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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 203, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2012/203\hfil Uniform attractors ]
{Existence and upper semi-continuity of uniform attractors for
 non-autonomous reaction diffusion equations on $\mathbb{R}^N$}

\author[T. Q. Bao \hfil EJDE-2012/203\hfilneg]
{Tang Quoc Bao}  % in alphabetical order

\address{Tang Quoc Bao \newline
School of Applied Mathematics and Informatics,
 Ha Noi University of Science and Technology\\
1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam}
\email{baotangquoc@gmail.com}

\thanks{Submitted April 10, 2012. Published November 24, 2012.}
\subjclass[2000]{34D45, 35B41, 35K57, 35B30}
\keywords{Uniform attractors; reaction diffusion equations;
\hfill\break\indent  unbounded domain;
upper semicontinuity}

\begin{abstract}
 We prove the existence of uniform attractors for the non-auto\-nomous
 reaction diffusion equation
 \begin{equation*}
 u_t - \Delta u + f(x,u) + \lambda u = g(t,x)
 \end{equation*}
 on $\mathbb{R}^N$, where the external force $g$ is translation bounded
 and the nonlinearity $f$ satisfies a polynomial growth condition.
 Also, we prove the upper semi-continuity of uniform attractors
 with respect to the nonlinearity.
\end{abstract}

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

\section{Introduction}

In this article, we study the following non-autonomous reaction
diffusion equation
\begin{equation}
\begin{gathered}
u_t - \Delta u + f(x,u) + \lambda u = g(t,x), \quad x\in \mathbb{R}^N,\\
u|_{t = \tau} = u_{\tau},
\end{gathered}\label{e1}
\end{equation}
where $\lambda>0$, the nonlinearity $f$ and the external force $g$
satisfy some specified conditions later.

Non-autonomous equation are of great importance and interest as they
appear in many applications in natural sciences.
One way to treat non-autonomous equations is that considering
its uniform attractors, which are extended from global attractors
for autonomous case. In the recent years, the existence of uniform
attractors for non-autonomous reaction diffusion equations or its
generalized forms is studied extensively by many authors
(see e.g. \cite{AnhQuang, Chen, Ma, Song}
for the case of bounded domains, and \cite{Yan} for the case of
unbounded domains). However, uniform attractors for \eqref{e1}
in the case of unbounded domains is not well understood.
In this paper, we prove the existence and the upper semicontinuity
of uniform attractors for \eqref{e1} in unbounded domains with a large
class of external force $g$.

To study problem \eqref{e1}, we assume the following hypotheses:
\begin{itemize}
\item[(H1)] The nonlinearity $f$ satisfies: there exists $p\geq 2$ such that
\begin{gather}
f(x,u)u \geq \alpha_1|u|^p - \phi_1(x), \label{e2}\\
|f(x,u)| \leq \alpha_2|u|^{p-1} + \phi_2(x), \label{e3} \\
f'_{u}(x,u)\geq -\ell, \label{e3_1}
\end{gather}
where $\phi_1\in L^{1}(\mathbb{R}^N)\cap L^{p/2}(\mathbb{R}^N)
\cap L^{\infty}(\mathbb{R}^N)$,
$\phi_2\in L^q(\mathbb{R}^N)\cap L^{2}(\mathbb{R}^N)$ with
$\frac{1}{p}+\frac{1}{q}=1$ and $\alpha_1, \alpha_2,\ell$
are positive constant. For the primitive $F(x,u) = \int_{0}^{u}f(x,\xi)d\xi$,
we assume that there are positive constants $\alpha_3, \alpha_4$ and
$\phi_3, \phi_4\in L^1(\mathbb{R}^N)$ satisfy
\begin{equation}
\alpha_3|u|^p -\phi_3(x)\leq F(x,u) \leq \alpha_4|u|^p + \phi_4(x).
\label{e4}
\end{equation}

\item[(H2)] The external force $g\in L^{2}_{\rm loc}\left(\mathbb{R}; L^2(\mathbb{R}^N)\right)$ satisfies
\begin{equation}
\sup_{t\in\mathbb{R}}\int_{t}^{t+1}\left(\|g(s)\|_{L^2(\mathbb{R}^N)}^2+\|\partial_tg(s)\|_{L^2(\mathbb{R}^N)}^2\right)ds <+\infty.
\label{e5}
\end{equation}
We borrow from \cite[Lemma 3.4]{Yan} the following result:
\begin{equation}
\limsup_{k\to +\infty}\Big(\sup_{t\in\mathbb{R}^N}\int_{t}^{t+1}
\int_{|x|\geq k}|g(s,x)|^2\,dx\,ds\Big) = 0.
\label{G}
\end{equation}
\end{itemize}

Since $\mathbb{R}^N$ is unbounded, the embedding
$H^1(\mathbb{R}^N)\subset L^2(\mathbb{R}^N)$ is no longer compact,
that causes the main difficulty. By using ``tail estimate"
 technique (see, e.g., \cite{Wang1, Wang2}), we overcome this
difficulty and thus prove the existence of a uniform attractor
$L^2(\mathbb{R}^N)$. For attractors in $L^p(\mathbb{R}^N)$,
we use some \emph{a priori} estimates (see, e.g., \cite{Song,Yan})
to prove the uniform asymptotic compactness of the family of processes.
Finally, the existence of a uniform attractor in $H^1(\mathbb{R}^N)$
is obtained by combining "tail estimate" method and useful estimates
of nonlinearity. The first main theorem is as follows.

\begin{theorem}\label{result1}
Suppose that $f$ and $g$ satisfy hypothesis {\rm (H1)--(H2)}.
Moreover, we assume that $g$ is normal (see Definition \ref{def_normal})
and $f$ satisfies
\begin{equation}\label{Fbis}
| \frac{\partial f}{\partial x}(x,s) | \leq \psi_5(x), \quad
\forall x\in \mathbb{R}^N, \forall s\in \mathbb{R},
\end{equation}
where $\psi_5\in L^2(\mathbb{R}^N)$. Then, the family of processes
$\{U_{\sigma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ has a unique uniform attractor in
 $H^1(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)$.
\end{theorem}

\begin{remark} \label{rmk1.1} \rm
To prove the existence of a uniform attractor in $L^2(\mathbb{R}^N)$
we only need $f$ and $g$ to satisfy (H1)-(H2). The addition conditions:
 $g$ is normal is needed to obtain the uniform attractor in
$L^p(\mathbb{R}^N)$; and \eqref{Fbis} of $f$ is to prove the asymptotic
compactness of family of processes in $H^1(\mathbb{R}^N)$.
\end{remark}

\begin{remark} \label{rmk1.2} \rm
In the case external force $g$ is bounded uniformly in $t\in \mathbb{R}$;
that is,
\begin{equation*}
\|g(t,\cdot)\|_{L^2(\mathbb{R}^N)} \leq M, \quad \forall t\in \mathbb{R},
\end{equation*}
where $M$ is independent of $t$, we can use arguments similar
to \cite{AnhQuang,Chen} to obtain the existence of a uniform attractor
in $H^1(\mathbb{R}^N)$ easily. In this paper, since $g$ only belongs
to $L^2_{b}(\mathbb{R}; L^2(\mathbb{R}^N))$
(see Definition \ref{translationbounded}), the required computations
are more complicated and involved.
\end{remark}

\begin{remark}\label{rmk1.3} \rm
The positivity of $\lambda$ is used for the dissipativity of the solution; 
that is, the solution of the equation should be bounded uniformly 
in all time $t>0$ (See Proposition \ref{absorbingset}).

If we replace $\mathbb {R}^n$ by a domain $\Omega$ (bounded or unbounded) 
that satisfies Poincare's inequality
\begin{equation*}
\int_{\Omega}|\nabla u|^2dx \geq C\int_{\Omega}|u|^2dx,
\end{equation*}
then we can let $\lambda = 0$ (or even $\lambda >-C$), and 
Proposition \ref{absorbingset} still follows the same way.

If $\lambda<0$ in general, solutions of \eqref{e1} can be unbounded when 
$t\to +\infty$ even in bounded domains. For example, consider 
the  one dimensional equation 
\begin{equation}\label{odd}
\begin{gathered}
u_t - u_{xx} + u - (2\pi^2+1)u = 0, \quad x\in (0,1), t>0,\\
u(t,0) = u(t,1) = 0,\quad t>0,\\
u(0,x) = \sin(\pi x), x\in (0,1).
\end{gathered}
\end{equation}
Here we have $f(u) \equiv u, g(t,x)\equiv 0$ and $\lambda = -(2\pi^2+1)<0$. 
It is easy to check that $u(t,x) = e^{\pi^2t}\sin(\pi x)$ is a solution 
to \eqref{odd} and
\begin{equation*}
\|u(t,\cdot)\|_{L^2(0,1)}^2 = \int_{0}^1e^{2\pi^2t}|\sin(\pi x)|^2dx \to +\infty
\quad \text{as } t\to +\infty.
\end{equation*}
\end{remark}




Another interesting feature of this paper is that we prove the upper
 semi-continuity of uniform attractors with respect to the nonlinearity.
Uniform attractors are not invariant under the family of processes,
this brings some difficulties in proving upper semi-continuous property.
In this work, in order to prove this kind of continuity, we use the structure
of uniform attractors, which says that each uniform attractor is a union
of kernels (see Definition \ref{def_kernel} and Theorem \ref{theo_struc_uniform}).

We consider a family of functions $f_{\gamma}, \gamma \in \Gamma$,
such that for each $\gamma \in \Gamma$, $f_{\gamma}$ satisfies
\eqref{e2}-\eqref{e4} and \eqref{Fbis} where the constants are
independent of $\gamma$. The topology $\mathcal T$ in $\Gamma$ can
be defined as follows:

If $\gamma_m \to \gamma$ in $\mathcal T$ then
 $f_{\gamma_{m}}(x, s) \to f_{\gamma}(x, s)$
for all $x\in \mathbb{R}^N$ and $s\in \mathbb{R}$.

Let $\{U_{\sigma}^{\gamma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ be the family of processes
corresponding to the problem
\begin{equation} \label{e1.9}
\begin{gathered}
u_t - \Delta u + f_{\gamma}(x, u) + \lambda u = g(t, x), \quad
x\in \mathbb{R}^N, t>\tau,\\
u(\tau) = u_{\tau}, \quad x\in \mathbb{R}^N.
\end{gathered}
\end{equation}
By Theorem \ref{result1}, for each $\gamma\in \Gamma$,
$\{U_{\sigma}^{\gamma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ has a compact uniform attractor
 $\mathcal A_{\gamma}$ in $H^1(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)$.
We have the second main result.

\begin{theorem}\label{result2}
The family of uniform attractors $\{\mathcal A_{\gamma}\}_{\gamma\in\Gamma}$
is upper semi-continuous in $L^2(\mathbb{R}^N)$ with respect to the
nonlinearity, that is,
\begin{equation*}
\lim_{\gamma_n\to \gamma}\operatorname{dist}_{L^2(\mathbb{R}^N)}(\mathcal A_{\gamma_n}, \mathcal A_{\gamma}) = 0,
\end{equation*}
whenever $\gamma_n\to \gamma$ in $\mathcal T$.
\end{theorem}

The rest of this article is organized as follows:
In section 2, for convenience to readers, we recall some basic
concepts related to uniform attractors and translation bounded functions.
The proof of Theorems \ref{result1} and \ref{result2} is showed in
Sections 3 and 4, respectively.

Throughout this article, we will denote by $\|\cdot\|$ and
$(\cdot, \cdot)$ the norm and the inner product in $L^2(\mathbb{R}^N)$, respectively.
For a Banach space $X$, $\|\cdot\|_{X}$ stands for its norm.
The letter $C$ denotes an arbitrary constant, which can be different
from line to line and even in a same line.

\section{Preliminaries}
\subsection{Uniform attractors}
Let $\Sigma$ be a parameter set, $X, Y$ be two Banach spaces.
 $\{U_\sigma(t,\tau),t\geq \tau,\tau\in \mathbb{R}\}$, $\sigma\in \Sigma$,
 is said to be a family of processes in $X$, if for each
$\sigma\in \Sigma, \{U_\sigma(t,\tau)\}$ is a process; that is,
the two-parameter family of mappings $\{U_\sigma(t,\tau)\}$ from $X$
to $X$ satisfies
\begin{gather*}
U_\sigma(t,s)U_\sigma(s,\tau) =U_\sigma(t,\tau), \quad \forall t\geq s\geq \tau,\;
\tau\in \mathbb{R},\\
U_\sigma(\tau,\tau) =Id, \quad  \tau \in \mathbb{R},
\end{gather*}
where $Id$ is the identity operator, $\sigma \in \Sigma$ is the symbol,
and  $\Sigma$ is called the symbol space.
Denote by $\mathcal {B}(X), \mathcal B(Y)$ the set of all bounded subsets
of $X$ and $Y$ respectively.

\begin{definition} \rm
A set $B_0\in \mathcal B(Y)$ is said to be a uniform absorbing set
in $Y$ for $\{U_\sigma(t,\tau)\}_{\sigma \in \Sigma}$,
if for any $\tau \in \mathbb{R}$ and any $B\in \mathcal B(X)$,
there exists $T_0\geq \tau$ such that
 $ \cup_{\sigma \in \Sigma}U_\sigma(t,\tau) B \subset B_0$
for all $t\geq T_0$.
\end{definition}

\begin{definition}\label{def_uniform_asypm} \rm
A family of processes $\{U_{\sigma}(t,\tau)\}_{\sigma \in \Sigma}$
is called uniform asymptotically compact in $Y$ if for any $\tau\in \mathbb{R}$
and any $B\in \mathcal B(X)$, we have $\{U_{\sigma_n}(t_n,\tau)x_n\}$
is relatively compact in $Y$, where $\{x_n\} \subset B$,
$\{t_n\}\subset [\tau,+\infty), t_n\to +\infty$ and
$\{\sigma_n\}\subset \Sigma$ are arbitrary.
\end{definition}

\begin{definition}\label{defUniform_att} \rm
A subset $\mathcal A_\Sigma\subset Y$ is said to be the uniform attractor
in $Y$ of the family of processes $\{U_\sigma(t,\tau)\}_{\sigma \in \Sigma}$ if
\begin{itemize}
\item[(i)] $\mathcal A_\Sigma$ is compact in $Y$;

\item[(ii)] for an arbitrary fixed $\tau\in \mathbb{R}$ and
$B\in \mathcal{B}(X)$ we have
\begin{equation*}
\lim_{t\to \infty}(\sup_{\sigma \in \Sigma}( \operatorname{dist}{}_{Y}
(U_\sigma (t,\tau)B, \mathcal A_{\Sigma}))=0,
\end{equation*}
where $  \operatorname{dist}_{Y}(A, B)
= \sup_{x\in A}\inf_{y\in B}\|x-y\|_{Y}$ for $A, B \subset Y$; and

\item[(iii)] if $\mathcal{A'}_{\Sigma}$ is a closed subset of $Y$ satisfying (i),
then $\mathcal{A}_{\Sigma}\subset \mathcal{A'}_{\Sigma}$.
\end{itemize}
\end{definition}

\begin{definition}\label{def_kernel} \rm
The kernel $\mathcal K$ of a process $\{U(t,\tau)\}$ acting on $X$
consists of all bounded complete trajectories of the process $\{U(t,\tau)\}$:
$$
\mathcal K = \{u(\cdot)| U(t,\tau)u(\tau) = u(t),
\operatorname{dist}(u(t),u(0))
\leq C_{u}, \forall t\geq \tau, \tau \in \mathbb{R}\}.
$$
The set $\mathcal K(s) = \{u(s)| u(\cdot)\in \mathcal K\}$ is said to be
kernel section at time $t=s,s\in \mathbb{R}$.
\end{definition}
We have the following result about the existence and structure of uniform
attractors.

\begin{theorem}[\cite{Chen}]\label{theo_struc_uniform}
Assume that the family of processes $\{U_{\sigma}(t,\tau)\}_{\sigma \in \Sigma}$
satisfies the following conditions:
\begin{itemize}
\item[(i)] $\Sigma$ is weakly compact, and
$\{U_{\sigma}(t,\tau)\}_{\sigma\in \Sigma}$ is $(X\times\Sigma, Y)$-weakly
continuous, that is, for any fixed $t\geq \tau$, the mapping
 $(u,\sigma)\mapsto U_{\sigma}(t,\tau)u$ is weakly continuous in $Y$.
 Moreover, there is a weakly continuous semigroup $\{T(h)\}_{h\geq 0}$
acting on $\Sigma$ satisfying
$$
T(h)\Sigma = \Sigma, U_{\sigma}(t+h,\tau+h)=U_{T(h)\sigma}(t,\tau), \quad
\forall \sigma \in \Sigma,t\geq \tau,h\geq 0;
$$
\item[(ii)] $\{U_{\sigma}(t,\tau)\}_{\sigma \in \Sigma}$
has a uniform absorbing set $B_{0}$ in $Y$;

\item[(iii)] $\{U_{\sigma}(t,\tau)\}_{\sigma \in \Sigma}$
is uniform asymptotically compact in $Y$.
 \end{itemize}
Then it possesses a uniform attractor $\mathcal A_{\Sigma}$ in $Y$, and
$$
\mathcal A_\Sigma = \cup_{\sigma \in \Sigma}\mathcal K_{\sigma}(s),\quad
\forall s\in \mathbb{R},
$$
where $\mathcal K_{\sigma}(s)$ is the section at time $s$ of the
process $\{U_{\sigma}(t,\tau)\}$.
\end{theorem}

\subsection{The translation bounded functions}

\begin{definition}\label{translationbounded} \rm
Let $\mathcal{E}$ be a reflexive Banach space.
A function $\varphi\in L^2_{loc}(\mathbb{R}; \mathcal{E})$ is said to
 be translation bounded if
$$
\|\varphi\|^2_{L^2_b}=\|\varphi\|_{L^2_b(\mathbb{R}; \mathcal{E})}^2
=\sup_{t\in\mathbb{R}}\int_t^{t+1}\|\varphi\|^2_{\mathcal{E}}ds<\infty.
$$
\end{definition}

Let $g\in L^{2}_{b}\left(\mathbb{R}, L^{2}(\mathbb{R}^N)\right)$,
we denote by $\mathcal {H}_w(g)$ be the closure of the set
$\{g(\cdot+h): h\in \mathbb{R}\}$ in $L^2_b(\mathbb{R}; L^2(\mathbb{R}^N))$
 with the weak topology.  The following results are proved in \cite{Chep}.

\begin{lemma}[{\cite[Proposition 4.2]{Chep}}] \label{lem0}
\begin{enumerate}
\item For all $\sigma \in \mathcal {H}_w(g)$,
 $\|\sigma\|_{L^2_b}^2\leq\|g\|^2_{L^2_b}$;

\item The translation group $\{T(h)\}$ is weakly continuous on $\mathcal {H}_w(g)$;

\item $T(h)\mathcal {H}_w(g)=\mathcal {H}_w(g)$ for $h\geq 0$;

\item $\mathcal {H}_w(g)$ is weakly compact.
\end{enumerate}
\end{lemma}

\section{Existence of uniform attractors}

In this section, we prove the existence of uniform attractors for the
family of processes corresponding to problem \eqref{e1}.
First, we state without proofs the results about the existence
 of a unique weak solution of \eqref{e1} and then prove there exists
a uniform absorbing set for $\{U_{\sigma}(t,\tau)u_{\tau}\}_{\sigma\in\mathcal{H}_w(g)}$.
Next, by a technique so called "tail estimate" we obtain a uniform
attractor in $L^2(\mathbb{R}^N)$. Then, using abstract result in \cite{Yan},
we prove the existence of a uniform attractor in $L^p(\mathbb{R}^N)$.
Finally, the existence of the uniform attractor in $H^1(\mathbb{R}^N)$
is obtained by combining "tail estimate" and arguments in \cite{Ma}.

\subsection{Existence of uniform absorbing set}

\begin{definition}\label{weaksol} \rm
A function $u(t,x)$ is called a weak solution of \eqref{e1} on
$(\tau, T)$, $T>\tau$, if
\begin{gather*}
u\in C\left([\tau,T];L^2(\mathbb{R}^N)\right)\cap
L^p\left(\tau,T;L^p(\mathbb{R}^N)\right) \cap L^2(\tau,T;H^1(\mathbb{R}^N)),
\\
u_t\in L^{2}(\tau,T;L^2(\mathbb{R}^N)),
\\
u(\tau, x) = u_{\tau}(x) \text{a.e. on } \mathbb{R}^N,
\end{gather*}
and for any $v\in C^{\infty}([\tau,T]\times \mathbb{R}^N)$,
\begin{equation*}
\int_{\tau}^{T}\int_{\mathbb{R}^N}
\left(u_t v + \nabla u \nabla v + f(x,u)v + \lambda uv \right)
 = \int_{\tau}^{T}\int_{\mathbb{R}^N}gv.
\end{equation*}
\end{definition}

By the standard Galerkin-Feado approximation, we can find the existence
 of unique weak solution for problem \eqref{e1} in the case of bounded domains.
 To overcome the difficulties of unboundedness of the domains,
following \cite{Mori}, one may take the domain to be a sequence of
balls with radius approaching $\infty$ to deduce the existence of a
weak solution to \eqref{e1} in $\mathbb{R}^N$. Here we state results only,
for the details of the proof, readers are referred to \cite{Mori}.

\begin{theorem}\label{solution}
Assume that $f$ and $g$ satisfy {\rm (H1)--(H2)}.
For any $u_\tau \in L^2(\mathbb{R}^N)$ and any $T>\tau$,
there exists a unique weak solution $u$ for problem \eqref{e1}, and
\[
u \in C\left([\tau,T];L^2(\mathbb{R}^N)\right); \quad
 u_t \in L^{2}\left(\tau,T;L^2(\mathbb{R}^N)\right).
\]
\end{theorem}

From Theorem \ref{solution}, we can define a family of processes
$\{U_{\sigma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ associated with \eqref{e1} acting
on $L^2(\mathbb{R}^N)$, where $U_{\sigma}(t,\tau)u_{\tau}$
is the solution of \eqref{e1} at time $t$ subject to initial condition
 $u(\tau) = u_{\tau}$ at time $\tau$ and with $\sigma$ in place of $g$.

\begin{proposition}\label{absorbingset}
There exists a uniform absorbing set $\mathcal B$ in
 $H^1(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)$ for the family of processes
 $\{U_{\sigma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ corresponding to \eqref{e1}.
\end{proposition}

\begin{proof}
Consider the equation
\begin{equation}
u_t - \Delta u + f(x,u) + \lambda u = \sigma(t,x). \label{e6}
\end{equation}
Taking the inner product of \eqref{e6} with $2u$ in $L^2(\mathbb{R}^N)$, we have
\begin{equation}
\frac{d}{dt}\|u\|^2 + 2\|\nabla u\|^2 + 2(f(x,u), u) + 2\lambda \|u\|^2
= 2(\sigma(t),u). \label{e7}
\end{equation}
Using \eqref{e2}, applying the Cauchy and Young's inequalities,
\begin{equation}
\frac{d}{dt}\|u\|^2 + \frac{3\lambda}{2}\|u\|^2 + 2\|\nabla u\|^2
 + 2\alpha_1\|u\|_{L^p(\mathbb{R}^N)}^p \leq \frac{2}{\lambda}\|\sigma(t)\|^2
+ 2\|\phi_1\|_{L^1(\mathbb{R}^N)}.
\label{e8}
\end{equation}
By Gronwall's lemma, we find
\begin{equation}
\|u(t)\|^2 \leq e^{-\lambda(t-\tau)}\|u_{\tau}\|^2
+ \frac{2\|\phi_1\|_{L^1(\mathbb{R}^N)}}{\lambda}
+ \frac{2}{\lambda}\int_{\tau}^{t}e^{-\lambda(t-s)}\|\sigma(s)\|^2ds.
\label{e9}
\end{equation}
For the last term of the right hand side,
\begin{equation}
\begin{aligned}
\int_{\tau}^{t}e^{-\lambda(t-s)}\|\sigma(s)\|^2ds
&\leq \Big(\int_{t-1}^{t}+\int_{t-2}^{t-1}+\int_{t-3}^{t-2}+\ldots\Big)
 e^{-\lambda (t-s)}\|\sigma(s)\|^2ds\\
&\leq \int_{t-1}^{t}\|\sigma(s)\|^2ds + e^{-\lambda}\int_{t-2}^{t-1}\|\sigma(s)\|^2 + \ldots\\
&\leq \big(1+e^{-\lambda}+e^{-2\lambda}+\ldots\big)\|\sigma\|_{L^{2}_{b}}^{2}\\
&\leq \frac{1}{1-e^{-\lambda}}\|g\|_{L^{2}_{b}}^2.
\end{aligned} \label{e10}
\end{equation}
Combining \eqref{e9}-\eqref{e10}, and noting that $u_{\tau}$ belongs to
a bounded set $B$, there exists a $T_0>0$ satisfies
\begin{equation}
\|u(t)\|^2 \leq \rho_0 = 1 + \frac{2\|\phi_1\|_{L^{1}(\mathbb{R}^N)}}{\lambda}
+ \frac{2e^{\lambda}}{\lambda(e^{\lambda}-1)}\|g\|_{L^{2}_{b}}^{2},
\label{e11}
\end{equation}
for all $t>T_0$, all $u_{\tau}\in B$ and all $\sigma\in\mathcal{H}_w(g)$. By integrating \eqref{e8},
we find that
\begin{equation}
\begin{aligned}
&\int_{t}^{t+1}\Big(\frac{\lambda}{2}\|u(s)\|^2 + 2\|\nabla u(s)\|^2
+ 2\alpha_1\|u(s)\|_{L^p(\mathbb{R}^N)}^p\Big)ds\\
&\leq \|u(t)\|^2 + \frac{2}{\lambda}\int_{t}^{t+1}\|\sigma(s)\|^2ds
+ \frac{2\|\phi_1\|_{L^1(\mathbb{R}^N)}}{\lambda}\\
&\leq \rho_0 + \frac{2}{\lambda}\|g\|_{L^{2}_{b}}^{2}
+ \frac{2\|\phi_1\|_{L^1(\mathbb{R}^N)}}{\lambda},
\end{aligned} \label{e12}
\end{equation}
for all $t\geq T_0$. From \eqref{e4},
\[
 \|u\|_{L^p(\mathbb{R}^N)}^p
\geq \frac{1}{\alpha_4}\Big(\int_{\mathbb{R}^N}F(x,u)dx
- \|\phi_4\|_{L^1(\mathbb{R}^N)}\Big),
\]
 and \eqref{e12}, it leads to
\begin{equation}
\int_{t}^{t+1}\Big(\lambda\|u(s)\|^2+\|\nabla u(s)\|^2
+ 2\int_{\mathbb{R}^N}F(x,u)dx\Big)ds \leq C, \quad \text{for all } t\geq T_0.
\label{e13}
\end{equation}
On the other hand, by multiplying \eqref{e6} by $2u_t$ then integrating
over $\mathbb{R}^N$, after using Cauchy's inequality,
\begin{equation}
\|u_t\|^2 + \frac{d}{dt}\Big( \lambda\|u\|^2+\|\nabla u\|^2
+ 2\int_{\mathbb{R}^N}F(x,u)dx\Big) \leq \|\sigma(t)\|^2.
 \label{e14}
\end{equation}
From \eqref{e13}-\eqref{e14} and the uniform Gronwall inequality, we obtain
\begin{equation}
\lambda\|u\|^2+\|\nabla u\|^2 + 2\int_{\mathbb{R}^N}F(x,u)dx
 \leq C, \text{ for all } t\geq T_0. \label{e15}
\end{equation}
Using \eqref{e4} again, there exists $\rho_1>0$ such that, for all $t\geq T_0$,
\begin{equation}
\|u(t)\|^2 + \|\nabla u(t)\|^2 + \|u(t)\|_{L^p(\mathbb{R}^N)}^p
 \leq \rho_1, \quad \forall u_{\tau}\in B, \forall \sigma\in\mathcal{H}_w(g).
\label{e16}
\end{equation}
This completes the proof.
\end{proof}

\begin{lemma}\label{weakcontinuity}
The family of processes associated with problem \eqref{e1} is
$(L^2(\mathbb{R}^N)\times \mathcal H_w(g), H^1(\mathbb{R}^N)\cap L^p(\mathbb{R}^N))$
weakly continuous, that is, for any $x_n \rightharpoonup x_0$ in $L^2(\mathbb{R}^N)$
and $\sigma_n \rightharpoonup \sigma$ in $\mathcal H_w(g)$, we have
\begin{equation}
U_{\sigma_n}(t,\tau)x_n \rightharpoonup U_{\sigma}(t,\tau)x\quad
\text{in } H^1(\mathbb{R}^N) \cap L^p(\mathbb{R}^N), \label{w1}
\end{equation}
for all $t>\tau$.
\end{lemma}

\begin{proof}
Denote by $u_n(t) = U_{\sigma_n}(t,\tau)x_n$, then $u_n$ solves
\begin{equation}
\partial_t u_n - \Delta u_n + f(x, u_n) + \lambda u_n = \sigma_n(t),
\label{w2}
\end{equation}
with initial condition $u_n(\tau) = x_n$. Using arguments in
Proposition \ref{absorbingset}, we can deduce that there exists a
function $w(t)$ such that
\begin{gather}
u_n \rightharpoonup w \text{ weak-star in } L^{\infty}(\tau,t;L^2(\mathbb{R}^N)),
\label{w7}
\\
u_n \rightharpoonup w \text{ in } L^p(\tau,t;L^p(\mathbb{R}^N)),
\label{w8}
\end{gather}
and the sequence
\begin{equation}
\{u_n(s)\}, \tau\leq s\leq t, \text{ is bounded in }
 H^1(\mathbb{R}^N)\cap L^p(\mathbb{R}^N).
\label{w9}
\end{equation}
By \eqref{e3} and \eqref{w8},
$$
\{f(x,u_n)\} \text{ is bounded } L^{q}(\tau,t;L^q(\mathbb{R}^N)),
$$
thus, by equation \eqref{w2},
$$
\{\partial_tu_n\} \text{ is bounded in } L^q(\tau,t;L^q(\mathbb{R}^N))
\cap L^{2}(\tau,t;H^{-1}(\mathbb{R}^N)).
$$
Therefore, one can pass to the limit (in the weak sense) of equation \eqref{w2}
to have
\begin{equation}
w_t - \Delta w + f(x, w) + \lambda w = \sigma(t),
\label{w10}
\end{equation}
with $w(\tau) = x$. In fact, there are some difficulties to overcome
when one wants to show $f(x, u_n)\rightharpoonup f(x,w)$, but it can be
solved by taking the domain to be a sequence of balls with radius
approaching $\infty$ as mentioned before Theorem \ref{solution}.
By the uniqueness of the weak solution, we obtain $U_{\sigma}(t,\tau)x = w(t)$
and thus complete the proof.
\end{proof}

\subsection{Existence of a uniform attractor in $L^2(\mathbb{R}^N)$}

\begin{lemma}\label{tail}
For any $\varepsilon >0$, any $\tau \in \mathbb{R}$ and any
$B\subset L^{2}(\mathbb{R}^N)$ is bounded, there exist
 $T_\varepsilon>\tau$ and $K_{\varepsilon}>0$ such that
\begin{equation}
\int_{|x|\geq K}|U_{\sigma}(t,\tau)u_{\tau}|^2dx \leq \varepsilon,
\label{e22}
\end{equation}
for all $K\geq K_{\varepsilon}$, $t\geq T_{\varepsilon}$, all $u_{\tau}\in B$ and all $\sigma\in\mathcal{H}_w(g)$.
\end{lemma}

\begin{proof}
Let $\phi: [0,+\infty) \to [0,1]$ be a smooth function such that
 $\phi(s) = 0$ for all $0\leq s\leq 1$ and $\phi(s) = 1$ for all $s\geq 2$.
It is easy to see that $\phi'(s)\leq C$, for all $s$, and $\phi'(s) = 0$
for all $s\geq 2$. Denote $u(t) = U_{\sigma}(t,\tau)u_{\tau}$ and
 multiply \eqref{e6} by $2\phi\big(\frac{|x|^2}{k^2}\big)u$, where $k>0$,
we obtain
\begin{equation} \label{e23}
\begin{aligned}
&\frac{d}{dt}\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)|u|^2dx
  + 2\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)|\nabla u|^2dx
 + 2\int_{\mathbb{R}^N}\phi'\Big(\frac{|x|^2}{k^2}\Big)u\frac{2x}{k^2}\cdot
  \nabla u\,dx\\
&+ 2\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)f(x,u)u\,dx
 + 2\lambda \int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)|u|^2dx\\
&= 2\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)u\sigma(t,x)dx.
\end{aligned}
\end{equation}
Now, we estimate terms in \eqref{e23}. First, using
condition \eqref{e2} of $f$, we find
\begin{equation}
\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)f(x,u)u\,dx
\geq -\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)\phi_1(x)dx
\geq -\int_{|x|\geq k}|\phi_1(x)|dx.
\label{e24}
\end{equation}
Next,
\begin{equation}
\begin{aligned}
\big|\int_{\mathbb{R}^N}\phi'\Big(\frac{|x|^2}{k^2}\Big)u\frac{2x}{k^2}\cdot
\nabla u\,dx\big|
&\leq \int_{|x|\leq k\sqrt{2}}\frac{C|x|}{k^2}|u||\nabla u|dx\\
&\leq \frac{C}{k}\int_{\mathbb{R}^N}|u||\nabla u|dx\\
&\leq \frac{C}{k}\Big(\|u\|^2 + \|\nabla u\|^2\Big)
\leq \frac{C}{k},
\end{aligned} \label{e25}
\end{equation}
for all $t\geq T_0$, since \eqref{e16}. Finally, for the right hand
side of \eqref{e23},
\begin{equation}
2\big|\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)\sigma(t,x)u\,dx\big|
\leq \frac{1}{\lambda}\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)
|\sigma(t,x)|^2dx + \lambda\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)
|u|^2dx. \label{e26}
\end{equation}
Combining \eqref{e23}-\eqref{e26}, we obtain
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)|u|^2dx
 +\lambda \int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)|u|^2dx
 + 2\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)|\nabla u|^2dx\\
&\leq \frac{C}{k} + 2\int_{|x|\geq k}|\phi_1(x)|dx
 + \frac{1}{\lambda}\int_{|x|\geq k}|\sigma(t,x)|^2dx.
\end{aligned}
\label{e27}
\end{equation}
By Gronwall's lemma, proceed as \eqref{e10}, we conclude that
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)|u(t)|^2dx
 +2\int_{\tau}^{t}e^{-\lambda(t-\tau)}\int_{\mathbb{R}^N}\phi
 \Big(\frac{|x|^2}{k^2}\Big)|\nabla u|^2\,dx\,ds\\
&\leq e^{-\lambda(t-\tau)}\int_{\mathbb{R}^N}\phi
 \Big(\frac{|x|^2}{k^2}\Big)|u_{\tau}|^2dx + \frac{C}{\lambda k}\\
&\quad + \frac{2}{\lambda}\int_{|x|\geq k}|\phi_1(x)|dx
 + \frac{1}{\lambda}\int_{\tau}^{t}e^{-\lambda(t-s)}\int_{|x|\geq k}|\sigma(s,x)|^2\,dx\,ds\\
&\leq e^{-\lambda(t-\tau)}\|u_{\tau}\|^2
 + C\Big(\frac{1}{k}+\int_{|x|\geq k}|\phi_1(x)|dx\Big)\\
&\quad + \frac{1}{\lambda(1-e^{-\lambda})}\sup_{t\in\mathbb{R}}
 \int_{t}^{t+1}\int_{|x|\geq k}|g(s,x)|^2\,dx\,ds.
\end{aligned} \label{e28}
\end{equation}
Using \eqref{G} and the fact that $\phi_1 \in L^1(\mathbb{R}^N)$,
it can be followed from \eqref{e28} that
\begin{equation}
\limsup_{t\to +\infty}\limsup_{k\to +\infty}\int_{|x|\geq k\sqrt{2}}|u(t)|^2dx = 0,
\label{e29}
\end{equation}
which completes the proof of \eqref{e22}.
\end{proof}

\begin{theorem}\label{L2}
Assume that assumptions {\rm (H1)--(H2)} hold. Then the family of processes
 $\{U_{\sigma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ possesses a uniform attractor
$\mathcal A_{2}$ in $L^2(\mathbb{R}^N)$. Moreover, we have
\begin{equation}
\mathcal A_2 = \cup_{\sigma\in\mathcal{H}_w(g)}\mathcal K_{\sigma}(s) \quad
\text{for all } s\in \mathbb{R}.
\label{e29_1}
\end{equation}
\end{theorem}

\begin{proof}
By Proposition \ref{absorbingset}, the family $\{U_{\sigma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ has a
uniform absorbing set in $L^2(\mathbb{R}^N)$. Thus, it is sufficient
 to prove the uniform asymptotic compactness of $\{U_{\sigma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$.
Let $\{x_n\}$ be a bounded set in $L^2(\mathbb{R}^N)$, $\{t_n\}$ be
a sequence such that $t_n \to +\infty$ as $n\to \infty$, and
$\{\sigma_n\}$ be an arbitrary sequence in $\mathcal H_w(g)$.
We have to show that $\{U_{\sigma_n}(t_n,\tau)x_n\}$ is precompact
in $L^2(\mathbb{R}^N)$. Let $\varepsilon>0$ arbitrary.
For $K>0$, we denote $B_K = \{x\in \mathbb{R}^N: |x|\leq K\}$.
From Lemma \ref{tail} and $\lim_{n\to \infty}t_n = +\infty$, there
exist $K>0$ and $N_0\in \mathbb N$ satisfy
\begin{equation}
\|U_{\sigma_n}(t_n,\tau)x_n\|_{L^{2}(B_K^c)}\leq \frac{\varepsilon}{3}, \quad
\forall n\geq N_0, \label{e30}
\end{equation}
where $B_K^c = \mathbb{R}^N\backslash B_{K}$. On the other hand,
 from Proposition \ref{absorbingset}, $\{U_{\sigma_n}(t_n,\tau)x_n\}$
is bounded in $H^1(\mathbb{R}^N)$, and then $\{U_{\sigma_n}(t_n,\tau)x_n\}$
restrict on $B_K$ is bounded in $H^1(B_K)$.
Since, $H^1(B_K)\hookrightarrow L^2(B_K)$ compactly,
$\{U_{\sigma_n}(t_n,\tau)x_n\}$ is precompact in $L^2(B_K)$, thus there
exist a subsequence $\{n'\}\subset \{n\}$ and $N_1$ such that
\begin{equation}
\|U_{\sigma_{m'}}(t_{m'},\tau)x_{m'} - U_{\sigma_{n'}}(t_{n'},
\tau)x_{n'}\|_{L^2(B_K)} \leq \frac{\varepsilon}{3}, \quad
\text{for all } m', n' \geq N_1. \label{e31}
\end{equation}
Taking $N = \max\{N_0, N_1\}$, then for all $m', n' \geq N$,
\begin{equation}
\begin{aligned}
&\|U_{\sigma_{m'}}(t_{m'},\tau)x_{m'}
 - U_{\sigma_{n'}}(t_{n'},\tau)x_{n'}\|_{L^{2}(\mathbb{R}^N)}\\
&\leq \|U_{\sigma_{m'}}(t_{m'},\tau)x_{m'}
 - U_{\sigma_{n'}}(t_{n'},\tau)x_{n'}\|_{L^{2}(B_K)}\\
&\quad +\|U_{\sigma_{m'}}(t_{m'},\tau)x_{m'}\|_{L^{2}(B_K^c)}
  + \|U_{\sigma_{n'}}(t_{n'},\tau)x_{n'}\|_{L^{2}(B_K^c)}
\leq \varepsilon,
\end{aligned} \label{e32}
\end{equation}
by \eqref{e30} and \eqref{e31}. This prove that
$\{U_{\sigma_{n}}(t_n,\tau)x_n\}$ is precompact in $L^2(\mathbb{R}^N)$.
Relation \eqref{e29_1} follows directly from Theorem \ref{theo_struc_uniform}
and Lemma \ref{weakcontinuity}. The proof is complete.
\end{proof}

\subsection{Existence of a uniform attractor in $L^p(\mathbb{R}^N)$}

To obtain the existence of a uniform attractor in $L^p(\mathbb{R}^N)$,
we assume that the external force $g$ belongs to $L_{n}^{2}$,
the space of normal functions, which is defined as follows.

\begin{definition}\label{def_normal}
A function $\varphi \in L^{2}_{\rm loc}(\mathbb{R}; L^{2}(\mathbb{R}^N))$
is said to be normal if for any $\varepsilon>0$ there exists $\eta>0$ such that
\begin{equation*}
\sup_{t\in \mathbb{R}^N}\int_{t}^{t+\eta}\|\varphi(s)\|_{L^{2}(\mathbb{R}^N)}^2ds
 \leq \varepsilon.
\end{equation*}
\end{definition}

\begin{lemma}[\cite{Lu}]\label{normal}
If $g\in L^{2}_{n}(\mathbb{R}; L^2(\mathbb{R}^N))$ then
$g\in L^2_b(\mathbb{R};L^2(\mathbb{R}^N))$ and for any $\tau \in \mathbb{R}^N$,
\begin{equation*}
\lim_{\gamma\to \infty}\sup_{t\geq \tau}\int_{\tau}^{t}e^{-\gamma(t-s)}
\|\sigma(s)\|_{L^2(\mathbb{R}^N)}^2ds = 0,
\end{equation*}
uniformly with respect to $\sigma \in \mathcal H_w(g)$.
\end{lemma}

We also need an additional result whose proof can be found in \cite{Yan}.

\begin{lemma}[\cite{Yan}]\label{L2Lp}
Assume $\{U_{\sigma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ is a family of processes
in $L^2(\mathbb{R}^N)$ and $L^p(\mathbb{R}^N)$, $p\geq 2$. If
\begin{itemize}
\item[(i)] $\{U_{\sigma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ possesses a uniform
attractor in $L^2(\mathbb{R}^N)$;

\item[(ii)] $\{U_{\sigma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ has a bounded uniform
absorbing set in $L^p(\mathbb{R}^N)$;

\item[(iii)] for any $\varepsilon>0$ and any bounded set
$B\subset L^2(\mathbb{R}^N)$, there exist $T = T(\varepsilon, B)$
and $M=M(\varepsilon, B)$ such that
\begin{equation}
\int_{\Omega(|U_{\sigma}(t,\tau)u_{\tau}|\geq M)}|U_{\sigma}(t,\tau)u_{\tau}|^pdx
\leq \varepsilon, \text{ for all } \sigma\in\mathcal{H}_w(g), t\geq T, u_{\tau}\in B,
\label{e33}
\end{equation}
where $\Omega(|U_{\sigma}(t,\tau)u_{\tau}|\geq M)
= \{x\in \mathbb{R}^N: U_{\sigma}(t,\tau)u_{\tau}(x)\geq M\}$;
\end{itemize}
then $\{U_{\sigma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ has a uniform attractor in $L^p(\mathbb{R}^N)$.
\end{lemma}

\begin{theorem}\label{Lp}
Assume that $f$ and $g$ satisfy {\rm (H1)--(H2)}. We also assume that
 $g$ is a normal function. Then the family of processes
$\{U_{\sigma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ has a uniform attractor $\mathcal A_p$
in $L^p(\mathbb{R}^N)$, moreover $\mathcal A_p$ coincides with $\mathcal A_2$.
\end{theorem}

\begin{proof}
By Proposition \ref{absorbingset}, Theorem \ref{L2} and Lemma \ref{L2Lp},
 we only have to prove that $\{U_{\sigma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ satisfies
 condition (iii) in Lemma \ref{L2Lp}. Let $B$ be a bounded subset
of $L^2(\mathbb{R}^N)$ and $\varepsilon>0$ arbitrary.
For $u(t) = U_{\sigma}(t,\tau)u_{\tau}$, we denote by $(u-M)_{+}$
the positive part of $u - M$; that is,
\begin{equation}
(u-M)_{+}=\begin{cases}
u - M&\text{if } u\geq M\\
0 &\text{otherwise},
\end{cases} \label{e34}
\end{equation}
Multiplying \eqref{e1} by $p(u-M)_{+}^{p-1}$, we obtain
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\|(u-M)_{+}\|_{L^p(\mathbb{R}^N)}^p
 + p(p-1)\int_{\mathbb{R}^N}|\nabla u|^2|(u-M)_{+}|^{p-2}dx\\
&+ p\int_{\mathbb{R}^N}f(u)(u-M)_{+}^{p-1}dx \\
&= \int_{\mathbb{R}^N}\sigma(t,x)(u-M)_{+}^{p-1}dx.
\end{aligned} \label{e35}
\end{equation}
By \eqref{e2}, we can take $M$ large enough to get
$f(x,u)\geq C|u|^{p-1}$ when $u\geq M$, and thus,
\begin{align*}
\int_{\mathbb{R}^N}f(u)(u-M)_{+}^{p-1}dx
&\geq C\int_{\mathbb{R}^N}|u|^{p-1}(u-M)_{+}^pdx\\
&\geq C\int_{\mathbb{R}^N}(u-M)_{+}^{2p-2}dx
 + CM^{p-2}\|(u-M)_{+}\|_{L^p(\mathbb{R}^N)}^p.
\end{align*} %\label{e36}
For the external force,
\begin{equation}
\int_{\mathbb{R}^N}\sigma(t,x)(u-M)_{+}^{p-1}dx
 \leq C\|\sigma(t)\|^2 + C\int_{\mathbb{R}^N}(u-M)_{+}^{2p-2}dx. \label{e37}
\end{equation}
Combining \eqref{e35}-\eqref{e37}, we obtain
\begin{equation}
\frac{d}{dt}\|(u-M)_{+}\|_{L^p(\mathbb{R}^N)}^p
+ CM^{p-2}\|(u-M)_{+}\|_{L^p(\mathbb{R}^N)}^p \leq C\|\sigma(t)\|^2.
\label{e38}
\end{equation}
By Gronwall's lemma,
\begin{equation}
\begin{aligned}
\|(u(t)-M)_{+}\|_{L^p(\mathbb{R}^N)}^p
&\leq e^{-CM^{p-2}(t-T_1)}\|(u(T_1)-M)_{+}\|_{L^p(\mathbb{R}^N)}^p\\
&\quad + C\int_{T_1}^{t}e^{-CM^{p-2}(t-s)}\|\sigma(s)\|^2ds,
\end{aligned}\label{e39}
\end{equation}
where $T_1$ is in \eqref{e16}. Applying \eqref{e16} and Lemma \ref{normal},
 we obtain
\begin{equation}
\int_{\Omega_1}|(u(t)-M)_{+}|^pdx \leq \varepsilon, \quad
\text{uniformly in } u_{\tau}\in B, \sigma\in\mathcal{H}_w(g),
\label{e40}
\end{equation}
when $t$ and $M$ are large enough. Repeat steps above,
just replace $(u-M)_{+}$ by $(u+M)_{-}$, where
\begin{equation}
(u+M)_{-}=
\begin{cases}
u + M&\text{if } u\leq -M\\
0 &\text{otherwise},
\end{cases}\label{e41}
\end{equation}
we can find $t$ and $M$ large enough such that
\begin{equation}
\int_{\Omega(u\leq -M)}|(u+M)_{-}|^pdx \leq \varepsilon, \quad
\forall u_{\tau}\in B, \forall \sigma\in\mathcal{H}_w(g). \label{e42}
\end{equation}
From \eqref{e40} and \eqref{e42}, we obtain \eqref{e33} and hence
complete the proof.
\end{proof}

\subsection{Existence of a uniform attractor in
$H^{1}(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)$}

In this section, we prove the uniform attractor in
$H^{1}(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)$. For this purpose,
we first assume an addition condition of the nonlinearity
\begin{equation}
\big|\frac{\partial f}{\partial x}(x,u)\big| \leq \phi_5(x),\label{e3_2}
\end{equation}
where $\phi_5\in L^2(\mathbb{R}^N)$. Next, we show that solutions
of \eqref{e1} is uniformly small when time and spatial variables
are large enough. Finally, combining this and arguments similar to
the ones used in \cite{Ma}, we can prove the uniform asymptotic
compactness of $\{U_{\sigma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ in $H^1(\mathbb{R}^N)$.

\begin{lemma}\label{u_t}
For any $\tau\in \mathbb{R}$ and any bounded set
$B\subset H^1(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)$,
there exist $\rho_2>0$ and $T_1\geq \tau$ such that
\begin{equation}
\|u_t(t)\|^2 \leq \rho_1, \forall t\geq T_1,\quad
\forall u_{\tau}\in B,\;\forall \sigma\in\mathcal{H}_w(g). \label{e17}
\end{equation}
\end{lemma}

\begin{proof}
Integrating \eqref{e14} from $t$ to $t+1$, where $t\geq T_0$,
using \eqref{e4} and \eqref{e16}, we have
\begin{equation}
\begin{aligned}
&\int_{t}^{t+1}\|u_t(s)\|^2ds + 2\|\phi_3\|_{L^1(\mathbb{R}^N)}\\
&\leq \lambda\|u(t)\|^2 + \|\nabla u(t)\|^2 + 2\int_{\mathbb{R}^N}F(x,u)dx + \int_{t}^{t+1}\|\sigma(s)\|^2ds\\
&\leq (\lambda+1+2\alpha_4)\rho_1 + 2\|\phi_4\|_{L^1(\mathbb{R}^N)} + \|g\|_{L^{2}_{b}}^{2},
\end{aligned}\label{e18}
\end{equation}
thus
\begin{equation}
\int_{t}^{t+1}\|u_t(s)\|^2ds \leq C, \quad\text{for all } t\geq T_0. \label{e19}
\end{equation}
Now, differentiate \eqref{e6} with respect to time, denote $v = u_t$,
then multiply by $2v$ in $L^2(\mathbb{R}^N)$, we see that
\begin{equation}
\frac{d}{dt}\|v\|^2 + 2\|\nabla v\|^2 + (f'(x,u)v, 2v) + 2\lambda\|v\|^2
 = (\sigma'(t), 2v). \label{e20}
\end{equation}
By \eqref{e3_1} and Cauchy's inequality,
\begin{equation}
\frac{d}{dt}\|v\|^2 \leq 2\ell\|v\|^2 + \frac{1}{2\lambda}\|\sigma'(t)\|^2.
\label{e21}
\end{equation}
Combining \eqref{e19} and \eqref{e21}, then using the uniform Gronwall lemma,
we obtain \eqref{e17}.
\end{proof}

\begin{lemma}\label{f(u)}
For any $\tau\in \mathbb{R}$, and any bounded set $B\subset L^2(\mathbb{R}^N)$,
\begin{equation}
\int_{\mathbb{R}}|f(x,U_{\sigma}(t,\tau)u_{\tau})|^2dx
\leq C(1+\|\sigma(t)\|_{L^2(\mathbb{R}^N)}^2), \label{e43}
\end{equation}
for all $t\geq T_1$,  all $u_{\tau}\in B$ and all $\sigma\in\mathcal{H}_w(g)$.
\end{lemma}

\begin{proof}
Multiply \eqref{e1} by $|u|^{p-2}u$ in $L^2(\mathbb{R}^N)$, we obtain
\begin{equation}
\begin{aligned}
&(u_t, |u|^{p-2}u) + (p-1)\int_{\mathbb{R}^N}|\nabla u|^2|u|^{p-2}dx \\
&+ \int_{\mathbb{R}^N}f(x, u)u|u|^{p-2}dx
 + \lambda\|u\|_{L^p(\mathbb{R}^N)}^p
= (\sigma(t,x), |u|^{p-2}u).
\end{aligned} \label{e44}
\end{equation}
By the Cauchy and Young's inequalities,
\begin{gather}
(u_t, |u|^{p-2}u) \leq C\|u_t\|^2
 + \frac{\alpha_1}{4}\int_{\mathbb{R}^N}|u|^{2p-2}dx, \label{e45}\\
(\sigma(t,x), |u|^{p-2}u) \leq C\|\sigma(t)\|^2
 + \frac{\alpha_1}{4}\int_{\mathbb{R}^N}|u|^{2p-2}dx.\label{e46}
\end{gather}
Using \eqref{e2}, we obtain
\begin{equation}
\begin{aligned}
\int_{\mathbb{R}^N}f(x,u)u|u|^{p-2}dx
&\geq \alpha_1\int_{\mathbb{R}^N}|u|^{2p-2}dx
 - \int_{\mathbb{R}^N}\phi_1(x)|u|^{p-2}dx\\
&\geq \alpha_1\int_{\mathbb{R}^N}|u|^{2p-2}dx
 - C\|\phi_1\|_{L^{p/2}(\mathbb{R}^N)}^{p/2} - C\|u\|_{L^p(\mathbb{R}^N)}^p.
\end{aligned}\label{e47}
\end{equation}
From \eqref{e44}--\eqref{e47}, we obtain
\begin{equation}
\begin{aligned}
\int_{\mathbb{R}^N}|u(t)|^{2p-2}dx
&\leq C(1+\|u_t(t)\|^2 + \|u(t)\|_{L^p(\mathbb{R}^N)}^{2}+ \|\sigma(t)\|^{2})\\
&\leq C(1+ \|\sigma(t)\|^2),
\end{aligned}\label{e48}
\end{equation}
for all $t\geq \max\{T_0, T_1\}$, since \eqref{e16} and \eqref{e17}.
On the other hand, by \eqref{e3},
\begin{equation}
\int_{\mathbb{R}^N}|f(x,u)|^2dx
\leq 2\alpha_2^2\int_{\mathbb{R}^N}|u|^{2p-2}dx + 2\|\phi_2\|^2.
\label{e49}
\end{equation}
This, combining with \eqref{e48}, completes the proof.
\end{proof}

\begin{lemma}\label{tail1}
For any $\varepsilon >0$, any $\tau \in \mathbb{R}$ and any
$B\subset L^{2}(\mathbb{R}^N)$ is bounded, there exist
$T_\varepsilon>\tau$ and $K_{\varepsilon}>0$ such that
\begin{equation}
\int_{|x|\geq K}|\nabla U_{\sigma}(t,\tau)u_{\tau}|^2dx \leq \varepsilon,
\label{50}
\end{equation}
for all $K\geq K_{\varepsilon}$, $t\geq T_{\varepsilon}$,
all $u_{\tau}\in B$ and all $\sigma\in\mathcal{H}_w(g)$.
\end{lemma}

\begin{proof}
By multiplying \eqref{e1} by $  -2\phi(|x|^2/k^2)\Delta u$,
 where $\phi$ is in Lemma \ref{tail}, we obtain
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)|\nabla u|^2dx
 + 2\int_{\mathbb{R}^N}\phi'\Big(\frac{|x|^2}{k^2}\Big)u_{t}\frac{2x}{k^2}\cdot
  \nabla u\,dx \\
&+ 2\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)|\Delta u|^2dx
 + 2\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)f'_{u}(x,u)|\nabla u|^2dx\\
&+ 2\int_{\mathbb{R}^N}\phi'\Big(\frac{|x|^2}{k^2}\Big)f(u)\frac{2x}{k^2}\cdot
  \nabla u\,dx
 +2\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)f'_{x}(x,u)\nabla u\\
&+ 2\lambda \int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)|\nabla u|^2
 +2\lambda \int_{\mathbb{R}^N}\phi'\Big(\frac{|x|^2}{k^2}\Big)u\frac{2x}{k^2}\cdot
  \nabla u\,dx \\
&= -\int_{\mathbb{R}^N}\sigma(t,x)\phi\Big(\frac{|x|^2}{k^2}\Big)\Delta u\,dx.
\end{aligned}\label{e51}
\end{equation}
Using arguments similar to Lemma \ref{tail}, taking into account \eqref{Fbis},
we find that
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)|\nabla u|^2dx
 + \lambda \int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)|\nabla u|^2dx\\
&\leq C\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)|\nabla u|^2dx
 + \frac{C}{k}\left(\|u_t\|^2 + \|u\|^2 + \|\nabla u\|^2
 + \int_{\mathbb{R}^N}|f(x,u)|^2dx\right) \\
&\quad +\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)|\phi_5(x)|^2dx
 + C\int_{|x|\geq k}|\sigma(t,x)|^2dx.
\end{aligned}\label{e51b}
\end{equation}
By Gronwall's lemma, Lemma \ref{tail} and Lemma \ref{f(u)},
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)|\nabla u(t)|^2dx \\
&\leq e^{-\lambda (t-T)}\|\nabla u(T)\|^2
 + C \int_{T}^{t}e^{-\lambda(t-s)}\int_{\mathbb{R}^N}\phi
 \Big(\frac{|x|^2}{k^2}\Big)|\nabla u(s)|^2\,dx\,ds\\
&\quad +\frac{C}{k}\int_{T}^{t}e^{-\lambda(t-s)}(1+\|u_t(s)\|^2
 + \|\nabla u(s)\|^2 + \|\sigma(s)\|^2)ds\\
&\quad + C\int_{|x|\geq k}|\phi_5(x)|^2dx
 + C\int_{T}^{t}e^{-\lambda(t-s)}\int_{|x|\geq k}|\sigma(t,x)|^2\,dx\,ds\\
&\leq e^{-\lambda (t-T)}\|\nabla u(T)\|^2
 + C \int_{T}^{t}e^{-\lambda(t-s)}\int_{\mathbb{R}^N}\phi
 \Big(\frac{|x|^2}{k^2}\Big)|\nabla u(s)|^2\,dx\,ds\\
&\quad + \frac{C}{k}\int_{T}^{t}e^{-\lambda(t-s)}(1+\rho_0+\rho_1
 + \|\sigma(s)\|^2)ds\\
&\quad + C\int_{|x|\geq k}|\phi_5(x)|^2dx
  + C\sup_{t\in \mathbb{R}^N}\int_{t}^{t+1}\int_{|x|\geq k}|g(t,x)|^2\,dx\,ds.
\end{aligned} \label{e52}
\end{equation}
From \eqref{e16}, \eqref{e28} and the fact that $\phi_5\in L^2(\mathbb{R}^N)$,
after detailed computations, we obtain from \eqref{e52} the desired result.
\end{proof}

Now, we define a smooth function $\psi = 1 - \phi$, and for a given positive
 number $k$, define $  v^k(t,x) = \psi(|x|^2/k^2)u(t,x)$.
Then, $v^k$ is a unique solution to the initial Cauchy problem
\begin{equation}
\begin{gathered}
\begin{aligned}
&v_t^k - \Delta v^k + \psi\Big(\frac{|x|^2}{k^2}\Big)f(x,u) + \lambda v^k\\
& = u\Delta \psi + \frac{4}{k^2}\psi'\Big(\frac{|x|^2}{k^2}\Big)x\cdot \nabla u
 + \psi\Big(\frac{|x|^2}{k^2}\Big)g(t),
\end{aligned}\\
v^k|_{\partial B_{2k}} = 0\\
v^k(\tau) = \psi\Big(\frac{|x|^2}{k^2}\Big)u_{\tau}.
\end{gathered}
\label{f1}
\end{equation}
Consider the eigenvalue problem
\begin{equation*}
-\Delta w = \lambda w \text{ in } B_{2k}, \quad
\text{with } w|_{\partial B_{2k}} = 0.
\end{equation*}
Then the problem has a family of eigenfunctions $\{e_{j}\}_{j\geq 1}$
 with corresponding eigenvalues $\{\lambda_j\}_{j\geq 1}$
such that $\{e_{j}\}_{j\geq 1}$ form an orthogonal basis of $H_0^1(B_{2k})$
and $0<\lambda_1\leq \lambda_2\leq \ldots\leq \lambda_n\to \infty$.
For given integer $m$, any $u\in H_0^1(B_{2k})$ has a unique
decomposition $u = u_1 + u_2 = P_mu + (Id - P_m)u$, where $P_m$
is the canonical projector from $H_0^1(B_{2k})$ onto the subspace
$\operatorname{span}\{e_1, e_2, \ldots, e_m\}$.

We have the following lemma about the precompactness of $v^k$.

\begin{lemma}\label{v}
Let $k>0$ is fixed. Then, for any $\tau\in \mathbb{R}$ and any
$\varepsilon>0$, there exist $T>\tau$, $m_0\in \mathbb N$ such that
\begin{equation}
\|(Id - P_m)v^k(t)\|_{H_0^1(B_{2k})}^2 \leq \varepsilon, \quad
\forall t\geq T, m\geq m_0 \text{ and } \forall \sigma\in\mathcal{H}_w(g). \label{f2}
\end{equation}
\end{lemma}

\begin{proof}
Let $v^k = P_mv^k + (Id - P_m)v^k = v_1 + v_2$, and then
 multiply \eqref{f1} by $-\Delta v_2$ in $L^2(B_{2k})$, we find that
\begin{equation}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|v_2\|_{H_0^1(B_{2k})}^2 + \|\Delta v_2\|_{L^2(B_{2k})}^2 \\
&- \int_{B_{2k}}\psi\Big(\frac{|x|^2}{k^2}\Big)\Delta v_2 f(x,u)dx
 + \lambda\|v_2\|_{H_0^1(B_{2k})}^2\\
&\leq -\int_{B_{2k}}u\Delta v_2\Delta \psi dx
 - \frac{4}{k^2}\int_{B_{2k}}\psi'\Big(\frac{|x|^2}{k^2}\Big)\Delta v_2 x\cdot
 \nabla u\,dx \\
&\quad - \int_{B_{2k}}\psi\Big(\frac{|x|^2}{k^2}\Big)g(t)\Delta v_2dx.
\end{aligned}\label{f3}
\end{equation}
From definition of $\psi$, we obtain
\begin{gather}
\big|\int_{B_{2k}}\psi\Big(\frac{|x|^2}{k^2}\Big)\Delta v_2 f(x,u)dx\big|
 \leq \frac{1}{8}\|\Delta v_2\|_{L^2(B_{2k})}^2 + C\int_{\mathbb{R}^N}|f(x,u)|^2dx,
\label{f4} \\
\int_{B_{2k}}u\Delta v_2\Delta \psi dx
\leq \frac{1}{8}\|\Delta v_2\|_{L^2(B_{2k})}^2 + C\|u\|^2,
\label{f5} \\
\int_{B_{2k}}\psi'\Big(\frac{|x|^2}{k^2}\Big)\Delta v_2 x\cdot \nabla u\,dx
\leq \frac{1}{8}\|\Delta v_2\|_{L^2(B_{2k})}^2 + C\|\nabla u\|^2,
\label{f6} \\
\int_{B_{2k}}\psi\Big(\frac{|x|^2}{k^2}\Big)g(t)\Delta v_2dx
 \leq \frac{1}{8}\|\Delta v_2\|_{L^{2}(B_{2k})}^2 + C\|g(t)\|^2.
\label{f7}
\end{gather}
From \eqref{f3}-\eqref{f7} and noting that
 $ \|\Delta v_2\|_{L^2(B_{2k})}^2 \geq \lambda_m\|v_2\|_{H_0^1(B_{2k})}^2$,
we obtain
\begin{equation}
\begin{split}
&\frac{d}{dt}\|v_2\|_{H_0^1(B_{2k})}^2 + \lambda_m\|v_2\|_{H_0^1(B_{2k})}^2\\
& \leq C\Big(\|u\|^2 + \|\nabla u\|^2 + \int_{\mathbb{R}^N}|f(x,u)|^2dx
 + \|\sigma(t)\|^2\Big).
\end{split}\label{f8}
\end{equation}
Take $T$ large enough such that \eqref{e16} and \eqref{e43} hold for
all $t\geq T$. Integrating \eqref{f8} from $T$ to $t\geq T$, and
 using \eqref{e16} and \eqref{e43}, we find that
\begin{equation}
\begin{aligned}
&\|v_2(t)\|_{H_0^1(B_{2k})}^2\\
&\leq e^{-\lambda_m(t-T)}\|v_2(T)\|_{H_0^1(B_{2k})}^2\\
&+C\int_{T}^{t}e^{-\lambda_m(t-s)}\Big(\|u(s)\|^2 + \|\nabla u(s)\|^2
 + \int_{\mathbb{R}}|f(x, u(s))|^2dx + \|\sigma(s)\|^2\Big)ds\\
&\leq e^{-\lambda_m(t-T)}\|v_2(T)\|_{H_0^1(B_{2k})}^2
+C\int_{T}^{t}e^{-\lambda_m(t-s)}\left(1+\rho_1 + \|\sigma(s)\|^2\right)ds.
\end{aligned}\label{f9}
\end{equation}
Noting that
\[
\|v_2(T)\|_{H_0^1(B_{2k})}^2
\leq \|v(T)\|_{H_0^1(B_{2k})}^2
\leq \|u(T)\|_{H^1(\mathbb{R}^N)}^2 \leq \rho_1
\]
 and taking into account Lemma \ref{normal}, we obtain \eqref{f2}
by letting $t$ and $m$ tend to infinity.
\end{proof}

\begin{proof}[Proof of Theorem \ref{result1}]
From Proposition \ref{absorbingset}, there is a bounded absorbing
set in $H^1(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)$ for
$\{U_{\sigma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$. Thus, by Theorem \ref{Lp}, it is sufficient
to prove the uniform asymptotic compactness of $\{U_{\sigma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$
in $H^1(\mathbb{R}^N)$.

For $\tau\in \mathbb{R}$, let $\{x_n\}$ be a bounded sequence in
$L^2(\mathbb{R}^N)$, $\{t_n\}$ such that $t_n \to +\infty$
and $\{\sigma_n\}\subset \mathcal H_{w}(g)$, we have to prove that
$\{U_{\sigma_n}(t_n,\tau)x_n\}_{n\geq 1}$ is precompact in
 $H^1(\mathbb{R}^N)$. Given $\varepsilon>0$, from Lemmas \ref{tail}
and \ref{tail1}, there exist $k_1>0$ and $N_1$ such that
\begin{equation}
\int_{|x|\geq 2k}\left(|U_{\sigma_n}(t_n, \tau)x_n|^2
+ |\nabla U_{\sigma_n}(t_n, \tau)x_n|^2\right)dx \leq \varepsilon,
\label{f10}
\end{equation}
as $n\geq N_1$ and $k\geq k_1$. Denote
\begin{equation}
  v^{k}(t_n) = \psi\Big(\frac{|x|^2}{k^2}\Big)U_{\sigma_n}(t_n, \tau)x_n.
\label{f10_1}
\end{equation}
From Lemma \ref{v}, we obtain $N_2$ and $m\in \mathbb N$ satisfying
\begin{equation}
\|\left(Id - P_m\right)v^{k}(t_n)\|_{H_0^1(B_{2k})}^2 \leq \varepsilon,
\label{f11}
\end{equation}
whenever $n\geq N_2$. By Proposition \ref{absorbingset},
we find that $\{P_m(v^{k}(t_n))\}_{n\geq 1}$ is bounded in a
finite dimensional space, which along with \eqref{f11} shows
that $\{v^{k}(t_n)\}_{n\geq 1}$ is precompact in $H_0^1(B_{2k})$.
Thus, we obtain by \eqref{f10_1} that $\{U_{\sigma_n}(t_n, \tau)x_n\}$
 is precompact in $H^1(B_{2k})$ since $  \psi(|x|^2/k^2) = 1$
as $|x|\leq k$. Combining this with inequality \eqref{f10} implies the
uniform asymptotic compactness of $\{U_{\sigma_n}(t_n, \tau)x_n\}$
in $H^1(\mathbb{R}^N)$. This completes the proof.
\end{proof}

\section{Continuous dependence of the attractor on the nonlinearity}

Recall that in this section, we consider a family of function
$f_{\gamma}, \gamma \in \Gamma$, such that for each $\gamma \in \Gamma$,
 $f_{\gamma}$ satisfies \eqref{e2}-\eqref{e4} and \eqref{Fbis} where the
constants are independent of $\gamma$. The topology $\mathcal T$ in $\Gamma$
can be defined as follows:

If $\gamma_m \to \gamma$ in $\mathcal T$ then
$f_{\gamma_{m}}(x, s) \to f_{\gamma}(x, s)$ for all
 $x\in \mathbb{R}^N$ and $s\in \mathbb{R}$.

Let $\{U_{\sigma}^{\gamma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ be the family of processes
corresponding to the problem
\begin{equation}
\begin{gathered}
u_t - \Delta u + f_{\gamma}(x, u) + \lambda u = g(t, x), \quad
 x\in \mathbb{R}^N, t>\tau,\\
u(\tau) = u_{\tau}, \quad x\in \mathbb{R}^N.
\end{gathered}\label{h1}
\end{equation}
From the previous section, for each $\gamma\in \Gamma$, the family
of processes $\{U_{\sigma}^{\gamma}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ has a compact uniform
attractor $\mathcal A_{\gamma}$ in $H^1(\mathbb{R}^N)$.
 Our aim in this section is proving the upper semicontinuity of a
family uniform attractors $\{\mathcal A_{\gamma}\}_{\gamma\in\Gamma}$; that is,
if $\gamma_{m} \to \gamma$ in $\mathcal T$ as $m\to \infty$,
then $\mathcal A_{\gamma_m}$ tends to $\mathcal A_{\gamma}$ in the sense that
\begin{equation}
\lim_{m \to \infty}\operatorname{dist}_{L^{2}(\mathbb{R}^N)}
(\mathcal A_{\gamma_m}, \mathcal A_{\gamma}) = 0.\label{h2}
\end{equation}
The following lemma is the key.

\begin{lemma}\label{key}
Let $\{x_n\}\subset L^2(\mathbb{R}^N), \{\sigma_n\}\in \mathcal H_w(g)$
and $\{\gamma_n\}\subset \Gamma$ such that
\begin{gather}
x_n \rightharpoonup x_0 \text{ weakly in } L^2(\mathbb{R}^N),
\label{h3}\\
\sigma_n\rightharpoonup \sigma \text{ weakly in } \mathcal H_w(g),
\label{h4} \\
\gamma_n \to \gamma \text{ in } \Gamma
\label{h5}
\end{gather}
as $n\to \infty$. Then, for any $t\geq \tau$, there exists a subsequence
 $\{j\}$ of $\{n\}$ such that
\begin{equation}
U_{\sigma_j}^{\gamma_j}(t,\tau)x_j \to U_{\sigma}^{\gamma}(t,\tau)x_0
\quad\text{strongly in } L^2(\mathbb{R}^N).\label{h6}
\end{equation}
\end{lemma}

\begin{proof}
Denote by $u_n(t) = U_{\sigma_n}^{\gamma_n}(t,\tau)x_n$, we find that
 $u_n$ solves the problem
\begin{equation}
\begin{gathered}
\partial_tu_n -\Delta u_n + f_{\gamma_n}(x, u_n) + \lambda u_n = \sigma_n(t),\\
u_n(\tau) = x_n.
\end{gathered}\label{h7}
\end{equation}
Using Proposition \ref{absorbingset} and noting that all constants are
independent of $n$, we obtain
\begin{equation}
\{u_n(t)\} \text{ is bounded in } H^1(\mathbb{R}^N) \text{ uniformly in } n.
\label{h8}
\end{equation}
Thus, there exists a function $v_0 \in L^2(\mathbb{R}^N)$ such that
$u_n(t)\rightharpoonup v_0$ weakly in $L^2(\mathbb{R}^N)$ (up to a subsequence).
For each $m>0$, take any $\psi \in L^2(B_m)$, we set $\bar{\psi}(x) = \psi(x)$
for all $x\in B_m$ and $\bar{\psi}(x) = 0$ for all $x>m$.
It is obviously that $\bar{\psi}\in L^2(\mathbb{R}^N)$ and
\begin{equation}
(u_n(t), \psi)_{L^{2}(B_m)} = (u_n(t), \bar{\psi})_{L^{2}(\mathbb{R}^N)}
\to (v_0, \bar{\psi})_{L^{2}(\mathbb{R}^N)} = (v_0, \psi)_{L^{2}(B_m)}.
\label{l1}
\end{equation}
It implies that $u_n(t) \rightharpoonup v_0$ in $L^2(B_m)$ for all $m>0$.
On the other hand, by \eqref{h8}, for $m>0$, $\{u_n(t)\}$ is bounded
in $H^1(B_m)$, then we find that $\{u_n(t)\}$ is precompact in $L^2(B_m)$
since $H^1(B_m) \hookrightarrow L^2(B_m)$ compactly. By a diagonalization
process, we can choose a subsequence $\{j\}$ of $\{n\}$  and $v_m \in L^2(B_m)$
such that $u_{j}(t) \to v_m$ strongly in $L^2(B_m)$ for all $m>0$.
Taking into account that $u_n(t) \rightharpoonup v_0$ weakly in $L^2(B_m)$
for all $m>0$, we obtain, by the uniqueness of weak limit,
\begin{equation}\label{l2}
u_j(t) \to v_0 \quad \text{ strongly in } L^2(B_m) \text{ for all } m>0.
\end{equation}
We will prove that $u_j(t) \to v_0$ in $L^2(\mathbb{R}^N)$. Indeed, we have
\begin{equation}
\int_{\mathbb{R}^N}\left|u_j(t) - v_0\right|^2
\leq \int_{B_m}|u_j(t) - v_0|^2 + 2\int_{B_m^c}|u_j(t)|^2 + 2\int_{B_m^c}|v_0|^2.
\label{k12}
\end{equation}
We now control terms of the right hand side of \eqref{k12}.
 First, by \eqref{l2} we obtain
\begin{equation}
\int_{B_m}|u_j(t) - v_0|^2 \to 0 \text{ as } n\to +\infty.
\label{k13}
\end{equation}
Next, using arguments in Lemma \ref{tail}, we easily deduce that
\begin{equation}
\begin{aligned}
\int_{B_m^c}|u_j(t)|^2dx
&\leq e^{-\lambda(t-\tau)}\int_{B_m^c}|x_j|^2dx
 + C\sup_{t\in\mathbb{R}}\int_{t}^{t+1}\int_{|x|\geq m}|g(s,x)|^2\,dx\,ds\\
&\quad + C\int_{B_m^c}|\phi_1(x)|dx
 + \frac{C}{m}\int_{\tau}^{t}\left(\|u_j(s)\|^2 + \|\nabla u_j(s)\|^2\right)ds.
\end{aligned}
\label{k14}
\end{equation}
Applying \eqref{G}, \eqref{h3}, $\phi_1\in L^1(\mathbb{R}^N)$ and
 Proposition \ref{absorbingset} in \eqref{k14} gives us
\begin{equation}
\int_{B_m^c}|u_j(t)|^2dx \to 0 \text{ as } j, m \to +\infty.
\label{k15}
\end{equation}
Because $v_0\in L^2(\mathbb{R}^N)$,
\begin{equation}
\int_{B_m^c}|v_0|^2dx \to 0 \text{ as } m\to +\infty.\label{k16}
\end{equation}
Combining \eqref{k12}-\eqref{k16}, we claim that
\begin{equation}
u_j(t) \to v_0 \text{ in } L^2(\mathbb{R}^N) \text{ as } n\to +\infty.\label{k17}
\end{equation}
On the other hand, doing similarly to Lemma \ref{weakcontinuity}, we have
\begin{equation}
U_{\sigma_j}^{\gamma_j}(t,\tau)x_j \rightharpoonup U^{\gamma}_{\sigma}(t,\tau)x_0 \quad
\text{ in } L^2(\mathbb{R}^N). \label{k18}
\end{equation}
From \eqref{k17} and \eqref{k18} we obtain the desired result.
\end{proof}

\begin{proof}[Proof of Theorem \ref{result2}]
Assume that $ \operatorname{dist}_{L^2(\mathbb{R}^N)}
(\mathcal A_{\gamma_n}, \mathcal A_{\gamma}) \not\to 0$. Hence,
by the compactness of $\mathcal A_{\gamma}$, we can choose a positive
constant $\delta>0$, a subsequence $\{m\}$ of $\{n\}$ and
 $\psi_m \in \mathcal A_{\gamma_m}$ satisfying
\begin{equation}
\operatorname{dist}_{L^2(\mathbb{R}^N)}(\psi_m, \mathcal A_{\gamma})
\geq \delta \quad \text{ for all } m\geq 1.
\label{k20}
\end{equation}
Since $\{U_{\sigma}^{\gamma_m}(t,\tau)\}_{\sigma\in\mathcal{H}_w(g)}$ has a uniform absorbing set,
which is independent of $m$, we see that the set
 $\mathfrak A = \cup_{m\geq 1}\mathcal A_{\gamma_m}$ is bounded in
$L^2(\mathbb{R}^N)$, and then by the uniform attracting property of
$\mathcal A_{\gamma}$, we can choose $t$ large enough such that
\begin{equation}
\operatorname{dist}_{L^2(\mathbb{R}^N)}\left(U_{\sigma}^{\gamma}(t,0)\mathfrak A,
 \mathcal A_{\gamma}\right) \leq \frac{\delta}{2}, \quad \text{for all } \sigma\in\mathcal{H}_w(g).
\label{k21}
\end{equation}
On the other hand,
\begin{equation}
\mathcal A_{\gamma_m} = \cup_{\sigma\in\mathcal{H}_w(g)}\mathcal K_{\sigma}^{\gamma_m}(t),
\label{k22}
\end{equation}
thus there exists a $\sigma_m \in \mathcal H_{w}(g)$ such that
$\psi_m \in \mathcal K_{\sigma_m}^{\gamma_m}(t)$.
By definition of $\mathcal K_{\sigma_m}^{\gamma_m}$, we obtain an
$x_m\in \mathcal K_{\sigma_m}^{\gamma_m}(0)$ that satisfies
$\psi_m = U_{\sigma_m}^{\gamma_m}(t,0)x_m$.
Since $\{x_n\}\subset \cup_{m\geq 1}\mathcal K_{\sigma_m}^{\gamma_m}(0)$
is bounded in $L^2(\mathbb{R}^N)$, $\mathcal H_{w}(g)$ is weakly compact,
we can assume without loss of generality that
\begin{gather}
x_m \rightharpoonup x_0 \text{ in } L^2(\mathbb{R}^N), \label{k23}\\
\sigma_m \rightharpoonup \sigma_0 \text{ in } \mathcal H_{w}(g).\label{k24}
\end{gather}
Now, applying Lemma \ref{key}, we deduce that
\begin{equation}
\psi_m = U_{\sigma_m}^{\gamma_m}(t,0)x_m\to U_{\sigma_0}^{\gamma}
(t,0)x_0 \in U_{\sigma_0}^{\gamma}(t,0)\mathfrak A,
\label{k25}
\end{equation}
which contradicts with \eqref{k20} and \eqref{k21}. This completes the proof.
\end{proof}

\subsection*{Acknowledgments}
The author would like to thank the anonymous referee for the
helpful comments and suggestions which improved the presentation of this
article.


\begin{thebibliography}{00}

\bibitem{AnhQuang} Cung The Anh, Nguyen Van Quang;
Uniform attractors for non-autonomous parabolic equations involving
 weighted $p$-Laplacian operators, \emph{Ann. Polon. Math} 98 (2010), 251-271.

\bibitem{Chen} G. Chen, C. K. Zhong;
 Uniform attractors for non-autonomous $p$-Laplacian equation,
\emph{Nonlinear Anal.} 68 (2008), 3349-3363.

\bibitem {Chep} V. V. Chepyzhov, M. I. Vishik;
 \emph{Attractors for Equations of Mathematical Physics},
Amer. Math. Soc. Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002.

\bibitem{Lu} S. S. Lu, H. Q. Wu, Chengkui Zhong;
 Attractors for non-autonomous $2D$ Navier-Stokes equations with normal
external forces, \emph{Discrete Contin. Dyn. Syst.} 13 (2005) 701-719.

\bibitem{Ma} Shan Ma, Haitao Song, Chengkui Zhong;
 Attractors of non-autonomous reaction-diffusion equations,
\emph{Nonlinearity} 22 (2009), 667-682.

\bibitem{Mori} F. Morillas, J. Valero;
 Attractors for reaction-diffusion equations in $\mathbb{R}^N$ with
continuous nonlinearity, \emph{Asymptotic Analysis} 44 (2005) 111-130.

\bibitem{Song} Haitao Song, Chengkui Zhong;
 Attractors of non-autonomous reaction-diffusion equations in $L^p$,
\emph{Nonlinear Anal.} 68 (2008), 1890-1897.

\bibitem{Wang1} Bixiang Wang;
 Attractors for reaction diffusion equations in unbounded domains,
\emph{Physics D} 128 (1999), 41 - 52.

\bibitem{Wang2} Bixiang Wang;
 Pullback attractors for non-autonomous Reaction-Diffusion equations on $R^n$,
\emph{Frontiers of Mathematics in China} 4 (2009), 563-583.

\bibitem{Yan} Xingjie Yan, Chengkui Zhong;
$L^p$-uniform attractor for nonautonomous reaction-diffusion equations
in unbounded domains, \emph{Journal of Mathematical Physics} 49 (2008), nl 10,
 1-17.

\end{thebibliography}

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