\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 185, pp. 1--8.\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{9mm}} \begin{document} \title[\hfilneg EJDE-2012/185\hfil Lower semicontinuity of pullback attractors] {Lower semicontinuity of pullback attractors for a singularly nonautonomous plate equation} \author[R. P. Silva \hfil EJDE-2012/185\hfilneg] {Ricardo Parreira da Silva} \address{Ricardo Parreira da Silva \newline Instituto de Geoci\^encias e Ci\^encias Exatas, UNESP - Univ. Estadual Paulista, Departamento de Matem\'atica, 13506-900 Rio Claro SP, Brazil} \email{rpsilva@rc.unesp.br} \thanks{Submitted April 9, 2012. Published October 28, 2012.} \thanks{Partially supported by FAPESP and PROPe/UNESP, Brazil.} \subjclass[2000]{35B41, 35L25, 35Q35} \keywords{Pullback attractors; nonautonomous systems; plate equation; \hfill\break\indent lower-semicontinuity} \begin{abstract} We show the lower semicontinuity of the family of pullback attractors for the singularly nonautonomous plate equation with structural damping \[ u_{tt} + a(t,x)u_{t} + (- \Delta) u_{t} + (-\Delta)^{2} u + \lambda u = f(u), \] in the energy space $H^2_0(\Omega) \times L^2(\Omega)$ under small perturbations of the damping term $a$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction}\label{sec:intr} In this paper, we shall continue the study started in \cite{CNSS} about the asymptotic behavior under perturbations of the nonautonomous plate equation \begin{equation}\label{eq:plate} \begin{gathered} u_{tt} + a_\epsilon(t,x)u_{t} + (- \Delta) u_{t} + (-\Delta)^{2} u + \lambda u = f(u) \quad \text{in } \Omega, \\ u= \Delta u = 0 \quad \text{on } \partial \Omega, \end{gathered} \end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $\lambda >0$ and $f \in C^2(\mathbb{R})$ is a nonlinearity satisfying % \begin{equation}\label{eq:non-grow-hyp1} \begin{split} &(i)\quad |f'(s)| \leqslant c(1 + |s|^{\rho-1}), \: \forall s \in \mathbb{R}, \text{ with } \begin{cases} 1< \rho < \frac{n+4}{n-4} &\text{if } n\geq 5, \\ \rho \in (1,\infty), &\text{if } n=1,2,3,4; \end{cases} \\ &(ii) \quad f(s)s<0, \; \forall s \in \mathbb{R} . \end{split} \end{equation} The map $\mathbb{R} \ni t \mapsto a_\epsilon(t,\cdot) \in L^\infty(\Omega)$ is supposed to be H\"older continuous with exponent $0<\beta <1$ and constant $C$ uniformly in $\epsilon \in [0,1]$, $0< \alpha_0 \leqslant a_\epsilon(t,x) \leqslant \alpha_1$, for $(t,x,\epsilon) \in \mathbb{R} \times \Omega\times [0,1]$, and $a_\epsilon(t,x) \stackrel{\epsilon \to 0}{\longrightarrow} a_0(t,x)$, uniformly in $\mathbb{R} \times \Omega$. Such problems arise on models of vibration of elastic systems, see for example \cite{Chen,Trig,DiBlasio,Huang,Xiao}. Writing $A:=(-\Delta)^{2}$ with domain $D(A)=\{u \in H^{4}(\Omega) \cap H^{1}_0(\Omega) : \Delta u_{|\partial \Omega}=~0\}$, it is well known that $A$ is a positive self-adjoint operator in $L^2(\Omega)$ with compact resolvent. For $\alpha \geqslant 0$, we consider the scale of Hilbert spaces $E^\alpha:= \big(D(A^\alpha), \|A^\alpha \cdot \|_{L^2(\Omega)} + \|\cdot \|_{L^2(\Omega)} \big)$, where $A^0=I$. It is of special interest the case $\alpha=\frac{1}{2}$, where $-A^{1/2}$ is the Laplace operator with homogeneous Dirichlet boundary conditions; i.e., $A^{1/2}= -\Delta$ with domain $E^{1/2} = H^2(\Omega) \cap H^1_0(\Omega)$. Setting the Hilbert space $X^0 := E^{1/2} \times E^0$, let $\mathcal{A}_\epsilon(t): D(\mathcal{A}_\epsilon(t)) \subset X^0 \to X^0$ be defined by $$ \mathcal{A}_\epsilon(t):= \begin{bmatrix} 0 & -I \\ A + \lambda I & A^{1/2} + a_\epsilon(t)I \end{bmatrix}, $$ with domain $D(\mathcal{A}_\epsilon (t)):= E^1 \times E^{1/2}$ (independent on $t$ and $\epsilon$). We also define $X^\alpha := E^\frac{\alpha+1}{2} \times E^\frac{\alpha}{2}$. In this framework was shown in \cite{CNSS} that the problem \eqref{eq:plate} can be written as an ordinary differential system \begin{equation}\label{eq:syst-nonl1} \frac{d}{dt} (u,v)+ \mathcal{A}_\epsilon(t) (u,v) = F ((u,v)),\quad (u(\tau),v(\tau))=(u_0,v_0) \in X^0, t\geqslant \tau \in \mathbb{R}, \end{equation} where $ F ((u,v)) =(0, f^e(u))$ and $f^e$ is the Nemitski\u{i} operator associated to $f$. This equation yields an evolution process $\{ S_\epsilon(t,\tau) : t\geqslant \tau \}$ in $X^0$ which is given by \begin{equation}\label{eq:evoper} S_\epsilon(t,\tau)x = L_\epsilon(t,\tau)x + \int_\tau^t L_\epsilon(t,s)F(S_\epsilon(s,\tau)x)\, ds, \quad \forall t \geqslant \tau \in \mathbb{R}, \; x\in X^0, \; \end{equation} being $\{L_\epsilon(t,\tau): t\geqslant \tau \in \mathbb{R} \}$ the linear evolution process associated to the homogeneous system \begin{equation}\label{eq:syst-lin1} \frac{d}{dt} (u,v)+ \mathcal{A}_\epsilon(t) (u,v) =(0,0),\quad (u(\tau),v(\tau))=(u_0,v_0) \in X^0, \; t \geqslant \tau. \end{equation} Furthermore the evolution process $\{ S_\epsilon(t,\tau) : t\geqslant \tau \}$ has a pullback attractor $\{ \mathbb{A}_\epsilon(t): t\in \mathbb{R} \}$ with the property that \begin{equation}\label{eq:bound-attr} \cup_{\epsilon \in [0,\epsilon_0]} \cup_{ t\in \mathbb{R}} \mathbb{A}_\epsilon(t)\subset X^0 \; \text{ is bounded. } \end{equation} Recalling the Hausdorff semi-distance of two subsets $A,B \subset X$ $$ \operatorname{dist}{}_H(A,B):=\sup_{a\in A} \inf_{b\in B} \| a-b \|_{X^0}, $$ also was shown the upper semicontinuity of the family $\{ \mathbb{A}_\epsilon(t): t\in \mathbb{R} \}$ at $\epsilon=0$; i.e., $$ \operatorname{dist}{}_H(\mathbb{A}_\epsilon(t),\mathbb{A}_0(t)) \stackrel{\epsilon \to 0}{\longrightarrow} 0. $$ Our aim in this paper is to prove its lower semicontinuity at $\epsilon=0$; i.e., $$ \operatorname{dist}{}_H(\mathbb{A}_0(t),\mathbb{A}_\epsilon(t)) \stackrel{\epsilon \to 0}{\longrightarrow} 0. $$ To achieve this propose we proceed in the following way: We assume there exists only a many finite number of equilibrium $e^*$ of \eqref{eq:syst-nonl1}, all of them hyperbolic in the sense that the linearized operator of \eqref{eq:syst-nonl1} around $e^*$ admits an exponential dichotomy. Then we write the limit attractor as an unstable manifold of the equilibria set, allowing us to obtain the lower semicontinuity as in \cite{CLR}. This article follows closely \cite{CCLR1,CCLR2}, and it is organized as follows: In Section \ref{sec:regular} we derive some addi\-tional stability properties of the solutions starting in the pullback attractors. In Section \ref{sec:structure} we get the characterization of the pullback attractor as a unstable manifold of the equilibria set, and in Section \ref{sec:lower-sem}, we show the hyperbolicity property of the equilibria of \eqref{eq:plate} and we derive the lower semicontinuity of the pullback attractors. \section{Stability of the process on the attractor}\label{sec:regular} In this section we prove an asymptotically stability result of the evolution processes starting on the attractors. First we recall from \eqref{eq:bound-attr} that $$ \{\mathbb{A}_\epsilon(t) : t \in \mathbb{R} \}= \{\xi \in C(\mathbb{R}, X^0): \xi \text{ is bounded and } S_\epsilon(t,\tau)\xi(\tau)=\xi(t) \}. $$ Therefore if $\xi(t) \in \mathbb{A}_\epsilon(t)$ for all $t \in \mathbb{R}$, then $$ \xi(t):=(u(t),u_t(t)) = L_\epsilon(t,\tau)\xi(\tau) + \int_\tau^t L_\epsilon(t,s)F(\xi(s))\, ds, $$ and by the exponential decay of $ L_\epsilon(t,\tau)$ \cite[Theorem 3.1]{CNSS}, we can write \begin{equation}\label{lim-sol} \xi(t) = \int_{-\infty}^t L_\epsilon(t,s)F(\xi(s))\, ds . \end{equation} For $w_0 =\xi(\tau)$ fixed, consider $$ U(t,\tau):=(w(t), w_t(t))=\int_\tau^t L_\epsilon(t,s)F(S_\epsilon(s,\tau)w_0)\, ds, $$ and note that \begin{equation} \label{e2.2} \begin{gathered} w_{tt} + a_\epsilon(t,x)w_{t} + (- \Delta) w_{t} + (-\Delta)^{2} w + \lambda w = f(u(t,\tau,w_0)), \\ w(\tau)= w_t(\tau)=0. \end{gathered} \end{equation} Also notice that by \cite[Theorem 3.2]{CNSS}, $\{U(t,\tau): t\geqslant \tau \}$ is a bounded subset of $X^0$. Therefore using the fact that $f^e$ maps bounded subsets of $E^{1/2}$ to bounded subsets of $E^{-\frac{1}{2} + \tilde{\gamma}}$, for some $\tilde{\gamma} >0$ \cite[Lemma 2.5]{CNSS}, we can state the problem \eqref{eq:syst-nonl1} in $X^{2\gamma}=E^{\frac{1}{2}+\gamma} \times E^{ \gamma}$ with $0<\gamma < \tilde{\gamma}$ (note that $U(0,0)=(0,0) \in E^{\frac{1}{2}+\gamma} \times E^{ \gamma}$), and we have \cite{CN} the estimate \begin{align*} \|U(t,\tau)\|_{X^{1+2\gamma}} & \leqslant \int_\tau^t \|L_\epsilon(t,s) \|_{\mathcal{L}(X^{1+2\gamma}, X^{-1+2\tilde{\gamma}})} \|F(S_\epsilon(s,\tau))w_0 \|_{E^{\frac{1}{2} + \gamma} \times E^{-\frac{1}{2} + \gamma}} \, ds \\ & \leqslant K \int_\tau^t (t-s)^{-1+2 \tilde{\gamma} - 2\gamma } e^{-\alpha(t-s)} \,ds. \end{align*} Noticing that $-1+2 \tilde{\gamma} > -1$, from \eqref{lim-sol} it follows that $$ \sup_{\epsilon \in [0,1]} \sup_{t\in \mathbb{R}} \sup_{\xi \in \mathcal{A}_\epsilon(t)} \|\xi(t)\|_{E^{\frac{1}{2}+\gamma} \times E^{ \gamma}} < \infty. $$ From the compact embedding $E^{\frac{1}{2}+\gamma} \times E^{ \gamma} \stackrel{cc}{\hookrightarrow} E^{1/2} \times E^{0}$, the set $\overline{\cup_{\epsilon \in [0,1]}\cup_{t\in \mathbb{R}} \mathbb{A_\epsilon}}$ is a compact subset of $X^0$. The rest of the section is dedicated to show asymptotically stability of those solutions starting on the attractors. Since the map $t \mapsto a_0(t,x)$ is a bounded and Lipschitz function uniform in $x \in \Omega$, given a sequence $\{ t_n\} \subset \mathbb{R}$, we have for each $t \in \mathbb{R}$ fixed, that the sequence $\{ a_n(t, x):=a_0(t+ t_n, x)\}$ has a subsequence convergent $a_n(t, x) \to \bar{a}(t, x)$, uniformly in compact subsets of $\mathbb{R}$ and $x \in \Omega$. Therefore $\bar{a}$ inherits the same boundedness and Lipschitz properties of $a_0$. This allows us to consider the following two problems: \begin{equation}\label{eq:perturbedd} \begin{gathered} u_{tt} + a_n(t,x)u_{t} + (- \Delta) u_{t} + (-\Delta)^{2} u + \lambda u = f(u) \quad \text{in } \Omega, \\ u= \Delta u = 0 \quad \text{on } \partial \Omega,\\ u(\tau)=u_0 \in H^2(\Omega)\cap H_0^1(\Omega), \quad u_t(\tau)=v_0 \in L^2(\Omega), \end{gathered} \end{equation} and \begin{equation}\label{eq:limittt} \begin{gathered} u_{tt} + \bar{a}(t,x)u_{t} + (- \Delta) u_{t} + (-\Delta)^{2} u + \lambda u = f(u) \quad \text{in } \Omega, \\ u= \Delta u = 0 \quad \text{on } \partial \Omega, \\ u(\tau)=u_0 \in H^2(\Omega)\cap H_0^1(\Omega), \quad u_t(\tau)=v_0 \in L^2(\Omega). \end{gathered} \end{equation} We want to compare solutions of the above problems with initial data $(u_0,v_0) \in \mathbb{A}_n(\tau)$, where $\{\mathbb{A}_n(t): t\in \mathbb{R} \}$ and $\{\mathbb{A}_\infty(t): t\in \mathbb{R} \}$ are the pullback attractors of \eqref{eq:perturbedd} and \eqref{eq:limittt} respectively. Proceeding as above we obtain that $$ \overline{\cup_{n\in \mathbb{N}}\cup_{t\in \mathbb{R}} \mathbb{A}_n(t) \cup \mathbb{A}_\infty(t)} \text{ is a compact subset of } X^0. $$ For $(u_0,v_0) \in \mathbb{A}_n(\tau)$, let $\xi_n(t)$ and $\bar{\xi}(t)$ be the solutions of \eqref{eq:perturbedd} and \eqref{eq:limittt} respectively. Defining $w(t):= \xi_n(t) - \bar{\xi}(t)$, we have % \begin{equation} \label{e2.5} \begin{gathered} w_{tt} = \bar{a}(t,x)\bar{\xi}_t - a_n(t,x)\xi_t + \Delta w_{t} - \Delta^{2} w - \lambda w + f(\xi)-f(\bar{\xi}) \\ w(\tau)=w_t(\tau)=0. \end{gathered} \end{equation} Define $Z (u,v)) = \frac{1}{2} ( \|u\|^2_{1/2} + \|v\|^2_{L^2(\Omega)})$. Since that $f^e$ is Lipschitz in bounded sets from $E^{1/2}$ to $E^0$, and $\xi$, $\bar{\xi}$, $\xi_t$, $\bar{\xi}_t$ are bounded, Young's Inequality leads to \begin{align*} &\frac{d}{dt} Z((w,w_t)) \\ & = \langle w, w_t \rangle_{E^{1/2}} + \langle w_t, w_{tt} \rangle_{L^2(\Omega)} \\ & = \langle \Delta w, \Delta w_{t} \rangle_{L^2(\Omega)} + \lambda \langle w, w_{t} \rangle_{L^2(\Omega)} + \langle w_t, w_{tt} \rangle_{L^2(\Omega)} \\ & = \langle \Delta^2 w + \lambda w + w_{tt}, w_t \rangle_{L^2(\Omega)} \\ & = \langle \bar{a}(t,x)\bar{\xi}_t - a_n(t,x)\xi_t + \Delta w_t + f(\xi) -f(\bar{\xi}), w_t \rangle_{L^2(\Omega)} \\ & = \langle - \bar{a}(t,x)w_t + (\bar{a}(t,x)-a_n(t,x))\xi_t, w_t \rangle_{L^2(\Omega)} - \|\nabla w_t \|^2_{L^2(\Omega)}\\ &\quad + \langle f(\xi)-f(\bar{\xi}), w_t \rangle_{L^2(\Omega)} \\ & \leqslant -\alpha_0 \| w_t \|^2_{L^2(\Omega)} + \|\bar{a} - a_n \|_{L^\infty([\tau,t] \times \Omega)} \|\xi_t \|_{L^2(\Omega)} \| w_t \|_{L^2(\Omega)} \\ &\quad + K(\|w\|^2_{L^2(\Omega)} + \|w_t\|^2_{L^2(\Omega)}) \\ & \leqslant \tilde{K} Z( (w,w_t)) + \tilde{K} \|\bar{a} - a_n \|_{L^\infty([\tau,t] \times \Omega)}. \end{align*} Therefore, \begin{align*} Z( (w,w_t) ) & \leqslant \tilde{K} \int_\tau^t Z((w(s),w_t(s))) ds + \tilde{K}(t-\tau)\|\bar{a} - a_n \|_{L^\infty([\tau,t] \times \Omega)}\\ &\quad + Z\big((w (\tau),w_t(\tau))\big) \\ & \leqslant \tilde{\tilde{K}} \int_\tau^t Z((w,w_t)) ds + \tilde{\tilde{K}} (t-\tau)\|\bar{a} - a_n \|_{L^\infty([\tau,t] \times \Omega)}, \end{align*} where $\tilde{\tilde{K}} = \max\big\{\tilde{{K}} , \frac{Z((w(\tau), w_t (\tau)))}{(\alpha_1 - \alpha_0)} \big\}$. Gronwall's Inequality yields \begin{equation}\label{eq:grownep} \|\xi_n(t) -\bar{\xi(t)}\|^2_{X^0} \leqslant \tilde{\tilde{\tilde{K}}} \|\bar{a} - a_n \|_{L^\infty([\tau,t] \times \Omega)} \int_\tau^t e^{K(t-s)} \, ds \to 0, \end{equation} as $n \to \infty$ in compact subsets of $\mathbb{R}$. \section{structure of the pullback attractor}\label{sec:structure} We will assume that there exist only finitely many $\{u^*_1, \dots, u^*_r\}$ solutions of the problem \begin{equation}\label{eq:equilibria} \begin{gathered} (-\Delta)^{2} u + \lambda u = f(u) \quad \text{in } \Omega, \\ u= \Delta u = 0 \quad \text{on } \partial \Omega, \end{gathered} \end{equation} Defining $\mathcal{E}=\{e^*_1, \dots, e^*_r \}$, where $e^*_i :=(u^*_i,0)$, we will show that \begin{equation}\label{eq:gradient-like} \mathbb{A}_0(t) = \cup_{i=1}^r W^u(e^*_i)(t), \quad \text{for all } t\in \mathbb{R}, \end{equation} where \begin{align*} W^u(e^*_i) = \big\{&(\tau, \zeta) \in \mathbb{R} \times X^0: \text{ there exists a backwards solution } \xi(t,\tau,\zeta) \text{ of \eqref{eq:syst-nonl1} }\\ & (\epsilon=0) \text{ satisfying } \xi(\tau,\tau,\zeta)=\zeta \text{ and } \|\xi(t,\tau,\zeta) -e^*_i \|_{X^0} \stackrel{t \to -\infty}{\longrightarrow} 0\big\}, \end{align*} and $W^u(e^*_i)(t)= \{\zeta \in X^0: (t,\zeta) \in W^u(e^*_i)\}$. Consider the norms in $E^{1/2}$ and $X^0$ given respectively by: $$ \|u\|_{1/2}:= [ \|\Delta u\|_{L^2(\Omega)}^2 + \lambda \| u \|_{L^2(\Omega)}^2 ]^{1/2} \text{ and } \|(u,v) \|_{X^0}=[\|u\|_{1/2}^2 + \|v\|_{L^2(\Omega)}^2 ]^{1/2}. $$ For any $00$. The choice $\eta= \frac{\lambda}{\alpha_{1}}$ leads to \begin{align*} \frac{d}{dt} {V}((u,u_t)) &\leqslant -(\alpha_{0} - 2b\lambda^{1/2} - b \lambda^{1/2} - \frac{b \alpha_{1}^2 }{\lambda^{1/2}} )\|u_{t}\|^2_{L^2(\Omega)} - b \lambda^{1/2} \|u\|_{1/2} \\ &\quad + 2b \lambda^{1/2} \int_{\Omega}f(u)udx \le 0, \end{align*} which means that ${V}$ is non-increasing on solutions of \eqref{eq:plate} and the global solutions where ${V}$ is constant must be an equilibrium. This implies in particular, that in $\mathcal{E}$ there is no homoclinic structure. Finally, we show that all solutions in the pullback attractor $\{\mathbb{A}_0: t\in \mathbb{R} \}$ are forwards and backwards asymptotic to equilibria. Let $\{\xi(t): t \in \mathbb{R} \} \subset \{\mathbb{A}_0(t): t\in \mathbb{R} \}$ a global solution in the attractor. Since it lies in a compact set of $X^0$, ${V}(\xi(t+r)) \stackrel{t \to -\infty}{\longrightarrow} \omega_1$ and ${V}(\xi(t+r)) \stackrel{t \to +\infty}{\longrightarrow} \omega_2$, for some $\omega_1$, $\omega_2 \in \mathbb{R}$ and $r \in \mathbb{R}$. We can choose a sequence $t_n\stackrel{n\to \infty}{\longrightarrow} \infty$ such that $a_0(t_n + r,x) \stackrel{n\to \infty}{\longrightarrow} \bar{a}(r,x)$, uniformly for $r$ in compact subsets of $\mathbb{R}$ and $x \in \Omega$. Therefore, the solution $(\zeta, \zeta_t)$ of the problem \begin{equation}\label{eq:aux-lim} \begin{gathered} u_{tt} + \bar{a}(t,x)u_{t} + (- \Delta) u_{t} + (-\Delta)^{2} u + \lambda u = f(u) \quad \text{in } \Omega, \\ u= \Delta u = 0 \quad \text{on } \partial \Omega, \end{gathered} \end{equation} satisfies ${V}((\zeta, \zeta_t ))= \omega_2$, for all $t \in \mathbb{R}$. Hence $(\zeta, \zeta_t ) \in \mathcal{E}$ and $\xi(t+r) \stackrel{t \to \infty}{\longrightarrow} (\zeta,\zeta_t )$. Taking $\tilde{t}_n\stackrel{n\to \infty}{\longrightarrow} -\infty$ we obtain a similar result. Now we show that this convergence does not depend on the particular choice of subsequences. In fact, suppose that there are sequences $\{t_n\}, \{s_n\} \stackrel{n\to \infty}{\longrightarrow}\infty$, such that $\xi(t_n) \stackrel{n\to \infty}{\longrightarrow} e^*_i \neq e^*_j \stackrel{n\to \infty}{\longleftarrow} \xi(s_n)$. Reindexing if necessary we can suppose that $t_{n+1} > s_n > t_n$, for all $n \in \mathbb{N}$. If $\tau_n \in (t_n, s_n)$, then $\tau_n \stackrel{n\to \infty}{\longrightarrow} \infty$ and (taking subsequence if necessary), $a_0(\tau_n + r)\stackrel{n\to \infty}{\longrightarrow} \bar{a}(r)$. Therefore we also have that $\xi(\tau_n + r) \stackrel{n\to \infty}{\longrightarrow} \bar{\zeta}(r)$, which is a solution of \begin{equation}\label{eq:aux-lim2} \begin{gathered} u_{tt} + \bar{a}(t,x)u_{t} + (- \Delta) u_{t} + (-\Delta)^{2} u + \lambda u = f(u) \quad \text{in } \Omega, \\ u= \Delta u = 0 \quad \text{on } \partial \Omega, \end{gathered} \end{equation} with ${V}( \bar{\zeta}, \bar{\zeta}_t)=\omega_2$ for all $t \in \mathbb{R}$. Consequently, $ \bar{\zeta}(t)\equiv e^*_m \in \mathcal{E}\setminus \{e^*_i, e^*_j \}$. Choosing $\tilde{\tau}_n \in (\tau_n,s_n)$ we can repeat the argument that leads to a contradiction with the fact that there are only finitely many equilibria. Therefore we can write the pullback attractor as in \eqref{eq:gradient-like}. \section{lower semicontinuity of attractors}\label{sec:lower-sem} \begin{definition} \label{def4.1} \rm We say that a linear evolution process $\{L(t,\tau): t \geqslant \tau \} \subset \mathcal{L}(X)$ in a Banach space $X$ has an exponential dichotomy with exponent $\omega$ and constant $M$ if there is a family of bounded linear projections $\{P(t): t\in \mathbb{R} \} \subset \mathcal{L}(X)$ such that % \begin{itemize} \item[(i)] $P(t)L(t,\tau) = L(t,\tau)P(\tau)$, for all $t\geqslant \tau$; \item[(ii)] The restriction $L(t,\tau)_{|P(\tau)X}$, is an isomorphism from $P(\tau)X$ into $P(t)X$, for all $t\geqslant \tau$; \item[(iii)] There are constants $\omega >0$ and $M >1$ such that \begin{gather*} \|L(t,\tau)(I-P(\tau)) \|_{\mathcal{L}(X)} \leqslant M e^{-\omega(t-\tau)}, \; t\geqslant \tau, \ \\ \|L(t,\tau)P(\tau) \|_{\mathcal{L}(X)} \leqslant M e^{\omega(t-\tau)}, \; t\leqslant \tau. \end{gather*} \end{itemize} \end{definition} To see that the linear process $\{ L_\epsilon(t,\tau): t \geqslant \tau \}$ has an exponential dichotomy, given $u_\epsilon$ the global solution of \eqref{eq:syst-nonl1}, define $z_\epsilon(t):= u_\epsilon(t) -e^*_j$, for any $e^*_j \in \mathcal{E}$. Then we have \begin{equation} \begin{gathered} {{z_\epsilon}_{tt}} + a_\epsilon(t,x){z_\epsilon}_{t}+ (- \Delta) {z_\epsilon}_{t} + (-\Delta)^{2} z_\epsilon + \lambda z_\epsilon - f'(e^*_j)z_\epsilon = h({z_\epsilon}) \\ z_\epsilon(\tau) = {z_\epsilon}_0,\quad {z_\epsilon}_t(\tau)={z_\epsilon}_1 \end{gathered} \end{equation} where $h(u)=f(u+e^*_j) - f(e^*_j) - f'(e^*_j)u$. Note that $h(0)=0$ as well $Dh(0)=0 \in \mathcal{L}(X^0)$. Let us consider the system \begin{eqnarray}\label{eq:syst-lineanon} \frac{d}{dt}(u,v) + \bar{A_\epsilon}(t)(u,v) = (0, h(u)), \end{eqnarray} where $$ \bar{A}_\epsilon(t):= \begin{bmatrix} 0 & -I \\ (-\Delta)^2- \lambda I - f'(e^*_j) & -\Delta + a_\epsilon(t) I \end{bmatrix}. $$ Under the hypothesis on the map $t\mapsto a_\epsilon(t)$, it follows from \cite[Theorem 7.6.11]{henry} that the process $\{L_{\epsilon}(t,\tau): t\geqslant \tau \}$ has an exponential dichotomy, for all $\epsilon \in [0,\epsilon_0]$, for some $\epsilon_0 >0$ sufficiently small. Therefore, the proof of the lower semicontinuity of the family $\{\mathbb{A}_\epsilon: t\in \mathbb{R} \}$, based on the proof of the continuity of the sets $W^u(e^*_i)$ and $W^u(e^*_{i,\epsilon})$, is achieved thanks to the following Theorem from \cite{CLR}. \begin{theorem}[{\cite[Theorem 3.1]{CLR}}] \label{theo:lower} Let ${X}$ be a Banach space and consider a family $\{S_\epsilon(t,\tau): t\geq \tau \}_{\epsilon \in [0,1]}$, of evolution process in ${X}$. Assume that for any $x$ in a compact subset of ${X}$, $\|S_\epsilon(t,\tau)x-S_0(t,\tau)x\|_{{X}} \stackrel{\epsilon \to 0}{\longrightarrow} 0$, for $[\tau, t] \subset \mathbb{R}$ and suppose that for each $\epsilon \in [0,1]$ there exist a pullback attractor $\{ \mathbb{A}_\epsilon(t): t \in \mathbb{R} \}$, such that $\cup_{t \in \mathbb{R}} \cup_{\epsilon \in [0, \epsilon_0]} \mathbb{A}_\epsilon(t) \subset X$ is relatively compact and $\{ \mathbb{A}_0(t): t \in \mathbb{R} \}$ is given as \eqref{eq:gradient-like}. Further, assume that for each $e^*_i \in \mathcal{E}_0$: \begin{itemize} \item[(i)] Given $\delta >0$, there exist $\epsilon_{i,\delta}$ such that for all $0<\epsilon < \epsilon_{i,\delta}$ there is a global hyperbolic solution $\xi_{i,\epsilon}$ of \eqref{eq:syst-nonl1} that satisfies $\sup_{t\in \mathbb{R}} \|\xi_{i,\epsilon}(t) - e^*_i \| < \delta$; \item[(ii)] The local unstable manifold of $\xi_{i,\epsilon}$ behaves continuously at $\epsilon= 0$; i.e., $$ \max[\operatorname{dist}{}_H(W^u_{0,\rm loc}(e^*_i), W^u_{\epsilon,\rm loc}(e^*_{i,\epsilon})), \operatorname{dist}{}_H(W^u_{\epsilon,\rm loc}(e^*_{i,\epsilon}), W^u_{0,\rm loc}(e^*_{i}))] \stackrel{\epsilon \to 0}{\longrightarrow} 0, $$ where $W^u_{\rm loc}(\cdot)=W^u(\cdot) \cap B_X(\cdot, \rho)$, for some $\rho >0$. \end{itemize} Then the family $\{ \mathbb{A}_\epsilon(t): t \in \mathbb{R}\}_{ \epsilon \in [0,\epsilon_0] }$ is lower semicontinuous at $\epsilon=0$. \end{theorem} \begin{thebibliography}{00} \bibitem{CCLR1} T. 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