsilon$. And we denote the solution and the semigroup of
the deterministic equation
\begin{equation} \label{5.1}
\frac{du}{dt}-\operatorname{div}(|\nabla u|^{p-2}\nabla u)+\lambda
u=f(x,u)+g(x),\quad  x\in \mathbb{R}^n,\; t>0
\end{equation}
by $u$ and $\Phi_0$ respectively.

Note that the existence of the global attractor $\mathcal{A}_0$ in
$L^2(\mathbb{R}^n)$ for the deterministic system \eqref{5.1} can similarly be
achieved by the discussion in section 4.

\begin{lemma} \label{precompact}
Let $0<\varepsilon\leq 1, g\in L^2(\mathbb{R}^n)$ and \eqref{3.3}-\eqref{3.5}
hold. Then the union $\cup_{0<\varepsilon \leq 1}
{\mathcal{A}}_\varepsilon (\omega)$ is precompact in $L^2(\mathbb{R}^n)$.
\end{lemma}

\begin{proof}
Given $\eta>0$, we want to show that the set
$\cup_{0<\varepsilon \leq 1} {\mathcal{A}}_\varepsilon
(\omega)$ has a finite covering of balls of radii less than $\eta$.
From Lemma \ref{estimate1}
\begin{equation} \label{5.2}
D_\varepsilon(\omega)=\{u\in L^2(\mathbb{R}^n):\|u\|^2\leq 1+ce^{2\varepsilon
z(\omega)}M_\varepsilon(\omega)\}
\end{equation}
is a closed and tempered random absorbing set for $\Phi_\varepsilon$
in $L^2(\mathbb{R}^n)$.
Let
\begin{align*} %\label{5.3}
&D(\omega)\\
&=\big\{&u\in L^2(\mathbb{R}^n):
\|u\|^2\leq 1+ce^{2|z(\omega)|} 
 \int_{-\infty}^0e^{2|z(\theta_s
\omega)|+2|\int_s^0z(\theta_\tau \omega)d\tau|+\lambda
s}ds<+\infty\big\} .
\end{align*}
It is easy to show that $D(\omega)$ is also a tempered set, that is,
$\{D(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$.

By Lemma \ref{estimate6}, we find that, given $\eta>0$ and
P-a.e.$\omega\in\Omega$, there is $T(B,\omega,\eta)>0$ and
$R(\omega,\eta)>0$, independent of $\varepsilon$, such that for all
$u_0(\theta_{-t}(\omega))\in D(\theta_{-t}(\omega))$ and $t\geq
T(B,\omega,\eta)$
\begin{equation}  \label{5.4}
\int_{|x|\geq
R}|u(t,\theta_{-t}\omega,u_0(\theta_{-t}\omega))|^2dx\leq\frac{\eta^2}{16} .
\end{equation}
For $\forall u_0(\theta_{-t}(\omega))\in
\mathcal{A}_\varepsilon(\theta_{-t}(\omega))$, note that
\begin{equation} \label{5.5}
 \cup_{0<\varepsilon \leq 1}
{\mathcal{A}}_\varepsilon (\omega)\subset
\cup_{0<\varepsilon \leq 1} D_\varepsilon (\omega)\subset
D(\omega) ,
\end{equation}
which implies that  $u_0(\theta_{-t}(\omega))\in
D(\theta_{-t}\omega)$. It follows that for every $0<\varepsilon\leq
1$, P-a.e. $\omega\in\Omega, t\geq T$ and for all
$u_0(\theta_{-t}(\omega))\in
\mathcal{A}_\varepsilon(\theta_{-t}(\omega))$
\begin{equation} \label{5.6}
\int_{|x|\geq R}|u(t,\theta_{-t}\omega,u_0(\theta_{-t}\omega))|^2dx
\leq\frac{\eta^2}{16} ,
\end{equation}
which along with the invariance of
$\{\mathcal{A}_\varepsilon(\omega)\}_{\omega\in\Omega}$ shows that
for P-a.e. $\omega\in\Omega$
\begin{equation} \label{5.7}
\int_{|x|\geq R}|\widetilde{u}(x)|^2dx\leq\frac{\eta^2}{16}, \quad
\forall\ \widetilde{u}\in \cup_{0<\varepsilon \leq 1} {\mathcal{A}}_\varepsilon
(\omega).
\end{equation}
Let
\begin{equation} \label{5.8}
Q_R=\{x\in \mathbb{R}^n: |x|<R\},\quad  Q_R ^c=\mathbb{R}^n- Q_R.
\end{equation}
Then \eqref{5.7} implies that for P-a.e. $\omega\in\Omega$,
\begin{equation} \label{5.9}
\|\widetilde{u}(x)\|_{L^2(Q_R^c)}\leq\frac{\eta}{4},\quad
\forall\ \widetilde{u}\in \cup_{0<\varepsilon
\leq 1} {\mathcal{A}}_\varepsilon (\omega).
\end{equation}
On the other hand, by Lemma \ref{estimate5}, we find that, for
every $0<\varepsilon\leq 1$ and P-a.e. $\omega\in\Omega$, there
exists $T(D,\omega)>0$, independent of $\varepsilon$, such that for
all $t\geq T(D,\omega)$,
\begin{equation} \label{5.10}
\|\Phi_\varepsilon(t,\theta_{-t}\omega,D(\theta_{-t}\omega))\|_{W^{1,p}
(Q_{R_3})}^p
\leq ce^{c\varepsilon K(\omega)}(1+\varepsilon
K(\omega))(1+M_\varepsilon(\omega)) ,
\end{equation}
which together with \eqref{5.5} implies that the set
$\cup_{0<\varepsilon \leq 1} {\mathcal{A}}_\varepsilon
(\omega)$ is bounded in $W^{1,p}(Q_R)$ for P-a.e. $\omega\in\Omega$.
By the compactness of embedding $W^{1,p}(Q_R)\hookrightarrow
L^2(Q_R)$, we find that, for the given $\eta$, the set
$\cup_{0<\varepsilon \leq 1} {\mathcal{A}}_\varepsilon
(\omega)$ has a finite covering of balls of radii less than
$\frac{\eta}{4}$ in $L^2(Q_R)$. This along with \eqref{5.9} shows
that $\cup_{0<\varepsilon \leq 1}
{\mathcal{A}}_\varepsilon (\omega)$ has a finite covering of balls
of radii less than $\eta$ in $L^2(\mathbb{R}^n)$.
\end{proof}


\begin{lemma} \label{continuity}
Let $g\in L^2(\mathbb{R}^n)$, \eqref{3.3}-\eqref{3.5} and 
\eqref{3.4}-\eqref{5.0} hold. Given $0<\varepsilon\leq 1$, 
let $u^\varepsilon$ and $u$ be the solutions of equation \eqref{3.1} 
and \eqref{5.1}
with initial conditions $u_0^\varepsilon$ and $u_0$, respectively.
Then for P-a.e. $\omega\in\Omega,u_0^\varepsilon\to u_0\
(\varepsilon\downarrow 0)$ and $t\geq 0$, we have
\begin{equation}
\lim_{\varepsilon\downarrow
0}u^\varepsilon(t,\omega,u_0^\varepsilon)=u(t,\omega,u_0) .
\end{equation}
\end{lemma}

\begin{proof}
Let
\[ 
v^\varepsilon(t,\omega,v_0^\varepsilon)=e^{-\varepsilon
z(\theta_t\omega)}u^\varepsilon(t,\omega,u_0^\varepsilon) ,
\quad
W=v^\varepsilon(t,\omega,v_0^\varepsilon)-u(t,\omega,u_0) .
\]
Then together with \eqref{3.13} and \eqref{5.1}, we obtain
\begin{equation}
\begin{aligned}
& \frac{\partial W}{\partial
t}+e^{\varepsilon(p-2)z(\theta_t\omega)}Av^\varepsilon-Au+\lambda
W\\
&= e^{-\varepsilon z(\theta_t\omega)}f(x,u^\varepsilon)-f(x,u)
+(e^{-\varepsilon z(\theta_t\omega)}-1)g(x)+\varepsilon v^\varepsilon
z(\theta_t\omega) ,
\end{aligned} \label{5.12}
\end{equation}
where $A\zeta=-\operatorname{div}(|\nabla \zeta|^{p-2}\nabla \zeta)$.

Taking the inner product of \eqref{5.12} with $W$ in $L^2(\mathbb{R}^n)$, we
have
\begin{equation}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|W\|^2 \\
&=-(e^{\varepsilon(p-2)z(\theta_t\omega)}Av^\varepsilon-Au,W)
 -\lambda\|W\|^2+(e^{-\varepsilon z(\theta_t\omega)}f(x,u^\varepsilon)-f(x,u),W)
\\
&\quad +((e^{-\varepsilon z(\theta_t\omega)}-1)g(x),W)+\varepsilon
z(\theta_t\omega)(v^\varepsilon,W) .
\end{aligned} \label{5.13}
\end{equation}
By the property of p-Laplacian operator for $p\geq 2$ and the Young
inequality, we obtain
\begin{equation}
\begin{aligned}
& -(e^{\varepsilon(p-2)z(\theta_t\omega)}Av^\varepsilon-Au,W)\\
&= -e^{\varepsilon(p-2)z(\theta_t\omega)}(Av^\varepsilon-Au,v^\varepsilon-u)
-(e^{\varepsilon(p-2)z(\theta_t\omega)}-1)(Au,W) \\
&\leq  |e^{\varepsilon(p-2)z(\theta_t\omega)}-1||(Au,v^\varepsilon-u)| \\
&\leq  c|e^{\varepsilon(p-2)z(\theta_t\omega)}-1|(\|\nabla u\|_p^p
+\|\nabla v^\varepsilon\|_p^p) .
\end{aligned} \label{5.14}
\end{equation}
For the nonlinear term on the right-hand side of \eqref{5.13}, by
\eqref{3.5}, \eqref{3.4} and \eqref{5.0},
\begin{align*}
&(e^{-\varepsilon
z(\theta_t\omega)}f(x,u^\varepsilon)-f(x,u),W)\\
&= e^{-\varepsilon
z(\theta_t\omega)}\int_{\mathbb{R}^n}(f(x,u^\varepsilon)-f(x,u))Wdx+(e^{-\varepsilon
z(\theta_t\omega)}-1)\int_{\mathbb{R}^n}f(x,u)Wdx \\
&=   e^{-\varepsilon z(\theta_t\omega)}\int_{\mathbb{R}^n}\frac{\partial
f}{\partial s}(x,s)(u^\varepsilon-u)Wdx+(e^{-\varepsilon
z(\theta_t\omega)}-1)\int_{\mathbb{R}^n}f(x,u)Wdx \\
&= e^{-\varepsilon z(\theta_t\omega)}\int_{\mathbb{R}^n}\frac{\partial
f}{\partial s}(x,s)[e^{\varepsilon
z(\theta_t\omega)}v^\varepsilon-e^{\varepsilon
z(\theta_t\omega)}u+(e^{\varepsilon
z(\theta_t\omega)}-1)u]Wdx\\&\quad +(e^{-\varepsilon
z(\theta_t\omega)}-1)\int_{\mathbb{R}^n}f(x,u)Wdx \\
&=  \int_{\mathbb{R}^n}\frac{\partial f}{\partial
 s}(x,s)W^2dx+(1-e^{-\varepsilon
 z(\theta_t\omega)})\int_{\mathbb{R}^n}\frac{\partial f}{\partial
 s}(x,s)uWdx\\
&\quad +(e^{-\varepsilon
 z(\theta_t\omega)}-1)\int_{\mathbb{R}^n}f(x,u)Wdx \\
&\leq  \beta \|W\|^2+|1-e^{-\varepsilon
 z(\theta_t\omega)}|\int_{\mathbb{R}^n}[\alpha_3(|u^\varepsilon|+|u|)^{p-2}|u||W|
 +|\psi_3(x)||u||W|\\
&\quad +|f(x,u)||W|]dx\\
&\leq  \beta \|W\|^2+c|1-e^{-\varepsilon
 z(\theta_t\omega)}|\int_{\mathbb{R}^n}[|u^\varepsilon|^{p-2}|u||W|+|u|^{p-1}|W|+|\psi_3(x)||u||W|\\&\quad +|u|^{p-1}|W|+|\psi_2(x)||W|]dx\\
&\leq \beta \|W\|^2+c|1-e^{-\varepsilon
 z(\theta_t\omega)}|(\|u^\varepsilon\|^p_p+\|u\|_p^p+\|W\|_p^p+\|\psi_3\|_{\frac{p}{p-2}}^{\frac{p}{p-2}}+\|\psi_2\|^2
 +\|W\|^2)\\
&\leq \beta \|W\|^2+c|1-e^{-\varepsilon
 z(\theta_t\omega)}|(\|u^\varepsilon\|^p_p+\|u\|_p^p+\|W\|_p^p
 +\|W\|^2+1) %\label{5.15} .
\end{align*}
By the Young inequality, the last two terms on the right-hand side
of \eqref{5.13} is bounded by
\begin{equation}
\begin{aligned}
&|e^{-\varepsilon z(\theta_t\omega)}-1||(g(x),W)|+\varepsilon
|z(\theta_t\omega)||(W+u,W)|\\
&\leq  |e^{-\varepsilon
z(\theta_t\omega)}-1|(\|g\|^2+\|W\|^2)+c\varepsilon
|z(\theta_t\omega)|(\|W\|^2+\|u\|^2) .
\end{aligned} \label{5.16}
\end{equation}
It follows from \eqref{5.13}-\eqref{5.16} that
\begin{align*}
&\frac{d}{dt}\|W\|^2 \\
&\leq c|e^{\varepsilon(p-2)z(\theta_t\omega)}-1|(\|\nabla u\|_p^p
+\|\nabla v^\varepsilon\|_p^p)+c \|W\|^2\\
&\quad +c|1-e^{-\varepsilon
z(\theta_t\omega)}|(\|u^\varepsilon\|^p_p+\|u\|_p^p+\|W\|_p^p
+\|W\|^2+1)\\
&\quad + |e^{-\varepsilon
z(\theta_t\omega)}-1|(\|g\|^2+\|W\|^2)+c\varepsilon
|z(\theta_t\omega)|(\|W\|^2+\|u\|^2)\\
&\leq c|e^{\varepsilon(p-2)z(\theta_t\omega)}-1|(\|\nabla u\|_p^p+\|\nabla v^\varepsilon\|_p^p)+c
\|W\|^2\\
&\quad +c|1-e^{-\varepsilon
z(\theta_t\omega)}|(\|u^\varepsilon\|^p_p+e^{-p\varepsilon
z(\theta_t\omega)}\|u^\varepsilon\|^p_p+\|u\|_p^p+\|W\|^2+1)\\
&\quad +c\varepsilon
|z(\theta_t\omega)|(\|W\|^2+\|u\|^2)\\
&= c(1+|1-e^{-\varepsilon
 z(\theta_t\omega)}|+\varepsilon |z(\theta_t\omega)|)\|W\|^2
 +c|e^{\varepsilon(p-2)z(\theta_t\omega)}-1|(\|\nabla u\|_p^p\\
&\quad +\|\nabla v^\varepsilon\|_p^p)+c|1-e^{-\varepsilon
z(\theta_t\omega)}|[(e^{-p\varepsilon
z(\theta_t\omega)}+1)\|u^\varepsilon\|^p_p+\|u\|_p^p+1]\\
&\quad +c\varepsilon |z(\theta_t\omega)|\|u\|^2\\
&\leq c(e^{|z(\theta_t\omega)|}+1+|z(\theta_t\omega)|)\|W\|^2+c|e^{\varepsilon(p-2)z(\theta_t\omega)}-1|(\|\nabla u\|_p^p+\|\nabla v^\varepsilon\|_p^p)\\
&\quad +c|1-e^{-\varepsilon z(\theta_t\omega)}|[(e^{-p\varepsilon
z(\theta_t\omega)}+1)\|u^\varepsilon\|^p_p+\|u\|_p^p+1]+c\varepsilon
|z(\theta_t\omega)|\|u\|^2,  %\label{5.17}
\end{align*}
where we used $0<\varepsilon\leq 1$. Then by the Gronwall Lemma, we
have for $\forall$ fixed $t\geq 0$,
\begin{equation}
\begin{aligned}
\|W(t)\|^2
&\leq e^{c\int_0^te^{(|z(\theta_s\omega)|}+1+|z(\theta_s\omega)|)ds}\|W(0)\|^2
+c\int_0^t\Big\{|e^{\varepsilon(p-2)z(\theta_s\omega)}-1|(\|\nabla u\|_p^p\\
&\quad +\|\nabla v^\varepsilon\|_p^p)
 +|1-e^{-\varepsilon z(\theta_s\omega)}|[(e^{-p\varepsilon
z(\theta_s\omega)}+1)\|u^\varepsilon(s)\|^p_p+\|u(s)\|_p^p+1]\\
&\quad +c\varepsilon |z(\theta_s\omega)|\|u(s)\|^2
\Big\}e^{c\int_s^te^{(|z(\theta_\tau\omega)|}+1+|z(\theta_\tau\omega)|)d\tau}ds\\
&\leq e^{c\int_0^te^{(|z(\theta_s\omega)|}+1+|z(\theta_s\omega)|)ds}[\|W(0)\|^2\\
&\quad + c\int_0^t|e^{\varepsilon(p-2)z(\theta_s\omega)}-1|(\|\nabla u\|_p^p
 +\|\nabla v^\varepsilon\|_p^p)ds\\
&\quad +c\int_0^t|1-e^{-\varepsilon z(\theta_s\omega)}|(e^{-p\varepsilon
z(\theta_s\omega)}+1)\|u^\varepsilon(s)\|^p_pds\\
&\quad +c\int_0^t|1-e^{-\varepsilon z(\theta_s\omega)}|(\|u(s)\|_p^p+1)ds
+c\varepsilon\int_0^t|z(\theta_s\omega)|\|u(s)\|^2ds] .
\end{aligned} \label{5.18}
\end{equation}
Since $u_0^\varepsilon\to u_0$ (as $\varepsilon\downarrow 0$),
we have
\begin{equation}
\begin{aligned}
\|W(0)\|^2
&= \|v_0^\varepsilon-u_0\|^2=\|e^{-\varepsilon
z(\omega)}u_0^\varepsilon-e^{-\varepsilon
z(\omega)}u_0+(e^{-\varepsilon
z(\omega)}-1)u_0\|^2\\
&\leq c(|e^{-\varepsilon
z(\omega)}|^2\|u_0^\varepsilon-u_0\|^2+|e^{-\varepsilon
z(\omega)}-1|^2\|u_0\|^2)\to 0\quad \text{as }
\varepsilon\downarrow 0  .
\end{aligned} \label{5.19}
\end{equation}
According to \eqref{4.15} with $T$ replaced by $0$, we have
\begin{equation}
\begin{aligned}
&\int_0^te^{-2\varepsilon
z(\theta_s\omega)+2\varepsilon\int_s^tz(\theta_\tau\omega)d\tau
 +\lambda(s-t)}\|u^\varepsilon(s,\omega,u_0^\varepsilon(\omega))\|_p^pds\\
&+ \int_0^te^{(p-2)\varepsilon
z(\theta_s\omega)+2\varepsilon\int_s^tz(\theta_\tau\omega)d\tau+\lambda(s-t)}
\|\nabla v^\varepsilon(s,\omega,v_0^\varepsilon(\omega))\|_p^pds\\
&\leq  c e^{2\varepsilon \int_0^tz(\theta_s\omega)ds-\lambda
t}\|v_0^\varepsilon(\omega)\|^2+c\int_0^te^{-2\varepsilon
z(\theta_s\omega)+2\varepsilon\int_s^tz(\theta_\tau\omega)d\tau+\lambda(s-t)}\\
&= c e^{2\varepsilon \int_0^tz(\theta_s\omega)ds-\lambda
t-2\varepsilon
z(\omega)}\|u_0^\varepsilon(\omega)\|^2+c\int_0^te^{-2\varepsilon
z(\theta_s\omega)+2\varepsilon\int_s^tz(\theta_\tau\omega)d\tau+\lambda(s-t)}\\
&\leq c e^{2 \int_0^t|z(\theta_s\omega)|ds-\lambda t+2|
z(\omega)|}\|u_0^\varepsilon(\omega)\|^2+c\int_0^te^{2|
z(\theta_s\omega)|+2\int_0^t|z(\theta_\tau\omega)|d\tau}\label{5.20} .
\end{aligned}
\end{equation}
Thus
\begin{align*}
& \int_0^t(e^{-p\varepsilon
z(\theta_s\omega)}+1)\|u^\varepsilon(s)\|^p_pds\\
&= \int_0^t(e^{-p\varepsilon z(\theta_s\omega)}+1)e^{2\varepsilon
z(\theta_s\omega)-2\varepsilon\int_s^tz(\theta_\tau\omega)d\tau-\lambda(s-t)}
 e^{-2\varepsilon z(\theta_s\omega)+2\varepsilon
 \int_s^tz(\theta_\tau\omega)d\tau+\lambda(s-t)}\\
&\quad\times \|u^\varepsilon(s,\omega,u_0^\varepsilon(\omega))\|_p^pds\\
&\leq (e^{p\max_{0\leq s\leq
t}|z(\theta_s\omega)|}+1)e^{2\max_{0\leq s\leq
t}|z(\theta_s\omega)| +2\int_0^t|z(\theta_\tau\omega)|d\tau+\lambda t} \\
&\quad\times \Big(c e^{2 \int_0^t|z(\theta_s\omega)|ds-\lambda t+2|
z(\omega)|}\|u_0^\varepsilon(\omega)\|^2
 +c\int_0^te^{2| z(\theta_s\omega)|+2\int_0^t|z(\theta_\tau\omega)|d\tau}ds\Big)
\\
&\leq  ce^{c(1+t)\max_{0\leq s\leq
t}|z(\theta_s\omega)|}\|u_0^\varepsilon(\omega)\|^2+cte^{c(1+t)\max_{0\leq
s\leq t}|z(\theta_s\omega)|+\lambda t} .% \label{5.21}
\end{align*}
Then it follows that
\begin{equation}
\begin{aligned}
& c\int_0^t|1-e^{-\varepsilon z(\theta_s\omega)}|(e^{-p\varepsilon
z(\theta_s\omega)}+1)\|u^\varepsilon(s)\|^p_pds\\
&= c\max_{0\leq s\leq t}|1-e^{-\varepsilon
z(\theta_s\omega)}|\int_0^t(e^{-p\varepsilon
z(\theta_s\omega)}+1)\|u^\varepsilon(s)\|^p_pds\to 0\quad
\text{as } \varepsilon\downarrow 0  .
\end{aligned} \label{5.22}
\end{equation}
Similarly, according to \eqref{5.20}, we can get
\begin{equation}\label{5.22'}
c\int_0^t|e^{\varepsilon(p-2)z(\theta_s\omega)}-1|
\|\nabla v^\varepsilon(s)\|_p^pds\to 0\quad
\text{as } \varepsilon\downarrow 0\,.
\end{equation}
And by \eqref{5.1} for $\varepsilon=0$, we can also get that
\begin{equation}\label{5.22''}
c\int_0^t|e^{\varepsilon(p-2)z(\theta_s\omega)}-1|\|\nabla u(s)\|_p^pds
+c\int_0^t|1-e^{-\varepsilon z(\theta_s\omega)}|\|u(s)\|_p^pds\to 0
\end{equation}
as $\varepsilon\downarrow 0$.
Furthermore, it is easy to see
\begin{equation}
\lim_{\varepsilon\downarrow 0}(c\int_0^t|1-e^{-\varepsilon
z(\theta_s\omega)}|ds+c\varepsilon\int_0^t|z(\theta_s\omega)|\|u(s)\|^2ds)=0.
\label{5.24}
\end{equation}
Thus, together \eqref{5.18}, \eqref{5.19}, \eqref{5.22}--\eqref{5.24}, imply
that for P-a.e. $\in\Omega,\forall$ fixed $t\geq 0,
u_0^\varepsilon\to u_0\ (\varepsilon\downarrow 0)$,
\begin{equation}  \label{5.25}
\|W(t)\|^2=\|v^\varepsilon-u\|^2\to 0\quad \text{as }
\varepsilon\downarrow 0 .
\end{equation}
Finally,  by \eqref{5.12} and \eqref{5.25}, we obtain
\begin{align*}
\|u^\varepsilon-u\|^2
&= \|e^{\varepsilon
z(\theta_t\omega)}v^\varepsilon-u\|^2=\|e^{\varepsilon
z(\theta_t\omega)}v^\varepsilon-e^{\varepsilon
z(\theta_t\omega)}u+(e^{\varepsilon
z(\theta_t\omega)}-1)u\|^2\\&\leq c(e^{2\varepsilon
z(\theta_t\omega)}\|v^\varepsilon-u\|^2+|e^{\varepsilon
z(\theta_t\omega)}-1|^2\|u\|^2)\to 0\quad \text{as }
\varepsilon\downarrow 0. % \label{5.26}
\end{align*}
This completes the proof.
\end{proof}


\begin{theorem}
Let $g\in L^2(\mathbb{R}^n)$, \eqref{3.3}-\eqref{3.5} and
\eqref{3.4}-\eqref{5.0} hold. Then for P-a.e. $\omega\in\Omega$
\begin{equation} \label{5.27}
\lim_{\varepsilon\downarrow 0}\operatorname{dist}{}_{L^2(\mathbb{R}^n)}
(\mathcal{A}_\varepsilon(\omega),\mathcal{A}_0)=0 .
\end{equation}
\end{theorem}

\begin{proof}
It suffices to show that conditions (i)--(iii) of Proposition
\ref{uppersemicontinuity} are all satisfied.
Note that $\{D_\varepsilon(\omega)\}_{\omega\in\Omega}$ is a closed
absorbing set for $\Phi_\varepsilon$ in $\mathcal{D}$, where
$ D_\varepsilon(\omega)$ is given by \eqref{5.1}. And it is easy to
get
\begin{equation}
\limsup_{\varepsilon\downarrow 0} \| D_\varepsilon (\omega) \| \leq
M=\sqrt{1+\frac{1}{\lambda}} ,
\end{equation}
where $\sqrt{1+\frac{1}{\lambda}}$ is a positive deterministic
constant. Thus condition $(ii)$ of Proposition
\ref{uppersemicontinuity} is satisfied. Lemma \ref{precompact} and
Lemma \ref{continuity} show that condition (iii) and (i) of
Proposition \ref{uppersemicontinuity} are satisfied respectively.
Hence, \eqref{5.27} follows from Proposition
\ref{uppersemicontinuity} immediately.
\end{proof}

\subsection{Acknowledgments}
This work was supported by National Natural
Science Foundation of China (11071199) and the doctor funds
of Southwest China university (SWU111068)

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\end{document}