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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 294, pp. 1--28.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/294\hfil Existences and upper semi-continuity]
{Existences and upper semi-continuity of pullback  attractors in
$H^1(\mathbb{R}^N )$ for non-autonomous reaction-diffusion equations
perturbed by multiplicative noise}

\author[W. Zhao \hfil EJDE-2016/294\hfilneg]
{Wenqiang Zhao}

\address{Wenqiang Zhao \newline
School of Mathematics and Statistics,
Chongqing Technology and Business University,
Chongqing 400067, China}
\email{gshzhao@sina.com}

\thanks{Submitted September 4, 2016. Published November 16, 2016.}
\subjclass[2010]{60H15, 60H30, 60H40, 35B40, 35B41}
\keywords{Random dynamical systems; upper semi-continuity;
\hfill\break\indent non-autonomous reaction-diffusion equation;
 pullback attractor}

\begin{abstract}
 In this article, we establish sufficient conditions on the existence and upper
 semi-continuity of pullback attractors in some \emph{non-initial spaces}
 for non-autonomous random dynamical systems. As an application, we prove
 the existence and upper semi-continuity  of pullback attractors
 in $H^1(\mathbb{R}^N)$ are proved for stochastic non-autonomous reaction-diffusion
 equation driven by  a Wiener type multiplicative noise as well as a non-autonomous
 forcing. The asymptotic compactness of solutions in $H^1(\mathbb{R}^N)$ is proved
 by the well-known tail estimate technique and the estimate of the integral of
 $L^{2p-2}$-norm of truncation of solutions over a compact interval.
\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}

In this paper, we consider the dynamics of solutions of
the  reaction-diffusion equation on $\mathbb{R}^N$ driven by a random
noise as well as  a deterministic non-autonomous forcing,
\begin{equation} \label{beq1}
du+(\lambda u-\Delta u)dt=f(x,u)dt+g(t,x)dt+\varepsilon u\circ d\omega(t),
\end{equation}
with the initial value
\begin{equation} \label{beq2}
u(\tau,x)=u_0(x), \quad  x\in\mathbb{R}^N,
\end{equation}
where $u_0\in L^2(\mathbb{R}^N)$, $\lambda$ is a positive constant,
$\varepsilon$ is the intensity of noise, the unknown $u=u(x,t)$ is a
real valued function of $x\in \mathbb{R}^N$ and $t> \tau$, $\omega(t)$
is a mutually independent two-sided real-valued Wiener process defined on a
 canonical Wiener probability space $(\Omega, \mathcal {F}, {P})$.

The notion of random attractor of random dynamical system, which is introduced
 in \cite{Rand1, Rand4, Rand3, Rand2} and systematically developed  in
\cite{Arn,Chues},  is an important tool to study the qualitative property of
stochastic partial differential equations (SPDE) . We can find a large body
of literature investigating the existence of random attractors in
\emph{an initial space} (the initial values located space) for  some concrete
SPDE, see \cite{Bates, And, Zhao6, Wang1,Wang4, Wangz, Zhang, Zhao} and the references
therein.  In particular, \cite{Wang1,Wang3,Wang5} discussed the upper
semi-continuity of a family of random attractors in the initial spaces.

As we know, the solutions of SPDE may possess some regularities, for example,
higher-order integrability or higher-order differentiability. In these cases,
the the solutions  may escape (or leave) the initial space and enter into another
space, which we call \emph{a non-initial space}.
Thus it is interesting for us to further investigate the existence and upper
semi-continuity of random attractors in a non-initial space, usually a
higher-regularity space, e.g., $L^p$($p>2$)  and $H^1$.

Recently in the case of bounded domain, Li \emph{et al} \cite{Liyangrong2, Lijia}
discussed the existence of random attractor of stochastic reaction-diffusion
equations in the non-initial spaces $L^p$, where $p$ is the growth exponent of
the nonlinearity. Zhao \cite{Zhao4} investigated the existence of random attractor
in $H^1_0$ for stochastic two dimensional micropolar fluid flows with coupled
additive noises. When the state space is unbounded,  Zhao and Li \cite{Zhao3}
proved the existence of random attractors for reaction-diffusion equations with
additive noises in $L^p(\mathbb{R}^N)$,  and for the same equation Li \emph{et al}
\cite{Liyangrong1} obtained  the upper semi-continuity of random attractor
in $L^p(\mathbb{R}^N)$.
Most recently Zhao \cite{Zhao0, Zhao5} proved the existence of random attractors
for semi-linear degenerate parabolic equations in $L^{2p-2}(D)\cap H^1(D)$,
where $D$ is a unbounded domain. By using the notion of omega-limit compactness,
Li \cite{Liyangrong3} obtained the existence of random attractors in
$L^q(\mathbb{R}^N)$ for semilinear Laplacian equations with multiplicative noise.
Tang \cite{Bao}  considered  the existence of  random attractors for non-autonomous
Fitzhugh-Nagumo system driven by additive noises in
$H^1(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$, and his work \cite{Bao1} investigated
the random dynamics of stochastic  reaction-diffusion equations with
\emph{additive noises} in $H^1(\mathbb{R}^N)$.
 However, it seems that the proofs in  \cite{Bao,Bao1} are essentially wrong,
see Li and Yin \cite{Liyangrong4} for the modified proof.

In this article, we study the existence and upper semi-continuity of pullback
(random) attractors in $H^1(\mathbb{R}^N)$ for stochastic reaction diffusion
equations  with \emph{multiplicative noise}  as well as  a non-autonomous forcing.
The nonlinearity $f$ and the deterministic non-autonomous function $g$ satisfy
almost the same conditions as in \cite{Wang1},
in which the author obtained the existence and upper-continuity of pullback
attractors in the initial space $L^2(\mathbb{R}^N)$.  Here, we develop their
results and show that such attractors  are also compact and attracting in
$H^1(\mathbb{R}^N)$.  Furthermore, we find that the upper continuity can also
 happen in $H^1(\mathbb{R}^N)$.
We recall that the existence of pullback attractors in an initial space  for
a non-autonomous SPDE is established in \cite{Wang3,Wang4}, where the measurability
of such attractors is proved. The applications we may see \cite{And,Wang1,Wang3,Wang4}
and so forth. For the theory of the upper semi-continuity of attractors, we may
refer to \cite{Wang1,Wang3,Wang5} for the stochastic case and to \cite{James,Hale}
for the deterministic case.

To solve our problem, we establish a sufficient criteria for
the existence and upper semi-continuity of  pullback attractors in a non-initial
space. It is showed that a family of such attractors obtained in an initial
space are compact, attracting and upper semi-continuous in a non-initial space
 if  some compactness conditions of the cocycle are satisfied, see
Theorems \ref{thm2.6}--\ref{thm2.8}.
This implies that  the continuity (or quasi-continuity
\cite{Liyangrong2}, norm-weak continuity \cite{Zhong}) and absorption in
the non-initial space are unnecessary things.
  This result is a  meaningful and convenient tool for us to consider the
existence and upper semi-continuity of pullback attractors in some related
 non-initial spaces for SPDE with a non-autonomous forcing term.

 Considering that the stochastic equation \eqref{beq1} is defined on unbounded domains,
the asymptotic compactness of solution in $H^1(\mathbb{R}^N)$ can not be
derived by the traditional technique. The reasons are as follows.
On the one hand, the equation \eqref{beq1} is stochastic and the Wiener process
$\omega$ is only continuous  in $t$ but not
differentiable. This leads to some difficulties for us to estimate the norm of
derivative  $u_t$ by the trick employed in \cite{ZhangY,Zhong} in the deterministic
case. Then the asymptotic compactness in $H^1(\mathbb{R}^N)$ can not be proved
by estimate of the difference of $\nabla u$  as in \cite{ZhangY}.
On the other hand, the estimate of $\Delta u$ is not available for our problem
(To our knoledge, actually we do not know how to estimate the norm $\Delta u$
of problem \eqref{beq1} and \eqref{beq2}, although this can be achieved by
estimate $u_t$ in the deterministic case, see \cite{Zhong}).
Hence  the Sobolev compact embeddings of $H^2\hookrightarrow H^1$ on bounded
domains is unavailable.

  Here we give a new method to prove the asymptotical compactness of solutions
in $H^1(\mathbb{R}^N)$. We first prove that the solutions vanish outside
a ball centred at zero in the state space $\mathbb{R}^N$ in the topology of $H^1$
when both the time and the radius of ball  are large enough, see
Proposition \ref{prop4.4}. Second by a new developed estimate (where the minus or plus
sign of nonlinearity is not required) we show that  the integral of $L^{2p-2}$-norm
of truncation of solutions over a compact interval is small for a large time,
see Proposition \ref{prop4.5}.   From these facts and along with some spectral arguments
the asymptotic compactness of solutions on bounded domains is followed,
and then the obstacles encountered in \cite{Bao,Bao1} are overcome.
The technique used here (without assumption that $\psi_1\in L^\infty$,
see \eqref{a1}) is different from that in  \cite{Liyangrong4} and  thus is optimal.

In the next section, we recall some notions  and prove a sufficient standard
for the existence and upper semi-continuity of pullback attractors of
non-autonomous system in a non-initial space. In section 3, we give the
assumptions on $g$ and $f$, and define a continuous cocycle for problem \eqref{beq1}
and \eqref{beq2}.
In section 4 and 5,  we prove the existence and upper semi-continuity
for this cocycle in $H^1(\mathbb{R}^N)$.


\section{Preliminaries and abstract results}


Let $(X, \|\cdot\|_X)$ and $(Y, \|\cdot\|_Y)$  be two completely separable
Banach spaces  with Borel
sigma-algebras $\mathcal {B}(X)$ and $\mathcal {B}(Y)$, respectively.
 $X\cap Y\neq \emptyset$. For convenience, we call $X$ \emph{an initial space}
(which contains all initial values of a SPDE) and $Y$ the associated
\emph{non-initial space} (usually the  regular solutions  located space).

In this section,  we give a sufficient standard for the existence and upper
semi-continuity  of pullback attractors
in the non-initial  space $Y$ for random dynamical system (RDS) over two
 parametric spaces.  The readers may refer to
\cite{Zhao0,Zhao3,Zhao4,Lijia,Liyangrong1,Liyangrong2,Liyangrong3,Yin}
for the existence and semi-continuity of such type attractors in the non-initial
space $Y$ for a RDS over one parametric space.
The existence of random attractors  in the initial space $X$ for the RDS over
one parametric space, the good references  are
\cite{Arn,Bates,Rand1,Rand2,Rand3,Rand4}. However, here we recall from
\cite{Wang4} some basic notions for RDS over two parametric spaces,
one of which is a  real numbers space and the other of which is a measurable
probability space.

\subsection{Preliminaries}

The basic notion in RDS is a metric dynamical system (MDS)
$\vartheta\equiv(\Omega,\mathcal{F},{P},\{\vartheta_t\}_{t\in
\mathbb{R}})$, which is  a probability space
$(\Omega,\mathcal{F},{P})$ incorporating  a group $\vartheta_t,t\in
\mathbb{R}$, of measure preserving transformations on
$(\Omega,\mathcal{F},{P})$. Sometimes, we call 
$\vartheta\equiv(\Omega,\mathcal{F},{P},\{\vartheta_t\}_{t\in
\mathbb{R}})$ a parametric dynamical system, see \cite{Wang1}.

A MDS $\vartheta$ is said to be
ergodic under ${P}$ if for any $\vartheta$-invariant set $F\in
\mathcal {F}$, we have either ${P}(F)=0$ or ${P}(F)=1$,
where the $\vartheta$-invariant set is in the sense that  $\vartheta_tF=F$ for
$F\in \mathcal {F}$ and all $t\in \mathbb{R}$.


\begin{definition} \label{def2.1} \rm
Let $(\Omega,\mathcal {F},{P},\{\vartheta_t\}_{t\in\mathbb{R}})$ be a metric
 dynamical system. A family of measurable mappings
 $\varphi: \mathbb{R}^+\times\mathbb{R}\times \Omega\times X\to X$
is called  a cocycle on $X$ over $\mathbb{R}$ and
$(\Omega,\mathcal {F},{P},\{\vartheta_t\}_{t\in\mathbb{R}})$
if for all $\tau\in \mathbb{R}, \omega\in\Omega$ and $t,s\in \mathbb{R}^+$,
the following conditions are satisfied:
\begin{gather*}
\varphi(0, \tau,\omega,\cdot)\quad \text{is the identity on X}, \\
\varphi(t+s,\tau, \omega,\cdot)=\varphi(t,\tau+s, \vartheta_s\omega, \cdot)
\circ \varphi(s,\tau, \omega, \cdot).
 \end{gather*}
In addition, if $\varphi(t,\tau, \omega,\cdot): X\to X$ is continuous for all $t\in
\mathbb{R}^+, \tau\in \mathbb{R}, \omega\in\Omega$, then $\varphi$ is called
a continuous cocycle on $X$ over $\mathbb{R}$ and
$(\Omega,\mathcal {F},{P},\{\vartheta_t\}_{t\in\mathbb{R}})$.
\end{definition}

\begin{definition} \label{def2.2} \rm
 Let $2^X$ be the collection of all subsets of $X$. A set-valued mapping
$K: \mathbb{R}\times \Omega\to 2^X$ is called measurable in $X$ with respect to
$\mathcal{F}$ in $\Omega$ if  the mapping
$\omega\in\Omega\mapsto \operatorname{dist}_X(x,K(\tau,\omega))$ is
($\mathcal{F},\mathcal{B}(\mathbb{R}))$-measurable for every fixed $x\in X$
and $\tau\in\mathbb{R}$, where $\operatorname{dist}_X$ is the Haustorff semi-metric in $X$.
In this case, we also say the family
$\{K(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\}$ is measurable in $X$
with respect to $\mathcal{F}$ in $\Omega$. Furthermore if the value
$K(\tau,\omega)$ is a closed nonempty subset of $X$ for all $\tau\in\mathbb{R}$
and $\omega\in \Omega$, then $\{K(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\}$
is called a closed measurable set of $X$ with respect to $\mathcal{F}$ in
$\Omega$.
\end{definition}

In this article,  the cocycle $\varphi$ acting on $X$ is further assumed to
take its values into the non-initial space $Y$ in the following sense:
\begin{itemize}
\item[(H1)] For every fixed $t>0, \tau\in\mathbb{R}$ and
 $\omega\in\Omega$, $\varphi(t,\tau,\omega,\cdot): X\to  Y$.
\end{itemize}

We use $\mathfrak{D}$ to denote a  collection  of some families of nonempty
subsets of $X$ parametrized by $\tau\in\mathbb{R}$ and $\omega\in \Omega$ such that
\begin{align*}
\mathfrak{D}= \big\{&B=\{B(\tau,\omega)\in 2^X; B(\tau,\omega)\neq\emptyset,
\tau\in\mathbb{R},\omega\in \Omega \big\}; \\
 &f_B \text{ satisfies certain conditions}\}.
\end{align*}
In particular, for $B_1,B_2\in\mathfrak{D}$ we say that $B_1=B_2$ if
$B_1(\tau,\omega)=B_2(\tau,\omega)$ for all $\tau\in\mathbb{R}$ and
$\omega\in \Omega$. The collection $\mathfrak{D}$ is called inclusion closed
if $\tilde{B}(\tau,\omega)\subset B(\tau,\omega)$ and $B\in\mathfrak{D}$
for every $\tau\in\mathbb{R}$ and $\omega\in\Omega$, then
 $\tilde{B}\in\mathfrak{D}$.

\begin{definition} \label{def2.3}  \rm
Let  $\mathfrak{D}$ be  a  collection  of some families of nonempty subsets of
$X$ and $K=\{K(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\}\in \mathfrak{D}$.
Then $K$ is  called a $\mathfrak{D}$-pullback absorbing set for
a cocycle  $\varphi$ in $X$ if for every $\tau\in \mathbb{R}, \omega\in\Omega$
and $B\in \mathfrak{D}$  there exists a absorbing time $T=T(\tau,\omega,B)>0$
such that
$$
\varphi (t,\tau-t, \vartheta_{-t}\omega, B(\tau-t,\vartheta_{-t}\omega))
\subseteq K(\tau, \omega)\quad \text{for all } t\geq T.
$$
If in addition $K$ is measurable in $X$ with respect to  $\mathcal{F}$ in
$\Omega$, then $K$ is said to a  measurable pullback absorbing set
for $\varphi$.
\end{definition}

 \begin{definition} \label{def2.4} \rm
Let  $\mathfrak{D}$ be  a  collection  of some families of nonempty subsets
of $X$. A cocycle  $\varphi$ is said to be
$\mathfrak{D}$-pullback asymptotically compact in $X$ (resp. in $Y$)
 if for each $\tau\in \mathbb{R}, \omega\in\Omega$
$$
\{\varphi(t_n, \tau-t_n, \vartheta_{-t_n}\omega, x_n)\} \text{ has a
convergent subsequence in } X(\text{resp. in}\ Y)
$$
whenever $t_n\to \infty$ and
$x_n\in B(\tau-t_n,\vartheta_{-t_n}\omega)$ with
$B=\{B(\tau, \omega);\tau\in \mathbb{R},\omega\in \Omega\} \in \mathfrak{D}$.
\end{definition}


\begin{definition} \label{def2.5} \rm
 Let  $\mathfrak{D}$ be  a  collection  of some families of nonempty subsets of
 $X$ and $\mathcal{A}=\{\mathcal{A}(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\}\in \mathfrak{D}$. $\mathcal{A}$ is called  a $\mathfrak{D}
$-pullback  attractor for a cocycle $\varphi$ in $X$ (resp. in $Y$)
 over $\mathbb{R}$ and $(\Omega,\mathcal{F},P,\{\vartheta_t\}_{t\in\mathbb{R}})$ if
\begin{itemize}
\item[(i)]  $\mathcal {A}$ is measurable in $X$ with respect to $\mathcal{F}$,
and $\mathcal {A}(\tau,\omega)$ is compact in $X$ (resp. in $Y$) for each
 $\tau\in \mathbb{R}, \omega\in\Omega$;

\item[(ii)] $\mathcal {A}$ is invariant, that is, for each $\tau\in \mathbb{R},
\omega\in\Omega$,
 $$
 \varphi(t,\tau,\omega,\mathcal{A}(\tau,\omega))=\mathcal{A}(\tau+t,\vartheta_t\omega),
\quad \forall\ t\geq0;
 $$

\item[(iii)]  $\mathcal {A}$  attracts every element
$B=\{B(\tau, \omega);\tau\in \mathbb{R},\omega\in \Omega\}
\in \mathfrak{D}$ in $X$ (resp. in $Y$), that is, for each
$\tau\in \mathbb{R}, \omega\in\Omega$,
\begin{gather*}
\lim_{t\to +\infty}\operatorname{dist}_X(\varphi(t,\tau-t,\vartheta_{-t}\omega,
B(\tau-t, \vartheta_{-t}\omega)),\mathcal{A}(\tau,\omega))= 0 \\
(\text{resp. }\lim _{t\to +\infty}\operatorname{dist}_Y(\varphi(t,\tau-t,
\vartheta_{-t}\omega, B(\tau-t, \vartheta_{-t}\omega)),
\mathcal{A}(\tau,\omega))= 0).
\end{gather*}
\end{itemize}
\end{definition}

\subsection{Existence of random attractors in a non-initial space}

 This subsection is concerned with the existence of $\mathfrak{D}$-pullback
attractor of the cocycle $\varphi$ in the non-initial space $Y$.
The continuity of $\varphi$ in $Y$ is not clear, and the embedding relation of
$X$ and $Y$ is also unknown except that the following  hypothesis
(H2)  holds:
\begin{itemize}
\item[(H2)]  If $\{x_n\}_n\subset X\cap Y$ such that $x_n\to x$ in
$X$ and $x_n\to y$ in $Y$ respectively, then $x=y$.
\end{itemize}



\begin{theorem} \label{thm2.6}
Let $\mathfrak{D}$ be a collection of some families of nonempty subsets
of $X$ which is inclusion closed. Let $\varphi$ be a continuous cocycle on
$X$ over $\mathbb{R}$ and $(\Omega,\mathcal{F},P,\{\vartheta_t\}_{t\in\mathbb{R}})$.
 Assume that
\begin{itemize}
\item[(i)] $\varphi$ has  a closed and measurable $\mathfrak{D}$-pullback
bounded absorbing set
$K=\{K(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\}\in \mathfrak{D}$ in $X$;

\item[(ii)] $\varphi$ is  $\mathfrak{D}$-pullback asymptotically compact in $X$.
\end{itemize}
Then the cocycle  $\varphi$ has a unique $\mathfrak{D}$-pullback  attractor
$\mathcal{A}_X=\{\mathcal{A}_X(\tau,\omega); \tau\in\mathbb{R},
\omega\in \Omega\}\in\mathfrak{D}$ in $X$,
structured by
\begin{equation}\label{bff01}
\mathcal{A}_X(\tau,\omega)=\cap_{s\geq0}\overline{\cup_{t\geq s}
\varphi(t,\tau-t,\vartheta_{-t}\omega, K(\tau-t,\vartheta_{-t}\omega))}^X, \quad
\tau\in \mathbb{R},\omega\in \Omega,
 \end{equation}
where the closure is taken in $X$.

If further {\rm (H1), (H2)} hold  and
\begin{itemize}
\item[(iii)] $\varphi$ is  $\mathfrak{D}$-pullback asymptotically compact in $Y$,
\end{itemize}
Then the cocycle  $\varphi$ has a unique $\mathfrak{D}$-pullback  attractor
$\mathcal{A}_Y=\{\mathcal{A}_Y(\tau,\omega); \tau\in\mathbb{R}, \omega\in \Omega\}$
in $Y$, given by
\begin{equation}\label{bff02}
\mathcal{A}_Y(\tau,\omega)=\cap_{s>0}\overline{\cup_{t\geq s} \varphi(t,\tau-t,
\vartheta_{-t}\omega, K(\tau-t,\vartheta_{-t}\omega))}^Y, \ \
\tau\in \mathbb{R},\omega\in \Omega.
 \end{equation}
In addition, we have $\mathcal{A}_Y=\mathcal{A}_X\subset X\cap Y$ in the sense
of set inclusion, i.e.,
 for each $\tau\in \mathbb{R},\omega\in \Omega$,
$\mathcal{A}_Y(\tau,\omega)=\mathcal{A}_X(\tau,\omega)$.
\end{theorem}

\begin{proof}
The first result is well known and thus we are interested in the
second result. Indeed, \eqref{bff02} makes sense by (H1) and
$\mathcal{A}_Y\neq \emptyset$
by the asymptotic compactness of the cocycle $\varphi$ in $Y$. In the following,
we  show that  $\mathcal{A}_Y$
satisfies  Definition \ref{def2.5} in the space $Y$.
\smallskip

\noindent\textbf{Step 1.} We claim that the set $\mathcal{A}_Y$ is measurable
in $X$ (with respect to  $\mathcal{F}$ in $\Omega$) and
$\mathcal{A}_Y\in\mathfrak{D}$ is invariant  by proving that
$\mathcal{A}_Y=\mathcal{A}_X$ since $\mathcal{A}_X$
is measurable (w.r.t $\mathcal{F}$ in $\Omega$)  and
$\mathcal{A}_X\in\mathfrak{D}$ is invariant (the measurability of $\mathcal{A}_X$
is proved by \cite[Theorem 2.14]{Wang3}).

For each fixed $\tau\in\mathbb{R}$ and $\omega\in\Omega$, taking
$x\in \mathcal{A}_X(\tau,\omega)$, by \eqref{bff01}, there exist two sequences
$t_n\to+\infty$ and $x_n\in K(\tau-t_n,\vartheta_{-t_n}\omega)$ such that
\begin{equation}  \label{bff3}
 \varphi(t_n, \tau-t_n, \vartheta_{-t_n}\omega, x_n)
\xrightarrow[n\to\infty]{\|\cdot\|_X}x.
\end{equation}
Since $\varphi$ is
$\mathfrak{D}$-asymptotically compact in $Y$, then there is a $y\in Y$ such
that up to a subsequence,
\begin{equation} \label{bff4}
 \varphi(t_n, \tau-t_n, \vartheta_{-t_n}\omega, x_n)
\xrightarrow[n\to\infty]{\|\cdot\|_Y}y.
\end{equation}
 It implies from \eqref{bff02} that $y\in \mathcal{A}_Y(\tau,\omega)$.
Then by (H2), along with \eqref{bff3} and \eqref{bff4}, we have
 $x=y\in \mathcal{A}_X(\tau,\omega)$ and thus
 $ \mathcal{A}_X(\tau,\omega)\subseteq  \mathcal{A}_Y(\tau,\omega)$ for every
fixed $\tau\in\mathbb{R}$ and $\omega\in\Omega$. The inverse inclusion can be
 proved in the same
 way then we omit it here. Thus $\mathcal{A}_X=\mathcal{A}_Y$ as required.
\smallskip


noindent\textbf{Step 2.}  We prove the attraction of  $\mathcal{A}_Y$
in $Y$ by a contradiction argument. Indeed,  if
there exist $\delta>0$, $x_n\in B(\tau-t_n,\vartheta_{-t_n}\omega)$ with
$B\in\mathfrak{D}$ and $t_n\to +\infty$  such that
\begin{equation}\label{ff5}
 \operatorname{dist}_Y\Big(\varphi(t_n,\tau-t_n,\vartheta_{-t_n}\omega,x_n),
 \mathcal{A}_Y(\tau,\omega)\Big)\geq \delta.
 \end{equation}
By the asymptotic compactness of $\varphi$ in $Y$, there exists $y_0\in Y$
 such that up to a subsequence,
\begin{equation}\label{ff6}
 \varphi(t_n,\tau-t_n,\vartheta_{-t_n}\omega,x_n)
\xrightarrow[n\to\infty]{\|\cdot\|_Y}y_0.
 \end{equation}
On the other hand, by condition (i), there exists a large time $T>0$ such that
\begin{equation} \label{ff7}
\begin{aligned}
 y_n&=\varphi(T,\tau-t_n,\vartheta_{-t_n}\omega,x_n) \\
&=\varphi(T,(\tau-t_n+T)-T,\vartheta_{-T}\vartheta_{-(t_n-T)}\omega,x_n) \\
 &\in K(\tau-t_n+T ,\vartheta_{-(t_n-T)}\omega).
\end{aligned}
 \end{equation}
Then by the cocycle property in Definition \ref{def2.1}, with \eqref{ff6} and
\eqref{ff7}, we infer that as $t_n\to\infty$,
\[
 \varphi(t_n,\tau-t_n,\vartheta_{-t_n}\omega,x_n)
=\varphi(t_n-T,\tau-(t_n-T),\vartheta_{-(t_n-T)}\omega,y_n)\to y_0
\quad \text{in } Y.
\]
 Therefore by \eqref{bff02}, $y_0\in \mathcal{A}_Y(\tau,\omega)$.
This implies
 \begin{equation}
 \operatorname{dist}_Y\Big(\varphi(t_n,\tau-t_n,\vartheta_{-t_n}\omega,x_n),
 \mathcal{A}_Y(\tau,\omega)\Big)\to 0
 \end{equation}
as $t_n\to\infty$, which is a contradiction to \eqref{ff5}.
\smallskip

\noindent\textbf{Step 3.}
It remains to prove the compactness of $A_Y$ in $Y$.  Let
$\{y_n\}_{n=1}^\infty$  be a sequence in
$A_Y(\tau,\omega)$. By the invariance of $A_Y(\tau,\omega)$ which is proved
in Step 1, we  have
 $$
 \varphi(t,\tau-t,\vartheta_{-t}\omega, \mathcal{A}_Y(\tau-t,\vartheta_{-t}\omega))
=\mathcal{A}_Y(\tau,\omega).
 $$
 Then it follows that there is a sequence $\{z_n\}_{n=1}^\infty$ with
$z_n\in \mathcal{A}_Y(\tau-t_n, \vartheta_{-t_n}\omega)$
such that for every $n\in\mathbb{Z}^+$,
 \begin{equation}
 y_n=\varphi(t_n,\tau-t_n,\vartheta_{-t_n}\omega,z_n).
 \end{equation}
Note that $A_Y\in \mathfrak{D}$. Then by the asymptotic compactness of $\varphi$
in $Y$, $\{y_n\}$  has a convergence subsequence in $Y$, \emph{i.e.},
 there is a $y_0\in Y$ such that
 $$
 \lim_{n\to\infty}y_n=y_0\quad \text{in }Y.
 $$
But  $A_Y(\tau,\omega)$ is closed in $Y$, so $y_0\in A_Y(\tau,\omega)$.

  The uniqueness is easily followed by the attraction property of $\varphi$ and
$A_Y\in \mathfrak{D}$. This completes the total proofs.
\end{proof}


\subsection{Upper semi-continuity of random attractors  in a non-initial space}

Assume that (H1) and (H2) hold.   Given
 the indexed set $I\subset \mathbb{R}$, for every $\varepsilon\in I$,
we use $\mathfrak{D}_\varepsilon$  to denote a
 a collection of some families of nonempty subsets of $X$.
Let $\varphi_\varepsilon (\varepsilon\in I)$ be a continuous cocycle on
$X$ over $\mathbb{R}$ and $(\Omega,\mathcal{F},P,\{\vartheta_t\}_{t\in\mathbb{R}})$.
We now consider the upper semi-continuous of pullback  attractors of a family of
  cocycle $\varphi_\varepsilon$ in $Y$.

Suppose first that  for every
$t\in\mathbb{R}^+, \tau\in \mathbb{R}, \omega\in\Omega, \varepsilon_n,
\varepsilon_0\in I$ with $\varepsilon_n\to \varepsilon_0$, and
 $x_n,x\in X$ with $x_n\to x$, there holds
 \begin{equation} \label{ff8}
\lim_{n\to \infty}\varphi_{\varepsilon_n}(t,\tau,\omega,x_n)
=\varphi_{\varepsilon_0}(t,\tau, \omega, x)\quad \text{in } X.
\end{equation}
Suppose second that there exists a map
$R_{\varepsilon_0}: \mathbb{R}\times \Omega\to \mathbb{R}^+$ such that
the family
\begin{equation} \label{ff9}
B_0=\{B_{0}(\tau,\omega)=\{x\in X;\|x\|_X\leq
R_{\varepsilon_0}(\tau,\omega)\}:\tau\in \mathbb{R},\omega\in\Omega\}
\end{equation}
belongs to $\mathfrak{D}_{\varepsilon_0}$.
And further for every $\varepsilon\in I$, $\varphi_\varepsilon$ has
$\mathfrak{D}_\varepsilon$-pullback  attractor
$\mathcal{A}_\varepsilon\in \mathfrak{D}_\varepsilon$ in $X\cap Y$ and
 a closed and measurable $\mathfrak{D}_\varepsilon$-pullback absorbing set
$K_\varepsilon\in \mathfrak{D}_\varepsilon$ in $X$ such that for every
$\tau\in \mathbb{R},\omega\in\Omega$,
\begin{equation}\label{ff10}
\limsup_{\varepsilon\to \varepsilon_0}\|K_\varepsilon(\tau,\omega)\|
\leq R_{\varepsilon_0}(\tau,\omega),
\end{equation}
 where $\|S\|_X=\sup_{x\in S}\|x\|_X$ for a set $S$.
We finally assume that for every  $\tau\in \mathbb{R}$, $\omega\in\Omega$,
\begin{gather}\label{ff11}
\cup_{\varepsilon\in I}\mathcal{A}_\varepsilon(\tau,\omega)
 \text{ is precompact in } X, \text{ and} \\
\label{ff12}
\cup_{\varepsilon\in I}\mathcal{A}_\varepsilon(\tau,\omega)
  \text{ is precompact in }Y.
\end{gather}

Then we have the upper semi-continuity in $Y$.

\begin{theorem} \label{thm2.7}
If \eqref{ff8}--\eqref{ff11} hold, then for each
$\tau\in \mathbb{R}$, $\omega\in\Omega$,
\[
\lim_{\varepsilon\to\varepsilon_0}\operatorname{dist}_X
(\mathcal{A}_\varepsilon(\tau,\omega),\mathcal{A}_{\varepsilon_0}(\tau,\omega))=0.
\]
If further (H1)-(H2) hold and conditions \eqref{ff8}-\eqref{ff12} are satisfied.
Then for each $\tau\in \mathbb{R}$, $\omega\in\Omega$,
\[
\lim_{\varepsilon\to\varepsilon_0}\operatorname{dist}_Y
(\mathcal{A}_\varepsilon(\tau,\omega),\mathcal{A}_{\varepsilon_0}(\tau,\omega))=0.
\]
\end{theorem}

\begin{proof}
   If \eqref{ff8}-\eqref{ff11} hold, the upper-continuous in $X$ is proved in
 \cite{Wang1}.  We only need to prove the upper semi-continuity of
$\mathcal{A}_\varepsilon$ at $\varepsilon=\varepsilon_0$ in $Y$.

 Suppose that there exist $\delta>0$,  $\varepsilon_n\to\varepsilon_0$ and a
sequence $\{y_n\}$ with $y_n\in \mathcal{A}_{\varepsilon_n}(\tau,\omega)$
such that for all
$n\in\mathbb{N}$,
\begin{equation}\label{ff18}
\lim_{\varepsilon\to\varepsilon_0}\operatorname{dist}_Y(y_n,
\mathcal{A}_{\varepsilon_0}(\tau,\omega))\geq 2\delta.
\end{equation}
Note that $y_n\in \mathcal{A}_{\varepsilon_n}(\tau,\omega)\subset
\mathbb{A}(\tau,\omega)=\cup_{\varepsilon\in I}\mathcal{A}_\varepsilon(\tau,\omega)$.
 Then by \eqref{ff11} and \eqref{ff12} and using  (H2), there exists a
 $y_0\in X\cap Y$ such that up to a subsequence,
\begin{equation}\label{ff19}
\lim_{n\to\infty}y_n=y_0\quad \text{in } X\cap Y.
\end{equation}
It suffices to show that
$\operatorname{dist}_Y(y_0, \mathcal{A}_{\varepsilon_0}(\tau,\omega))<\delta$.
Given  a positive sequence $\{t_m\}$ with $t_m\uparrow +\infty$ as $m\to\infty$.
For $m=1$, by the invariance of $\mathcal{A}_{\varepsilon_n}$,
there exists a sequence $\{y_{1,n}\}$ with
$y_{1,n}\in \mathcal{A}_{\varepsilon_n}(\tau-t_1,\vartheta_{-t_1}\omega)$
such that
\begin{equation}\label{ff20}
y_n=\varphi_{\varepsilon_n}(t_1,\tau-t_1,\vartheta_{-t_1}\omega,y_{1,n}),
\end{equation} for each $n\in\mathbb{N}$.
Since $y_{1,n}\in \mathcal{A}_{\varepsilon_n}(\tau-t_1,\vartheta_{-t_1}\omega)
\subset \mathbb{A}(\tau-t_1,\vartheta_{-t_1}\omega)$, then by by \eqref{ff11}
and \eqref{ff12} and using (H2),
there is a $z_1\in X\cap Y$ and a subsequence of $\{y_{1,n}\}$ such that
\begin{equation}\label{ff21}
\lim_{n\to\infty}y_{1,n}=z_1\quad \text{in } X\cap Y.
\end{equation}
Then \eqref{ff8} and \eqref{ff21} imply
\begin{equation}\label{ff22}
\lim_{n\to\infty}\varphi_{\varepsilon_n}(t_1,\tau-t_1,\vartheta_{-t_1}\omega,y_{1,n})
=\varphi_{\varepsilon_0}(t_1,\tau-t_1,\vartheta_{-t_1}\omega,z_1)\quad
\text{in } X.
\end{equation}
Thus by  combining  \eqref{ff19}, \eqref{ff20} and \eqref{ff22} we obtain
\begin{equation}\label{ff23}
y_0=\varphi_{\varepsilon_0}(t_1,\tau-t_1,\vartheta_{-t_1}\omega,z_1).
\end{equation}
Note that $K_{\varepsilon_n}$ as a $\mathfrak{D}_{\varepsilon_n}$-pullback
absorbing set in $X$ absorbs $\mathcal{A}_{\varepsilon_n}\in
\mathfrak{D}_{\varepsilon_n}$,
\emph{i.e.}, there is a $T=T(\tau,\omega,\mathcal{A}_{\varepsilon_n})$
such that for all $t\geq T$,
\begin{equation}\label{ff24}
\varphi(t,\tau-t,\vartheta_{-t}\omega,\mathcal{A}_{\varepsilon_n}(\tau-t,
\vartheta_{-t}\omega))\subseteq K_{\varepsilon_n}(\tau,\omega).
\end{equation}
Then by the invariance of $\mathcal{A}_{\varepsilon_n}(\tau,\omega)$,
it follows from \eqref{ff24} that
\begin{equation}\label{ff25}
\mathcal{A}_{\varepsilon_n}(\tau,\omega)\subseteq K_{\varepsilon_n}(\tau,\omega).
\end{equation}
Since $y_{1,n}\in \mathcal{A}_{\varepsilon_n}(\tau-t_1,\vartheta_{-t_1}\omega)
\subseteq K_{\varepsilon_n}(\tau-t_1,\vartheta_{-t_1}\omega)$,
then by \eqref{ff21} and \eqref{ff10}, we obtain
\begin{equation}\label{ff26}
\begin{aligned}
\|z_1\|_X&=\limsup_{n\to\infty}\|y_{1,n}\|_X \\
&\leq \limsup_{n\to\infty}\|K_{\varepsilon_n}(\tau-t_1,\vartheta_{-t_1}\omega)
\|_X \\
&\leq R_{\varepsilon_0}(\tau-t_1,\vartheta_{-t_1}\omega).
\end{aligned}
\end{equation}
By an induction argument, for each $m\geq1$, there is $z_m\in X\cap Y$
 such that for all $m\in\mathbb{N}$,
\begin{gather}\label{ff27}
y_0=\varphi_{\varepsilon_0}(t_m,\tau-t_m,\vartheta_{-t_m}\omega,z_m), \\
\label{ff28}
\|z_m\|_X\leq R_{\varepsilon_0}(\tau-t_m,\vartheta_{-t_m}\omega).
\end{gather}
Thus from \eqref{ff9} and \eqref{ff28}, for each $m\in\mathbb{N}$,
\begin{equation}\label{ff29}
z_m\in B_0(\tau-t_m,\vartheta_{-t_m}\omega).
\end{equation}
We consider that the pullback  attractor $\mathcal{A}_{\varepsilon_0}$ attracts every element in $\mathfrak{D}_{\varepsilon_0}$ in the topology of $Y$
 and connection with $B_0\in\mathfrak{D}_{\varepsilon_0}$.
 Then $\mathcal{A}_{\varepsilon_0}$ attracts $B_0$ in the topology of $Y$.
Therefore by \eqref{ff27} and \eqref{ff29} we have
\begin{equation}
\operatorname{dist}_Y(y_0, \mathcal{A}_{\varepsilon_0}(\tau,\omega))
=\operatorname{dist}_Y(\varphi_{\varepsilon_0}(t_m,\tau-t_m,
\vartheta_{-t_m}\omega,z_m), \mathcal{A}_{\varepsilon_0}(\tau,\omega))\to0,
\end{equation}
as $m\to\infty$. That is to say,
$\operatorname{dist}_Y(y_0, \mathcal{A}_{\varepsilon_0}(\tau,\omega))
=\inf_{u\in\mathcal{A}_{\varepsilon_0}(\tau,\omega)}\|y_0-u\|_Y=0$
and thus we can choose a $u_0\in\mathcal{A}_{\varepsilon_0}(\tau,\omega)$
such that
\begin{equation}\label{ff30}
\|y_0-u_0\|_Y\leq \delta.
\end{equation}
Therefore, by \eqref{ff19} and \eqref{ff30}, as $n\to\infty$,
$$
\operatorname{dist}_Y(y_n, \mathcal{A}_{\varepsilon_0}(\tau,\omega))
\leq \|y_n-u_0\|_Y\leq  \|y_n-y_0\|_Y+\delta\to\delta,
$$
which is a contradiction to \eqref{ff18}. This concludes the proof.
\end{proof}

We next consider a special case of Theorem \ref{thm2.7}, in which case
the limit cocycle $\varphi_{\varepsilon_0}$ is independent of the parameter
$\omega\in\Omega$.
We call such  $\varphi_{\varepsilon_0}$ a deterministic  non-autonomous
cocycle on $X$ over $\mathbb{R}$. That is to say, $\varphi_{\varepsilon_0}$
satisfies the following two statements:
\begin{itemize}
\item[(i)] $\varphi_0(0,\tau,\cdot)$ is the identity on $X$;

\item[(ii)] $\varphi_0(t+s,\tau,\cdot)=\varphi_0(t,\tau+s,\cdot)
\circ \varphi_0(s,\tau,\cdot)$.
\end{itemize}
If $\varphi_0(t, \tau, .): X\to X$ is continuous for every $t\in\mathbb{R}^+$
and $\tau\in\mathbb{R}$, then  $\varphi_{\varepsilon_0}$ is called a
deterministic  non-autonomous  continuous cocycle on $X$ over $\mathbb{R}$.

 Let $\mathfrak{D}_{\varepsilon_0}$ be a collection  of some
families of nonempty subsets of $X$ denoted by
$$
\mathfrak{D}_{\varepsilon_0}
=\{B=\{B(\tau)\neq\emptyset; B(\tau)\in 2^X, \tau\in\mathbb{R}\};
f_B\ \text{satisfies certain conditions}\}.
$$
A family $\mathcal{A}_{\varepsilon_0}\in \mathfrak{D}_{\varepsilon_0}$ is called a $\mathfrak{D}_{\varepsilon_0}$-pullback attractor of $\varphi_{\varepsilon_0}$
in $X$ (resp. in $Y$) if
\begin{itemize}
\item[(i)] for each $\tau\in\mathbb{R}$, $\mathcal{A}_{\varepsilon_0}(\tau)$
is compact in $X$(resp. of $Y$);

\item[(ii)] $\varphi_{\varepsilon_0}(t,\tau,\mathcal{A}_{\varepsilon_0}(\tau))
 =\mathcal{A}_{\varepsilon_0}(\tau+t)$ for all $t\in\mathbb{R}^+$ and
 $\tau\in\mathbb{R}$;

\item[(iii)]  $\mathcal{A}_{\varepsilon_0}$ pullback attracts every element of
 $\mathfrak{D}_{\varepsilon_0}$ under the Hausdorff semi-metric of
$X$ (resp. of Y).
\end{itemize}

To obtain the convergence at $\varepsilon=\varepsilon_0$ in $Y$, we make some
 modifications of the conditions used in random case.
We assume that  for every $t\in\mathbb{R}^+, \tau\in \mathbb{R}, \omega\in\Omega,
\varepsilon_n\in I$ with $\varepsilon_n\to \varepsilon_0$, and
 $x_n,x\in X$ with $x_n\to x$, it  holds
 \begin{equation}\label{ff31}
\lim_{n\to \infty}\varphi_{\varepsilon_n}(t,\tau,\omega,x_n)
=\varphi_{\varepsilon_0}(t,\tau, x)\quad \text{in } X.
\end{equation}
There exists a map $R'_{\varepsilon_0}: \mathbb{R}\to \mathbb{R}$ such that
the family
\begin{equation}\label{ff32}
B'_0=\{B'_{0}(\tau)=\{x\in X;\|x\|_X\leq R'_{\varepsilon_0}(\tau)\};
\tau\in \mathbb{R}\} \text{ belongs to } \mathfrak{D}_{\varepsilon_0}.
\end{equation}
For every $\varepsilon\in I$,  $\varphi_\varepsilon$ has a closed measurable
$\mathfrak{D}_{\varepsilon}$-pullback absorbing set
$K_\varepsilon=\{K_\varepsilon(\tau,\omega);\omega\in\Omega\}
\in \mathfrak{D}_\varepsilon$ in $X$ such that for every
$\tau\in \mathbb{R},\omega\in\Omega$,
\begin{equation}\label{ff33}
\limsup_{\varepsilon\to \varepsilon_0}\|K_\varepsilon(\tau,\omega)\|
\leq R'_{\varepsilon_0}(\tau).
\end{equation}

Then we have the following, which can be proved by a similar argument as
Theorem \ref{thm2.7} and so the proof is omitted.

 \begin{theorem} \label{thm2.8}
 If \eqref{ff11} and \eqref{ff31}-\eqref{ff33} hold, then
  for each $\tau\in \mathbb{R}$, $\omega\in\Omega$,
\[
\lim_{\varepsilon\to\varepsilon_0}\operatorname{dist}_X
(\mathcal{A}_\varepsilon(\tau,\omega),\mathcal{A}_{\varepsilon_0}(\tau))=0.
\]
If  further (H1)-(H2) hold and conditions \eqref{ff12} and
\eqref{ff31}-\eqref{ff33} are satisfied, then  for each
$\tau\in \mathbb{R}, \omega\in\Omega$,
\begin{equation}\label{}
\lim_{\varepsilon\to\varepsilon_0}\operatorname{dist}_Y
(\mathcal{A}_\varepsilon(\tau,\omega),\mathcal{A}_{\varepsilon_0}(\tau))=0.
\end{equation}
\end{theorem}

\section{Non-autonomous reaction-diffusion equation on $\mathbb{R}^N$
 with multiplicative noise}

For the non-autonomous reaction-diffusion equations \eqref{beq1} and \eqref{beq2},
the nonlinearity  $f(x,s)$ satisfies almost the same assumptions as in \cite{Wang1},
i.e., for $ x\in\mathbb{R}^N$ and $s\in\mathbb{R}$,
\begin{gather}\label {a1}
f(x,s)s\leq -\alpha_1|s|^p+\psi_1(x), \\
\label {a2}
|f(x,s)|\leq \alpha_2|s|^{p-1}+\psi_2(x), \\
\label {a3}
\frac{\partial f}{\partial s}f(x,s)\leq \alpha_3 , \\
\label {a4}
\Big|\frac{\partial f}{\partial x}f(x,s)\Big|\leq \psi_3(x),
\end{gather}
 where $\alpha_i> 0$ $(i=1,2,3)$ are determined constants, $p\geq2$,
$\psi_1\in L^{1}(\mathbb{R}^N)\cap L^{p/2}(\mathbb{R}^N)$,
$\psi_2\in L^2(\mathbb{R}^N)$ and $\psi_3\in L^2(\mathbb{R}^N)$.
And the non-autonomous term $g$  satisfies that for every $\tau\in\mathbb{R}$
and some $\delta\in [0,\lambda)$,
\begin{equation}\label {a5}
\int_{-\infty}^\tau e^{\delta s} \|g(s,\cdot)\|_{L^2(\mathbb{R}^N)}^2ds<+\infty,
\end{equation}
where $\lambda$ is as in \eqref{beq1}, which implies that
\begin{equation}\label {a66}
\int_{-\infty}^0 e^{\delta s} \|g(s+\tau,\cdot)\|_{L^2(\mathbb{R}^N)}^2ds<+\infty,
\quad g\in L^2_{Loc}(\mathbb{R},L^2(\mathbb{R}^N)).
\end{equation}

For the probability space $(\Omega,\mathcal{F},P)$, we write
 $\Omega=\{\omega\in C(\mathbb{R},\mathbb{R}); \omega(0) =0\}$. Let
 $\mathcal {F}$ be the  Borel $\sigma$-algebra induced by the compact-open
topology of $\Omega$ and ${P}$ be  the corresponding
 Wiener measure on $(\Omega,\mathcal{F})$.
  We define a shift operator $\vartheta$ on $\Omega$ by
$$
\vartheta_t\omega(s)=\omega(s+t)-\omega(t), \quad \text{for every }
 \omega\in\Omega, t,s\in\mathbb{R}.
$$
 Then $(\Omega,\mathcal{F},P,\{\vartheta_t\}_{t\in\mathbb{R}})$ which is the model
for random noise is called a metric dynamical system. Furthermore
$(\Omega,\mathcal{F},P,\{\vartheta_t\}_{t\in\mathbb{R}})$ is ergodic
with respect to $\{\vartheta_t\}_{t\in\mathbb{R}}$ under $P$, which means that
every $\vartheta_t$-invariant set has measure zero or one, $t\in\mathbb{R}$.
By the law of the iterated logarithm (see \cite{Rand1}), we know that
\begin{equation} \label{b3.6}
\frac{\omega(t)}{t}\to0, \quad \text{as } |t|\to+\infty.
\end{equation}

For $\omega\in \Omega$, put
$z(t,\omega)=z_\varepsilon(t,\omega)=e^{-\varepsilon \omega (t)}$. Then we have
$dz+\varepsilon z\circ d\omega(t)=0.$
Put $v(t,\tau,\omega, v_0)=z(t,\omega)u(t,\tau,\omega,u_0)$, where $u$ is a
solution of problem
\eqref{beq1} and \eqref{beq2} with the initial value $u_0$. Then $v$ solves
the  non-autonomous equation
\begin{equation} \label{pr1}
\frac{dv}{dt}+\lambda v-\Delta v=z(t,\omega)f(x,z^{-1}(t,\omega)v)
+z(t,\omega)g(t,x),
\end{equation}
with the initial value
\begin{equation} \label{pr2}
 v(\tau,x)=v_0(x)=z(\tau,\omega)u_0(x).
\end{equation}

As pointed out in \cite{Wang1}, for every $v_0\in L^2(\mathbb{R}^N)$
we may show that the problem \eqref{pr1}-\eqref{pr2}  possesses a continuous
solution $v(\cdot)$ on $L^2(\mathbb{R}^N)$ such that
$v(\cdot)\in C([\tau,+\infty),L^2(\mathbb{R}^N))\cap L^2{\rm loc}((\tau,+\infty),
 H^1(\mathbb{R}^N))\cap L^p{\rm loc}((\tau,+\infty), L^p(\mathbb{R}^N))$.
In addition, the solution $v$ is
 $(\mathcal{F},\mathcal{B}(L^2(\mathbb{R}^N)))$-measurable in $\Omega$.
Then formally $u(\cdot)=z^{-1}(.,\omega)v(\cdot)$ is a
$(\mathcal{F},\mathcal{B}(L^2(\mathbb{R}^N)))$-measurable and continuous
solution of problem \eqref{beq1} and \eqref{beq2} on $L^2(\mathbb{R}^N)$
with $u_0=z^{-1}(\tau,\omega)v_0$.

Define the mapping $\varphi: \mathbb{R}^+\times\mathbb{R}\times \Omega \times
L^2(\mathbb{R}^N)\to L^2(\mathbb{R}^N)$  such that
\begin{equation}\label{eq0}
\begin{aligned}
 \varphi(t,\tau,\omega, u_0)
&=u(t+\tau,\tau,\vartheta_{-\tau}\omega, u_0) \\
&=z^{-1} (t+\tau,\vartheta_{-\tau}\omega)
 v(t+\tau,\tau,\vartheta_{-\tau}\omega, z(\tau,\vartheta_{-\tau}\omega)u_0),
\end{aligned}
\end{equation}
where $u_0=u_{\tau}\in L^2(\mathbb{R}^N)$ and
 $t\in\mathbb{R}^+, \tau\in\mathbb{R}, \omega\in \Omega$.
Then by the measurability and  continuity of $v$ in  $v_0\in L^2(\mathbb{R}^N)$
and $t\in \mathbb{R}^+$, we see  that the mappings $\varphi$  is
$(\mathcal{B}(\mathbb{R}^+)\times \mathcal{F}\times
\mathcal{B}(L^2(\mathbb{R}^N)))\to \mathcal{B}(L^2(\mathbb{R}^N))$-measurable.
That is to say, the mappings $\varphi$ defined by \eqref{eq0} is
 a continuous cocycle on $L^2(\mathbb{R}^N)$  over $\mathbb{R}$ and
$(\Omega,\mathcal{F},P,\{\vartheta_t\}_{t\in\mathbb{R}})$.
 Furthermore, from \eqref{eq0} we infer that
\begin{equation} \label{eq00}
\begin{aligned}
 \varphi(t,\tau-t,\vartheta_{-t}\omega, u_0)
&=u(\tau,\tau-t,\vartheta_{-\tau}\omega, u_0) \\
&=z(-\tau,\omega)v(\tau,\tau-t,\vartheta_{-\tau}\omega,
 z(\tau-t,\vartheta_{-\tau}\omega)u_0),
\end{aligned}
\end{equation}
 where $u_0=u_{\tau-t}$.

 We define the collection  $\mathfrak{D}$ as
\begin{equation} \label{D}
\begin{aligned}
 \mathfrak{D}=\{&B=\{B(\tau,\omega)\subseteq L^2(\mathbb{R}^N);
 \tau\in\mathbb{R}, \omega\in \Omega\}; \\
&\lim_{t\to+\infty}e^{-\lambda t}z^2(-t,\omega)\|B(\tau-t,\vartheta_{-t}\omega)\|^2=0
\text{ for } \tau\in\mathbb{R}, \omega\in \Omega\}
\end{aligned}
\end{equation}
where $\|B\|=\sup_{v\in B}\|v\|_{L^2(\mathbb{R}^N)}$ and $\lambda$ is in \eqref{pr1}.
Note that this collection $\mathfrak{D}$ is much larger that
the collection defined by \cite{Wang1}. That  is to say, the collection
$\mathfrak{D}$ defined above includes all tempered families of bounded nonempty
subsets of $L^2(\mathbb{R}^N)$.

We can show that all the results in \cite{Wang1} hold for the collection
$\mathfrak{D}$ defined by \eqref{D}.  Thus, the existence and upper
semi-continuous of $\mathfrak{D}$-pullback  attractors for the cocycle
$\varphi_\varepsilon$ in \emph{the initial space} $L^2(\mathbb{R}^N)$
have been proved by \cite{Wang1}.

\begin{theorem}[\cite{Wang1}] \label{thm3.1}
Assume that \eqref{a1}-\eqref{a5} hold. Then the cocycle $\varphi_\varepsilon$
has a unique $\mathfrak{D}$-pullback  attractor
$\mathcal{A}_\varepsilon=\{\mathcal{A}_\varepsilon(\tau,\omega),\tau\in\mathbb{R},
 \omega\in \Omega\}$ in $L^2(\mathbb{R}^N)$, given by
\begin{equation}\label{L2}
\mathcal{A}_{\varepsilon}(\tau,\omega)=\cap_{s\geq0}
\overline{\cup_{t\geq s} \varphi(t,\tau-t,\vartheta_{-t}\omega,
K_\varepsilon(\tau-t,\vartheta_{-t}\omega))}^{L^2(\mathbb{R}^N)},
\end{equation}
for $\tau\in \mathbb{R}$ and $\omega\in \Omega$,
where $K_\varepsilon$ is a closed and measurable $\mathfrak{D}$-pullback bounded
absorbing set of $\varphi_\varepsilon$ in $L^2(\mathbb{R}^N)$. Furthermore,
$\mathcal{A}_\varepsilon$ is upper semi-continuous in $L^2(\mathbb{R}^N)$
at $\varepsilon=0$.
\end{theorem}

Note that in  most cases, we  write $v$ (resp. $\varphi$ and $z$) as
 the abbreviation of $v_\varepsilon$ (resp. $\varphi_\varepsilon$ and
$z_\varepsilon$).
Next, we consider some applications of Theorems \ref{thm2.6}--\ref{thm2.8}
 to the non-autonomous
stochastic reaction-diffusions \eqref{beq1} and \eqref{beq2}.
We emphasize that the result of
 Theorem \ref{thm3.1} holds in  the smooth functions space $H^1(\mathbb{R}^N)$.
In particular, we prove  the upper semi-continuity of the obtained attractors
 $\mathcal{A}_\varepsilon$ in $H^1(\mathbb{R}^N)$.

\section{Existence of pullback attractor in $H^{1}(\mathbb{R}^N)$}

In this section, we apply Theorem \ref{thm2.6} to  prove the existence  of
$\mathfrak{D}$-pullback  attractors in $H^1(\mathbb{R}^N)$ for the cocycle
defined in \eqref{eq0}.
 To this end, we need to prove the uniform smallness of solutions outside a
large ball under  $H^1(\mathbb{R}^N)$ norm (see Proposition \ref{prop4.4}),
and in the bounded ball of $\mathbb{R}^N$ we will prove the asymptotic
compactness of solutions by  space-splitting  and function-truncation
techniques (see Proposition \ref{prop4.5} and Lemma \ref{lem4.6}).


We consider that
$e^{-|\omega(s)|}\leq z(s,\omega)=e^{-\varepsilon \omega(s)}\leq e^{|\omega(s)|}$
for $\varepsilon\in(0,1]$, and that $\omega(s)$ is continuous function in $s$.
Then there exist two positive random constants
 $E=E(\omega)$ and  $F=F(\omega)$ depending only on $\omega$ such that for all
$s\in[-2,0]$ and $\varepsilon\in(0,1]$.
\begin{equation} \label{EE}
0<E\leq z(s,\omega)\leq F,\quad \omega\in\Omega.
\end{equation}

Hereafter, we denote by $\|\cdot\|, \|\cdot\|_p$ and $\|\cdot\|_{H^1}$ the
 norms in $L^2(\mathbb{R}^N),L^p(\mathbb{R}^N)$ and $H^1(\mathbb{R}^N)$,
respectively. The numbers $c$ and $C(\tau,\omega)$  are  two generic positive
constants which may have different values in different places even in the same line.
The first one depends only on $p, \lambda$ and $\alpha_{i}(i=1,2,3)$, and the
second one depends on $\tau, \omega, p, \lambda$ and $\alpha_{i}(i=1,2,3)$.
We always assume $p>2$ in the following discussions.

\subsection{$H^1$-tail estimate of solutions}

This can be achieved by a series of previously proved lemmas.
First we stress that \cite[Lemma 5.1]{Wang1} holds on the compact
interval $[\tau-1,\tau]$, which is necessary for
us to estimate of the tail of solutions in $H^1(\mathbb{R}^N)$.

\begin{lemma} \label{lem4.1}
 Assume that \eqref{a1} and \eqref{a3}-\eqref{a5} hold. Let
$\tau\in\mathbb{R}, \omega\in\Omega$,
 $B=\{B(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\}\in\mathfrak{D}$ and
$u_{0}\in B(\tau-t,\vartheta_{-t}\omega)$. Then there exists a constant
$T=T(\tau,\omega,B)\geq2$
such that for all $t\geq T$,  the solution $v$ of problem \eqref{pr1} and
\eqref{pr2} satisfies that for every $\zeta\in[\tau-1,\tau]$,
\begin{gather} \label{ee1}
\|v(\zeta,\tau-t,\vartheta_{-\tau}\omega, v_0)\|_{H^1(\mathbb{R}^N)}^2
\leq L_1(\tau,\omega,\varepsilon), \\
 \label{ee2}
\int_{\tau-2}^\tau \|v(s,\tau-t,\vartheta_{-\tau}\omega, v_0)\|_p^pds
\leq  L_1(\tau,\omega,\varepsilon),
\end{gather}
where $v_0=z(\tau-t,\vartheta_{-\tau}\omega)u_0$ and
$L_1(\tau,\omega,\varepsilon)=:cz^{-2}(-\tau,\omega)\int_{-\infty}^0
e^{\lambda s}z^2(s,\omega)( \|g(s+\tau,\cdot)\|^2+1)ds$.
\end{lemma}

The proof of the above lemma is similar to that of \cite[Lemma 5.1]{Wang1},
 with a small modification, using $\zeta\in[\tau-1,\tau]$ instead of $\tau$.


\begin{lemma} \label{lem4.2}
Assume that \eqref{a1} and \eqref{a3}-\eqref{a5} hold. Let
$\tau\in\mathbb{R}, \omega\in\Omega$ and
$B=\{B(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\}\in\mathfrak{D}$.
Then for every $\eta>0$, there exist two constants $T=T(\tau,\omega,\eta,B)\geq2$
and $R=R(\tau,\omega,\eta)>1$ such that the weak solution $v$ of  \eqref{pr1}
 and \eqref{pr2} satisfies that for all $t\geq T$ and $k\geq R$,
\begin{align*}
&\int_{|x|\geq k}| v(\tau,\tau-t,\vartheta_{-\tau}\omega,
 z(\tau-t,\vartheta_{-\tau}\omega)u_0)|^2dx \\
&+\int_{\tau-1}^\tau\int_{|x|\geq k}|\nabla v(s,\tau-t,
 \vartheta_{-\tau}\omega, z(\tau-t,\vartheta_{-\tau}\omega)u_0)|^2\,dx\,ds\leq\eta,
\end{align*}
where $u_0\in B(\tau-t,\vartheta_{-t}\omega)$,  $R$ and $T$ are independent
of $\varepsilon$.
\end{lemma}

The proof of the above lemma is a simple modification of the proof
of \cite[Lemma 5.5]{Wang1}.

\begin{lemma} \label{lem4.3}
Assume that \eqref{a1} and \eqref{a3}-\eqref{a5} hold.
Let $\tau\in\mathbb{R}, \omega\in\Omega$ and
$B=\{B(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\}\in\mathfrak{D}$.
Then  there exists $T=T(\tau,\omega,B)\geq 2$
such that the weak solution $v$ of problem \eqref{pr1}-\eqref{pr2} satisfies
that for all $t\geq T$,
\begin{gather}\label{4b1}
\int_{\tau-1}^\tau\| v(s,\tau-t,\vartheta_{-\tau}\omega,
z(\tau-t,\vartheta_{-\tau}\omega)u_0)\|_{2p-2}^{2p-2}ds
\leq L_2(\tau,\omega,\varepsilon), \\
\label{4b2}
\int_{\tau-1}^\tau \| v_s(s,\tau-t,\vartheta_{-\tau}\omega,
 z(\tau-t,\vartheta_{-\tau}\omega)u_0)\|^2ds\leq L_2(\tau,\omega,\varepsilon),
\end{gather}
where $v_s=\frac{\partial v}{\partial s}$, $u_0\in B(\tau-t,\vartheta_{-t}\omega)$
and
\begin{equation}
L_2(\tau,\omega,\varepsilon)=: C(\tau,\omega)
\int_{-\infty}^{0}e^{\lambda s}(z^2(s,\omega)+z^p(s, \omega))
(\|g(s+\tau,\cdot)\|^2+1)ds.
\end{equation}
\end{lemma}


\begin{proof} In the sequel, we always regard $v$ as a solution at the time $t$
with the initial value $v_0=v_{\tau-t}$ at the initial time $\tau-t$.
 We multiply \eqref{pr1} by $|v|^{p-2}v$  and then integrate over $\mathbb{R}^N$
to yield  that
\begin{equation} \label{b4.12}
\begin{aligned}
&\frac{1}{p}\frac{d}{dt}\|v\|_{p}^p +\lambda\|v\|_p^p \\
&\leq z(t,\omega)\int_{\mathbb{R}^N}f(x,z^{-1}v)
|v|^{p-2}v\,dx+z(t,\omega)\int_{\mathbb{R}^N}|v|^{p-2}v{g}dx.
\end{aligned}
\end{equation}
By using \eqref{a1},  we see that
\begin{equation} \label{b4.13}
\begin{aligned}
&z(t,\omega)\int_{\mathbb{R}^N}f(x,z^{-1}v) |v|^{p-2}v\,dx \\
&\leq -\alpha_1z^{2-p}(t,\omega)\int_{\mathbb{R}^N}|v|^{2p-2}dx
 +z^2(t,\omega)\int_{\mathbb{R}^N}\psi_1(x)|v|^{p-2}dx \\
& \leq-\alpha_1z^{2-p}(t,\omega)\int_{\mathbb{R}^N}|v|^{2p-2}dx
 +\frac{\lambda}{2}\|v\|_p^p
 +\Big(\frac{2}{\lambda}\Big)^{-\frac{p-2}{2}}z^{p}(t,\omega)\|\psi_1\|^{p/2}_{p/2},
\end{aligned}
\end{equation}
where  the $\epsilon$-Young's inequality are repeatedly used:
\begin{equation} \label{Young}
|ab|\leq \epsilon |a|^m+\epsilon^{-q/p}|b|^n,\quad \epsilon>0,\;
 m>1,n>1,\; \frac{1}{m}+\frac{1}{n}=1.
\end{equation}
At the same time, the last term on the right hand side  of \eqref{b4.12}
is bounded as
\begin{equation} \label{b4.14}
\begin{aligned}
&z(t,\omega)\int_{\mathbb{R}^N}|v|^{p-2}v{g}dx\\
&\leq \frac{1}{2}\alpha_1z^{2-p}(t,\omega)\int_{\mathbb{R}^N}|v|^{2p-2}dx
 +\frac{1}{2\alpha_1}z^p(t,\omega)\|g(t,\cdot)\|^2.
\end{aligned}
\end{equation}
 By a combination of \eqref{b4.12}-\eqref{b4.14},  noticing that $p>2$,
 we obtain that
\begin{equation} \label{b4.15}
\frac{d}{dt}\|v\|_{p}^p+\lambda \|v\|_{p}^p
+\alpha_1z^{2-p}(t,\omega)\|v\|_{2p-2}^{2p-2}\leq c
z^p(t,\omega)(\|g(t,\cdot)\|^2+1),
\end{equation}
where $c$ only depends $p,\lambda$ and $\alpha_1$.
 Applying \cite[Lemma 5.1]{Zhao0}(or \cite{Zhao5}) over the interval
$[\tau-2,\zeta]$, $\zeta\in[\tau-1,\tau]$, along with  $\omega$
being replaced by $\vartheta_{-\tau}\omega$, we deduce that
\begin{equation} \label{b4.16}
\begin{aligned}
&\|v(\zeta, \tau-t,\vartheta_{-\tau}\omega, v_0)\|_{p}^p \\
&\leq \frac{e^\lambda}{\zeta-\tau+2}\int_{\tau-2}^{\tau} e^{\lambda (s-\tau)}
 \|v(s,\tau-t,\vartheta_{-\tau}\omega,v_0)\|_{p}^pds \\
&\quad +ce^{\lambda}z^{-p}(-\tau,\omega)\int_{-\infty}^{0} e^{\lambda s}
  z^p(s, \omega)(\|g(s+\tau,\cdot)\|^2+1)ds.
\end{aligned}
\end{equation}
Since $\frac{e^\lambda}{\zeta-\tau+2}\leq 1$ for $\zeta\in[\tau-1,\tau]$,
then by \eqref{ee2} and \eqref{b4.16} we find that there exists $T>2$ such that
for all $t\geq T$,
\begin{equation} \label{b4.17}
\begin{aligned}
&\|v(\zeta, \tau-t, \vartheta_{-\tau}\omega, v_0)\|_{p}^p \\
&\leq C(\tau,\omega)\int_{-\infty}^{0}e^{\lambda s}(z^2(s,\omega)
 +z^p(s, \omega))(\|g(s+\tau,\cdot)\|^2+1)ds.
\end{aligned}
\end{equation}
Integrating \eqref{b4.15} over the interval $[\tau-1,\tau]$,
with $\omega$ replaced by $\vartheta_{-\tau}\omega$, yields
\begin{equation} \label{b4.20}
\begin{aligned}
&\alpha_1\int_{\tau-1}^\tau z^{2-p}(s,\vartheta_{-\tau}\omega)
\|v(s,\tau-t,\vartheta_{-\tau}\omega, v_0)\|_{2p-2}^{2p-2}ds \\
&\leq \|v(\tau-1, \tau-t,\vartheta_{-\tau}\omega, v_0)\|_{p}^p
 +c\int_{\tau-1}^\tau z^p(s,\vartheta_{-\tau}\omega)(\|{g}(s,\cdot)\|^2+1)ds \\
&\leq\|v(\tau-1, \tau-t,\vartheta_{-\tau}\omega, v_0)\|_{p}^p \\
&\quad +ce^{-\lambda}\int_{\tau-1}^\tau e^{\lambda(s-\tau)}z^p
 (s,\vartheta_{-\tau}\omega)(\|{g}(s,\cdot)\|^2+1)ds.
\end{aligned}
\end{equation}
Then from \eqref{EE}, \eqref{b4.17} and \eqref{b4.20} we deduce for all $t\geq T$,
\begin{align*}
&\int_{\tau-1}^\tau \|v(s,\tau-t,\vartheta_{-\tau}\omega, v_0)\|_{2p-2}^{2p-2}ds \\
&\leq C(\tau,\omega)\int_{-\infty}^{0}e^{\lambda s}(z^2(s,\omega)+z^p
 (s, \omega))(\|g(s+\tau,\cdot)\|^2+1)ds,
\end{align*}
which proves \eqref{4b1}.

To estimate the derivative $v_t$ in $L^2{\rm loc}(\mathbb{R},L^2(\mathbb{R}^N))$,
we multiply \eqref{pr1} by $v_t$ and integrate
over $\mathbb{R}^N$ to produce
\begin{align*}
&\|v_t\|^2+\frac{1}{2}\frac{d}{dt}(\lambda\|v\|^2+\|\nabla v\|^2) \\
&=z(t,\omega)\int_{\mathbb{R}^N}f(x,z^{-1}v)v_tdx+z(t,\omega)
 \int_{\mathbb{R}^N}gv_tdx \\
&\leq\frac{1}{2}\|v_t\|^2+c\alpha_2^2z^{4-2p}(t,\omega)\|v\|^{2p-2}_{2p-2}
 +cz^2(t,\omega)\|\psi_2\|^2+cz^2(t,\omega)\|g(t,\cdot)\|^2,
\end{align*}
i.e., we have
\begin{equation}\label{b4.22}
\begin{aligned}
&\|v_t\|^2+\frac{d}{dt}(\lambda\|v\|^2+\|\nabla v\|^2) \\
&\leq cz^{4-2p}(t,\omega)\|v\|^{2p-2}_{2p-2}+cz^2(t,\omega)
 (\|g(t,\cdot)\|^2+\|\psi_2\|^2).
\end{aligned}
\end{equation}
Integrate \eqref{b4.22} over the interval $[\tau-1,\tau]$ to obtain
\begin{equation} \label{b4.305}
\begin{aligned}
&\int_{\tau-1}^\tau\|v_s(s,\tau-t,\vartheta_{-\tau}\omega,v_0)\|^2ds \\
&\leq c\int_{\tau-1}^\tau z^{4-2p}(s,\vartheta_{-\tau}\omega)
 \|v(s,\tau-t,\vartheta_{-\tau}\omega,v_0)\|^{2p-2}_{2p-2}ds \\
&\quad +c\int_{\tau-1}^\tau z^2(s,\vartheta_{-\tau})(\|g(s,\cdot)\|^2+1)ds
 +c\|v(\tau-1,\tau-t,\vartheta_{-\tau}\omega,v_0)\|_{H^1}^2.
\end{aligned}
\end{equation}
Then by \eqref{EE}, \eqref{ee1}, \eqref{4b1} and \eqref{b4.305} we get that for
all $t\geq T$,
\begin{equation}\label{b4.3b}
\begin{aligned}
&\int_{\tau-1}^\tau \|v_s(s,\tau-t,\vartheta_{-\tau}\omega,v_0)\|^2ds \\
&\leq C(\tau,\omega)\int_{-\infty}^{0}e^{\lambda s}(z^2(s,\omega)
 +z^p(s, \omega))(\|g(s+\tau,\cdot)\|^2+1)ds,
\end{aligned}
\end{equation}
where $T$ is as in Lemma \ref{lem4.1}. This completes the proof.
\end{proof}

We now can give the $H^1$-tail estimate of solutions of problem \eqref{pr1}
and \eqref{pr2}, which is one crucial condition for proving the asymptotic
compactness in $H^1(\mathbb{R}^N)$.


\begin{proposition} \label{prop4.4}
Assume that \eqref{a1}-\eqref{a5} hold. Let  $\tau\in\mathbb{R}, \omega\in\Omega$
and  $B=\{B(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\}\in\mathfrak{D}$.
Then for every $\eta>0$, there exist two constants $T=T(\tau,\omega, \eta,B)\geq2$
and $R=R(\tau,\omega,\eta)>1$ such that the weak solution $v$ of  \eqref{pr1}
and \eqref{pr2} satisfies that for all $t\geq T$,
\begin{align*}
&\int_{|x|\geq R}\Big(|v(\tau,\tau-t,\vartheta_{-\tau}\omega,
  z(\tau-t,\vartheta_{-\tau}\omega)u_0)|^2 \\
&+|\nabla v(\tau,\tau-t,\vartheta_{-\tau}\omega,
 z(\tau-t,\vartheta_{-\tau}\omega)u_0)|^2\Big)dx \leq\eta,
\end{align*}
where $u_0\in B(\tau-t,\vartheta_{-t}\omega)$ and $R,T$ are independent of
$\varepsilon$.
\end{proposition}

\begin{proof}
 We first need to define a smooth function  $\xi(\cdot)$ on $\mathbb{R}^+$ such that
\[
\xi(s)=
\begin{cases}
0,& \text{if } 0\leq s\leq 1,\\
0\leq\xi(s)\leq1, &\text{if }  1\leq s\leq 2,\\
1,&\text{if } s\geq 2,
\end{cases}
\]
which obviously implies that there is a  positive constant $C_1$ such that
 the $|\xi'(s)|+|\xi''(s)|\leq C_1$ for all $s\geq0$.
For convenience, we write $\xi=\xi(\frac{|x|^2}{k^2})$.

We multiply \eqref{pr1} by $-\xi\Delta v$ and integrate  over $\mathbb{R}^N$
to find that
\begin{equation}\label{b4.25}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^N}\xi|\nabla v|^2dx
 +\int_{\mathbb{R}^N}(\nabla\xi.\nabla v)v_tdx
 +\lambda\int_{\mathbb{R}^N}\xi|\nabla v|^2dx \\
&+\lambda\int_{\mathbb{R}^N}(\nabla\xi.\nabla v)v\,dx
 +\int_{\mathbb{R}^N}\xi|\Delta v|^2dx \\
&=-z(t,\omega)\int_{\mathbb{R}^N}f(x,z^{-1}v)\xi\Delta v\,dx
 -z(t,\omega)\int_{\mathbb{R}^N}g\xi\Delta v\,dx.
\end{aligned}
\end{equation}
Now, we estimate each term in \eqref{b4.25} as follows. First  we have
\begin{equation}\label{b4.26}
\begin{aligned}
\Big|\int_{\mathbb{R}^N}(\nabla\xi.\nabla v)v_tdx
 +\lambda\int_{\mathbb{R}^N}(\nabla\xi.\nabla v)v\,dx\Big|
&=\Big|\int_{\mathbb{R}^N}(v_t+\lambda v)(\frac{2x}{k^2}.\nabla v)\xi' dx\Big| \\
&\leq \frac{c}{k}(\|v_t\|^2+\|v\|_{H^1}^2).
\end{aligned}
\end{equation}
For the  nonlinearity in \eqref{b4.25},  we see that
\begin{equation}\label{b4.27}
\begin{aligned}
&-z\int_{\mathbb{R}^N}f(x,z^{-1}v)\xi \Delta v\,dx \\
&=z\int_{\mathbb{R}^N}f(x,z^{-1}v)(\nabla\xi.\nabla v)dx
 +z\int_{\mathbb{R}^N}(\frac{\partial}{\partial x} f(x,z^{-1}v).\nabla v)\xi dx \\
&\quad +\int_{\mathbb{R}^N}\frac{\partial}{\partial u}f(x,z^{-1}v)|\nabla v|^2\xi dx.
\end{aligned}
\end{equation}
On the other hand, by using\ \eqref{a2}, \eqref{a3} and  \eqref{a4}, respectively,
 we calculate that
\begin{gather}\label{b4.28}
\begin{aligned}
\Big|z\int_{\mathbb{R}^N}f(x,z^{-1}v)(\nabla\xi.\nabla v)dx\Big|
&\leq \frac{2z\sqrt{2}C_1}{k}\int_{k\leq |x|\leq \sqrt{2}k}|f(x,z^{-1}v)||\nabla v|dx
\\
&\leq \frac{c}{k}(z^{4-2p}\|v\|_{2p-2}^{2p-2}+z^2\|\psi_2\|^2+\|\nabla v\|^2) ,
\end{aligned} \\
\label{b4.29}
\int_{\mathbb{R}^N}\frac{\partial}{\partial u}f(x,z^{-1}v)|\nabla v|^2\xi dx
\leq \alpha_3\int_{\mathbb{R}^N}\xi|\nabla v|^2 dx, \\
\label{b4.30}
\begin{aligned}
\Big|z\int_{\mathbb{R}^N}(\frac{\partial}{\partial x} f(x,z^{-1}v).\nabla v)\xi dx\Big|
&\leq \Big|z\int_{\mathbb{R}^N}|\psi_3||\nabla v|\xi dx\Big| \\
&\leq \frac{\lambda}{2}\int_{\mathbb{R}^N}\xi|\nabla v|^2 dx
 +cz^2\int_{\mathbb{R}^N}\xi|\psi_3|^2dx.
\end{aligned}
\end{gather}
Then from \ref{b4.27})-\eqref{b4.30} it follows that
\begin{equation}\label{b4.31}
\begin{aligned}
&-z\int_{\mathbb{R}^N}f(x,z^{-1}v)\xi \Delta v\,dx \\
&\leq \frac{c}{k}(z^{4-2p}\|v\|_{2p-2}^{2p-2}+z^2\|\psi_2\|^2+\|\nabla v\|^2) \\
&+\frac{\lambda}{2}\int_{\mathbb{R}^N}\xi|\nabla v|^2 dx
 +cz^2\int_{\mathbb{R}^N}\xi|\psi_3|^2dx+\alpha_3\int_{\mathbb{R}^N}\xi|\nabla v|^2 dx.
\end{aligned}
\end{equation}
For the last term on the right-hand side of \eqref{b4.25},  we have
\begin{equation}\label{b4.32}
\Big|z\int_{\mathbb{R}^N}g\xi \Delta v\,dx\Big|
\leq \frac{\lambda}{2}\int_{\mathbb{R}^N}\xi |\Delta v|^2dx
+\frac{1}{2\lambda}z^2\int_{\mathbb{R}^N}\xi |{g}|^2dx.
\end{equation}
Then  we use \eqref{b4.26} and \eqref{b4.31}--\eqref{b4.32} in \eqref{b4.25}
to find that
\begin{equation}\label{b4.33}
\begin{aligned}
&\frac{d}{dt}\int_{\mathbb{R}^N}\xi |\nabla v|^2dx
+\lambda\int_{\mathbb{R}^N}\xi |\nabla v|^2dx \\
&\leq \frac{c}{k}(\|v_t\|^2+\|v\|_{H^1}^2+z^{4-2p}\|v\|_{2p-2}^{2p-2}
 +z^2\|\psi_2\|^2) \\
&+2\alpha_3\int_{\mathbb{R}^N}\xi|\nabla v|^2 dx
 +cz^2\int_{\mathbb{R}^N}\xi(|\psi_3|^2+|g|^2)dx.
\end{aligned}
\end{equation}
Applying \cite[Lemma 5.1]{Zhao0} to  \eqref{b4.33} over the interval
$[\tau-1,\tau]$, along with  $\omega$ being replaced by $\vartheta_{-\tau}\omega$,
we deduce that
\begin{equation}\label{b4.34}
\begin{aligned}
&\int_{\mathbb{R}^N}\xi |\nabla v(\tau,\tau-t,\vartheta_{-\tau}\omega,v_0)|^2dx \\
&\leq \frac{c}{k}\int_{\tau-1}^\tau e^{\lambda(s-\tau)}
 \Big(\|v_s(s)\|^2+\|v(s)\|_{H^1}^2+z^{4-2p}(s,\vartheta_{-\tau}\omega)
 \|v(s)\|_{2p-2}^{2p-2} \\
&\quad +z^2(s,\vartheta_{-\tau}\omega)\|\psi_2\|^2\Big)ds
 +c\int_{\tau-1}^\tau e^{\lambda(s-\tau)} \int_{|x|\geq k}|\nabla v(s)|^2 \,dx\,ds \\
& \quad  +cz^{-2}(\tau,\omega)\int_{-1}^0 e^{\lambda s}z^2
(s,\omega)\int_{|x|\geq k}(|\psi_3|^2+|g(s+\tau,x)|^2)\,dx\,ds,
\end{aligned}
\end{equation}
where $v(s)=v(s,\tau-t,\vartheta_{-\tau}\omega,z(\tau-t,\vartheta_{-\tau}\omega)u_0)$.
Our task in the following is to show that
each term on the right hand side of \eqref{b4.34} vanishes when $t$ and $k$ are
larger. First,
by Lemma \ref{lem4.2},  there are two constants $T_1=T_1(\tau,\omega,B,\eta)\geq2$ and
$R_1=R_1(\tau,\omega,\eta)\geq1$ such that
for all $t\geq T_1$ and $k\geq R_1$,
\begin{equation} \label{b4.36}
\begin{aligned}
&c\int_{\tau-1}^\tau e^{\lambda(s-\tau)} \int_{|x|\geq k}|\nabla v(s)|^2 \,dx\,ds\\
&\leq c\int_{\tau-1}^\tau \int_{|x|\geq k}|\nabla v(s,\tau-t,\vartheta_{-\tau}
\omega,v_0)|^2 \,dx\,ds
\leq \frac{\eta}{6}.
\end{aligned}
\end{equation}
By \eqref{ee1} in Lemma \ref{lem4.1},  there exist $T_2=T_2(\tau,\omega,B)\geq2$ and
$R_2=R_2(\tau,\omega,\eta)\geq1$ such that  for all $t\geq T_2$ and $k\geq R_2$,
\begin{equation} \label{b4.38}
\begin{aligned}
&\frac{c}{k}\int_{\tau-1}^\tau  e^{\lambda(s-\tau)}
\|v(s,\tau-t,\vartheta_{-\tau}\omega,v_0)\|_{H^1}^2ds \\
&\leq\frac{c}{k}\int_{\tau-1}^\tau  \|v(s,\tau-t,\vartheta_{-\tau}
\omega,v_0)\|_{H^1}^2ds
\leq \frac{\eta}{6}.
\end{aligned}
\end{equation}
By Lemma \ref{lem4.3},  there exist $T_3=T_3(\tau,\omega,B)\geq2$ and
$R_3=R_3(\tau,\omega,\eta)\geq1$ such that  for all $t\geq T_3$ and  $k\geq R_3$,
\begin{equation} \label{b4.39}
\begin{aligned}
&\frac{c}{k}\int_{\tau-1}^\tau e^{\lambda(s-\tau)} z^{4-2p}
 (s,\vartheta_{-\tau}\omega)\|v(s,\tau-t,\vartheta_{-\tau}
 \omega,v_0)\|_{2p-2}^{2p-2}ds  \\
&\leq \frac{c}{k}z^{2p-4}(-\tau,\omega)E^{4-2p}L_2(\tau,\omega,\varepsilon)
\leq \frac{\eta}{6},
\end{aligned}
\end{equation}
and
\begin{equation} \label{b4.40}
\frac{c}{k}\int_{\tau-1}^\tau  e^{\lambda(s-\tau)}
\|v_s(s,\tau-t,\vartheta_{-\tau}\omega,v_0)\|^2ds
\leq \frac{c}{k}L_2(\tau,\omega,\varepsilon)\leq \frac{\eta}{6}.
\end{equation}
By the assumptions on $\psi_3$ and $g$, we deduce that there exist
$R_4=R_4(\tau,\omega,\eta)$ such that for all $k\geq R_4$,
\begin{equation} \label{b4.37}
cz^{-2}(\tau,\omega)\int_{-1}^0 e^{\lambda s}z^2(s,\omega)
\int_{|x|\geq k}(|\psi_3|^2+|g(s+\tau,x)|^2)\,dx\,ds\leq \frac{\eta}{6}.
\end{equation}
Obviously,  there exists $R_5=R_5(\tau,\omega,\eta)$ such that  for all
$k\geq R_5$,
\begin{equation} \label{b4.41}
\begin{aligned}
&\frac{c}{k}\int_{\tau-1}^\tau e^{\lambda(s-\tau)} z^2(s,\vartheta_{-\tau}\omega)
\|\psi_2\|^2ds \\
&\leq\frac{c}{k}\|\psi_2\|^2z^{-2}(-\tau,\omega)\int_{-1}^0z^2(s, \omega)ds
\leq \frac{\eta}{6},
\end{aligned}
\end{equation}
where $\int_{-1}^0z^2(s, \omega)ds<+\infty$. Finally,
take
$$
T=\{T_1,T_2,T_3\},\quad   {R}=\max\{R_1,R_2,R_3,R_4,R_5\}.
$$
 It is obvious that $R$ and $T$ are independent of the intension $\varepsilon$.
Then \eqref{b4.36}-\eqref{b4.41} are integrated  into \eqref{b4.34} to get
that for all $t\geq T$ and $k\geq R$,
 \begin{equation}
\int_{|x|\geq \sqrt{2} k} |\nabla v(\tau,\tau-t,\vartheta_{-\tau}\omega,v_0)|^2dx
\leq \eta.
\end{equation}
  Then in connection with Lemma  \ref{lem4.2}, the desired result is achieved.
\end{proof}


\subsection{Estimate of the truncation of solutions in $L^{2p-2}$}

Given $u$ the solution of problem \eqref{beq1} and \eqref{beq2}, for each
fixed $\tau\in\mathbb{R},\omega\in\Omega$,
we write $M=M(\tau,\omega)>1$ and
\begin{equation} \label{}
\mathbb{R}^N(|u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_0)|\geq M)
=\{x\in \mathbb{R}^N; |u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_0)|\geq M|\}.
\end{equation}

We introduce the truncation version of solutions of problem \eqref{pr1}-\eqref{pr2}.
Let  $(v-M)_+$ be the positive part of $v-M$, i.e.,
$$
(v-M)_+= \begin{cases}
    v-M, &\text{if } v> M;\\
  0, & \text{if } v\leq M.
       \end{cases}
$$
The next lemma shows that the integral of  $L^{2p-2}$-norm of $|u|$
over the interval $[\tau-1,\tau]$  vanishes   on the state domain
$\mathbb{R}^N(|u(\tau,\tau-t,\vartheta_{-\tau}\omega),u_0)|\geq M)$ for $M$
large enough, which is the second crucial condition
for proving the asymptotic compactness of solutions in $H^1(\mathbb{R}^N)$.

\begin{proposition} \label{prop4.5}
Assume that \eqref{a1}-\eqref{a5} hold. Let $\tau\in\mathbb{R}, \omega\in\Omega$,
\[
B=\{B(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\}\in\mathfrak{D}
\]
 and $u_0\in B(\tau-t,\vartheta_{-t}\omega)$. Then for any $\eta>0$, there exist
constants $\tilde{M}=\tilde{M}(\tau,\omega,\eta)>1$  and $T=T(\tau,\omega,B)\geq 2$
such that the solution $u$ of problem \eqref{pr1} and \eqref{pr2} satisfies
that for all $t\geq T$ and  all $\varepsilon\in(0,1]$,
\[
\int_{\tau-1}^\tau e^{\tilde{\varrho}(s-\tau)}
\int_{\mathcal{O}}|v(s,\tau-t,\vartheta_{-\tau}\omega,
 z(\tau-t,\vartheta_{-\tau}\omega)u_0)|^{2p-2} \,dx\,ds\leq \eta,
\]
 where $p>2$,  $\tilde{M}$ and $T$ are independent of  $\varepsilon$,
\[
\mathcal{O}=\mathbb{R}^N(|v(s,\tau-t,\vartheta_{-\tau}\omega,
z(\tau-t,\vartheta_{-\tau}\omega)u_0)|\geq \tilde{M})
\]
 and
$$
\tilde{\varrho}=\tilde{\varrho}(\tau,\omega,\tilde{M})
=\alpha_1 F^{2-p}e^{-(p-2)|\omega(-\tau)|}\tilde{M}^{p-2}.
$$
\end{proposition}

\begin{proof}
First,  we replace  $\omega$ by $\vartheta_{-\tau}\omega$ in \eqref{pr1}
to see that
$$
v=v(s)=:v(s,\tau-t,\vartheta_{-\tau}\omega,v_0),\quad  s\in[\tau-1,\tau],
$$ is a solution of the SPDE
\begin{equation} \label{p011}
\frac{dv}{ds}+\lambda v-\Delta v=\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}f(x,u)
+\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}{g}(s,x),
\end{equation}
 with the initial data $v_0=z(\tau-t,\vartheta_{-\tau}\omega)u_0$,
where we have used
$z(s,\vartheta_{-\tau}\omega)=\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}>0$.

We  multiply \eqref{p011} by  $(v-M)_+^{p-1}$  and integrate over $\mathbb{R}^N$
to obtain that for every $s\in[\tau-1,\tau]$,
\begin{equation} \label{p01}
\begin{aligned}
&\frac{1}{p}\frac{d}{ds}\int_{\mathbb{R}^N} (v-M)_+^{p}dx
 +\lambda\int_{\mathbb{R}^N} v(v-M)_+^{p-1}dx
 -\int_{\mathbb{R}^N}\Delta v(v-M)_+^{p-1}dx \\
&=\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}
 \int_{\mathbb{R}^N}f(x,u)(v-M)_+^{p-1}dx
 +\frac{z(s-\tau,\omega)}{z(-\tau,\omega)} \int_{\mathbb{R}^N}{g}
 (s,x)(v-M)_+^{p-1}dx.
\end{aligned}
\end{equation}
We now need to estimate every term in \eqref{p01}. First, it is obvious that
\begin{gather} \label{p02}
-\int_{\mathbb{R}^N}\Delta v(v-M)_+^{p-1}dx
=(p-1)\int_{\mathbb{R}^N}(v-M)_+^{p-2}|\nabla v|^2dx\geq0, \\
 \label{p03}
\lambda\int_{\mathbb{R}^N} v(v-M)_+^{p-1}dx
\geq \lambda\int_{\mathbb{R}^N} (v-M)_+^{p}dx.
\end{gather}
If $v> M$, then $u=z^{-1}(s,\vartheta_{-\tau}\omega)v>0$. Therefore
by  assumption \eqref{a1},  we have
\begin{equation} \label{er}
\begin{aligned}
 f(x,u)
&\leq -\alpha_1u^{p-1}+\frac{\psi_1(x)}{u} \\
&=-\alpha_1\Big(\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}\Big)^{1-p}v^{p-1}
 +\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}\frac{\psi_1(x)}{v}.
\end{aligned}
\end{equation}
 Since $s\in[\tau-1,\tau]$ and $p>2$, then by \eqref{EE} we have
 $$
F^{2-p}\leq z^{2-p}(s-\tau,\omega)\leq E^{2-p},
$$
from which and \eqref{er} it follows that
\begin{align*}
&\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}f(x,u) \\
&\leq -\alpha_1\Big(\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}
 \Big)^{2-p}v^{p-1}+\frac{z^2(s-\tau,\omega)}{z^2(-\tau,\omega)}\frac{\psi_1(x)}{v} \\
&=-\frac{\alpha_1}{2}\Big(\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}
 \Big)^{2-p}v^{p-1}-\frac{\alpha_1}{2}
 \Big(\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}\Big)^{2-p}v^{p-1}
 +\frac{z^2(s-\tau,\omega)}{z^2(-\tau,\omega)}\frac{\psi_1(x)}{v} \\
&\leq -\frac{\alpha_1}{2}\frac{F^{2-p}}{z^{2-p}(-\tau,\omega)}M^{p-2}(v-M)
 -\frac{\alpha_1}{2}\frac{F^{2-p}}{z^{2-p}(-\tau,\omega)}(v-M)^{p-1} \\
&\quad +\frac{F^2}{z^2(-\tau,\omega)}|\psi_1(x)|(v-M)^{-1},
\end{align*}
which by the nonlinearity in \eqref{p01} is estimated as
\begin{equation} \label{p04}
\begin{aligned}
&\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}
 \int_{\mathbb{R}^N}f(x,u)(v-M)_+^{p-1}dx \\
& \leq-\frac{\alpha_1}{2}\frac{F^{2-p}}{z^{2-p}(-\tau,\omega)}M^{p-2}
 \int_{\mathbb{R}^N}(v-M)_+^{p}dx 
 -\frac{\alpha_1}{2}\frac{F^{2-p}}{z^{2-p}(-\tau,\omega)} \\
&\quad \times  \int_{\mathbb{R}^N}(v-M)_+^{2p-2}dx 
 + \frac{F^2}{z^2(-\tau,\omega)}\int_{\mathbb{R}^N}|\psi_1(x)|(v-M)_+^{p-2}dx \\
&\leq-\frac{\alpha_1}{2}\frac{F^{2-p}}{z^{2-p}(-\tau,\omega)}M^{p-2}
 \int_{\mathbb{R}^N}(v-M)_+^{p}dx 
  -\frac{\alpha_1}{2}\frac{F^{2-p}}{z^{2-p}(-\tau,\omega)} \\
&\quad\times  \int_{\mathbb{R}^N}(v-M)_+^{2p-2}dx 
 +\frac{1}{2}\lambda\int_{\mathbb{R}^N}(v-M)_+^{p}dx \\
&\quad +\frac{cF^p}{z^p(-\tau,\omega)}\int_{\mathbb{R}^N(v\geq M)}|\psi_1(x)|^{p/2}dx,
\end{aligned}
\end{equation}
where the last term the $\epsilon$-Young's inequality
\eqref{Young} is used.
The second term on the right-hand side of \eqref{p01} is bounded as
\begin{equation} \label{p05}
\begin{aligned}
&\frac{F}{z(-\tau,\omega)}\Big| \int_{\mathbb{R}^N}{g}(s,x)(v(s)-M)_+^{p-1}dx\Big| \\
&\leq \frac{\alpha_1}{4}\frac{F^{2-p}}{z^{2-p}(-\tau,\omega)}
\int_{\mathbb{R}^N}(v-M)_+^{2p-2}dx \\
&\quad +\frac{1} {\alpha_1}\frac{F^p}{z^p(-\tau,\omega)}
\int_{\mathbb{R}^N(v(s)\geq M)}{g}^2(s,x)dx.
\end{aligned}
\end{equation}
By a combination of  \eqref{p01}--\eqref{p05}, we obtain
\begin{equation} \label{p060}
\begin{aligned}
&\frac{d}{ds}\int_{\mathbb{R}^N} (v(s)-M)_+^{p}dx
 +\frac{\alpha_1 F^{2-p}}{z^{2-p}(-\tau,\omega)}M^{p-2}
 \int_{\mathbb{R}^N}(v(s)-M)_+^{p}dx \\
&+\frac{\alpha_1F^{2-p}}{z^{2-p}(-\tau,\omega)}
 \int_{\mathbb{R}^N}(v-M)_+^{2p-2}dx \\
&\leq \frac{cF^p}{z^p(-\tau,\omega)}\Big(\|{g}(s,\cdot)\|^2+
\|\psi_1\|_{p/2}^{p/2}\Big),
\end{aligned}
\end{equation}
where the positive constant $c$  is independent of $\varepsilon,\tau,\omega$ and $M$.
  Note that for each $\tau\in \mathbb{R}$ and $\varepsilon\in(0,1]$,
\begin{equation} \label{TT}
 e^{-|\omega(-\tau)|} \leq z(-\tau,\omega)
=e^{-\varepsilon \omega(-\tau)}\leq e^{|\omega(-\tau)|}.
 \end{equation}
Here for convenience, we put
\begin{gather*}
 \varrho=\varrho(\tau,\omega,M)=\alpha_1 F^{2-p}e^{-(p-2)|\omega(-\tau)|}M^{p-2}>0,\\
d=d(\tau,\omega)=\alpha_1F^{2-p}e^{-(p-2)|\omega(-\tau)|}>0,
\end{gather*}
where $d$ is unchanged  and $\varrho\to+\infty$ as $M\to+\infty$.
Then  from \eqref{p060} and $\eqref{TT}$  we infer that
\begin{equation} \label{p06}
\begin{aligned}
&\frac{d}{ds}\int_{\mathbb{R}^N} (v(s)-M)_+^{p}dx
+\varrho\int_{\mathbb{R}^N}(v(s)-M)_+^{p}dx
 +d\int_{\mathbb{R}^N}(v-M)_+^{2p-2}dx \\
&\leq cF^pe^{p|\omega(-\tau)|}\Big(\|{g}(s,\cdot)\|^2+1\Big),
\end{aligned}
\end{equation}
 where $s\in[\tau-1,\tau]$ and $\varrho,E,F$ are independent of $\varepsilon$
and $t$. By using \cite[Lemma 5.1]{Zhao0}  to \eqref{p06} over the interval
$[\tau-1,\tau]$, we find that
 \begin{equation} \label{p07}
\begin{aligned}
&\int_{\tau-1}^\tau e^{\varrho(s-\tau)}\int_{\mathbb{R}^N}(v(s)-M)_+^{2p-2}\,dx\,ds \\
&\leq \frac{1}{d}\int_{\tau-1}^\tau e^{\varrho(s-\tau)}
 \int_{\mathbb{R}^N} \Big(v(s,\tau-t,\vartheta_{-\tau}\omega,v_0)
 -M\Big)_+^{p}\,dx\,ds \\
&\quad + \frac{cF^pe^{p|\omega(-\tau)|}}{d}\int_{\tau-1}^\tau
 e^{\varrho(s-\tau)} \Big(\|g(s,\cdot)\|^2+1\Big)ds.
\end{aligned}
\end{equation}
First by \eqref{b4.17}, there exists $T_1=T_1(\tau,\omega,B)\geq2$ such that for
all $t\geq T_1$,
\begin{equation} \label{xxdf}
\begin{aligned}
&\frac{1}{d}\int_{\tau-1}^\tau e^{\varrho(s-\tau)}
 \int_{\mathbb{R}^N} \Big(v(s,\tau-t,\vartheta_{-\tau}\omega,v_0)
 -M\Big)_+^{p}\,dx\,ds \\
&\leq \frac{1}{d}\int_{\tau-1}^\tau e^{\varrho(s-\tau)}\|v(s,\tau-t,
 \vartheta_{-\tau}\omega,v_0)\|_p^pds \\
&\leq N(\tau,\omega)\frac{1}{d\varrho}\to0,
\end{aligned}
\end{equation}
as $\varrho\to+\infty$, where $N(\tau,\omega)$ is the bound of the right hand
side of \eqref{b4.17}.
  We then  show  that the second term on the right hand side of \eqref{p07}
is also small as $\varrho\to+\infty$.  Indeed,
choosing $\varrho>\delta$(where $\delta\in(0,\lambda)$ is in \eqref{a5}) and
taking $\varsigma\in(0,1)$, we have
\begin{align*}
&\int_{\tau-1}^\tau e^{\varrho(s-\tau)} \Big(\|g(s,\cdot)\|^2+1\Big)ds \\
&=\int_{\tau-1}^{\tau-\varsigma} e^{\varrho(s-\tau)} (\|g(s,\cdot)\|^2+1)ds
 +\int_{\tau-\varsigma}^{\tau} e^{\varrho(s-\tau)}(\|g(s,\cdot)\|^2+1)ds \\
&=e^{-\varrho\tau}\int_{\tau-1}^{\tau-\varsigma} e^{(\varrho-\delta)s}e^{\delta s}
 (\|g(s,\cdot)\|^2+1)ds+e^{-\varrho\tau}\int_{\tau-\varsigma}^{\tau}
 e^{\varrho s}(\|g(s,\cdot)\|^2+1)ds \\
&\leq e^{-\varrho\varsigma} e^{\delta(\varsigma-\tau)}\int_{-\infty}^{\tau}
 e^{\delta s}(\|g(s,\cdot)\|^2+1)ds+\int_{\tau-\varsigma}^{\tau}
 (\|g(s,\cdot)\|^2+1)ds.
\end{align*}
By \eqref{a5}, the first term above vanishes as $\varrho\to+\infty$, and by
$g\in L^2{\rm loc}(\mathbb{R}, L^2(\mathbb{R}^N))$  we can choose
$\varsigma$ small enough such that the second term is small. Hence when
$\varrho\to +\infty$, we have
  \begin{equation} \label{p0771}
 \frac{cF^pe^{p|\omega(-\tau)|}}{d}\int_{\tau-1}^\tau e^{\varrho_2(s-\tau)}
\Big(\|g(s,\cdot)+\|^2+1\Big)ds\to0.
\end{equation}
Then by \eqref{p07}--\eqref{p0771}, there exist two large positive constants
$M_1=M_1(\tau,\omega)$ and $T_1=T_1(\tau,\omega,B)\geq 2$ such that all
$t\geq T_1$,
 \begin{equation} \label{p0772}
 \int_{\tau-1}^\tau e^{\varrho_1(s-\tau)}\int_{\mathbb{R}^N}(v(s)-M_1)_+^{2p-2}
\,dx\,ds\leq \eta,
 \end{equation}
where $\varrho_1=\alpha_1 F^{2-p}e^{-(p-2)|\omega(-\tau)|}M_1$.
 Note that $v-M_1\geq \frac{v}{2}$ for $v\geq2M_1$. Then  \eqref{p0772}
gives that for all $t\geq T_1$,
 \begin{equation} \label{2p1}
\begin{aligned}
&\int_{\tau-1}^\tau e^{\varrho_1(s-\tau)}\int_{\mathbb{R}^N(v(s)\geq 2M_1)}
 |v(s)|^{2p-2}\,dx\,ds \\
&\leq 2^{2p-2}\int_{\tau-1}^\tau e^{\varrho_1(s-\tau)}\int_{\mathbb{R}^N}
 (v(s)-M_1)_+^{2p-2}\,dx\,ds
\leq 2^{2p-2}\eta.
\end{aligned}
 \end{equation}
 By a similar argument, we can show that there exist two large positive constants
 $M_2=M_2(\tau,\omega)$  and $T_2=T_2(\tau,\omega,B)\geq2$  such that for all
$t\geq T_2$,
 \begin{equation} \label{2p2}
 \int_{\tau-1}^\tau e^{\varrho_2(s-\tau)}
\int_{\mathbb{R}^N(v(s)\leq -2M_2)}|v(s)|^{2p-2}\,dx\,ds\leq 2^{2p-2}\eta,
 \end{equation}
where $\varrho_2=\alpha_1 F^{2-p}e^{-(p-2)|\omega(-\tau)|}M_2$. Put
 $\tilde{M}=2\times\max\{M_1,M_2\}$ and $T=\max\{T_1,T_2\}$.
  Then \eqref{2p1} and \eqref{2p2} together  imply the desired.
\end{proof}

\subsection{Asymptotic compactness  on bounded domains}

In this subsection,  by using Proposition \ref{prop4.5}, we prove the asymptotic
compactness of the cocyle $\varphi$ defined by \eqref{eq0} in
$H_0^1(\mathcal{O}_R)$
for any $R>0$, where $\mathcal{O}_R=\{x\in \mathbb{R}^N; |x|\leq R\}$.
For this purpose, we define $\phi(\cdot)=1-\xi(\cdot)$, where $\xi$ is the
cut-off function as in \eqref{b4.305}. Then we know that $0\leq\phi(s)\leq1$,
 and $\phi(s)=1$ if $s\in[0,1]$ and $\phi(s)=0$ if $s\geq2$.
Fix a positive constant $k$, we define
\begin{equation} \label{vuu}
\tilde{v}(t,\tau,\omega,v_0)=\phi(\frac{x^2}{k^2})v(t,\tau,\omega,v_0),\quad
\tilde{u}(t,\tau,\omega,u_0)=\phi(\frac{x^2}{k^2})u(t,\tau,\omega,u_0),
\end{equation}
where $v$ is the solution of problem \eqref{pr1}-\eqref{pr2} and $u$ is the solution of problem ({\ref{beq1}})-({\ref{beq2}}) with $v=z(t,\omega)u$. Then we have
\begin{equation} \label{vu}
\tilde{u}(t,\tau,\omega,u_0)=z^{-1}(t,\omega)\tilde{v}(t,\tau,\omega,v_0).
\end{equation}
It is obvious that $\tilde{v}$
solves the following equations:
\begin{equation}
\begin{gathered}
\tilde{v}_t+\lambda\tilde{v}-\Delta \tilde{v}=\phi zf(x,z^{-1}v)
+\phi zg-v\Delta \phi-2\nabla \phi.\nabla v,\\
\tilde{v}|_{\partial\mathcal{O}_{k\sqrt{2}}}=0,\\
\tilde{v}(\tau,x)=\tilde{v}_0(x)=\phi v_0(x),
\end{gathered}
\end{equation}
 where $\phi=\phi(x^2/k^2)$.

It is well-known that the eigenvalue problem on bounded domains
$\mathcal{O}_{k\sqrt{2}}$ with Dirichlet boundary condition:
\begin{gather*}
-\Delta \tilde{v}=\lambda \tilde{v},\\
\tilde{v}|_{\partial\mathcal{O}_{k\sqrt{2}}}=0
\end{gather*}
has a family of orthogonal eigenfunctions $\{e_j\}_{=1}^{+\infty}$ in both
$L^2(\mathcal{O}_{k\sqrt{2}})$ and
$H_0^1(\mathcal{O}_{k\sqrt{2}})$ such that the corresponding eigenvalue
$\{\lambda_j\}_{j=1}^{+\infty}$ is non-decreasing in $j$.

Let $H_m=\text{Span}\{e_1,e_2,\dots ,e_m\}\subset H_0^1(\mathcal{O}_{k\sqrt{2}})$
and $P_m:  H^1_0(\mathcal{O}_{k\sqrt{2}})\to H_m$ be the canonical
projector and $I$ be the identity. Then for every $\tilde{u}\in
H^1_0(\mathcal{O}_{k\sqrt{2}})$, $\tilde{u}$ has a unique decomposition:
$\tilde{u}=\tilde{u}_1+\tilde{u}_2$,
where $\tilde{u}_1=P_m\tilde{u}\in H_m$
 and $\tilde{u}_2=(I-P_m)\tilde{u}\in H_m^{\bot}$, i.e.,
 $ H^1_0(\mathcal{O}_{k\sqrt{2}})=H_m\oplus H_m^{\bot}$.

\begin{lemma} \label{lem4.6}
Assume that \eqref{a1}--\eqref{a5} hold. Let $\tau\in\mathbb{R}$, $\omega\in\Omega$
and  
\[
B=\{B(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\}\in\mathfrak{D}.
\]
Then for every $\eta>0$, there are $N_0=N_0(\tau,\omega, k,\eta)\in Z^+$ and
$T=T(\tau,\omega,B,\eta)\geq 2$ such that for all $t\geq T$ and
$m> N_0$,
\[
 \| (I-P_m)\tilde{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,
\tilde{u}_0)\|_{H^1_0(\mathcal{O}_{k\sqrt{2}})}\leq\eta,
\]
  where $\tilde{u}_0=\phi{u}_0$ with ${u}_0\in B(\tau-t,\vartheta_{-\tau}\omega)$.
Here $\tilde{u}$ is as in \eqref{vu} and $N,T$ are independent of $\varepsilon$.
\end{lemma}

\begin{proof} By \eqref{vu}, we start at the estimate of $\tilde{v}$.
For $\tilde{v}\in H^1_0(\mathcal{O}_{k\sqrt{2}})$,
we write $\tilde{v}=\tilde{v}_1+\tilde{v}_2$ where $\tilde{v}_1=P_m\tilde{v}$ and
$\tilde{v}_2=(I-P_m)\tilde{v}$.
 Then naturally, we have a splitting about $\tilde{u}=\tilde{u}_1+\tilde{u}_2$
 where $\tilde{u}_1=P_m\tilde{u}$ and
$\tilde{u}_2=(I-P_m)\tilde{u}$.
 Multiplying (4.47) by $\Delta \tilde{v}_2$ we get that
\begin{equation} \label{b4.44}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|\nabla \tilde{v}_2\|_{L^2(\mathcal{O}_{k\sqrt{2}})}^2
 +\lambda\|\nabla \tilde{v}_2\|_{L^2(\mathcal{O}_{k\sqrt{2}})}^2
 +\|\Delta\tilde{v}_2\|_{L^2(\mathcal{O}_{k\sqrt{2}})}^2 \\
&=-z\int_{\mathcal{O}_{k\sqrt{2}}}\phi f(x,z^{-1}v)\Delta \tilde{v}_2dx
 +\int_{\mathcal{O}_{k\sqrt{2}}}(\phi zg-v\Delta \phi-2\nabla \phi.\nabla v)
 \Delta \tilde{v}_2dx,
\end{aligned}
\end{equation}
where $z$ is the abbreviation of $z(t,\omega)$.
By \eqref{a2}, we deduce that
\begin{equation} \label{b4.45}
z\int_{\mathcal{O}_{k\sqrt{2}}}\phi f(x,z^{-1}v)\Delta \tilde{v}_2dx
\leq \frac{1}{4}\|\Delta\tilde{v}_2\|_{L^2(\mathcal{O}_{k\sqrt{2}})}^2
+cz^{4-2p}\|v\|_{L^{2p-2}(\mathcal{O}_{k\sqrt{2}})}^{2p-2}+z^2\|\psi_2\|^2.
\end{equation}
On the other hand,
\begin{equation} \label{b4.46}
\begin{aligned}
&\int_{\mathcal{O}_{k\sqrt{2}}}(\phi zg-v\Delta \phi-2\nabla \phi.\nabla v)
\Delta \tilde{v}_2dx \\
&\leq \frac{1}{4}\|\Delta\tilde{v}_2\|_{L^2(\mathcal{O}_{k\sqrt{2}})}^2
+c(z^2\|g\|^2+\|{v}\|^2+\|\nabla{v}\|^2).
\end{aligned}
\end{equation}
Then by \eqref{b4.44}--\eqref{b4.46} we find that
\begin{align*}
&\frac{d}{dt}\|\nabla \tilde{v}_2\|_{L^2(\mathcal{O}_{k\sqrt{2}})}^2
 +\|\Delta\tilde{v}_2\|_{L^2(\mathcal{O}_{k\sqrt{2}})}^2 \\
&\leq c(z^{4-2p}\|v\|_{L^{2p-2}(\mathcal{O}_{k\sqrt{2}})}^{2p-2}
 +z^2\|\psi_2\|^2+z^2\|g\|^2+\|{v}\|_{H^1}^2).
\end{align*}
from which and Poincar\'{e}'s inequality
$$
\|\Delta\tilde{v}_2\|_{L^2(\mathcal{O}_{k\sqrt{2}})}^2
\geq \lambda_{m+1}\|\nabla\tilde{v}_2\|_{L^2(\mathcal{O}_{k\sqrt{2}})}^2,
$$
it follows that
\begin{equation} \label{b4.47}
\begin{aligned}
&\frac{d}{dt}\|\nabla \tilde{v}_2\|_{L^2(\mathcal{O}_{k\sqrt{2}})}^2
+\lambda_{m+1}\|\nabla\tilde{v}_2\|_{L^2(\mathcal{O}_{k\sqrt{2}})}^2 \\
&\leq c(z^{4-2p}\|v\|_{L^{2p-2}(\mathcal{O}_{k\sqrt{2}})}^{2p-2}
+z^2\|\psi_2\|^2+z^2\|g\|^2+\|{v}\|_{H^1}^2).
\end{aligned}
\end{equation}
 Applying \cite[Lemma 5.1]{Zhao0} to \eqref{b4.47} over the interval
 $[\tau-1,\tau]$, along with $\omega$  being replaced by $\vartheta_{-\tau}\omega$,
we find that
\begin{align}
&\|\nabla \tilde{v}_2(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{v}_0)
\|_{L^2(\mathcal{O}_{k\sqrt{2}})}^2 \nonumber  \\
&\leq \int_{\tau-1}^\tau e^{\lambda_{m+1}(s-\tau)}\|\nabla
 \tilde{v}_2(s,\tau-t,\vartheta_{-\tau}\omega,\tilde{v}_0)
 \|_{L^2(\mathcal{O}_{k\sqrt{2}})}^2ds \nonumber  \\
&\quad +c\int_{\tau-1}^\tau e^{\lambda_{m+1}(s-\tau)}z^{4-2p}
(s,\vartheta_{-\tau}\omega)\|v(s,\tau-t,\vartheta_{-\tau}\omega,
 \tilde{v}_0)\|_{L^{2p-2}(\mathcal{O}_{k\sqrt{2}})}^{2p-2}ds \nonumber \\
&\quad +c\int_{\tau-1}^\tau e^{\lambda_{m+1}(s-\tau)}z^2(s,
 \vartheta_{-\tau}\omega)(\|\psi_2\|^2+\|g(s,\cdot)\|^2)ds \nonumber \\
&\quad +c\int_{\tau-1}^\tau e^{\lambda_{m+1}(s-\tau)}\|{v}(s,\tau-t,
 \vartheta_{-\tau}\omega,\tilde{v}_0)\|_{H^1}^2ds \nonumber \\
&\leq c\int_{\tau-1}^\tau e^{\lambda_{m+1}(s-\tau)}z^{4-2p}(s,
 \vartheta_{-\tau}\omega)\|v(s,\tau-t,\vartheta_{-\tau}\omega,
 \tilde{v}_0)\|_{L^{2p-2}(\mathcal{O}_{k\sqrt{2}})}^{2p-2}ds \nonumber \\
&\quad +c\int_{\tau-1}^\tau e^{\lambda_{m+1}(s-\tau)}\|v(s,\tau-t,
 \vartheta_{-\tau}\omega,\tilde{v}_0)\|_{H^1}^2ds \nonumber \\
&\quad +c\int_{\tau-1}^\tau e^{\lambda_{m+1}(s-\tau)}z^2(s,
 \vartheta_{-\tau}\omega)\Big(\|g(s,\cdot)\|^2+1\Big)ds \nonumber  \\
&=I_1+I_2+I_3. \label{b4.48}
\end{align}
We next to show that $I_1,I_2$ and $I_3$ converge to zero as $m$
increases to infinite. First since by \eqref{EE},
$z^{4-2p}(s-\tau,\omega)\leq E^{4-2p}$ for $s\in [-1,0]$, then we have
\begin{equation} \label{b4.50}
\begin{aligned}
&I_1 \\
&=z^{2p-4}(-\tau,\omega)\int_{\tau-1}^\tau e^{\lambda_{m+1}(s-\tau)
 }z^{4-2p}(s-\tau,\omega) \\
&\quad\times \|v(s,\tau-t,\vartheta_{-\tau}\omega,\tilde{v}_0)\|_{L^{2p-2}
 (\mathcal{O}_{k\sqrt{2}})}^{2p-2}ds \\
&\leq z^{2p-4}(-\tau,\omega)E^{4-2p}
\int_{\tau-1}^\tau e^{\lambda_{m+1}(s-\tau)}\|v(s,\tau-t,\vartheta_{-\tau}\omega,
\tilde{v}_0)\|_{L^{2p-2}(\mathcal{O}_{k\sqrt{2}})}^{2p-2}ds \\
&\leq z^{2p-4}(-\tau,\omega)E^{4-2p}\Big(\int_{\tau-1}^\tau
 e^{\lambda_{m+1}(s-\tau)} \\
&\quad\times \int_{\mathcal{O}_{k\sqrt{2}}(|v(s)|\geq M)}|v(s,\tau-t,
 \vartheta_{-\tau}\omega,\tilde{v}_0)|^{2p-2}\,dx\,ds \\
&\quad +\int_{\tau-1}^\tau e^{\lambda_{m+1}(s-\tau)}
 \int_{\mathcal{O}_{k\sqrt{2}}(|v(s)|\leq M)}
 |v(s,\tau-t,\vartheta_{-\tau}\omega,\tilde{v}_0)|^{2p-2}\,dx\,ds\Big).
\end{aligned}
\end{equation}
By Proposition \ref{prop4.5},  there exist $T_1=T_1(\tau,\omega,B,\eta)\geq2$,
$\tilde{M}=\tilde{M}(\tau,\omega,\eta)$  such that for all  $t\geq T_1$,
\begin{equation} \label{b4.501}
\begin{aligned}
&z^{2p-4}(-\tau,\omega)E^{4-2p}\int_{\tau-1}^\tau e^{\tilde{\varrho}(s-\tau)} \\
&\times \int_{\mathcal{O}_{k\sqrt{2}}(|v(s)|\geq \tilde{M})}|v(s,\tau-t,
\vartheta_{-\tau}\omega,\tilde{v}_0)|^{2p-2} \,dx\,ds\leq \eta.
\end{aligned}
\end{equation}
But $\lambda_{m+1}\to +\infty$, then there exists $N'=N'(\tau,\omega,\eta)>0$ 
such that  for all $m>N'$, $\lambda_{m+1}>\tilde{\varrho}$. Hence
by \eqref{b4.501} it gives us that  for all  $t\geq T_1$ and $m>N'$ there holds
\begin{equation}  \label{b4.5010}
\begin{aligned}
&z^{2p-4}(-\tau,\omega)E^{4-2p}\int_{\tau-1}^\tau e^{\lambda_{m+1}(s-\tau)} \\
&\quad\times \int_{\mathcal{O}_{k\sqrt{2}}(|v(s)|\geq \tilde{M})}
|v(s,\tau-t,\vartheta_{-\tau}\omega,\tilde{v}_0)|^{2p-2} \,dx\,ds\leq \eta.
\end{aligned}
\end{equation}
For the second term on the right hand side of \eqref{b4.50}, since
 $\mathcal{O}_{k\sqrt{2}}(|v(s)|\leq \tilde{M})$ is a bounded domain, then
there exists  $N''=N''(\tau,\omega,\eta)>0$ such that
for all $m>N''$,

\begin{equation} \label{b4.502}
\begin{aligned}
&z^{2p-4}(-\tau,\omega)E^{4-2p}\int_{\tau-1}^\tau e^{\lambda_{m+1}(s-\tau)}\\
&\times \int_{\mathcal{O}_{k\sqrt{2}}(|v(s)|\leq \tilde{M})}|v(s,\tau-t,
 \vartheta_{-\tau}\omega,\tilde{v}_0)|^{2p-2}\,dx\,ds \\
&\leq z^{2p-4}(-\tau,\omega)E^{4-2p}\frac{\tilde{M}^{2p-2}}{\lambda_{m+1}}
 |(\mathcal{O}_{k\sqrt{2}}(|v(s)|
 \leq\tilde{ M}))|\leq\eta,
\end{aligned}
\end{equation}
where $|(\mathcal{O}_{k\sqrt{2}}(|v(s)|\leq \tilde{M}))|$ is the finite measure
of the bounded domain \break $\mathcal{O}_{k\sqrt{2}}(|v(s)|\leq \tilde{M})$.
Put $N_1=\max\{N^{\prime},N''\}$. It follows from \eqref{b4.50}-\eqref{b4.502}
that for all $m>N_1$ and $t\geq T_1$,
\begin{equation} \label{b4.55}
I_1\leq 2\eta.
\end{equation}
By Lemma \ref{lem4.1}, there exists $T_2=T_2(\tau,\omega,B)$ and
$N_2=N_2(\tau,\omega,\eta)>0$ such that for all  $m>N_2$ and $t\geq T_2$,
\begin{equation} \label{b4.56}
I_2\leq \frac{L_1(\tau,\omega,\varepsilon)}{\lambda_{m+1}}\leq\eta.
\end{equation}
By a same technique as \eqref{p0771}, we can show that there exists
$N_3=N_3(\tau,\omega,\eta)>0$ such that for all  $m>N_3$,
\begin{equation} \label{b4.551}
I_3=c\int_{\tau-1}^\tau e^{\lambda_{m+1}(s-\tau)}z^2
(s,\vartheta_{-\tau}\omega)\Big(\|g(s,\cdot)\|^2+1\Big)ds\leq \eta.
\end{equation}
Let $N_0=\max\{N_1,N_2,N_3\}$ and $T=\max\{T_1,T_2\}$.
Then \eqref{b4.55}--\eqref{b4.551} are integrated into \eqref{b4.48}
to get that  for all  $m>N_0$ and $t\geq T$,
\begin{equation} \label{b4.126}
\|\nabla \tilde{v}_2(\tau,\tau-t,\vartheta_{-\tau}\omega,
\tilde{v}_0)\|_{L^2(\mathcal{O}_{k\sqrt{2}})}\leq 4\eta.
\end{equation}
 Then by \eqref{eq00} and  \eqref{b4.126}, we have
\begin{align*}
\|\nabla \tilde{u}_2(\tau,\tau-t,\vartheta_{-\tau}\omega,
\tilde{u}_0)\|_{L^2(\mathcal{O}_{k\sqrt{2}})} \\
&=z(-\tau,\omega)\|\nabla\tilde{v}_2(\tau,\tau-t,\vartheta_{-\tau}\omega,
 \tilde{v}_0)\|_{L^2(\mathcal{O}_{k\sqrt{2}})} \\
&\leq C(\tau,\omega)\eta,
\end{align*}
for all  $m>N_0$ and $t\geq T$, which completes the proof.
\end{proof}


 \begin{lemma} \label{lem4.7}
Assume that \eqref{a1}--\eqref{a5} hold.  Let  $\tau\in\mathbb{R}, \omega\in\Omega$.
Then for every $k>0$, the sequence
$\{\tilde{u}(\tau, \tau-t_n,\vartheta_{-\tau}\omega,
\phi(\frac{x^2}{k^2})u_{0,n})\}_{n=1}^\infty$ has a convergent subsequence
 in $H_0^1(\mathcal{O}_{k\sqrt{2}})$ whenever $t_n\to+\infty$ and
$u_{0,n}\in B(\tau-t_n,\vartheta_{-t_n}\omega)$.
\end{lemma}

\begin{proof}
 Given $\eta>0$, by Lemma \ref{lem4.6}, there exists $N_0\in \mathbb{Z}^+$ such that
 as $t_n\to+\infty$,
 \begin{equation} \label{L101}
\|(I-P_{N_0})\tilde{u}(\tau,\tau-t_n,\vartheta_{-\tau}\omega,
\phi(\frac{x^2}{k^2})u_{0,n})\|_{H^1(\mathcal{O}_{k\sqrt{2}})}\leq \eta.
\end{equation}
By Lemma \ref{lem4.1}, we deduce that for $t_n$ large enough,
 \begin{equation} \label{L102}
\|P_{N_0}\tilde{u}(\tau,\tau-t_n,\vartheta_{-\tau}\omega,
\phi(\frac{x^2}{k^2})u_{0,n})\|_{H^1(\mathcal{O}_{k\sqrt{2}})}
\leq L_1(\tau,\omega,\varepsilon).
\end{equation}
 Note that
\[
H^1(\mathcal{O}_{k\sqrt{2}})=P_{N_0}H^1(\mathcal{O}_{k\sqrt{2}})+(I-P_{N_0})
H^1(\mathcal{O}_{k\sqrt{2}}),
\]
 but $P_{N_0}H^1(\mathcal{O}_{k\sqrt{2}})$ is a finite
 dimensional space, which is compact. Then by \eqref{L102},  if $n,m$ large enough,
  \begin{equation} \label{L103}
\begin{aligned}
&\big\|P_{N_0}\tilde{u}(\tau,\tau-t_n,\vartheta_{-\tau}\omega,
\phi(\frac{x^2}{k^2})u_{0,n}) \\
&-P_{N_0}\tilde{u}(\tau,\tau-t_m,\vartheta_{-\tau}\omega,
\phi(\frac{x^2}{k^2})u_{0,m}\big\|_{H^1(\mathcal{O}_{k\sqrt{2}})}
\leq \eta.
\end{aligned}
\end{equation}
Then it is easy to complete the proof using \eqref{L101} and \eqref{L103}
and a standard argument.
\end{proof}


\subsection{Existence of pullback attractor in $H^{1}(\mathbb{R}^N)$}

In this subsection, we prove the existences of pullback attractors in
$H^{1}(\mathbb{R}^N)$ for problem \eqref{beq1} and \eqref{beq2} for every
$\varepsilon\in(0,1]$.

\begin{proposition} \label{prop4.8}
Assume that \eqref{a1}-\eqref{a5} hold. Then the cocycle $\varphi$ defined
by \eqref{eq0} is asymptotically compact in $H^1(\mathbb{R}^N)$;
 i.e., for every  $\tau\in\mathbb{R}$ and $\omega\in\Omega$,  the sequence
$\{\varphi(t, \tau-t_n,\vartheta_{-t}\omega, u_{0,n})\}_{n=1}^\infty$
 has a convergent subsequence  in $H^1(\mathbb{R}^N)$ whenever $t_n\to+\infty$
and $u_{0,n}\in B=B(\tau-t_n,\vartheta_{-t_n}\omega)$ with $B\in\mathfrak{D}$.
\end{proposition}

\begin{proof}
 Given $R>0$, we denote $\mathcal{O}^c_{R}=\mathbb{R}^N-\mathcal{O}_{R}$,
 where $\mathcal{O}_R=\{x\in \mathbb{R}^N; |x|\leq R\}$.
By  Proposition \ref{prop4.4},  for  any $\eta>0$, there exist $R=R(\tau,\omega,\eta)>0$
and $N_1=N_1(\tau,\omega,B,\eta)\in \mathbb{Z}^+$ such that for all $n\geq N_1$,
\begin{equation} \label{n08}
\|v(\tau, \tau-t_n,\vartheta_{-\tau}\omega,
z(\tau-t_n,\vartheta_{-\tau}\omega)u_{0,n})\|_{H^1(\mathcal{O}^c_{R})}
\leq \frac{\eta}{8} e^{-|\omega(-\tau)|},
\end{equation}
for every $u_{0,n}\in B=B(\tau-t_n,\vartheta_{-t_n}\omega)$.
By \eqref{eq00} and \eqref{n08}, we have
\begin{equation} \label{n0800}
\|u(\tau, \tau-t_n,\vartheta_{-\tau}\omega, z(\tau-t_n,
\vartheta_{-\tau}\omega)u_{0,n})\|_{H^1(\mathcal{O}^c_{R})}\leq \frac{\eta}{8}.
\end{equation}
On the other hand, for this  $R$, by Lemma \ref{lem4.7}, there exists
$N_2=N_2(\tau,\omega,B,\eta)\geq N_1$ such that for all $m,n\geq N_2$,
\begin{equation} \label{n081}
\begin{aligned}
&\big\|u(\tau, \tau-t_n,\vartheta_{-\tau}\omega,
\phi(\frac{x^2}{R^2})u_{0,n}) \\
&\quad -u(\tau, \tau-t_m,\vartheta_{-\tau}\omega, \phi(\frac{x^2}{R^2})u_{0,m})
\big\|_{H^1_0(\mathcal{O}_{R\sqrt{2}})} \\
&\leq \frac{\eta}{8}.
\end{aligned}
\end{equation}
Then the desired result follows from \eqref{n0800} and \eqref{n081}
by a standard argument.
\end{proof}

Given $\varepsilon\in (0,1]$, by Lemma \ref{lem4.1}, we deduce that the
$\mathfrak{D}$-pullback absorbing set $K_\varepsilon$ of $\varphi_{\varepsilon}$
in $L^2(\mathbb{R}^N)$ is defined by
\begin{equation}\label{}
K_\varepsilon=\{K_\varepsilon(\tau,\omega)=\{u\in L^2(\mathbb{R}^N);
\|u\|\leq L_\varepsilon(\tau,\omega)\};\tau\in\mathbb{R},\omega\in
\Omega\},
\end{equation}
 where
\[
L_\varepsilon(\tau,\omega)=\Big(c\int_{-\infty}^0e^{\lambda s}
e^{-2\varepsilon \omega(s)}(\|g(s+\tau,\cdot)\|^2+1)\Big)^{1/2}.
\]


By Proposition \ref{prop4.8} and  Theorem \ref{thm2.6},  we  have the following result.

\begin{theorem} \label{thm4.9}
Assume that \eqref{a1}--\eqref{a5} hold. Then for every fixed
$\varepsilon\in(0,1]$, the cocycle $\varphi_\varepsilon$ defined by \eqref{eq0}
possesses a unique $\mathfrak{D}$-pullback attractor
$\mathcal{A}_{\varepsilon,H^1}=\{\mathcal{A}_{\varepsilon,H^1}
(\tau,\omega);\tau\in\mathbb{R},\omega\in \Omega\}$ in $H^1(\mathbb{R}^N)$,
given by
\[
\mathcal{A}_{\varepsilon,H^1}(\tau,\omega)
=\cap_{s>0}\overline{\cup_{t\geq s} \varphi_\varepsilon(t,\tau-t,
\vartheta_{-t}\omega, K_\varepsilon(\tau-t,\vartheta_{-t}\omega))}^{H^1(\mathbb{R}^N)},
 \quad \tau\in \mathbb{R},\;\omega\in \Omega.
 \]
Furthermore,  $\mathcal{A}_{\varepsilon,H^1}$ is consistent with the
$\mathfrak{D}$-pullback random attractor $\mathcal{A}_\varepsilon$ in
the space $L^2(\mathbb{R}^N)$, which is defined as in \eqref{L2}.
\end{theorem}


\section{Upper semi-continuity  of pullback attractor in $H^{1}(\mathbb{R}^N)$}

From Theorem \ref{thm4.9}, for every $\varepsilon\in(0,1]$, the cocycle $\varphi_\varepsilon$
admits a common $\mathfrak{D}$-pullback  attractor $\mathcal{A}_\varepsilon$
in both $L^2(\mathbb{R}^N)$ and $H^1(\mathbb{R}^N)$, where $\mathfrak{D}$ is
defined by \eqref{eq00}.  Then we may investigate the upper semi-continuity
of $\mathcal{A}_\varepsilon$ in both $L^2(\mathbb{R}^N)$ and $H^1(\mathbb{R}^N)$.
Note that \cite{Wang1} only proved the upper semi-continuity in $L^2(\mathbb{R}^N)$
at $\varepsilon=0$. In this section, we strengthen this study and  prove that
the upper semi-continuity of $\mathcal{A}_\varepsilon$  may happen in the topology
of $H^1(\mathbb{R}^N)$ at $\varepsilon=0$.

For the upper semi-continuity, we  also give a  further assumption as in \cite{Wang1},
 that is, $f$ satisfies that for all $x\in\mathbb{R}^N$ and
$s\in\mathbb{R}$,
\begin{equation}\label{a6}
 \big|\frac{\partial}{\partial s}f(x,s)\big|\leq \alpha_4|s|^{p-2}+\psi_4(x),
\end{equation}
where $\alpha_4>0$,  $\psi_4\in L^\infty(\mathbb{R}^N)$ if $p=2$ and
$\psi_4\in L^{\frac{p}{p-2}}(\mathbb{R}^N)$ if $p>2$.

Let $\varphi_0$ be the continuous cocycle associated with the problem \eqref{beq1}
and \eqref{beq2} for $\varepsilon=0$. That is to say,
$\varphi_0$ is a deterministic non-autonomous cocycle over $\mathbb{R}$.
Denote by $\mathfrak{D}_0$ the collection of some families of deterministic
nonempty subsets of $L^2(\mathbb{R}^N)$:
$$
\mathfrak{D}_0=\{B=\{B(\tau)\subseteq L^2(\mathbb{R}^N);\tau\in \mathbb{R}\};
\lim_{t\to+\infty}e^{-\delta t}\|B(\tau-t)\|=0,\tau\in\mathbb{R},\delta<\lambda\},
$$
where $\lambda$ is as in \eqref{pr1}.
As a special case of Theorem \ref{thm4.9}, under the  assumptions \eqref{a1}-\eqref{a5},
$\varphi_0$ has a common $\mathfrak{D}_0$-pullback attractor
$\mathcal{A}_0=\{ \mathcal{A}_0(\tau);\tau\in\mathbb{R}\}$ in both
$ L^2(\mathbb{R}^N)$ and $H^1(\mathbb{R}^N)$.

To prove the upper semi-continuity of $\mathcal{A}_\varepsilon$ at
$\varepsilon=0$, we have to check that the conditions \eqref{ff8}-\eqref{ff12} in
Theorem \ref{thm2.8} hold in $L^2(\mathbb{R}^N)$ and $H^1(\mathbb{R}^N)$ point by point.
But \eqref{ff8}-\eqref{ff11} have been achieved,
see \cite[Corollary 7.2, Lemma 7.5 and equality (7.31)]{Wang1}.
We only need to prove the condition \eqref{ff12} holds in
$H^1(\mathbb{R}^N)$.

\begin{lemma} \label{lem5.1}
Assume that \eqref{a1}-\eqref{a5} hold. Then for every  $\tau\in\mathbb{R}$ and
$\omega\in \Omega$, the union
$\cup_{\varepsilon\in(0,1]}\mathcal{A}_{\varepsilon}(\tau,\omega)$
is precompact in $H^1(\mathbb{R}^N)$.
\end{lemma}

\begin{proof}
For any $\eta>0$, it suffices to show that for  every fixed $\tau\in\mathbb{R}$
and $\omega\in \Omega$, the set
$\cup_{\varepsilon\in(0,1]}\mathcal{A}_{\varepsilon}(\tau,\omega)$
has finite $\eta$-nets in  $H^1(\mathbb{R}^N)$.
Let $\chi=\chi(\tau,\omega)\in \cup_{\varepsilon\in(0,1]}
\mathcal{A}_{\varepsilon}(\tau,\omega)$. Then there exists a
$\varepsilon\in (0,1]$ such that
 $\chi(\tau,\omega)\in \mathcal{A}_{\varepsilon}(\tau,\omega)$.
By the invariance of $\mathcal{A}_{\varepsilon}(\tau,\omega)$,
it follows that there is a
$u_0\in \mathcal{A}_{\varepsilon}(\tau-t,\vartheta_{-t}\omega)$ such that
(by \eqref{eq00})
\begin{equation}\label{mm00}
\chi(\tau,\omega)=\varphi_{\varepsilon}(t,\tau-t,\vartheta_{-t}\omega,u_0)
=u_\varepsilon(\tau,\tau-t,\vartheta_{-\tau}\omega,u_0)\quad\forall t\geq0\,.
\end{equation}
 Given $R>0$, denote $\mathcal{O}^c_{R}=\mathbb{R}^N-\mathcal{O}_{R}$,
where $\mathcal{O}_R=\{x\in \mathbb{R}^N; |x|\leq R\}$.
 Note that $\mathcal{A}_{\varepsilon}(\tau,\omega)\in\mathfrak{D}$.
Then by Proposition \ref{prop4.4}, for every $\eta>0$, there exist
$T=T(\tau,\omega,\eta)\geq2$ and $R=R(\tau,\omega,\eta)>1$ such that the
solution $u$ of problem \eqref{beq1} and \eqref{beq2}  satisfies
 \begin{equation}\label{mm01}
\|u_\varepsilon(\tau,\tau-t,\vartheta_{-\tau}\omega,u_0)\|_{H^1(\mathcal{O}^c_R)}
\leq \eta,\quad \forall t\geq T\,.
\end{equation}
Then by \eqref{mm00}-\eqref{mm01}, we have
\begin{equation}\label{mm02}
\|\chi(\tau,\omega)\|_{H^1(\mathcal{O}^c_R)}\leq \eta,\quad \text{for all }
 \chi\in\cup_{\varepsilon\in(0,1]}\mathcal{A}_{\varepsilon}(\tau,\omega).
\end{equation}
On the other hand, by Lemma \ref{lem4.6}, there
exist a projector $P_{N_0}$ and   $T=T(\tau,\omega,\eta)\geq2$
such that for all $t\geq T$,
\begin{equation} \label{mm03}
 \| (I-P_{N_0})\tilde{u}_\varepsilon(\tau,\tau-t,
\vartheta_{-\tau}\omega, \tilde{u}_0)\|_{H^1_0(\mathcal{O}_{R\sqrt{2}})}\leq\eta,
\end{equation}
where $\tilde{u}_\varepsilon$ is the cut-off of $u_\varepsilon$ on the domain
$\mathcal{O}_{R\sqrt{2}}$, by \eqref{vuu}. Because
$P_{N_0}\tilde{u}_\varepsilon\in H_{N_0}$,
where $H_{N_0}=\text{span}\{e_1,_2,\dots ,e_{N_0}\}$ is a finite dimension
space and $P_{N_0}\tilde{u}_\varepsilon(\tau,\tau-t,\vartheta_{-\tau}\omega,
\tilde{u}_0)$ is bounded
in $H_{N_0}$ which is compact.
Therefore there exist  finite points  $v_1,v_2,\dots ,v_s\in H_{N_0}$ such that
\begin{equation} \label{mm04}
 \| P_{N_0}\tilde{u}_\varepsilon(\tau,\tau-t,\vartheta_{-\tau}\omega,
\tilde{u}_0)-v_i\|_{H^1_0(\mathcal{O}_{R\sqrt{2}})}\leq\eta.
\end{equation}
Thus by \eqref{mm00}, the inequalities \eqref{mm03} and \eqref{mm04} are rewritten as
\begin{equation} \label{mm05}
 \| (I-P_{N_0})\chi(\tau,\omega)\|_{H^1_0(\mathcal{O}_{R\sqrt{2}})}\leq\eta,\quad
 \| P_{N_0}\chi(\tau,\omega)-v_i\|_{H^1_0(\mathcal{O}_{R\sqrt{2}})}\leq\eta,
\end{equation}
for all $\chi\in\cup_{\varepsilon\in(0,1]}\mathcal{A}_{\varepsilon}(\tau,\omega)$.
We now define $\tilde{v}_i=\tilde{v}_i(x)=0$ if $x\in \mathcal{O}^c_{R\sqrt{2}}$
and $\tilde{v}_i=v_i$ if $x\in \mathcal{O}_{R\sqrt{2}}$. Then for every
$i=1,2,\dots ,s$, $\tilde{v}_i\in H^1(\mathbb{R}^N)$. Furthermore, by \eqref{mm02}
and \eqref{mm05}, we have
\begin{align*}
 \|\chi(\tau,\omega)-\tilde{v}_i\|_{H^1(\mathbb{R}^N)}
&\leq \|\chi(\tau,\omega)-\tilde{v}_i\|_{H^1(\mathcal{O}^c_{R\sqrt{2}})}
 +\|\chi(\tau,\omega)-\tilde{v}_i\|_{H^1_0(\mathcal{O}_{R\sqrt{2}})} \\
 &\leq\|\chi(\tau,\omega)\|_{H^1(\mathcal{O}^c_{R\sqrt{2}})}
+\|P_{N_0}\chi(\tau,\omega)-\tilde{v}_i\|_{H^1_0(\mathcal{O}_{R\sqrt{2}})} \\
&\quad +\|(I-P_{N_0})\chi(\tau,\omega)\|_{H^1_0(\mathcal{O}_{R\sqrt{2}})}\leq 3\eta,
\end{align*}
for all $\chi\in\cup_{\varepsilon\in(0,1]}\mathcal{A}_{\varepsilon}(\tau,\omega)$.
Thus $\cup_{\varepsilon\in (0,1]}\mathcal{A}_\varepsilon(\tau,\omega)$
 has finite $\eta$-nets in $H^1(\mathbb{R}^N)$, which implies that the union
$\cup_{\varepsilon\in(0,1]}\mathcal{A}_{\varepsilon}(\tau,\omega)$
is precompact in $H^1(\mathbb{R}^N)$.
\end{proof}

Then we obtain that the family of random attractors $\mathcal{A}_\varepsilon$
indexed by $\varepsilon$ converges to the deterministic $\mathcal{A}_0$ in
$H^1(\mathbb{R}^N)$ in the following sense.

\begin{theorem} \label{thm5.2}
Assume that \eqref{a1}-\eqref{a5} and \eqref{a6} hold. Then for each
$\tau\in\mathbb{R}$ and $\omega\in \Omega$,
$$
\lim_{\varepsilon\downarrow0}\operatorname{dist}_{H^1}
(\mathcal{A}_\varepsilon(\tau,\omega), \mathcal{A}_0(\tau))=0,
$$
where $\operatorname{dist}_{H^1}$ is the Haustorff semi-metric in
$H^1(\mathbb{R}^N)$.
\end{theorem}

\subsection*{Acknowledgments}
 This work was supported by the Chongqing Basis and Frontier Research Project
no. cstc2014jcyjA00035 and by the  National Natural Science Foundation of China
(no. 11271388).

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\end{thebibliography}

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