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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 138, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/38\hfil Existence and upper semicontinuity]
{Existence and upper semicontinuity of global attractors
for neural fields in an unbounded domain}

\author[S. H. da Silva \hfil EJDE-2010/38\hfilneg]
{Severino Hor\'acio da Silva} 

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

\thanks{Submitted March 16, 2010. Published September 27, 2010.}
\thanks{Supported by  grants 620150/2008 from CNPq-Brazil Casadinho,
 and 5733523/2008-8 \hfill\break\indent from INCTMat}
\subjclass[2000]{45J05, 45M05, 34D45}
\keywords{Well-posedness; global attractor; upper
semicontinuity of attractors}

\begin{abstract}
 In this article, we prove the existence and upper semicontinuity of
 compact global attractors for the flow of the equation
 $$
 \frac{\partial u(x,t)}{\partial t}=-u(x,t)+ J*(f\circ u)(x,t)+ h,
 \quad h  > 0,
 $$
 in $L^{2}$ weighted spaces.
\end{abstract}

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

\section{Introduction}

We consider here the non local evolution equation
\begin{equation}
\frac{\partial u(x,t)}{\partial t}=-u(x,t)+ J*(f\circ u)(x,t)+ h,
\quad h  > 0, \label{1.1}
\end{equation}
where $u(x,t)$ is a real-valued function on $\mathbb{R}\times
\mathbb{R}_{+}$, $h$ is a positive constant,
$J \in C^{1}(\mathbb{R})$ is a non negative even function supported in the
interval $[-1,1]$, and, $f$ is a non negative nondecreasing
function. The $*$ above denotes convolution product, namely:
\begin{equation}
(J*u)(x)=\int_{\mathbb{R}}J(x-y)u(y)dy \label{1.2}.
\end{equation}

Equation \eqref{1.1} was derived  by Wilson and Cowan,
\cite{Wilson}, to model a single layer of neurons in 1972. The function
$u(x,t)$ denotes the mean membrane potential of a patch of tissue
located at position $x\in (-\infty, \infty)$ at time $t\geq 0$. The
connection function $J(x)$ determines the coupling between the
elements at position $x$ and position $y$. The non
negative nondecreasing function $f(u)$ gives the neural firing rate,
or averages rate at which spikes are generated, corresponding to an
activity level $u$. The neurons at a point $x$ are said to be active
if $f(u(x,t))>0$. The parameter $h$ denotes a constant external
stimulus applied uniformly to the entire neural field, (see
\cite{Amari}, \cite{Chen}, \cite{Ermentrout}, \cite{Kishimoto},
\cite{Krisner}, \cite{Laing}, \cite{Rubin} and \cite{Silva}).

An equilibrium of \eqref{1.1} is  a solution for \eqref{1.1} that is
constant with respect to $t$. Thus, if $u$ is an equilibrium for
\eqref{1.1} then $u$ satisfies
\begin{equation}
u(x)= J*(f\circ u)(x)+ h.\label{1.3}
\end{equation}


In the literature, there are already several works dedicated to the
analysis of this model. In \cite{Amari} lateral inhibition type
coupling is studied. Furthermore, when $f$ is a Heaviside step
function, \cite{Amari} also treats the behavior of time dependent
periodic solutions as well as traveling waves for systems of
equations. Existence and uniqueness of monotone traveling waves was
investigated in \cite{Ermentrout}. An another prove of existence of 
monotone travelling waves is given in \cite{Chen}. In \cite{Kishimoto}, the
existence of a non-homogeneous stationary solution referred to as
``bump" is proved.  One link between the integral equations given by
\eqref{1.3} and ODEs is given in \cite{Krisner}. In \cite{Laing},
the existence of a non-homogeneous stationary solution of the type
``double-bump" is proved. In \cite{Rubin} is proved that solutions
as ``bump" can exist and be linearly stable in neural population
models without recurrent excitation. In \cite{Silva}, assuming that
 $f$ is Lipschitz and bounded, is proved
the existence of global attractor, for the flow generated by
\eqref{1.1}, in weighted space.

We consider here the unique additional condition on $f$ which will
is used as hypothesis in our results when necessary.
\begin{itemize}
\item[(H1)] The function $f:\mathbb{R}\to \mathbb{R}$ is Lipschitz,
that is, there exists $k_{1}>0$ such that
\begin{equation}
|f(x)-f(y)|\leq k_{1}|x-y|, \quad  \forall \, x,y \in
\mathbb{R},\label{1.4}
\end{equation}
 From \eqref{1.4}, follows that there exists constant $k_{2}\geq 0$
such that
\begin{equation}
|f(x)|\leq k_{1}|x|+k_{2}.\label{1.5}
\end{equation}
\end{itemize}

This paper is organized as follows. In Section 2 we prove that, under hypothesis (H1), in the phase space
$L^{2}(\mathbb{R},\rho)=\{u\in L^{1}_{\rm loc}(\mathbb{R}) : \int
u^{2}\rho(x)dx<\infty\}$, the Cauchy problem for \eqref{1.1} is well
posed with globally defined solutions. In Section 3 we prove that the system is dissipative in the
sense of \cite{Hale},  that is, it has a global compact attractor.
Our proof is stronger  of what the given one in \cite{Silva} because
we do not use no hypothesis of limitation on $f$. In our proof, we
only use the Sobolev's compact embedding
$H^{1}([-l,l])\hookrightarrow L^{2}([-l,l])$ and some ideias from
\cite{Pereira}, where the equation $u_{t}=-u+\tanh(\beta J*u +h)$ is
considered (see also \cite{Barros}, \cite{Masi1}, \cite{Severino}
and \cite{Severino2} for related work). In Section 4, we prove an 
uniform estimate for the
attractor and finally, in Section 5, after obtaining some estimates
for the flow of \eqref{1.1}, we prove the upper semicontinuity
property of the attractors with respect to function $J$ present in
\eqref{1.1}.


\section{Well-posedness}

In this section we consider the flow generated by \eqref{1.1} in the
space $L^{2}(\mathbb{R},\rho)$ defined by
\begin{align*}
L^{2}(\mathbb{R},\rho)=\big\{u\in L^{1}_{\rm loc}(\mathbb{R}) :
\int_{\mathbb{R}}u^{2}(x)\rho(x)dx <+\infty\big\},
\end{align*}
with norm
$\|u\|_{L^{2}}(\mathbb{R},\rho)=\left(\int_{\mathbb{R}}u^{2}(x)\rho(x)dx
\right)^{1/2}$. Here $\rho$ is an integrable positive even function
 with $\int_{\mathbb{R}}\rho(x)dx=1$. Note that in
this space the constant function equal to 1 has norm 1. The
corresponding higher-order Sobolev space $H^{k}(\mathbb{R},\rho)$ is
the space of functions $u\in L^{2}(\mathbb{R},\rho)$ whose
distributional derivatives up to order $k$ are also in
$L^{2}(\mathbb{R}, \rho)$, with norm
$$
\|u\|_{H^{k}(\mathbb{R},\rho)}=\Big(\sum_{i=1}^{k}\|\frac{\partial^{i}
u}{\partial
x^{i}}\|_{L^{2}(\mathbb{R},\rho)}^{2}\Big)^{1/2}.
$$

To obtain some convenient estimates we will need the following additional
  hypothesis  on the function $\rho$.
\begin{itemize}
\item[(H2)]  There exists constant $K>0$ such that
$$
\sup\{\rho(x) :x\in \mathbb{R},\; y-1 \leq x
\leq y+1\}\leq K\rho(y), \quad \forall \,y\in
\mathbb{R}.
$$
\end{itemize}

\begin{remark} \label{rmk2.1} \rm
  When $\rho(x)=\frac{1}{\pi}(1+x^{2})^{-1}$, the
hypothesis (H2), is verified with
$K=3$, (see, \cite{Pereira}).
\end{remark}


\begin{lemma} \label{lem5.1}
Suppose that {\rm  (H2)} holds. Then
$$
\|J*u\|_{L^{2}(\mathbb{R},\rho)}\leq \sqrt{K}
\|J\|_{L^{1}}\|u\|_{L^{2}(\mathbb{R},\rho)}.
$$
\end{lemma}

\begin{proof}
 Since $J$ is bounded and compact supported, $(J*u)(x)$ is well
defined for $u\in L^{1}_{\rm loc}(\mathbb{R})$. Thus, using \eqref{1.2}
and Holder's inequality (see \cite{Brezis}), we obtain
\begin{align*}
\|J*u\|_{L^{2}(\mathbb{R},\rho)}^{2}&= \int_{\mathbb{R}}|(J*u)(x)|^{2}\rho(x)dx\\
&\leq \int_{\mathbb{R}}\Big(\int_{\mathbb{R}}(J(x-y))^{1/2}(J(x-y))^{1/2}
|u(y)|dy\Big)^{2}\rho(x)dx\\
&\leq \int_{\mathbb{R}}\Big( \Big[\int_{\mathbb{R}}J(x-y)dy
\Big]^{1/2} \Big[\int_{\mathbb{R}}J(x-y)|u(y)|^{2}dy
\Big]^{1/2}\Big)^{2}\rho(x)dx\\
&= \|J\|_{L^{1}}\int_{\mathbb{R}}
\Big(\int_{\mathbb{R}}J(x-y)|u(y)|^{2}dy \Big)\rho(x)dx\\
&= \|J\|_{L^{1}}\int_{\mathbb{R}}
\Big(\int_{\mathbb{R}}J(x-y)\rho(x)dx\Big)|u(y)|^{2}dy
\\
&\leq \|J\|_{L^{1}}\int_{\mathbb{R}}\Big(\int_{x=y-1}^{x=y+1}J(x)\rho(x)dx
\Big)|u(y)|^{2}dy\\
&\leq \|J\|_{L^{1}}\int_{\mathbb{R}}
\Big(K\rho(y)\int_{x=y-1}^{x=y+1}J(x)dx
\Big)|u(y)|^{2}dy\\
&\leq K\|J\|_{L^{1}}^{2}\int_{\mathbb{R}}|u(y)|^{2}\rho(y)dy\\
&= K\|J\|_{L^{1}}^{2}\|u\|^{2}_{L^{2}(\mathbb{R},\rho)}.
\end{align*}
It conclude the result.
\end{proof}


\begin{remark}\label{rmk2.3} \rm
Under hypothesis (H1), for each $u\in L^{2}(\mathbb{R}, \rho)$, we have
\begin{equation}
|J*(f \circ u)(x)|\leq k_{1} (J*|u|)(x)
+k_{2}\|J\|_{L^{1}}.\label{estL_inf}
\end{equation}
 In fact, using \eqref{1.5} we obtain
\begin{align*}
|J*(f \circ u)(x)|
&\leq  \int_{\mathbb{R}}J(x-y)[k_{1}|u(y)|+k_{2}]dy\\
&= k_{1}\int_{\mathbb{R}}J(x-y)|u(y)|dy +k_{2}\int_{\mathbb{R}}J(x-y)dy\\
&=  k_{1}J*|u|(x)+k_{2}\|J\|_{L^{1}}.
\end{align*}
\end{remark}

\begin{proposition} \label{prop5.2}
Suppose that the hypotheses {\rm (H1)} and {\rm (H2)} hold. Then the
function
$$
F(u)=-u+J*(f \circ u) +h
$$
is globally Lipschitz in $L^{2}(\mathbb{R}, \rho)$.
\end{proposition}

\begin{proof}
  From triangle inequality and Lemma \ref{lem5.1}, it
follows that
\begin{align*}
\|F(u)-F(v)\|_{L^{2}(\mathbb{R}, \rho)}
&\leq \|v-u\|_{L^{2}(\mathbb{R}, \rho)}+\|J*(f\circ u)-J*(f\circ
v)\|_{L^{2}(\mathbb{R}, \rho)}\\
&\leq  \|v-u\|_{L^{2}(\mathbb{R},
\rho)}+\sqrt{K}\|J\|_{L^{1}}\|(f\circ u)-(f\circ
v)\|_{L^{2}(\mathbb{R}, \rho)}.
\end{align*}
Using \eqref{1.4}, we have
\[
\|(f\circ u)-(f\circ v)\|_{L^{2}(\mathbb{R},\rho)}^{2}
\leq  \int_{\mathbb{R}}k_{1}^{2}|u(x)-v(x)|^{2}\rho(x)dx
= k_{1}^{2}\|u-v\|_{L^{2}(\mathbb{R}, \rho)}^{2}.
\]
Then
\[
\|F(u)-F(v)\|_{L^{2}(\mathbb{R},\rho)}\leq
(1+\sqrt{K}\|J\|_{L^{1}}k_{1})\|u-v\|_{L^{2}(\mathbb{R}, \rho)}.
\]
Therefore, $F$ is globally Lipschitz in $L^{2}(\mathbb{R}, \rho)$.
\end{proof}

\begin{remark} \label{rmk2.5} \rm
Since the right-hand side of \eqref{1.1} defines a Lipschitz map 
in $L^{2}(\mathbb{R}, \rho)$, from standard results of ODEs in 
Banach spaces, follows that the Cauchy problem for \eqref{1.1} is well
posed in $L^{2}(\mathbb{R}, \rho)$ with globally defined solutions, 
(see \cite{Brezis} and
\cite{Daleckii}).
\end{remark}


\section{Existence of a global attractor}

In this section, we prove the existence of a global maximal
invariant compact set $\mathcal{A}\subset L^{2}(\mathbb{R}, \rho)$ for
the flow of \eqref{1.1}, which attracts each bounded set of
$L^{2}(\mathbb{R}, \rho)$ (the global attractor, see \cite{Hale} and
\cite{Teman}).

To obtain the existence of a global attractor we will need
the following additional hypothesis on the function $J$.
\begin{itemize}
\item[(H3)]
 The function $J$ satisfies $k_{1}\sqrt{K}\|J\|_{L^{1}}<1$.
\end{itemize}

\begin{remark} \label{rmk2.6} \rm
In the particular case that $\rho(x)=\frac{1}{\pi}(1+x^{2})^{-1}$
and $f=\tanh$, whenever $\|J\|_{L^{1}}<\frac{1}{\sqrt{3}}$,
the hypothesis (H3) is satisfied.
\end{remark}

In what follows, we denote by $S(t)$ the flow generated by
\eqref{1.1}.

We recall that a set $\mathcal{B} \subset L^{2}(\mathbb{R}, \rho)$ is
an absorbing set for the flow $S(t)$ in $L^{2}(\mathbb{R}, \rho)$
if, for any bounded set $B \subset L^{2}(\mathbb{R}, \rho)$, there
is a $t_{1}>0$ such that $S(t)B \subset \mathcal{B}$ for any $t\geq
t_{1}$, (see \cite{Teman}).

\begin{lemma} \label{lem6.1}
Assume that {\rm (H1), (H2), (H3)} hold. Let
\[
R=\frac{2(k_{2}\|J\|_{L^{1}}+h)}{1-k_{1}\sqrt{K}\|J\|_{L^{1}}}.
\]
Then the ball with center at the origin of $L^{2}(\mathbb{R}, \rho)$
and radius $R$ is an absorbing set for the flow $S(t)$.
\end{lemma}

\begin{proof}
Let $u(x,t)$ be the solution of \eqref{1.1}, then
\begin{align*}
&\frac{d}{dt} \int_{\mathbb{R}} |u(x,t)|^{2}\rho(x)dx\\
&= \int_{\mathbb{R}}2u(x,t)\frac{d}{dt}u(x,t)\rho(x)dx\\
&= -2\int_{\mathbb{R}}u^{2}(x,t)\rho(x)dx+2\int_{\mathbb{R}}u(x,t)[J*(f
\circ u)(x,t)+h]\rho(x)dx.
\end{align*}
Using Holder inequalit's, \eqref{estL_inf} and Lemma \ref{lem5.1},
we obtain
\begin{align*}
&\int_{\mathbb{R}}u(x,t)[J*(f \circ u)(x,t)+h]\rho(x)dx\\
&\leq \Big(\int_{\mathbb{R}}u(x,t)^{2}\rho(x)dx\Big)^{1/2}
\Big( \int_{\mathbb{R}}|J*(f \circ u)(x,t)+h|^{2}\rho(x)dx\Big)^{1/2}\\
&\leq \|u(\cdot ,t)\|_{L^{2}({\mathbb{R}},\rho)}
\Big( \int_{\mathbb{R}}[k_{1}J*|u(x,t)|+k_{2}\|J\|_{L^{1}}+h]^{2}
 \rho(x)dx\Big)^{1/2}\\
&= \|u(\cdot,t)\|_{L^{2}(\mathbb{R},\rho)}\|k_{1}J*|u(\cdot,t)|
 +k_{2}\|J\|_{L^{1}}+h\|_{L^{2}(\mathbb{R},\rho)}\\
&\leq k_{1}\sqrt{K}\|J\|_{L^{1}}\|u(\cdot,t)\|_{L^{2}({\mathbb{R}},\rho)}^{2}
+ (k_{2}\|J\|_{L^{1}}+h)\|u(\cdot,t)\|_{L^{2}({\mathbb{R}},\rho)}.
\end{align*}
Hence
\[
\frac{d}{dt}\int_{\mathbb{R}}|u(x,t)|^{2}\rho(x)dx
\leq 2\|u(\cdot,t)\|_{L^{2}(\mathbb{R})}^{2} \big[-1 +
k_{1}\sqrt{K}\|J\|_{L^{1}}+
\frac{(k_{2}\|J\|_{L^{1}}+h)}{\|u(\cdot,t)\|_{L^{2}({\mathbb{R}},\rho)}}
\big].
\]
Since $k_{1}\sqrt{K}\|J\|_{L^{1}}<1$, let $\varepsilon
=1-k_{1}\sqrt{K}\|J\|_{L^{1}} >0$. Then, while
$\|u(\cdot,t)\|_{L^{2}({\mathbb{R}},\rho)} >
\frac{2(k_{2}\|J\|_{L^{1}}+h)}{\varepsilon}$, we have
\[
\frac{d}{dt}\|u(\cdot,t)\|_{L^{2}(\mathbb{R}, \rho)}^{2}
\leq  2\|u(\cdot,t)\|_{L^{2}(\mathbb{R}, \rho)}^{2}(-\varepsilon
+\frac{\varepsilon}{2})
= -\varepsilon\|u(\cdot,t)\|_{L^{2}(\mathbb{R}, \rho)}^{2}.
\]
Therefore,
\begin{align*}
\|u(\cdot,t)\|_{L^{2}(\mathbb{R}, \rho)}
&\leq  e^{-\varepsilon t}\|u(\cdot,0)\|_{L^{2}(\mathbb{R},
\rho)}\\
&= e^{-(1-k_{1}\sqrt{K}\|J\|_{L^{1}})
t}\|u(\cdot,0)\|_{L^{2}(\mathbb{R}, \rho)}.
\end{align*}
This concludes the proof.
\end{proof}

\begin{remark} \label{rmk3.3} \rm
 From Lemma \ref{lem6.1}, follows that the ball of center in the
origin and radius $R$ is invariant set under flow $S(t)$.
\end{remark}

\begin{lemma} \label{lem6.2}
Besides the assumptions from Lemma \ref{lem6.1} we also suppose
that the functions $J$ and $\rho$ satisfy the relation $J(x) \leq C
\rho(x)$, $\forall x\in [-1,1]$, for some constant $C>0$. Let
$R=\frac{2(k_{2}\|J\|_{L^{1}}+h)}{1-k_{1}\sqrt{K}\|J\|_{L^{1}}}$ be,
then, for any $\eta>0$, there exists $t_{\eta}$ such that
$S(t_{\eta})B(0,R)$ has a finite covering by balls of
$L^{2}(\mathbb{R}, \rho)$ with radius smaller than
$\eta$.
\end{lemma}

\begin{proof}
   From Lemma \ref{lem6.1}, it follows that $B(0,R)$ is
invariant. Now, the solutions of \eqref{1.1} with initial
condition $u_{0}\in B(0,R)$ is given, by the variation of constant
formula, by
$$
u(x,t)=e^{-t}u_{0}(x)+\int_{0}^{t}e^{-(t-s)}[(J*(f \circ
u))(x,s)+h]ds.
$$
Write
$$
v(x,t)=e^{-t}u_{0}(x), \quad
w(x,t)=\int_{0}^{t}e^{-(t-s)}[(J*(f \circ u))(x,s)+h]ds.
$$
Let $\eta >0$ given. We may find $t(\eta)$ such that if $t \geq
t(\eta)$ then $\|v(\cdot,t)\|_{L^{2}(\mathbb{R}, \rho)} \leq
\frac{\eta}{2}$. In fact,
$$
\|v(\cdot,t)\|_{L^{2}(\mathbb{R},
\rho)}=e^{-t}\|u_{0}\|_{L^{2}(\mathbb{R}, \rho)},
$$
then for $t>\ln(\frac{2 \|u_{0}\|_{L^{2}(\mathbb{R}, \rho)}}{\eta})$, we
have $\|v(\cdot,t)\|_{L^{2}(\mathbb{R}, \rho)}< \frac{\eta}{2}$ for
any $u_{0}\in B(0,R)$.

Now, from (H1) it follows that
\begin{align*}
|J*(f \circ u)(x,s)|
& \leq  k_{1}\int J(x-y)|u(y, s)|dy + k_{2}\int J(x-y)dy\\
 &=  k_{1}\int J(y-x)|u(y, s)|dy + k_{2}\|J\|_{L^{1}}\\
 &= k_{1}\int_{y=x-1}^{y=x+1}J(y)|u(y,s)|dy + k_{2}\|J\|_{L^{1}}.
\end{align*}
Since that $\rho$ is a positive function, $J$ is supported in the
interval $[-1,1]$ and  $J(x)\leq C\rho(x)$, $\forall\, x\in [-1,1]$,
we obtain
\begin{align*}
|J*(f \circ u)(x,s)|
& \leq  C k_{1}\int_{y=x-1}^{y=x+1}\rho(y)|u(y,s)|dy + k_{2}\|J\|_{L^{1}}\\
& \leq  C k_{1}\int\rho(y)|u(y,s)|dy + k_{2}\|J\|_{L^{1}}\\
&= C k_{1}\int\rho^{1/2}(y)|u(y,s)|\rho^{1/2}(y)dy + k_{2}\|J\|_{L^{1}}\\
&\leq C k_{1}\Big(\int\rho(y)|u(y,s)|^{2}dy
\Big)^{1/2}\Big(\int\rho(y)dy\Big)^{1/2} +
k_{2}\|J\|_{L^{1}}.
\end{align*}
Then
\begin{equation}
|J*(f \circ u)(x,s)| \leq Ck_{1}\|u(\cdot,s)\|_{L^{2}(\mathbb{R},
\rho)}+k_{2}\|J\|_{L^{1}}.\label{6.11}
\end{equation}
Thus, using \eqref{6.11} and that
$\|u(\cdot,s)\|_{L^{2}(\mathbb{R},\rho)}\leq R$, results
\begin{align*}
|w(x,t)|&\leq  \int_{0}^{t}e^{-(t-s)}[|J*(f \circ u)(x,s)|+h]ds\\
&\leq  \int_{0}^{t}e^{-(t-s)}(Ck_{1}R+k_{2}\|J\|_{L^{1}}+h).
\end{align*}
Hence
\begin{equation}
|w(x,t)|\leq Ck_{1}R+k_{2}\|J\|_{L^{1}}+h. \label{6.12}
\end{equation}
Now, since
\begin{align*}
J'*|u|(x,s)&= \int_{x-1}^{x+1}J'(x-y)|u(y,s)|ds\\
&\leq
\Big(\int_{x-1}^{x+1}|J'(x-y)|^{2}dy\Big)^{1/2}
\Big(\int_{x-1}^{x+1}|u(y,s)|^{2}dy \Big)^{1/2}\\
&\leq  \|J'\|_{L^{2}}\Big(\int_{x-1}^{x+1}|u(y,s)|^{2}dy
\Big)^{1/2},
\end{align*}
if $x\in [-l,l]$, we obtain
\begin{align*}
J'*|u|(x,s)&\leq
\|J'\|_{L^{2}}\Big(\int_{l-1}^{l+1}|u(y,s)|^{2}dy \Big)^{1/2}\\
&\leq \|J'\|_{L^{2}}\Big(\int_{\mathbb{R}}|u(y,s)|^{2}\chi_{l+1}\rho(y)
\frac{1}{\rho_{l}}dy
\Big)^{1/2}
\end{align*}
where $\chi_{l}$ is the characteristic function of the interval
$[-l,l]$ and $\rho_{l}=\inf\{|\rho(x)| : x\in [-l-1,l+1]\}$.
Then if $u_{0}\in B(0,R)$, then
\begin{equation}
J'*|u|(x,s)\leq
\frac{R\|J'\|_{L^{2}}}{\sqrt{\rho_{l}}}.\label{estL_inf3}
\end{equation}
Furthermore, differentiating $w$ with respect to $x$, for $t\geq 0$, we
have
$$
\frac{\partial w}{\partial
x}(x,t)=\int_{0}^{t}e^{-(t-s)}\left(J'*(f\circ u)\right)(x,s)ds.
$$
Thus
\begin{align*}
\big|\frac{\partial w (x, t)}{\partial x} \big|
&\leq \int_{0}^{t}e^{-(t-s)}|J'*(f\circ u)(x,
s)|_{L^{2}(\mathbb{R}, \rho)}ds\\
&\leq \int_{0}^{t}e^{-(t-s)}[k_{1}J'*|u(x,s)|+k_{2}\|J'\|_{L^{1}}]ds
.
\end{align*}
But, proceeding as in the proof of \eqref{estL_inf}, we obtain
$$
|J'*(f \circ u)(x,s)|\leq k_{1}(J'|u|)(x,s)+k_{2}\|J'\|_{L^{1}}.
$$
Hence, using \eqref{estL_inf3}, results
\begin{equation}
\big|\frac{\partial w (x, t)}{\partial x} \big|\leq
k_{1}\frac{R}{\sqrt{\rho_{l}}}\|J'\|_{L^{2}} +k_{2}\|J'\|_{L^{2}}
.\label{6.9}
\end{equation}
 From \eqref{6.12} and \eqref{6.9} follows that the restriction of
$w(\cdot,t)$ to the interval $[-l,l]$ is bounded in $H^{1}([-l,l])$
(by a constant independent of $u_{0}\in B(0,R)$ and of $t$), and
therefore the set $\{\chi_{l}w(\cdot,t)\}$ with $w(\cdot,0)\in
B(0,R)$ is relatively compact subset of $L^{2}(\mathbb{R}, \rho)$
for any $t>0$ and, hence, it can be covered by a finite number of
balls with radius smaller than $\frac{\eta}{4}$.

Now, from Lemma \ref{lem6.1}, follows  that, for all $t \geq 0$
and any $u_{0}\in B(0,R)$,
\begin{equation}
\|w(\cdot,t)\|_{L^{2}(\mathbb{R}, \rho)}\leq 2R.\label{6.10}
\end{equation}
Then, let $l>0$ be such that
\begin{equation}
2R(Ck_{1}R+k_{2}\|J\|_{L^{1}}+h)
\Big(\int_{\mathbb{R}}(1-\chi_{l}(x))^{4}\rho(x)dx
\Big)^{1/2}\leq \frac{\eta}{4}.\label{6.13}
\end{equation}
Hence, using \eqref{6.12}, \eqref{6.10} and \eqref{6.13}, we obtain
\begin{align*}
&\|(1-\chi_{l})w(\cdot,t)\|_{L^{2}(\mathbb{R}, \rho)}^{2}\\
&= \int_{\mathbb{R}}\left[w(x,t)\rho(x)^{1/2}(1-\chi_{l})^{2}
(x)w(x,t)\rho(x)^{1/2}\right]dx\\
&\leq \Big(\int_{\mathbb{R}}|w(x,t)|^{2}\rho(x)dx
\Big)^{1/2}\Big(\int_{\mathbb{R}}(1-\chi_{l})^{4}(x)|w(x,t)|^{2}\rho(x)dx
\Big)^{1/2}\\
&\leq  \|w(\cdot,t)\|_{L^{2}(\mathbb{R}, \rho)}
\Big((Ck_{1}R+k_{2}\|J\|_{L^{1}}+h)^{2}\int_{\mathbb{R}}
(1-\chi_{l})^{4}(x)\rho(x)dx
\Big)^{1/2}\\
&\leq  2R(Ck_{1}R+k_{2}\|J\|_{L^{1}}+h)
\left(\int_{\mathbb{R}}(1-\chi_{l})^{4}(x)\rho(x)dx
\right)^{1/2}
 \leq  \frac{\eta}{4}.
\end{align*}
Therefore, since
$$
u(\cdot,t)=v(\cdot,t)+\chi_{l}w(\cdot,t)+(1-\chi_{l})w(\cdot,t),
$$
it follows that $S(t_{\eta})B(0,R)$ has a finite covering by balls
of $L^{2}(\mathbb{R},\rho)$ with radius smaller than $\eta$ because
$$
\|u(\cdot,t)\|_{L^{2}(\mathbb{R},
\rho)}=\|v(\cdot,t)\|_{L^{2}(\mathbb{R},
\rho)}+\|\chi_{l}w(\cdot,t)\|_{L^{2}(\mathbb{R},
\rho)}+\|(1-\chi_{l})w(\cdot,t)\|_{L^{2}(\mathbb{R}, \rho)}.
$$
\end{proof}

We denote by $\omega(D)$ the $\omega$-limit of a set $D$.

\begin{theorem} \label{thm6.3}
Assume the  hypotheses in Lemma \ref{lem6.2}.
Then  $\mathcal{A}=\omega(B(0,R))$, is a global attractor
for the flow $S(t)$
generated by \eqref{1.1} in $L^{2}(\mathbb{R}, \rho)$ which is
contained in the ball of radius $R$.
\end{theorem}

\begin{proof}
  From Lemma \ref{lem6.1}, it follows that $\mathcal{A}$ is
contained in the ball of radius $R$ and center in the origin of
$L^{2}(\mathbb{R}, \rho)$. Now, being $\mathcal{A}$ invariant by flow
$S(t)$, it follows that $\mathcal{A}\subset S(t)B(0,R)$, for any
$t\geq 0$ and then, from Lemma \ref{lem6.2}, it results that
the measure of noncompactness of $\mathcal{A}$ is zero.
Hence $\mathcal{A}$ is relatively
compact and, since $\mathcal{A}$ is closed, follows that $\mathcal{A}$ 
is also compact. Finally, if $D$ is bounded set in $L^{2}(\mathbb{R}, \rho)$
then $S(t_{0})D\subset B(0,R)$ for $t_{0}$ big enough and,
therefore, $\omega(D)\subset \omega(B(0,R))$.
\end{proof}

\section{Boundedness results}

In this section we prove uniform estimates for the attractor whose
existence was proved in the Theorem \ref{thm6.3}.

\begin{theorem} \label{thm6.4}
Assume the same hypotheses from Theorem \ref{thm6.3}, and $J\in
C^{r}(\mathbb{R})$, for some integer $r>0$. Then the attractor
$\mathcal{A}$ is bounded in $C^{r}_{\rho}(\mathbb{R})$.
\end{theorem}

\begin{proof}
 Let $u(x,t)$ be a solution of \eqref{1.1} in $\mathcal{A}$. Then, by
the variation of constants formula
$$
u(x,t)=e^{-(t-t_{0})}u(x,t_{0})+\int_{t_{0}}^{t}e^{-(t-s)}[J*(f\circ
u)(x,s)+h]ds.
$$
 From Theorem \ref{thm6.3} follows that
$\|u(\cdot,t)\|_{L^{2}(\mathbb{R},\rho)}\leq R$, where
$R=\frac{2(k_{2}\|J\|_{L^{1}}+h)}{1-k_{1}\sqrt{K}\|J\|_{L^{1}}}$.
Since $\|u(\cdot,t_{0})\|_{L^{2}(\mathbb{R}, \rho)}\leq R$, letting
$t_{0}\to -\infty$, we obtain
\begin{equation}
u(x,t)=\int_{-\infty}^{t}e^{-(t-s)}[J*(f\circ
u)(x,s)+h]ds,\label{emL2a}
\end{equation}
where the equality in \eqref{emL2a} is in the sense of
$L^{2}(\mathbb{R}, \rho)$.

Using that $J\in C^{1}(\mathbb{R})$ follows, from
\eqref{emL2a}, that $u(x,t)$ is differentiable with respect to $x$ and
\begin{equation}
\frac{\partial u(x,t)}{\partial x}= \int_{-\infty}^{t} e^{-(t-s)}
J'*(f\circ u)(x,s)ds.\label{emL2b}
\end{equation}
Now, using that $J'\in C^{1}(\mathbb{R})$ follows, from
\eqref{emL2b}, that $\frac{\partial u(x,t)}{\partial x}$
is differentiable with respect to $x$ and
\begin{align*}
\frac{\partial^{2} u(x,t)}{\partial x^{2}}= \int_{-\infty}^{t}
e^{-(t-s)}  J''*(f\circ u)(x,s)ds.
\end{align*}
Following this idea, using that $J^{(r-1)}\in C^{1}(\mathbb{R})$, we
have that $\frac{\partial^{r-1} u(x,t)}{\partial x^{r-1}}$ is
differentiable with respect to $x$ and
\begin{equation}
\frac{\partial^{r} u(x,t)}{\partial x^{r}}= \int_{-\infty}^{t}
e^{-(t-s)}  J^{r}*(f\circ u)(x,s)ds.\label{emL2c}
\end{equation}
Now, since $J$ is bounded and compact supported, it also follows
that $J^{(r)}$ is bounded and compact supported. Thus $J^{(r)}*v$ is
well defined for $v\in L^{1}_{\rm loc}(\mathbb{R})$. Hence, proceeding
as in the Lemma \ref{lem5.1}, obtain
$$
\|J^{(r)}*v \|_{L^{2}(\mathbb{R},\rho)}\leq
\sqrt{K}\|J^{(r)}\|_{L^{1}}\|v\|_{L^{2}(\mathbb{R},\rho)}.
$$
Thus,
\begin{align*}
\|J^{(r)}*(f\circ u)(\cdot,t)\|_{L^{2}(\mathbb{R},\rho)} \leq
\sqrt{K}\|J^{(r)}\|_{L^{1}}\|(f \circ u)(\cdot,
t)\|_{L^{2}(\mathbb{R},\rho)}.
\end{align*}
Using \eqref{1.5}, we have
\begin{equation}
\|f(u(\cdot,s))\|_{L^{2}(\mathbb{R}, \rho)}\leq k_{1}
\|u(\cdot,s)\|_{L^{2}(\mathbb{R}, \rho)}+k_{2}.\label{6.7}
\end{equation}
Since the ball $B(0,R)$ is invariant,
$\|u(\cdot,t)\|_{L^{2}(\mathbb{R},\rho)}\leq R$, from \eqref{6.7}
results
$$
\|(f\circ u)(\cdot,t)\|_{L^{2}(\mathbb{R},\rho)} \leq k_{1}R+k_{2}.
$$
Hence
\begin{equation}
\|J^{(r)}*(f\circ u)(\cdot,t)\|_{L^{2}(\mathbb{R},\rho)} \leq
\sqrt{K}\|J^{(r)}\|_{L^{1}}(k_{1}R+k_{2}) .\label{emL2d}
\end{equation}
Therefore, from \eqref{emL2c} and \eqref{emL2d}, follows that
\begin{align*}
\big\|\frac{\partial^{r} u(x,t)}{\partial
x^{r}}\big\|_{L^{2}(\mathbb{R},\rho)}
&\leq \int_{-\infty}^{t}e^{-(t-s)}\|J^{(r)}*(f\circ u)
(\cdot,t)\|_{L^{2}(\mathbb{R},\rho)}ds\\
&\leq  \sqrt{K}\|J^{(r)}\|_{L^{1}}(k_{1}R+k_{2})
\int_{-\infty}^{t}e^{-(t-s)}ds\\
&=  \sqrt{K}\|J^{(r)}\|_{L^{1}}(k_{1}R+k_{2}).
\end{align*}
Therefore, we can obtain boundedness for the derivatives of $u$ of any
order, in terms only of $J$ and of the derivatives of $J$,
concluding the proof.
\end{proof}

\begin{theorem} \label{thm6.5}
Assume the same hypotheses from Theorem \ref{thm6.3}. Then the
attractor $\mathcal{A}$ belongs to the ball $\|\cdot\|_{\infty}\leq a$,
where $a=Ck_{1}R +k_{2}\|J\|_{L^{1}}+h$.
\end{theorem}

\begin{proof}
  Let $u(x,t)$ be a solution of \eqref{1.1} in $\mathcal{A}$. Then as
we see in \eqref{emL2a}
$$
u(x,t)=\int_{-\infty}^{t}e^{-(t-s)}[J*(f\circ u)(x,s)+h]ds,
$$
where the equality above is in the sense of $L^{2}(\mathbb{R},
\rho)$. Thus, using \eqref{6.11}, obtain
\begin{align*}
|u(x,t)|
&\leq  \int_{-\infty}^{t}e^{-(t-s)}[|J*(f\circ u)(x,s)|+h]ds\\
&\leq \int_{-\infty}^{t}(Ck_{1}R+k_{2}\|J\|_{L^{1}}+h)e^{-(t-s)}ds\\
&= \int_{-\infty}^{t}ae^{-(t-s)}ds
= a.
\end{align*}
\end{proof}


\section{Upper semicontinuity of attractors with respect to $J$ }

A natural  question to examine is the dependence of this attractors
on the function $J$ present in \eqref{1.1}.
We denote by $ \mathcal{A}_{J}$ the global  attractor  whose existence
 was proved in the Theorem \ref{thm6.3}

Let us recall that a family of subsets $\{\mathcal{A}_{J}\}$, is upper
semicontinuous at $J_{0}$ if
$$
\operatorname{dist}(\mathcal{A}_{J}, \mathcal{A}_{J_{0}})\to 0,
\quad\text{as }J\to J_{0},
$$
where
$$
\operatorname{dist}(\mathcal{A}_{J}, \mathcal{A}_{J_{0}})
 =\sup_{x\in \mathcal{A}_{J}}\operatorname{dist}(x,
\mathcal{A}_{J_{0}})=\sup_{x\in \mathcal{A}_{J}} \inf_{y\in
\mathcal{A}_{J_{0}}}\|x-y\|_{L^{2}(\mathbb{R}, \rho)}.
$$

In this section, we prove that the family of attractors is upper
semicontinuous, in $L^{2}(\mathbb{R},\rho)$, with respect to
function $J$ at $J_{0}$ with $J\in C^{1}(\mathbb{R})$ non negative
even and supported in the interval $[-1,1]$ and $J(x)\leq C\rho(x)$,
$\forall \, x \in [-1,1]$, where $C$ is the constant given in the
Lemma \ref{lem6.2}.

\begin{lemma} \label{lem7.1}
Assume {\rm (H1), (H2), (H3)} hold.
Then the flow
$S_{J}(t)$ is continuous with respect to variations of $J$, in the
$L^{1}-norm$, at $J_{0}$, uniformly for $t\in [0,b]$ with $b<\infty$
and $u$ in bounded sets.
\end{lemma}

\begin{proof}
  As shown above  the solutions of
 \eqref{1.1} satisfy the variations of constants formula,
$$
S_{J}(t)u=e^{-t}u+\int_{0}^{t}e^{-(t-s)}[J*(f \circ S_{J}(s)u+ h]ds.
$$
  Let $J_{0} \in C^{1}(\mathbb{R})$ be a non negative even function
supported in the interval $[-1,1]$, $b>0$ and $D$ a bounded set
in $L^{2}(\mathbb{R},\rho)$, for example the ball $B(0,R)$
(Although $R$ depends on $J$, it can be uniformly chosen in a
neighborhood of $J_{0}$) .
  Given  $\varepsilon > 0$, we want to  find  $\delta
>0$ such that $\|J - J_{0}\|_{L^{1}} < \delta$ implies
$$
\|S_{J}(t)u-S_{J_{0}}(t)u\|_{L^{2}(\mathbb{R}, \rho)} < \varepsilon,
$$
for $t\in [0,b]$  and $u\in D$. Note that
\[
\|S_{J}(t)u-S_{J_{0}}(t)u\|_{L^{2}(\mathbb{R}, \rho)} \leq
\int_{0}^{t}e^{-(t-s)} \| J*(f \circ S_{J}(s)u) - J_{0}*(f \circ
S_{J_{0}}(s)u\|_{L^{2}(\mathbb{R}, \rho)}ds.
\]
Subtracting and summing the term $J_{0}*(f\circ S_{J}(s)u)$ and
using Lemma \ref{lem5.1}, for any $t>0$,   we obtain
\begin{align*}
\|S_{J}(t)u-S_{J_{0}}(t)u\|_{L^{2}(\mathbb{R}, \rho)} &\leq
\int_{0}^{t}e^{-(t-s)}[ \| (J-J_{0})*(f \circ
S_{J}(s)u)\|_{L^{2}(\mathbb{R},
\rho)}\\
&\quad + \|J_{0}*[f \circ S_{J}(s)u-f \circ
S_{J_{0}}(s)u]\|_{L^{2}(\mathbb{R},
\rho)}]ds\\
&\leq \int_{0}^{t}e^{-(t-s)}[ \sqrt{K} \|J-J_{0}\|_{L^{1}}\|f\circ
S_{J}(s)u\|_{L^{2}(\mathbb{R},
\rho)}\\
&\quad + \sqrt{K} \|J_{0}\|_{L^{1}}\|f\circ S_{J}(s)u-f\circ
S_{J_{0}}(s)u\|_{L^{2}(\mathbb{R}, \rho)}]ds.
\end{align*}
Using \eqref{6.7}, we obtain
$$
\|f\circ S_{J}(s)u\|_{L^{2}(\mathbb{R}, \rho)}\leq
k_{1}\|u(\cdot,s)\|_{L^{2}(\mathbb{R}, \rho)}+k_{2}\leq k_{1}R+k_{2}
$$
and, using (H1), we obtain
$$
\|f\circ S_{J}(s)u-f\circ S_{J_{0}}(s)u\|_{L^{2}(\mathbb{R},
\rho)}\leq k_{1}\|S_{J}(s)u-S_{J_{0}}(s)u\|_{L^{2}(\mathbb{R},
\rho)}.
$$
Therefore,
\begin{align*}
\|S_{J}(t)u-S_{J_{0}}(t)u\|_{L^{2}(\mathbb{R}, \rho)}
&\leq (k_{1}R+k_{2})\sqrt{K}\|J-J_{0}\|_{L^{1}}\\
&\quad+\int_{0}^{t}e^{-(t-s)}\sqrt{K}\|J_{0}\|_{L^{1}}k_{1}\|S_{J}(s)u
 -S_{J_{0}}(s)u\|_{L^{2}(\mathbb{R},
\rho)}.
\end{align*}
Hence
\begin{align*}
e^{t}\|S_{J}(t)u-S_{J_{0}}(t)u\|_{L^{2}(\mathbb{R}, \rho)}
&\leq (k_{1}R+k_{2})\sqrt{K}\|J-J_{0}\|_{L^{1}}e^{t}\\
&\quad +\int_{0}^{t}e^{s}\sqrt{K}\|J_{0}\|_{L^{1}}k_{1}\|S_{J}(s)u
 -S_{J_{0}}(s)u\|_{L^{2}(\mathbb{R},
\rho)}.
\end{align*}
Therefore, by Gronwall's Lemma, it follows that
$$
\|S_{J}(t)u-S_{J_{0}}(t)u\|_{L^{2}(\mathbb{R}, \rho)} \leq
(k_{1}R+k_{2})\sqrt{K}\|J-J_{0}\|_{L^{1}}
e^{(\sqrt{K}\|J_{0}\|_{L^{1}}k_{1})t}.
$$
 From this, the results follows immediately.
\end{proof}

\begin{theorem} \label{thm7.2}
Assume the same  hypotheses as in Lemma \ref{lem7.1}. Then the
family of attractors $\mathcal{A}_{J}$ is upper semicontinuous with
respect to $J$ at $J_{0}$.
\end{theorem}

\begin{proof}
   From hypotheses of the theorem, it follows that, for every $J
\in C^{1}(\mathbb{R})$, sufficiently close to $J_{0}$ in the
$L^{1}$-norm, non negative even supported in $[-1,1]$ and satisfying
$J(x)\leq C\rho(x)$, for all $x\in[-1,1]$, the attractor,
$\mathcal{A}_{J}$, given by Theorem \ref{thm6.3} is in the closed ball
$B[0,R]$ in $L^{2}(\mathbb{R},\rho)$. Therefore
$$
\cup_{J}\mathcal{A}_{J}\subset B[0,R].
$$
Since $\mathcal{A}_{J_{0}}$ is global attractor and $B[0,R]$ is a
bounded set then, for every $\varepsilon >0$, there exists $t^{*}>0$
such that
$S_{J_{0}}(t)B[0,R]\subset \mathcal{A}_{J_{0}}^{\varepsilon/2}$,
for all $t\geq t^{*}$, where
$\mathcal{A}_{J_{0}}^{\frac{\varepsilon}{2}}$ is
$\frac{\varepsilon}{2}$-neighborhood of $\mathcal{A}_{J_{0}}$.

 From Lemma \ref{lem7.1}, it follows that $S_{J}(t)$ is continuous
at $J_{0}$, uniformly for $u$ in a bounded set and $t$ in compacts.
Thus, there exists $\delta>0$ such that
\[
\|J-J_{0}\|_{L^{1}}< \delta \Rightarrow
\|S_{J}(t^{*})u-S_{J_{0}}(t^{*})u\|_{L^{2}(\mathbb{R},\rho)} <
\frac{\varepsilon}{2}, \quad \forall \, u\in B[0,R].
\]
We will show that if $\|J - J_{0}\|< \delta$ then
$\mathcal{A}_{J} \subset \mathcal{A}_{J_{0}}^{\varepsilon}$.
In fact, let $u\in \mathcal{A}_{J}$. Since $\mathcal{A}_{J}$ is invariant,
 $v=S_{J}(-t^{*})u\in
\mathcal{A}_{J} \subset B[0,R]$. Therefore,
\begin{gather}
S_{J_{0}}(t^{*})v\in \mathcal{A}_{J_{0}}^{\varepsilon/2}, \label{*}\\
\|S_{J}(t^{*})v
-S_{J_{0}}(t^{*})v\|_{L^{2}(\mathbb{R},\rho)}
<\frac{\varepsilon}{2}.\label{**}
\end{gather}
 From \eqref{*} and \eqref{**}, it follows that
$$
u=S_{J}(t^{*})S_{J}(-t^{*})u=S_{J}(t^{*})v
\in \mathcal{A}_{J_{0}}^{\varepsilon}
$$
and the upper semicontinuity of $\mathcal{A}_{J}$ follows.
\end{proof}

\begin{remark} \label{rmk5.3} \rm
Similar results can be obtained for the flow of \eqref{1.1} in
$$
C_{\rho}(\mathbb{R})\equiv \{f:\mathbb{R}\to \mathbb{R} \,\,
\mbox{continuous with the norm}\,\, \|\cdot\|_{\rho}\},
$$
where
$$
\|u\|_{\rho}=\sup_{x\in \mathbb{R}}\{|u(x)|\rho(x)\}<\infty ,
$$
being $\rho$ a positive continuous function on $\mathbb{R}$.
\end{remark}


\subsection*{Acknowledgments}
The author would like to thank the anonymous referee for his/her
 careful reading of the manuscript. He also would like to thank
 professors Antonio L. Pereira,  for his suggestions,
and Vandik E. Barbosa for the encouragement received.

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\end{document}