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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 221, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/221\hfil Finite fractal dimensionality]
{Finite fractal dimensionality of attractors for nonlocal
evolution equations}

\author[S. H. da Silva, F. D. M. Bezerra \hfil EJDE-2013/221\hfilneg]
{Severino Hor\'acio da Silva, Flank D. M. Bezerra}  % in alphabetical order

\address{Severino Hor\'acio da Silva \newline
Unidade Acad\^emica de Matem\'atica e Estat\'istica, UAME/CCT/UFCG \newline
 Rua Apr\'igio Veloso, 882,  Bairro Universit\'ario,
 Campina Grande-PB 58429-900, Brazil}
\email{horaciousp@gmail.com, horacio@dme.ufcg.edu.br}

\address{Flank D. M. Bezerra \newline
Departamento de Matem\'atica, UFPB,
Cidade Universit\'aria, Campus I \newline
Jo\~ao Pessoa-PB 58051-900, Brazil}
\email{flank@mat.ufpb.br}

\thanks{Submitted November 1, 2012. Published September 4, 2013.}
\subjclass[2000]{34G20, 47H15}
\keywords{Exponential attractor; global attractor;
fractal dimension; \hfill\break\indent non local evolution equations}

\begin{abstract}
 In this work we consider the Dirichlet problem governed by
 a non local evolution equation. We prove the existence of
 exponential attractors for the flow generated by this problem,
 and as a consequence we obtain the finite dimensionality of the global
 attractor whose existence was proved in \cite{severino}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

Global attractors for dynamical systems generated by non local
evolution equations in infinite dimensional Hilbert space have been
considered in the literature within past few years, see
\cite{severino, Pereira, severino2} and references therein.

In the literature, there are some works on existence of exponential
attractors for the flow governed by  non local
evolution equations and on the problem of determining upper bounds
for the fractal dimension of these attractors (see for instance
\cite{CCM, Xiao,T}).

In this paper, we consider the  non linear Dirichlet
problem  with non local terms
\begin{equation}\label{e1.1}
\begin{gathered}
 \partial_t u ( x , t )  =
 - u ( x , t ) + g \big( \beta (Ku) ( x , t )   \big),  \quad
 x \in \Omega, t> 0    \\
 u(x,0) = u_0 (x), \quad x  \in \Omega\\
 u (x,t)  = 0, \quad x \not\in \Omega, \;t> 0
\end{gathered}
\end{equation}
where  $\Omega\subset\mathbb{R}^N$ ($N\geq1$) is a bounded smooth
domain, $ \beta>0$ and $K$ is an integral operator with symmetric
kernel
\[
(Ku)(x):=\int_{{\mathbb{R}^{N}}} J(x,y)u(y)dy.
\]

Here, $g:\mathbb{R}\to\mathbb{R} $ is a non linear real function of
class $C^1$ with $g(0)=0$, $J$ is a non negative, symmetric bounded
function with bounded derivative, satisfying
$\int_{{\mathbb{R}^{N}}}J(x,y)dy=1$ and
\[ %\label{norma-derivada-J}
\sup_{x\in \mathbb{R}^{N}}\int_{\mathbb{R}^{N}}|\partial_{x}J(x,y)|dy\leq S,\quad
\sup_{y\in \mathbb{R}^{N}}\int_{\mathbb{R}^{N}}|\partial_{ x}J(x,y)|dx\leq S,
\] 
for some constant $0<S<\infty$.

For the sake of clarity and future reference, we
list  the hypotheses on $g$ that were  used  in \cite{severino}.
\begin{itemize}
\item[(H1)]
The function $g:\mathbb{R}\to \mathbb{R}$, is globally Lipschitz continuous 
with constant $k_1$.

\item[(H2)] The function $g\in C^{1}(\mathbb{R})$ and $g'$ is
Lipschitz continuous with constant $k_{2}$.

\item[(H3)] There exists $a>0$ such that $|g(x)|< a<\infty$, for all
$x\in \mathbb{R}$.

\end{itemize}
Note that if (H1) and (H2) hold then
$$
|g'(x)|\leq k_1, \quad \forall x\in \mathbb{R}.
$$


\begin{remark} \label{rmk1.1} \rm
To prevent the flow generated by \eqref{e1.1} becomes a
contraction (see Theorem \ref{estT}) and, hence, the global attractor
 be reduced to single point, we assume $k_1\beta > 1$.
\end{remark}


In this article, $\|\cdot\|=\langle\cdot,\cdot\rangle^{1/2}$
denotes the $L^2(\mathbb{R}^{N})$ norm.  We use
$\|J\|_\infty$ to denote
$\|J\|_{L^\infty(\mathbb{R}^N\times\mathbb{R}^N;\mathbb{R})}$ and
$\|J'\|_\infty$ to denote
$\|J'\|_{L^\infty(\mathbb{R}^N\times\mathbb{R}^N;L
(\mathbb{R}^N\times\mathbb{R}^N;\mathbb{R}))}$.




Under hypothesis (H1)--(H3), it was proved in \cite{severino} that 
problem \eqref{e1.1} has a global compact attractor $\mathcal{A}$
which is contained in ball centered at the origin of radius
$a\sqrt{|\Omega|}$ in $L^2(\mathbb{R}^N)$. Also, under some
additional hypotheses on $g$, the continuity of the global
attractors and the existence of nonhomogeneous equilibria for
\eqref{e1.1} were proved in \cite{severino}. In \cite{CCM} bi-space
global and exponential attractors for the time continuous dynamical
systems are considered and the bounds on their fractal dimension are
discussed in the context of the smoothing properties of the system
between appropriately chosen function spaces and applications to the
sample problems are given, but no remark is made on problems
governed by operators of the type Hilbert-Schmidt, where the
symmetry of the problem is an extra difficulty inherent in the
evolution equations with non local terms.

Our goal is to investigate, under the  above conditions, the
existence of an exponential attractor  for the flow  generated by
\eqref{e1.1} and consequently to obtain bounds on the fractal
dimension of the global attractor associated to problem \eqref{e1.1}.

This article is organized as follows. In Section \ref{DynSys} we
prove Lipschitz continuity of the dynamical system generated by
\eqref{e1.1}, whose well posedness in $X=\{u\in
L^{2}(\mathbb{R}^{N}):u(x)=0, \ if \ x\notin \Omega \}$ has
been established in \cite{severino}. In Section \ref{existence} we
prove that, in this phase space, the system has an exponential
attractor and, as a consequence , we conclude that the global
attractor has finite fractal dimension.

\section{Dynamical system generated by \eqref{e1.1}}\label{DynSys}

It is known from previous work, see \cite{severino}, that
 under hypotheses (H1) and (H2) the function
 \[
[F(u)](x)=\begin{cases}
 -u(x)+g(\beta (Ku)(x)), & x\in \Omega\\
 0, &x\notin \Omega,
 \end{cases}
\]
is globally Lipschitz and continuously Fr\'echet differentiable in
$X=\{u\in L^{2}(\mathbb{R}^{N}):u(x)=0, \text{ if } x\notin \Omega
\}$. Therefore, the autonomous problem
\begin{equation}
\partial_t u=F(u)\label{Prob}
\end{equation}
with initial condition $u(x,0)=u_0(x)$ generates a $C^1$ flow in $X$
which is given by $T(t)u_0=u(x,t)$ where $u(x,t)$ is given by the
variation of constants formula by
\[
u(x,t)=e^{-t}u(x,0)+\int_0^{t}e^{-(t-s)}g(\beta(Ku)(x,s))ds.
\]
Furthermore, as consequence from Lemma \ref{WPH1} below, the problem
\eqref{Prob} is also well posed in $H^1$.

\begin{lemma}\label{WPH1}
Under  hypotheses {\rm (H1), (H2)}, the subset $H^{1}$ of $X$
given by $H^{1}=\{u\in H^{1}(\mathbb{R}^{N}):u(x)=0, \text{ if }
x\notin \Omega \}$ is invariant under the map $F$.
\end{lemma}

\begin{proof} 
If $u\in H^{1}$, from hypothesis (H2) it follows that $F(u)$ is
differentiable and
\begin{align*}
\partial_{x_i} F(u)(x)
&= -\partial_{x_i}u(x)+g'(\beta (Ku)(x)) \beta \partial_{x_i}(Ku)(x)\\
&= -\partial_{x_i}u(x)+g'(\beta (Ku)(x)) \beta
\int_{\mathbb{R}^{N}}\partial_{x_i}J(x,y)u(y)dy.
\end{align*}
Using  hypotheses (H1) and (H2) and Generalized Young's
Inequality (see \cite{Folland}), we obtain
\begin{align*}
\|\partial_{x_i}F(u)\|
&\leq \|\partial_{x_i}u\|+\|g'(\beta (Ku)) \beta
\int_{\mathbb{R}^{N}}\partial_{x_i}J(x,y)u(y)dy\|\\
&\leq  \|\partial_{x_i}u\|+ k_1\beta S\|u\|.
\end{align*}
It implies $F(u)\in H^{1}$, as claimed.
\end{proof}

In the next result we prove the Lipschitz continuity of the flow
$T(t)$ generated by problem \eqref{Prob}.

\begin{theorem}\label{estT}
Assume  hypothesis {\rm (H1)} holds. Then, for $u_1,u_2\in X$ and
$t\geq 0$, we have
\begin{equation}\label{Ineq1}
\|T(t)u_1-T(t)u_2\| \leq e^{c t}\|u_1-u_2\|,
\end{equation}
for $c=k_1\beta -1>0$.
\end{theorem}

\begin{proof} 
Let $u_1,u_{2}\in X$. Suppose that $T(t)u_1(x)$ and
$T(t)u_2(x)$ are solutions of \eqref{Prob} with initial conditions
$u_1$ and $u_2$, respectively. Then
\begin{equation}\label{EEE}
\|T(t)u_1-T(t)u_2\|\leq e^{-t}\|u_1-u_2\|+\int_0^t
e^{-(t-s)}\|g(\beta(KT(s)u_1))-g(\beta(KT(s)u_2))\|ds, 
\end{equation}
for $t\geq0$.
Using (H1) and Young's inequality, it follows that
\begin{align*}
\|g(\beta(KT(s)u_1))-g(\beta(KT(s)u_2))\|
&\leq k_1\beta\|K(T(s)u_1-T(s)u_2)\|\\
&\leq k_1\beta\|T(s)u_1-T(s)u_2\|.
\end{align*}
Then
\[
\|T(t)u_1-T(t)u_2\|\leq e^{-t}\|u_1 -
u_2\|+\int_0^{t}e^{-(t-s)}k_1\beta \|T(s)u_1-T(s)u_2\|ds.
\]
Thus, using Gronwall's inequality, we obtain
\[
\|T(t)u_1-T(t)u_2\|\leq e^{(k_1\beta -1)t}\|u_1 - u_2\|.
\]
Hence \eqref{Ineq1} is satisfied.
\end{proof}

\section{Existence of an exponential attractor}\label{existence}

First we need to introduce some terminology. For a general 
introduction to theory of exponential
attractors and fractal dimension see, for example
\cite{CCM,Efendiev}.

Recall that if $B\neq \emptyset$ is a precompact set in the Banach
space ${\mathcal{X}}$ then its \textit{fractal dimension} is given by
\[
d^{\mathcal{X}}_f(B)=\limsup_{\epsilon\to0}\log_{1/\epsilon}N_\epsilon(B),
\]
where $N_\epsilon(B)$ denotes the smallest number of
$\epsilon$-balls in ${\mathcal{X}}$ needed to cover $B$.

We recall that a set $B\subset {\mathcal{X}}$ is an absorbing set for the
flow $T(t)$ if, for any bounded set $C$ in $\mathcal{X}$, there is a
$t_1=t_1(C)$ such that $T(t)C \subset B$ for any $t\geq t_1$
(see \cite{T}).


Let $\mathcal{Y}$ and $\mathcal{X}$ be Banach spaces such that $\mathcal{Y}$ is
compactly embedded in $\mathcal{X}$. Recalling the generalization of the
notion of an exponential attractor, see \cite{CCM}, we will say that
a nonvoid set $\mathcal{M}\subset {\mathcal{Y}} $ is called an
\textit{exponential (${\mathcal{Y}}-{\mathcal{X}}$) attractor} for $T(t)$ if
$\mathcal{M}$ is \textit{positively invariant} under $T(t)$, closed
in $\mathcal{Y}$, compact in $\mathcal{X}$, $d^{\mathcal{X}}_{f}(\mathcal{M})<\infty$
and there exists $\omega>0$ such that for all $B$ bounded in $\mathcal{Y}$,
\[
\lim_{t\to \infty}e^{\omega t}\operatorname{dist}_{\mathcal{X}}(T(t)B,\mathcal{M})=0.
\]

\begin{remark} \label{rmk3.1} \rm
If $u\in X$, proceeding as in \cite{severino1}, using H\"{o}lder's
inequality, we obtain
\begin{equation}
\begin{aligned}
|K(u)(x,s)|
&\leq  \int_{\mathbb{R}^{N}}J(x,y)|u(y,s)|dy  \\
&\leq  \int_{\mathbb{R}^{N}}\|J\|_{\infty}|u(y,s)|dy  \\
&\leq  \|J\|_{\infty}\sqrt{|\Omega|}\|u(\cdot,s)\|.
\end{aligned}\label{EstKinf}
\end{equation}
\end{remark}


\begin{theorem}\label{absorbing} % \label{teo 3.1}
Assume that {\rm (H1)--(H3)} hold. Then , for any
$\varepsilon >0$, the ball centered at origin and radius $\rho=(1+
k_1\beta \|J'\|_{\infty}|\Omega|)a\sqrt{|\Omega|} +\varepsilon$ in
$H^{1}=\{u\in H^{1}(\mathbb{R}^{N}) :  u(x)=0, \text{ if } x \neq
\Omega\}$ absorbs bounded subsets of $H^{1}$.
\end{theorem}

\begin{proof} Let $u(x,t)$ be the solution of \eqref{Prob} with initial
condition $u(\cdot,0)\in B$, where $B$ is a bounded subset of $H^1$. 
Then, if $x \not \in \Omega$ we have $u(x,t)=0$, and if $x
\in \Omega$ we obtain, by the variation of constants formula,
\begin{equation} \label{FVC}
u(x,t)=e^{-t}u(x,0)+\int_0^{t}e^{-(t-s)}g(\beta(Ku)(x,s))ds.
\end{equation}
By (H3) we have
\begin{equation}\label{Test1}
\|u(\cdot,t)\|\leq e^{-t} \|u(\cdot,0)\| + a\sqrt{|\Omega|}.
\end{equation}
Hence, given $\varepsilon>0$ there exists
$t_1=\ln(\frac{2\|u(\cdot,0)\|}{\varepsilon})$ such that, for
$t>t_1$, we obtain
\begin{equation}\label{In1}
\|u(\cdot,t)\|< \frac{\varepsilon}{2}+ a\sqrt{|\Omega|}.
\end{equation}
Furthermore, using hypothesis (H2), from \eqref{FVC} we obtain
\[
\partial_x u (x,t)=e^{-t}\partial_x u  (x,0)+\int_0^{t}e^{-(t-s)}g'(\beta(Ku)(x,s))\beta
\partial_x (Ku)(x,s)ds.
\]
From  (H1) and (H2) it follows that $|g'(x)|\leq k_1$,
for all $x\in \mathbb{R}$. Then
\[
|\partial_x u (x,t)|\leq e^{-t}|\partial_x u
(x,0)|+\int_0^{t}e^{-(t-s)}k_1\beta |\partial_{x} (Ku)(x,s)|ds.
\]
But, as in \eqref{EstKinf}, using H\"{o}lder's inequality, we obtain
\begin{equation}
\begin{aligned}
|\partial_{x} K(u)(x,s)|
&\leq  \int_{\mathbb{R}^{N}} |\partial_{x} J(x,y)||u(y,s)|dy  \\
&\leq  \int_{\mathbb{R}^{N}}\|J'\|_{\infty}|u(y,s)|dy  \\
&\leq  \|J'\|_{\infty}\sqrt{|\Omega|}\|u(\cdot,s)\|.
\end{aligned} \label{EstK'inf}
\end{equation}
Thus, using  \eqref{Test1} and \eqref{EstK'inf}, we obtain
\begin{align*}
|\partial_x u (x,t)|
&\leq  e^{-t}|\partial_x u (x,0)|+k_1\beta
\|J'\|_{\infty}\sqrt{|\Omega|}\int_0^{t}e^{-(t-s)}(e^{-s}\|u(\cdot,0)\|
+ a\sqrt{|\Omega|})ds\\
&\leq  e^{-t}|\partial_x u (x,0)|+k_1\beta
\|J'\|_{\infty}\sqrt{|\Omega|}\|u(\cdot,0)\| e^{-t}t +  k_1\beta
\|J'\|_{\infty}a|\Omega|.
\end{align*}
Hence
\begin{equation}\label{In2}
\|\partial_x u (\cdot,t)\| \leq e^{-t}\|\partial_x u
(\cdot,0)\|+k_1\beta \|J'\|_{\infty}|\Omega|\|u(\cdot,0)\| e^{-t}t
+  k_1\beta \|J'\|_{\infty}|\Omega|a\sqrt{|\Omega|}.
\end{equation}
But, there exists $t_2=\ln(4\|\partial_x u(\cdot,0)\|/\varepsilon)$
such that for $t>t_2$ we have
\begin{equation}\label{In4}
e^{-t}\|\partial_x u(\cdot,0)\|<\frac{\varepsilon}{4},
\end{equation}
and, since $\lim_{t \to \infty}e^{-t}t=0$, there exists
$t_{3}>0$ such that, for $t>t_3$, we have
\begin{equation}\label{In3}
k_1\beta \|J'\|_{\infty}|\Omega|\|u(\cdot,0)\|
e^{-t}t<\frac{\varepsilon}{4}.
\end{equation}
Then, using \eqref{In2}, \eqref{In4} and \eqref{In3}, we obtain
\begin{align*}
\|\partial_x u (\cdot,t)\| < \frac{\varepsilon}{2} +  k_1\beta
\|J'\|_{\infty}|\Omega|a\sqrt{|\Omega|}.
\end{align*}
for all $t> t^*:= \max\{t_2,t_3\}$.
It follows that for $t> \max\{t_1,t^{*}\}$,
\begin{align*}
\|u(\cdot,t)\|+\|\partial_x u (\cdot,t)\| \leq  (1+ k_1\beta
\|J'\|_{\infty}|\Omega|)a\sqrt{|\Omega|} +\varepsilon.
\end{align*}
From this, the result follows immediately.
\end{proof}

For the rest of this article, we denote by $B_0$ the ball in $H^{1}$
centered at origin and radius $\rho=(1+ k_1\beta
\|J'\|_{\infty}|\Omega|)a\sqrt{|\Omega|} +\varepsilon$, (with
$\varepsilon > 0$ fixed arbitrarily).

\begin{theorem}\label{estPM}
Assume {\rm (H1)--(H3)}, and let $B_0$ be the set that
absorbs bounded subsets of $H^{1}$, given in Theorem
\ref{absorbing}. Then, there exists $t_0\geq t_1(B_0)$ such
that the following conditions hold: $T(t_0)$ admits a
decomposition
$$
T(t_0)=P(t_0)+M(t_0)
$$
where $P(t_0):B_0\to X\subset L^{2}(\mathbb{R}^{N})$ is a
contraction on $B_0$; that is,
\begin{equation}
\|P(t_0)u_1-P(t_0)u_{2}\|\leq \delta\|u_1-u_{2}\|, \quad
\forall  u_1, u_{2} \in B_0,
\label{estP}
\end{equation}
for some $0\leq \delta < 1/2$, and $M(t_0):B_0\to H^{1}$
satisfies
\begin{equation}
\|M(t_0)u_1-M(t_0)u_{2}\|_{H^{1}}\leq k\|u_1-u_{2}\|, \quad
\forall  u_1, u_{2} \in B_0,
\label{estM}
\end{equation}
for some $k>0$.
\end{theorem}

\begin{proof} Let $u(x,t)$ be the solution of \eqref{Prob} with initial
condition $u$, then
$$
T(t)u=e^{-t}u+\int_0^{t}e^{-(t-s)}g(\beta (KT(s))u)ds.
$$
Write $P(t)u=e^{-t}u$ and $M(t)u=\int_0^{t}e^{-(t-s)}g(\beta
(KT(s))u)ds$. Note that, choosing $t_0> \ln 2$, it follows that
$$
 \|P(t_0)u_1-P(t_0)u_{2}\|\leq e^{-t_0}\|u_1-u_{2}\|, \quad
\forall   u_1, u_{2} \in B_0,
$$
then  \eqref{estP} is satisfied with $\delta=e^{-t_0}$.
Now, from (H3), it follows that
\[
\|M(t)u\|
\leq \int_0^{t}e^{-(t-s)}\|g(\beta( K T(s)u)\|ds\\
\leq \int_0^{t}e^{-(t-s)}a\sqrt{|\Omega|}ds\\
\leq a\sqrt{|\Omega|}.
\]

Using  (H1),(H2), and
\eqref{EstK'inf}, we have
\begin{align*}
|\partial_{x} M(t)u(x)|
&\leq \int_0^{t}e^{-(t-s)}\beta|g'(\beta (K T(s)u)(x))||\partial_{x} (K T(s)u)(x)|ds\\
&\leq \int_0^{t}e^{-(t-s)}k_1\beta
\|J'\|_{\infty}\sqrt{|\Omega|}\| T(s)u\|ds.
\end{align*}
Since $u\in B(0,\rho)$, it follows that $\|T(s)u\|\leq
\rho+a\sqrt{|\Omega|}$. Hence
\[
|\partial_{x} M(t)u(x)|
\leq  k_1\beta \|J'\|_{\infty}\sqrt{|\Omega|}(\rho + a
\sqrt{|\Omega|}).
\]
Therefore, $M(t):B_0\to H^{1}$ for all $t\geq 0$.

Also, using (H1) and Theorem \ref{estT}, we obtain
\begin{align*}
\|M(t)u_1-M(t)u_{2}\|
&\leq  \int_0^{t}e^{-(t-s)}\|g(\beta K T(s)u_1)-g(\beta K
T(s)u_{2})\|ds\\
&\leq  \int_0^{t}e^{-(t-s)}\beta k_1 \| T(s)u_1- T(s)u_{2}\|ds\\
&\leq  \int_0^{t}e^{-(t-s)}\beta k_1 e^{(k_1\beta -1)s}\| u_1- u_{2}\|ds\\
&=  k_1\beta \| u_1- u_{2}\|\int_0^{t}e^{-(t-s)}e^{(k_1\beta
-1)s}ds\\
&\leq  \| u_1- u_{2}\|e^{(k_1\beta -1) t}.
\end{align*}
Using hypothesis (H1), (H2), \eqref{EstKinf} and
\eqref{EstK'inf},
it follows that
\begin{align*}
&|\partial_{x}M(t)u_1(x)-\partial_{x} M(t)u_{2}(x)|\\
&\leq \int_0^{t}e^{-(t-s)}\beta|g'(\beta (K T(s)u_1)(x))-g'(\beta (K
T(s)u_{2})(x))||\partial _{x}(KT(s)u_1)(x)|ds\\
&\quad + \int_0^{t}e^{-(t-s)}\beta|g'(\beta (K T(s)u_{2})(x))
||\partial_{x}[ K(T(s)u_1 -T(s)u_{2})](x)|ds
\\
&\leq \int_0^{t}e^{-(t-s)}k_{2}\beta^2
\|J\|_{\infty}\sqrt{|\Omega|}\|T(s)u_1-T(s)u_{2}\|\|J'\|_{\infty} 
\sqrt{|\Omega|}\| T(s)u_1\|ds\\
&\quad +\int_0^{t}e^{-(t-s)}k_1\beta \|J'\|_{\infty}
\sqrt{|\Omega|}\|T(s)u_1 -T(s)u_{2}\|ds.
\end{align*}
Thus, using  Theorem \ref{estT}, we have
\begin{align*}
&|\partial_{x}M(t)u_1(x)-\partial_{x} M(t)u_{2}(x)|\\
&\leq  \int_0^{t}e^{-(t-s)}k_{2}\beta^{2}\|J\|_{\infty} |\Omega|
\|J'\|_{\infty}(\rho+a\sqrt{|\Omega|})
 e^{(k_1\beta -1)s}\|u_1-u_{2}\|ds\\
&\quad + \int_0^{t}e^{-(t-s)}k_1\beta \|J'\|_{\infty}\sqrt{|\Omega|}
e^{(k_1\beta -1)s}\|u_1-u_{2}\|ds\\
&\leq \bigg[\frac{k_{2}\beta^{2}
\|J\|_{\infty}|\Omega|\|J'\|_{\infty}(\rho+a\sqrt{|\Omega|})+k_1\beta
\|J'\|_{\infty}\sqrt{|\Omega|}}{k_1\beta -1}\bigg] e^{(k_1\beta
-1)t}\|u_1-u_{2}\|.
\end{align*}
Therefore, \eqref{estM} is satisfied with $t_0>\ln 2$ and
\begin{equation}
\begin{aligned}
k=\max\Big\{&e^{(k_1\beta -1) t_0},
\big[\frac{k_{2}\beta^{2} \|J\|_{\infty}|\Omega|\|J'\|_{\infty}
 (\rho+a\sqrt{|\Omega|})+k_1\beta \|J'\|_{\infty}
 \sqrt{|\Omega|}}{k_1\beta -1}\big]\\
&\times \sqrt{|\Omega|}e^{(k_1\beta -1)t_0}\Big\}.
\end{aligned}\label{constk}
\end{equation}
It completes the proof. 
\end{proof}

The proof of the  following lemma is very easy and  it will be omitted.

\begin{lemma} \label{lem3.4}
For any bounded interval $I$, there exist $0<\theta<1$ and $c>0$ such
that for $t,s\in I$ we have
\begin{equation}
|e^{-t}-e^{-s}|\leq c|t-s|^{\theta}.  \label{estTheta}
\end{equation}
In particular, when $I=[t_0,2t_0]$ the inequality above is
obtained with $c=1$.
\end{lemma}

\begin{theorem}\label{estFinal}
Assume  {\rm (H1)} and {\rm (H3)}. Then, for $t_1,
t_{2}\in [t_0,2t_0]$ and $u_1,u_{2}\in B_0$, $T(t)$
satisfies
\begin{equation}
\|T(t_1)u_1-T(t_{2})u_{2}\| \leq \mu
(|t_1-t_{2}|^{\theta}+\|u_1-u_{2}\|)
\end{equation}
with some $\mu>0$ and $0<\theta <1$.
\end{theorem}

\begin{proof} 
Note that
\begin{align*}
T(t_1)u_1 - T(t_{2})u_{2}
&= (e^{-t_1}u_1 - e^{-t_2}u_2) 
+ \Big( \int_0^{t_1}e^{-(t_1-s)}g(\beta KT(s)u_1)ds \\
&\quad - \int_0^{t_{2}}e^{-(t_{2}-s)}g(\beta KT(s)u_{2})ds\Big).
\end{align*}
Using Lemma \ref{lem3.4}, we obtain
\begin{align*}
\|e^{-t_1}u_1 - e^{-t_2}u_2 \| 
&\leq  |e^{-t_1} - e^{-t_2}|\|u_1\|+e^{-t_{2}}\|u_1-u_{2}\|\\
&\leq \|u_1\||t_1-t_{2}|^{\theta}+\|u_1-u_{2}\|\\
&= \mu_1(|t_1-t_{2}|^{\theta}+\|u_1-u_{2}\|),
\end{align*}
where $\mu_1 =\max\{\rho,1\}$.
Now,
\begin{align*}
&\|\int_0^{t_1}e^{-(t_1-s)}g(\beta KT(s)u_1)ds -
\int_0^{t_{2}}e^{-(t_{2}-s)}g(\beta KT(s)u_{2})ds\|\\
&\leq  \|\int_0^{t_1}e^{-(t_1-s)}g(\beta KT(s)u_1)ds -
\int_0^{t_1}e^{-(t_{2}-s)}g(\beta KT(s)u_{2})ds\|\\
&\quad + \|\int_0^{t_1}e^{-(t_{2}-s)}g(\beta KT(s)u_{2})ds -
\int_0^{t_{2}}e^{-(t_{2}-s)}g(\beta KT(s)u_{2})ds\|.
\end{align*}
Using  (H1) and (H3), Young's inequality, Lemma
\ref{estTheta} and Theorem \ref{lem3.4}, we have
\begin{align*}
&\|\int_0^{t_1}e^{-(t_1-s)}g(\beta K T(s)u_1)ds -
\int_0^{t_1}e^{-(t_{2}-s)}g(\beta K T(s)u_{2})ds\|\\
&\leq  \int_0^{t_1}\|e^{-(t_1-s)}g(\beta K
T(s)u_1)-e^{-(t_{2}-s)}g(\beta
K T(s)u_{2})\|ds\\
&\leq  \int_0^{t_1}|e^{-(t_1-s)}-e^{-(t_{2}-s)}|\|g(\beta K T(s)u_1)\|ds \\
&\quad + \int_0^{t_1}e^{-(t_{2}-s)} \|g(\beta K T(s)u_1)-
g(\beta K T(s)u_{2})\|ds\\
&\leq  \int_0^{t_1}a \sqrt{|\Omega|}c|t_1-t_{2}|^{\theta}ds
+ \int_0^{t_1}e^{-(t_{2}-s)} k_1\beta e^{(k_1\beta -1)s}\|u_1- u_{2} \|ds\\
&\leq  2 t_0a\sqrt{|\Omega|} c|t_1-t_{2}|^{\theta} +
\|u_1-u_{2} \| e^{k_1\beta 2t_0}\\
&= \mu_{2}(|t_1-t_{2}|^{\theta} + \|u_1-u_{2} \|),
\end{align*}
where $\mu_{2}=\max\{2t_0ac\sqrt{|\Omega|}, \, e^{k_1\beta 2t_0}\}$.

Without loss of generality assuming that $t_1<t_{2}$,  using
hypothesis (H3), we obtain
\begin{align*}
&\|\int_0^{t_1}e^{-(t_{2}-s)}g(\beta KT(s)u_{2})ds -
\int_0^{t_{2}}e^{-(t_{2}-s)}g(\beta KT(s)u_{2})ds\|\\
&\leq  \int_{t_1}^{t_{2}}e^{-(t_{2}-s)}\|g(\beta KT(s)u_{2})\|ds\\
&\leq  \int_{t_1}^{t_{2}}e^{-(t_{2}-s)}a\sqrt{|\Omega|}ds\\
&\leq  (t_{2}-t_1)a\sqrt{|\Omega|}\\
&\leq  t_0^{1-\theta}a\sqrt{|\Omega|}(t_{2}-t_1)^{\theta}\\
&=  \mu_{3}(t_{2}-t_1)^{\theta},
\end{align*}
where $\mu_{3}=t_0^{1-\theta}a\sqrt{|\Omega|}$.
Therefore,
$$
\|T(t_1)u_1-T(t_{2})u_{2}\|\leq
\mu(|t_{2}-t_1|^{\theta}+\|u_1-u_{2}\|)
$$
for some $\mu >0$ and $0<\theta <1$.
\end{proof}

Since $W^{1,2}(\Omega)\hookrightarrow L^{2}(\Omega)$ it follows that
$H^{1}\hookrightarrow X$. Then, from Theorems \ref{absorbing},
\ref{estPM} and \ref{estFinal}, it follows that the assumptions in
the Proposition 2.7 of \cite{CCM} are satisfied. Therefore, we have
the following result.

\begin{theorem}\label{exppp}
Under the hypotheses {\rm (H1)-(H3)}, for any $\nu\in
(0,\frac{1}{2}-\delta)$, there exists a nonvoid set
$\mathcal{M}=\mathcal{M}_{\nu}\subset B(0,\rho)$, positively
invariant under $T(t)$ and  precompact in $X$, with the following properties:
\begin{itemize}
\item[(1)] there exists $\omega >0$ such that for any bounded set 
$B\subset H^1$ we have
\[
\lim_{t\to \infty}e^{\omega t}\operatorname{dist}(T(t)B,\mathcal{M})=0
\]
where $\operatorname{dist}(\cdot,\cdot)$ denotes the Hausdorff
semi-distance in $X$ (see \cite{Hale}).

\item[(2)] $\mathcal{M}$ possesses finite fractal dimension in $X$; 
more precisely, we have for any $\nu\in(0,\frac{1}{2}-\delta)$
\[
d^X_f(\mathcal{M})\leq\frac{1}{\theta}
(1+\log_{\frac{1}{2(\delta+\nu)}}N_{\nu/k}(B(0,1)),
\]
where $d_f(\mathcal{M})$ denotes the fractal dimension of $\mathcal{M}$,
$B(0,1)$ denotes the open ball centered at $0$ and radius $1$ in
$H^1=\{u\in H^{1}(\mathbb{R}^{N}): u(x)=0, \text{ if }
x\notin \Omega\}$, $N_{\nu/k}(B)$ denotes the smallest
number of $\frac{\nu}{k}-$balls in $L^2(\mathbb{R}^N)$ needed to
cover $B(0,1)$ and $k$ is the constant given in \eqref{constk}.
\end{itemize}
\end{theorem}

Denoting by $\overline{\mathcal{M}_{\nu}}$ the closure in $X$ of the
set $\mathcal{M}_{\nu}$, from \cite[Corollary 2.8]{CCM} we have the
following result.

\begin{corollary} \label{coro3.7}
Under the  hypotheses of Theorem \ref{exppp}, for any $\nu\in
(0,\frac{1}{2}-\delta)$:
\begin{itemize}
\item[(1)] $\overline{\mathcal{M}_{\nu}}$ is an exponential
 ($H^{1}-X$) attractor bounded in $H^1$;
\item[(2)] there exists a finite dimensional global 
($H^{1}-X$) attractor $\mathcal{A}\subset \overline{\mathcal{M}_{\nu}}$.
\end{itemize}
\end{corollary}

\subsection*{Acknowledgments}
The authors would like to thank the anonymous referee for his/her
reading of the manuscript and valuable suggestions  that have
improved the quality of our initial manuscript. Our gratitude goes
also to professors Ant\^onio L. Pereira from USP and Jesualdo G. das
Chagas from UFCG for your suggestions and discussions.

S. H. da Silva was partially supported by CNPq-Brazil grants
Casadinho/Procad-552.464/2012-2, INCTMat-5733523/2008-8.
F. D. M. Bezerra was partially supported by grant FAPESP 11/04166-5.

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\end{document}