\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 $$\begin{gathered} u_t - \Delta u + f(x,u) + \lambda u = g(t,x), \quad x\in \mathbb{R}^N,\\ u|_{t = \tau} = u_{\tau}, \end{gathered}\label{e1}$$ 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 $$\alpha_3|u|^p -\phi_3(x)\leq F(x,u) \leq \alpha_4|u|^p + \phi_4(x). \label{e4}$$ \item[(H2)] The external force $g\in L^{2}_{\rm loc}\left(\mathbb{R}; L^2(\mathbb{R}^N)\right)$ satisfies $$\sup_{t\in\mathbb{R}}\int_{t}^{t+1}\left(\|g(s)\|_{L^2(\mathbb{R}^N)}^2+\|\partial_tg(s)\|_{L^2(\mathbb{R}^N)}^2\right)ds <+\infty. \label{e5}$$ We borrow from \cite[Lemma 3.4]{Yan} the following result: $$\limsup_{k\to +\infty}\Big(\sup_{t\in\mathbb{R}^N}\int_{t}^{t+1} \int_{|x|\geq k}|g(s,x)|^2\,dx\,ds\Big) = 0. \label{G}$$ \end{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 $$\label{Fbis} | \frac{\partial f}{\partial x}(x,s) | \leq \psi_5(x), \quad \forall x\in \mathbb{R}^N, \forall s\in \mathbb{R},$$ 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 $$\label{odd} \begin{gathered} u_t - u_{xx} + u - (2\pi^2+1)u = 0, \quad x\in (0,1), t>0,\\ u(t,0) = u(t,1) = 0,\quad t>0,\\ u(0,x) = \sin(\pi x), x\in (0,1). \end{gathered}$$ 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 $$\label{e1.9} \begin{gathered} u_t - \Delta u + f_{\gamma}(x, u) + \lambda u = g(t, x), \quad x\in \mathbb{R}^N, t>\tau,\\ u(\tau) = u_{\tau}, \quad x\in \mathbb{R}^N. \end{gathered}$$ 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 $$u_t - \Delta u + f(x,u) + \lambda u = \sigma(t,x). \label{e6}$$ Taking the inner product of \eqref{e6} with $2u$ in $L^2(\mathbb{R}^N)$, we have $$\frac{d}{dt}\|u\|^2 + 2\|\nabla u\|^2 + 2(f(x,u), u) + 2\lambda \|u\|^2 = 2(\sigma(t),u). \label{e7}$$ Using \eqref{e2}, applying the Cauchy and Young's inequalities, $$\frac{d}{dt}\|u\|^2 + \frac{3\lambda}{2}\|u\|^2 + 2\|\nabla u\|^2 + 2\alpha_1\|u\|_{L^p(\mathbb{R}^N)}^p \leq \frac{2}{\lambda}\|\sigma(t)\|^2 + 2\|\phi_1\|_{L^1(\mathbb{R}^N)}. \label{e8}$$ By Gronwall's lemma, we find $$\|u(t)\|^2 \leq e^{-\lambda(t-\tau)}\|u_{\tau}\|^2 + \frac{2\|\phi_1\|_{L^1(\mathbb{R}^N)}}{\lambda} + \frac{2}{\lambda}\int_{\tau}^{t}e^{-\lambda(t-s)}\|\sigma(s)\|^2ds. \label{e9}$$ For the last term of the right hand side, \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} Combining \eqref{e9}-\eqref{e10}, and noting that $u_{\tau}$ belongs to a bounded set $B$, there exists a $T_0>0$ satisfies $$\|u(t)\|^2 \leq \rho_0 = 1 + \frac{2\|\phi_1\|_{L^{1}(\mathbb{R}^N)}}{\lambda} + \frac{2e^{\lambda}}{\lambda(e^{\lambda}-1)}\|g\|_{L^{2}_{b}}^{2}, \label{e11}$$ 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{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} 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 $$\int_{t}^{t+1}\Big(\lambda\|u(s)\|^2+\|\nabla u(s)\|^2 + 2\int_{\mathbb{R}^N}F(x,u)dx\Big)ds \leq C, \quad \text{for all } t\geq T_0. \label{e13}$$ On the other hand, by multiplying \eqref{e6} by $2u_t$ then integrating over $\mathbb{R}^N$, after using Cauchy's inequality, $$\|u_t\|^2 + \frac{d}{dt}\Big( \lambda\|u\|^2+\|\nabla u\|^2 + 2\int_{\mathbb{R}^N}F(x,u)dx\Big) \leq \|\sigma(t)\|^2. \label{e14}$$ From \eqref{e13}-\eqref{e14} and the uniform Gronwall inequality, we obtain $$\lambda\|u\|^2+\|\nabla u\|^2 + 2\int_{\mathbb{R}^N}F(x,u)dx \leq C, \text{ for all } t\geq T_0. \label{e15}$$ Using \eqref{e4} again, there exists $\rho_1>0$ such that, for all $t\geq T_0$, $$\|u(t)\|^2 + \|\nabla u(t)\|^2 + \|u(t)\|_{L^p(\mathbb{R}^N)}^p \leq \rho_1, \quad \forall u_{\tau}\in B, \forall \sigma\in\mathcal{H}_w(g). \label{e16}$$ 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 $$U_{\sigma_n}(t,\tau)x_n \rightharpoonup U_{\sigma}(t,\tau)x\quad \text{in } H^1(\mathbb{R}^N) \cap L^p(\mathbb{R}^N), \label{w1}$$ for all $t>\tau$. \end{lemma} \begin{proof} Denote by $u_n(t) = U_{\sigma_n}(t,\tau)x_n$, then $u_n$ solves $$\partial_t u_n - \Delta u_n + f(x, u_n) + \lambda u_n = \sigma_n(t), \label{w2}$$ 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 $$\{u_n(s)\}, \tau\leq s\leq t, \text{ is bounded in } H^1(\mathbb{R}^N)\cap L^p(\mathbb{R}^N). \label{w9}$$ 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 $$w_t - \Delta w + f(x, w) + \lambda w = \sigma(t), \label{w10}$$ 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 $$\int_{|x|\geq K}|U_{\sigma}(t,\tau)u_{\tau}|^2dx \leq \varepsilon, \label{e22}$$ 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 \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} Now, we estimate terms in \eqref{e23}. First, using condition \eqref{e2} of $f$, we find $$\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)f(x,u)u\,dx \geq -\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)\phi_1(x)dx \geq -\int_{|x|\geq k}|\phi_1(x)|dx. \label{e24}$$ Next, \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} for all $t\geq T_0$, since \eqref{e16}. Finally, for the right hand side of \eqref{e23}, $$2\big|\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big)\sigma(t,x)u\,dx\big| \leq \frac{1}{\lambda}\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big) |\sigma(t,x)|^2dx + \lambda\int_{\mathbb{R}^N}\phi\Big(\frac{|x|^2}{k^2}\Big) |u|^2dx. \label{e26}$$ Combining \eqref{e23}-\eqref{e26}, we obtain \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} By Gronwall's lemma, proceed as \eqref{e10}, we conclude that \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} Using \eqref{G} and the fact that $\phi_1 \in L^1(\mathbb{R}^N)$, it can be followed from \eqref{e28} that $$\limsup_{t\to +\infty}\limsup_{k\to +\infty}\int_{|x|\geq k\sqrt{2}}|u(t)|^2dx = 0, \label{e29}$$ 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 $$\mathcal A_2 = \cup_{\sigma\in\mathcal{H}_w(g)}\mathcal K_{\sigma}(s) \quad \text{for all } s\in \mathbb{R}. \label{e29_1}$$ \end{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 $$\|U_{\sigma_n}(t_n,\tau)x_n\|_{L^{2}(B_K^c)}\leq \frac{\varepsilon}{3}, \quad \forall n\geq N_0, \label{e30}$$ 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 $$\|U_{\sigma_{m'}}(t_{m'},\tau)x_{m'} - U_{\sigma_{n'}}(t_{n'}, \tau)x_{n'}\|_{L^2(B_K)} \leq \frac{\varepsilon}{3}, \quad \text{for all } m', n' \geq N_1. \label{e31}$$ Taking $N = \max\{N_0, N_1\}$, then for all $m', n' \geq N$, \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} 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 $$\int_{\Omega(|U_{\sigma}(t,\tau)u_{\tau}|\geq M)}|U_{\sigma}(t,\tau)u_{\tau}|^pdx \leq \varepsilon, \text{ for all } \sigma\in\mathcal{H}_w(g), t\geq T, u_{\tau}\in B, \label{e33}$$ 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, $$(u-M)_{+}=\begin{cases} u - M&\text{if } u\geq M\\ 0 &\text{otherwise}, \end{cases} \label{e34}$$ Multiplying \eqref{e1} by $p(u-M)_{+}^{p-1}$, we obtain \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} 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, $$\int_{\mathbb{R}^N}\sigma(t,x)(u-M)_{+}^{p-1}dx \leq C\|\sigma(t)\|^2 + C\int_{\mathbb{R}^N}(u-M)_{+}^{2p-2}dx. \label{e37}$$ Combining \eqref{e35}-\eqref{e37}, we obtain $$\frac{d}{dt}\|(u-M)_{+}\|_{L^p(\mathbb{R}^N)}^p + CM^{p-2}\|(u-M)_{+}\|_{L^p(\mathbb{R}^N)}^p \leq C\|\sigma(t)\|^2. \label{e38}$$ By Gronwall's lemma, \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} where $T_1$ is in \eqref{e16}. Applying \eqref{e16} and Lemma \ref{normal}, we obtain $$\int_{\Omega_1}|(u(t)-M)_{+}|^pdx \leq \varepsilon, \quad \text{uniformly in } u_{\tau}\in B, \sigma\in\mathcal{H}_w(g), \label{e40}$$ when $t$ and $M$ are large enough. Repeat steps above, just replace $(u-M)_{+}$ by $(u+M)_{-}$, where $$(u+M)_{-}= \begin{cases} u + M&\text{if } u\leq -M\\ 0 &\text{otherwise}, \end{cases}\label{e41}$$ we can find $t$ and $M$ large enough such that $$\int_{\Omega(u\leq -M)}|(u+M)_{-}|^pdx \leq \varepsilon, \quad \forall u_{\tau}\in B, \forall \sigma\in\mathcal{H}_w(g). \label{e42}$$ 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 $$\big|\frac{\partial f}{\partial x}(x,u)\big| \leq \phi_5(x),\label{e3_2}$$ 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 $$\|u_t(t)\|^2 \leq \rho_1, \forall t\geq T_1,\quad \forall u_{\tau}\in B,\;\forall \sigma\in\mathcal{H}_w(g). \label{e17}$$ \end{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{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} thus $$\int_{t}^{t+1}\|u_t(s)\|^2ds \leq C, \quad\text{for all } t\geq T_0. \label{e19}$$ 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 $$\frac{d}{dt}\|v\|^2 + 2\|\nabla v\|^2 + (f'(x,u)v, 2v) + 2\lambda\|v\|^2 = (\sigma'(t), 2v). \label{e20}$$ By \eqref{e3_1} and Cauchy's inequality, $$\frac{d}{dt}\|v\|^2 \leq 2\ell\|v\|^2 + \frac{1}{2\lambda}\|\sigma'(t)\|^2. \label{e21}$$ 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 $B\subset L^2(\mathbb{R}^N)$, $$\int_{\mathbb{R}}|f(x,U_{\sigma}(t,\tau)u_{\tau})|^2dx \leq C(1+\|\sigma(t)\|_{L^2(\mathbb{R}^N)}^2), \label{e43}$$ 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{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} 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{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} From \eqref{e44}--\eqref{e47}, we obtain \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} for all $t\geq \max\{T_0, T_1\}$, since \eqref{e16} and \eqref{e17}. On the other hand, by \eqref{e3}, $$\int_{\mathbb{R}^N}|f(x,u)|^2dx \leq 2\alpha_2^2\int_{\mathbb{R}^N}|u|^{2p-2}dx + 2\|\phi_2\|^2. \label{e49}$$ 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 $$\int_{|x|\geq K}|\nabla U_{\sigma}(t,\tau)u_{\tau}|^2dx \leq \varepsilon, \label{50}$$ 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{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} Using arguments similar to Lemma \ref{tail}, taking into account \eqref{Fbis}, we find that \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} By Gronwall's lemma, Lemma \ref{tail} and Lemma \ref{f(u)}, \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} 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{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} 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 $$\|(Id - P_m)v^k(t)\|_{H_0^1(B_{2k})}^2 \leq \varepsilon, \quad \forall t\geq T, m\geq m_0 \text{ and } \forall \sigma\in\mathcal{H}_w(g). \label{f2}$$ \end{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{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} 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 $$\begin{split} &\frac{d}{dt}\|v_2\|_{H_0^1(B_{2k})}^2 + \lambda_m\|v_2\|_{H_0^1(B_{2k})}^2\\ & \leq C\Big(\|u\|^2 + \|\nabla u\|^2 + \int_{\mathbb{R}^N}|f(x,u)|^2dx + \|\sigma(t)\|^2\Big). \end{split}\label{f8}$$ 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{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} 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 $$\int_{|x|\geq 2k}\left(|U_{\sigma_n}(t_n, \tau)x_n|^2 + |\nabla U_{\sigma_n}(t_n, \tau)x_n|^2\right)dx \leq \varepsilon, \label{f10}$$ as $n\geq N_1$ and $k\geq k_1$. Denote $$v^{k}(t_n) = \psi\Big(\frac{|x|^2}{k^2}\Big)U_{\sigma_n}(t_n, \tau)x_n. \label{f10_1}$$ From Lemma \ref{v}, we obtain $N_2$ and $m\in \mathbb N$ satisfying $$\|\left(Id - P_m\right)v^{k}(t_n)\|_{H_0^1(B_{2k})}^2 \leq \varepsilon, \label{f11}$$ 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 $$\begin{gathered} u_t - \Delta u + f_{\gamma}(x, u) + \lambda u = g(t, x), \quad x\in \mathbb{R}^N, t>\tau,\\ u(\tau) = u_{\tau}, \quad x\in \mathbb{R}^N. \end{gathered}\label{h1}$$ 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 $$\lim_{m \to \infty}\operatorname{dist}_{L^{2}(\mathbb{R}^N)} (\mathcal A_{\gamma_m}, \mathcal A_{\gamma}) = 0.\label{h2}$$ 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 $$U_{\sigma_j}^{\gamma_j}(t,\tau)x_j \to U_{\sigma}^{\gamma}(t,\tau)x_0 \quad\text{strongly in } L^2(\mathbb{R}^N).\label{h6}$$ \end{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 $$\begin{gathered} \partial_tu_n -\Delta u_n + f_{\gamma_n}(x, u_n) + \lambda u_n = \sigma_n(t),\\ u_n(\tau) = x_n. \end{gathered}\label{h7}$$ Using Proposition \ref{absorbingset} and noting that all constants are independent of $n$, we obtain $$\{u_n(t)\} \text{ is bounded in } H^1(\mathbb{R}^N) \text{ uniformly in } n. \label{h8}$$ 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 $$(u_n(t), \psi)_{L^{2}(B_m)} = (u_n(t), \bar{\psi})_{L^{2}(\mathbb{R}^N)} \to (v_0, \bar{\psi})_{L^{2}(\mathbb{R}^N)} = (v_0, \psi)_{L^{2}(B_m)}. \label{l1}$$ 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, $$\label{l2} u_j(t) \to v_0 \quad \text{ strongly in } L^2(B_m) \text{ for all } m>0.$$ We will prove that $u_j(t) \to v_0$ in $L^2(\mathbb{R}^N)$. Indeed, we have $$\int_{\mathbb{R}^N}\left|u_j(t) - v_0\right|^2 \leq \int_{B_m}|u_j(t) - v_0|^2 + 2\int_{B_m^c}|u_j(t)|^2 + 2\int_{B_m^c}|v_0|^2. \label{k12}$$ We now control terms of the right hand side of \eqref{k12}. First, by \eqref{l2} we obtain $$\int_{B_m}|u_j(t) - v_0|^2 \to 0 \text{ as } n\to +\infty. \label{k13}$$ Next, using arguments in Lemma \ref{tail}, we easily deduce that \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} Applying \eqref{G}, \eqref{h3}, $\phi_1\in L^1(\mathbb{R}^N)$ and Proposition \ref{absorbingset} in \eqref{k14} gives us $$\int_{B_m^c}|u_j(t)|^2dx \to 0 \text{ as } j, m \to +\infty. \label{k15}$$ Because $v_0\in L^2(\mathbb{R}^N)$, $$\int_{B_m^c}|v_0|^2dx \to 0 \text{ as } m\to +\infty.\label{k16}$$ Combining \eqref{k12}-\eqref{k16}, we claim that $$u_j(t) \to v_0 \text{ in } L^2(\mathbb{R}^N) \text{ as } n\to +\infty.\label{k17}$$ On the other hand, doing similarly to Lemma \ref{weakcontinuity}, we have $$U_{\sigma_j}^{\gamma_j}(t,\tau)x_j \rightharpoonup U^{\gamma}_{\sigma}(t,\tau)x_0 \quad \text{ in } L^2(\mathbb{R}^N). \label{k18}$$ 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 $$\operatorname{dist}_{L^2(\mathbb{R}^N)}(\psi_m, \mathcal A_{\gamma}) \geq \delta \quad \text{ for all } m\geq 1. \label{k20}$$ 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 $$\operatorname{dist}_{L^2(\mathbb{R}^N)}\left(U_{\sigma}^{\gamma}(t,0)\mathfrak A, \mathcal A_{\gamma}\right) \leq \frac{\delta}{2}, \quad \text{for all } \sigma\in\mathcal{H}_w(g). \label{k21}$$ On the other hand, $$\mathcal A_{\gamma_m} = \cup_{\sigma\in\mathcal{H}_w(g)}\mathcal K_{\sigma}^{\gamma_m}(t), \label{k22}$$ 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 $$\psi_m = U_{\sigma_m}^{\gamma_m}(t,0)x_m\to U_{\sigma_0}^{\gamma} (t,0)x_0 \in U_{\sigma_0}^{\gamma}(t,0)\mathfrak A, \label{k25}$$ which contradicts with \eqref{k20} and \eqref{k21}. 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