\documentclass[twoside]{article}
\usepackage{amssymb, amsmath}
\pagestyle{myheadings}

\markboth{\hfil Viscoelastic wave equations with localized damping
 \hfil EJDE--2002/44}
{EJDE--2002/44\hfil M. M. Cavalcanti, V. N. D. Cavalcanti  \& J. A. Soriano \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 44, pp. 1--14. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Exponential decay for the solution of semilinear viscoelastic wave
  equations with localized damping
 %
\thanks{ {\em Mathematics Subject Classifications:} 35L05, 35L70, 35B40, 74D10.
\hfil\break\indent
{\em Key words:} semilinear wave equation, memory, localized damping.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted March 07, 2002. Published May 22, 2002. \hfil\break\indent
Partially supported by a grant from CNPq, Brazil.} }
\date{}
%
\author{Marcelo M. Cavalcanti, Val\'eria N. Domingos Cavalcanti, \\
\& Juan A. Soriano}
\maketitle

\begin{abstract}
 In this paper we obtain an exponential rate of decay for
 the solution of the viscoelastic nonlinear wave equation
 $$
 u_{tt}-\Delta u+f(x,t,u)+\int_0^tg(t-\tau )\Delta u(
 \tau )\,d\tau +a(x)u_t=0\quad \hbox{in }\Omega\times (0,\infty ).
 $$
 Here the damping term $a(x)u_t$ may be null for some part of the domain
 $\Omega$. By assuming that the kernel $g$ in the
 memory term  decays exponentially, the damping effect allows
 us to avoid compactness arguments and and to reduce number of
 the energy estimates considered in the prior literature. We
 construct a suitable Liapunov functional and make use of
 the perturbed energy method.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks
\numberwithin{equation}{section}
\section{Introduction}

This manuscript is devoted to the study of the exponential
decay of the solutions of the viscoelastic nonlinear wave equation
\begin{gather}
u_{tt}-\Delta u+f(x,t,u)+\int_0^tg(t-\tau )\Delta u(
\tau )\,d\tau +a(x)u_t=0\quad\mbox{in }\Omega\times (0,\infty ) \notag\\
u=0\quad \hbox{on }\Gamma\times (0,\infty )  \label{P}\\
u(x,0)=u^0(x);\quad u_t(x,0)=u^1(x),\quad x\in\Omega , \notag
\end{gather}
where $\Omega$ is a bounded domain of $\mathbb{R}^n$ whose boundary
$\Gamma$ is assumed regular.  Let $x^0\in \mathbb{R}^n$ be an arbitrary
point of $\mathbb{R}^n$ and we set
\begin{equation} \label{e1.1}
\Gamma (x^0)=\{x\in\Gamma ;\; m(x)\cdot\nu (x)>0\}
\end{equation}
where $\nu$ represents the unit normal vector pointing towards the exterior
of $\Omega$ and
\begin{equation}
m(x)=x-x^0. \label{e1.2}
\end{equation}
Let $\omega$ be a neighborhood of $\Gamma (x^0)$ in $\Omega$ and consider
$\delta >0$ sufficiently small such that
\begin{gather}
Q_0=\{x\in\Omega ;\; d(x,\Gamma(x^0))<\delta\}\subset\omega ,\label{e1.3} \\
Q_1=\{x\in\Omega ;\,\,d(x,\Gamma(x^0))<2\delta\}\subset\omega .\label{e1.4}
\end{gather}
Here, if $A\subset \mathbb{R}^n$, and $x\in \mathbb{R}^n$,
$d(x,A)=\inf_{y\in A}|x-y|$. Then,
$Q_0\subset Q_1\subset\omega$.

Next, we make some remarks about early works in connection
with problem (\ref{P}).  When $g=0$ and $f \neq 0$ and the feedback
term depends
on the velocity in a linear way, as in the present paper,
Zuazua \cite{z2} proved that the energy related to problem (\ref{P})
decays exponentially if the damping region contains a
neighbourhood of the boundary $\Gamma$ or, at least, contains a
neibourhood of the particular part given by (1.1). In the same
direction and considering $g=f=0$, it is
important to mention the result of  Bardos ,  Lebeau and
Rauch \cite{b}, based on microlocal analysis, that ensures a necessary
and sufficient condition to obtain exponential decay, namely, the
damping region satisfies the well known {\em geometric control condition.   }
The classical example of an open subset $\omega$ verifying this
condition  is when $\omega$ is a neighbourhood of the
boundary. Later, still considering $g=0$ and $f=0$,  Nakao \cite{n1,n2}
extended the results of Zuazua \cite{z2} treating
first the case of a linear degenerate equation, and then the
case of a nonlinear dissipation $\rho (x,u_t)$ assuming, as usually, that
the function $\rho$ has a polynomial growth near the origin.
More recently, Martinez \cite{m1} improved the previous results
mentioned above in what concerns the linear wave equation
subject to a nonlinear dissipation $\rho (x,u_t)$, avoiding the polynomial
growth of the function $\rho (x,s)$ in zero. His proof is based on
the piecewise multiplier technique developed by Liu \cite{l2}
combined with nonlinear integral inequalities  to
show that the energy of the system decays to zero with a
precise decay rate estimate if the damping region satisfies
some geometrical conditions. It is important to mention that
Lasiecka and  Tataru \cite{l1} studied the nonlinear wave equation subject
to a nonlinear feedback acting on a part of the boundary of the
system and they were the first to prove that the energy decays to zero
as fast as the solution of some associated differential equation and
without assuming that the feedback has a polynomial growth in
zero, although no decay rate has been showed.

On the other hand, when $a=0$, that is, when the unique
damping mechanism is given by the memory term, then, following
ideas introduced by Munoz Rivera \cite{m2}, it is possible to prove
that the exponential decay holds for small initial data. Indeed, we
presume that the result above mentioned is a direct consequence
of his proof ( see \cite{m2} for details ). In this context we can cite the works
of  Dafermos \cite{d1}, Dafermos and Nohel \cite{d1,d2},
Jiang and  Munoz Rivera \cite{j},
among others.

To end, we would like to mention the works from the authors
Cavalcanti,  Domingos Cavalcanti, Prates Filho and Soriano \cite{c}
in connection with the viscoelastic linear
wave equation and nonlinear boundary dissipation and
Aassila, Cavalcanti and Soriano \cite{a}
concerned with the wave equation subject to viscoelastic
effects on the boundary and nonlinear boundary feedback.


The purpose of this paper is to obtain an exponential decay
rate to the solutions of a viscoelastic nonlinear wave equation subject to a
locally distributed dissipation as in Zuazua \cite{z2}. We would like
to emphasize that the additional damping effect given by the kernel
of the memory of the material allows us to avoid the compactness
arguments and the amount of estimates and fields considered
in the prior literature  by constructing a suitable Liapunov
functional and making use of the perturbed energy method. For
simplicity, we will consider the linear localized effect, although we
could consider the nonlinear one $\rho (x,u_t)$. In this direction, our work
complements the previous ones for the viscoelastic wave
equation.

Our paper is organized as follows.  In section 2 we state the
notation, assumptions and the main result and in section 3 we
give the proof of the uniform decay.

\section{Notation and Statment of Results}

We begin by introducing some notation that will be used throughout
this work. For the standard $L^p(\Omega )$ space we write
$$(u,v)=\int_{\Omega}u(x)v(x)\,dx,\quad\|u\|_p^p
=\int_{\Omega}|u(x)|^pdx.$$
Next, we give the precise assumptions on the functions $a(x)$, $f(x,t,u)$
and on the memory term $g(t)$.
\begin{enumerate}
\item[(A.1)]
Assume that $a:\Omega\to \mathbb{R}$ is a nonnegative and bounded function
such that
\begin{equation}
a(x)\geq a_0>0\quad \hbox{a.e.\quad in}\quad \omega .\label{H.1}
\end{equation}

\item[(A.2)]
Assume that $f:\overline {\Omega}\times [0,\infty )\times \mathbb{R}\to
\mathbb{R}$ is an element of the space
$C^1(\overline {\Omega}\times [0,\infty )\times \mathbb{R})$ and satisfies
\begin{equation}
|f(x,t,\xi )|\leq C_0(1+|\xi|^{\gamma+1})\label{H.2}
\end{equation}
where $C_0$ is a positive constant.
\end{enumerate}
 Let $\gamma$ be a constant such that $\gamma >0$ for $n=1,2$ and $
0<\gamma\leq 2/(n-2)$ for $n\geq 3$ and let $C_0^{\prime}$ be a positive
constant such that
\begin{equation}
f(x,t,\xi )\eta\geq|\xi|^{\gamma}\xi\eta-C_0^{\prime}|\xi \|
\eta| ;\quad\forall \eta\in \mathbb{R}.\label{H.3}
\end{equation}
Assume that there exist positive constants $C_1$ and $C_2$ such that
\begin{gather}
|f_t(x,t,\xi )|\leq C_1(1+|\xi|^{\gamma+1}),\label{H.4}\\
|f_{\xi}(x,t,\xi )|\leq C_2(1+|\xi|^{\gamma}).\label{H.5}
\end{gather}
We also assume that there is a positive constant $D_1$ such that for all
$\eta$ $,\hat{\eta}$ in $\mathbb{R}^n$ and for all $\xi$, $\hat{\xi}$
in $\mathbb{R}$,
\begin{equation}
(f(x,t,\xi )-f(x,t,\hat\xi ))(\eta -\hat\eta)
\geq -D_1(|\xi|^{\gamma}+|\hat\xi|^{
\gamma})|\xi -\hat\xi||\eta -\hat\eta|.\label{H.6}
\end{equation}
A simple example of a function $f$ that satisfies the above conditions
is given by $f(x,t,\xi)=|\xi|^{\gamma}\xi + \varphi (x,t) sin (\xi)$,
where $\varphi \in W^{1, +\infty}(\Omega \times \infty)$.

\paragraph{Remark:}
The variational formulation associated with problem (\ref{P}) leads us to
the identity
\begin{align*}
 (u''(t),v ) +(\nabla u(t) , \nabla v )+ ( f(x,t,u(t)), v)&\\
 - \int_0^t g(t-\tau) (\nabla u(\tau) , \nabla v ) d\tau + (a u'(t) , v )&
 =0\quad \hbox{for all } v\in H_0^1(\Omega).
\end{align*}
Assumption (\ref{H.2}) is required to prove that the nonlinear term in
the above identity is well defined. The hypothesis (\ref{H.3}) is used
in the first a priori estimate and also and in the asymptotic stability.
Now, hypotheses (\ref{H.4})-(\ref{H.5}) are needed for the second a priori
estimate while assumption (\ref{H.6}) is necessary for proving the uniqueness
of solutions.


For simplicity, we will consider the asymptotic stability for the simple
case when $ f(x,t,\xi)=| \xi |^{\gamma}\xi$, although the general case
follows exactly making use of the same procedure.

To obtain the existence of regular and weak solutions we
assume that kernel $g$ satisfies the following.

\begin{enumerate}
\item[(A.3)]  The function $g:\mathbb{R}_{+}\to \mathbb{R}_{+}$ is in
$W^{2,1}(0,\infty )\cap W^{1,\infty}(0,\infty )$.
Now, to obtain the uniform decay, we suppose that
\begin{equation}
1-\int_0^\infty g(s)\,ds=l>0,\label{H.7}
\end{equation}
and there exist $\xi_1,\xi_2>0$ such that
\begin{equation}
g(0)>0\quad \hbox{and}\quad -\xi_1g(t)\leq g'(t)\leq -\xi_2g(t);
\quad\forall t\geq 0.\label{H.8}
\end{equation}
\end{enumerate}
Note that (\ref{H.8}) implies
\begin{equation}
g(0)e^{-\xi_1t}\leq g(t)\leq g(0)e^{-\xi_2t};\quad\forall t\geq
0.\label{e2.1}
\end{equation}
Consequently from (\ref{e2.1}) we have that the kernel $g(t)$ is between two
exponential functions.

We observe that given $\{u^0,u^1\}\in H_0^1(\Omega )\cap
H^2(\Omega )\times H_0^1(\Omega )$, problem (\ref{P})
possesses a unique solution in the class
$$
u\in L_{{\rm loc}}^{\infty}(0,\infty ;H_0^1(\Omega )\cap H^2(\Omega )
),u'\in L_{{\rm loc}}^{\infty}(0,\infty ;H_0^1(\Omega )),\,u''
\in L_{{\rm loc}}^{\infty}(0,\infty ;L^2(\Omega ))
$$
which can easily obtained making use, for instance, of the Faedo-Galerkin
method. Now, if $\{u^0,u^1\}\in H_0^1(\Omega )\times L^
2(\Omega )$ and considering standard
arguments of density, we can prove that problem (\ref{P}) has a unique
solution
\begin{equation}
u\in C^0([0,\infty );H_0^1(\Omega ))\cap C^1([0
,\infty );L^2(\Omega )).\label{e2.3}
\end{equation}
To see the proof of pexitence and uniqueness of a similar
problem to the viscoelastic wave equation subject to nonlinear
boundary feedback, we refer the reader to \cite{c}.

The energy related to problem (\ref{P}) is
\begin{equation}
E(t)=\frac{1} {2}\|u'(t)\|_2^2+\frac{1} {2}\|\nabla u(t)\|_2^2
+ \frac{1} {\gamma +2}\|u(t)\|_{\gamma +2}^{\gamma +2}.\label{e2.4}
\end{equation}
Now, we are in a position to state our main result.

\begin{theorem} \label{thm2.1}
Given $\{u^0,u^1\}\in H_0^1(\Omega)\times L^2(\Omega )$ and assuming that
Hypotheses (A.1)-(A.3) hold, then, the unique solution of problem (\ref{P})
in the class given in (\ref{e2.3}) decays exponentially, that is, there exist
$C$, $\beta$ positive constants such that
\begin{equation}
E(t)\leq Ce^{-\beta t}\label{e2.5}
\end{equation}
provided that $\int_0^{\infty}g(s)ds$ is sufficiently small.
\end{theorem}

\section{Uniform decay}

In this section we prove the exponential decay for regular solutions
of problem (\ref{P}), and by using standard arguments of density we also
can extend the same result to weak solutions. From (\ref{P}) we deduce
that
\begin{equation}
E'(t) \leq -\int_{\Omega}a(x)|u'(x,t)|^2dx+\int_0^tg(t-\tau
)(\nabla u(\tau ),\nabla u'(t))d\tau .\label{e3.1}
\end{equation}
A direct computation shows that
\begin{equation}
\begin{aligned}
\int_0^t g(t-\tau )(\nabla u(\tau ),\nabla u'(t))d\tau \\
=\frac{1}{2}(g'\diamond\nabla u)(t)-\frac{1}{2}(g\diamond \nabla u)'(t)
{+\frac{d}{dt}\{\frac{1}{2}\int_0^t g(s)\,ds\|\nabla u(t)\|_2^2\}
-\frac{1}{2}g(t)\|\nabla u(t)\|_2^2,}
\end{aligned} \label{e3.2}
\end{equation}
where
$(g\diamond v)(t)=\int_0^tg(t-\tau )\|v(t)-v(\tau )\|_2^2d\tau$.
Define the modified energy as
\begin{equation}
e(t)=\frac{1}{2}\|u'(t)\|_2^2+\frac{1}{2}(1-\int_0^tg(s)\,ds)
\|\nabla u(t)\|_2^2
+\frac{1}{\gamma +2}\|u(t)\|_{\gamma +2}^{\gamma +2}
+\frac{1}{2}(g\diamond\nabla u)(t).\label{e3.3}
\end{equation}
From (\ref{H.7}) we deduce that $e(t)\geq 0$.  Moreover,
\begin{equation}
E(t)\leq l^{-1}e(t);\quad\forall t\geq 0.\label{e3.4}
\end{equation}
Consequently, the uniform decay of $E(t)$ is a consequence of the decay of
$e(t)$.  On the other hand, from (\ref{e2.4}), (\ref{e3.1}), (\ref{e3.2})
and (\ref{e3.3}) we deduce
\begin{equation}
e'(t)\leq -\int_{\Omega}a(x)|u'(x,t)|^2dx+\frac{1}{2}
(g'\diamond\nabla u)(t)-\frac{1}{2}g(t)\|\nabla u(t)\|_2^2.\label{e3.5}
\end{equation}
Considering the assumptions (\ref{H.1}) and (\ref{H.8}), from (\ref{e3.5})
 we obtain
\begin{equation}
e'(t)\leq - a_0\|u'(t)\|_{L^2(\omega )}^2-\frac{
\xi_2} {2}(g\diamond\nabla u)(t)-\frac{1}{2}g(t)
\|\nabla u(t)\|_2^2.\label{e3.6}
\end{equation}
Then  $e'(t)\leq 0$ and consequently $e(t)$ are non-increasing functions.
Having in mind (\ref{e1.3})-(\ref{e1.4}), we consider
$\psi\in C_0^{\infty}(\mathbb{R}^n)$ such that
\begin{equation}
\begin{gathered}
0\leq\psi\leq 1 \\
\psi =1\quad \hbox{in } \overline {\Omega}\backslash Q_1 \\
\psi =0\quad \hbox{in } Q_0.
\end{gathered} \label{e3.7}
\end{equation}
For an arbitrary $\varepsilon >0$, define the peturbed energy
\begin{equation}
e_{\varepsilon}(t)=e(t)+\varepsilon\rho (t)\label{e3.8}
\end{equation}
where
\begin{gather}
\rho (t)=2\int_{\Omega}u'(h\cdot\nabla z)dx+\theta\int_{
\Omega}u'zdx,\label{e3.9} \\
h=m\psi ,\label{e3.10} \\
z(t)=u(t)-\int_0^tg(t-\tau )u(\tau )\,d\tau ,\label{e3.11}
\end{gather}
and $\theta\in ]n-2,n[$, $\theta >\frac{2n}{\gamma +2}$.
For short notation, put
\begin{equation}
k_1=\min\big\{2(\theta -n+2),2(n-\theta ),(\gamma +2)(\theta
-\frac{2n}{\gamma +2})\big\}>0.\label{e3.12}
\end{equation}

\begin{proposition} \label{prop3.1}
 There exists $\delta_0>0$ such that
$$|e_{\varepsilon}(t)-e(t)|\leq\varepsilon\delta_0e(t)
\quad\forall t\geq 0\quad \mbox{and}\quad \forall\varepsilon >0.
$$
\end{proposition}

\paragraph{Proof.} From the Cauchy-Schwarz inequality
\begin{equation}
|\rho (t)|\leq 2R(x^0)\|u'(t)\|_2 \|\nabla
z(t)\|_2+\theta\lambda\|u'(t)\|_2\|\nabla z(t)\|_2, \label{e3.13}
\end{equation}
where $\lambda >0$ and satisfies $\|v|\|_2\leq\lambda\|\nabla v\|_2$ for all
$v\in H_0^1(\Omega )$, and
\begin{equation}
R(x^0)=\max_{x\in\overline {\Omega}}|x-x^0|.\label{e3.14}
\end{equation}
From (3.13) we obtain
\begin{equation}
|\rho (t)|\leq(2R(x^0)+\theta\lambda)
\big\{\frac{1}{2}\|u'(t)\|_2^2+\frac{1}{2}\|\nabla z(t)\|_2^2
\big\}.\label{e3.15}
\end{equation}
We note that
\begin{equation}
\begin{aligned}
\|\nabla z(t)\|_2^2=&\|\nabla u(t)\|_2^2-2\int_0^tg(t-\tau )
(\nabla u(\tau ),\nabla u(t))d\tau \notag\\
&+\int_0^tg(t-\tau )\Big(\int_0^tg(t-s)(\nabla u(\tau ),\nabla
u(s))ds\Big)d\tau
\end{aligned} \label{e3.16}
\end{equation}
Now we estimate each term on the right-hand side of this inequality.
\\
Estimate for $J_1:=-2\int_0^tg(t-\tau )(\nabla u(\tau ),\nabla u(t))d\tau$.
We have,
\begin{align*}
|J_1|\leq& \|\nabla u(t)\|_2^2+(\int_0^tg(t-\tau )\|\nabla u(\tau )\|_2)^2\\
\leq&\|\nabla u(t)\|_2^2+2\|g\|_{L^1(0,\infty )}\int_0^tg(t-\tau )
\big[\|\nabla u(\tau )-\nabla u(t)\|_2^2+\|\nabla u(t)\|_2^2\big]d\tau.
\end{align*}
Having in mind that $\|g\|_{L^1(0,\infty )}<1$, we obtain
\begin{equation}
|J_1|\leq 3\|\nabla u(t)\|_2^2
+2(g\diamond\nabla u)(t).\label{e3.17}
\end{equation}
Estimate for
$J_2:=\int_0^tg(t-\tau )(\int_0^tg(t-s)(\nabla u(\tau ),\nabla u(s))ds)d\tau$.
We infer
\begin{align*}
|J_2|\leq&(\int_0^tg(t-\tau )\|\nabla u(\tau )\|_2^2)^2\\
\leq& 2\|g\|_{L^1(0,\infty )} \int_0^tg(t-\tau )\big[\|\nabla u(\tau )
-\nabla u(t)\|_2^2+\|\nabla u(t)\|_2^2\big]d\tau .
\end{align*}
Consequently
\begin{equation}
|J_2|\leq 2(g\diamond\nabla u)(t)+2\|\nabla u(t)\|_2^2.\label{e3.18}
\end{equation}
Combining (\ref{e3.15})-(\ref{e3.18})  taking (\ref{e3.4}) into account,
it follows that
$$|\rho (t)|\leq 20(l^{-1}+1)(2R(x^0)+ \theta\lambda)e(t)$$
which, in view of (3.8) allow us to conclude the desired result.  This
completes the proof. \hfill 

\begin{proposition} \label{prop3.2}
There exists $\delta_1=\delta_1(\varepsilon)$ such that
$$ e_{\varepsilon}'(t)\leq -\delta_1e(t)\quad\forall t\geq 0
$$
provided that $\|g\|_{L^1(0,\infty )}$ is small enough.
\end{proposition}

\paragraph{Proof.} Getting the derivative of $\rho (t)$ given in
(\ref{e3.9}) with respect to $t$ and substituting
$u''=\Delta z-f(x,t,u)-a(x)u'$ in the obtained expression,
\begin{equation}
\begin{aligned}
\rho'(t)=&2(\Delta z(t),h\cdot\nabla z(t))-2(f(x,t,u(t)),h\cdot\nabla z(t))\\
&-2(au'(t),h\cdot\nabla z(t))+2(u'(t),h\cdot\nabla z'(t))
+\theta(\Delta z(t),z(t)) \\
&-\theta(f(x,t,u(t)),z(t)) -\theta(au'(t),z(t))+\theta\|u'(t)\|_2^2 \\
&-\theta g(0) ( u'(t), u(t) ) - \theta \int_0^t g'(t-\tau) (u'(t), u(\tau) )
d\tau.
\end{aligned}\label{e3.19}
\end{equation}
Next, we will estimate some terms on the right-hand side of identity
(\ref{e3.19}). \\
Estimate for $I_1:=2(\Delta z(t),h\cdot\nabla z(t))$.
Employing Gauss and Green formulas, we deduce
\begin{equation} \begin{aligned}
I_1=&-2\sum_{i,k=1}^n\int_{\Omega}\frac{\partial z} {\partial x_
k}\frac{\partial h_i} {\partial x_k}\frac{\partial z} {\partial
x_i}dx+\int_{\Omega}(\mathop{\rm div}h)|\nabla z|^2dx\\
&-\int_{\Gamma}(h\cdot\nu )|\nabla z|^2d\Gamma +\int_{
\Gamma}\frac{\partial z}{\partial\nu}(2h\cdot\nabla z)d\Gamma.
\end{aligned}\label{e3.20}
\end{equation}
Estimate for $I_2:=-2(u'(t),h\cdot\nabla z'(t))$.
Making use of Gauss formula and noting that $u'=0$ on $\Gamma$,
\begin{equation}
I_2=-\int_{\Omega}(\mathop{\rm div}h)|u'|^2dx-2g(0)(u'(t),
h\cdot\nabla u(t))
-2\int_0^tg'(t-\tau )(u'(t),h\cdot\nabla u(\tau ))d\tau.
\label{e3.21}
\end{equation}
Estimate for $I_3:=\theta(\Delta z(t),z(t))$.
Considering Green formula and observing that $z=0$ on $\Gamma$,
\begin{equation}
I_3=-\theta\|\nabla z(t)\|_2^2.\label{e3.22}
\end{equation}
Estimate for $I_4:=-2(f(x,t,u(t)),h\cdot\nabla z(t))$.
Considering assumption (\ref{H.3}), taking (\ref{e3.11}) and Gauss
formula into account and noting that $u=0$ on $\Gamma$, it follows that
\begin{equation}
\begin{aligned}
I_4\leq&\frac{2n}{\gamma +2}\|u(t)\|_{\gamma+2}^
{\gamma+2}+\frac{2}{\gamma +2}\int_{\Omega}(\nabla\psi\cdot m)
|u|^{\gamma +2}dx\\
&+2\int_0^tg(t-\tau )\int_{\Omega}|u(x,t)|^{\gamma +1}\psi\sum_{
k=1}^n|m_k||\frac{\partial u}{\partial x_k}(x,\tau )|dx\,d\tau.
\end{aligned}\label{e3.23}
\end{equation}
Estimate for $I_5:=-\theta(f(x,t,u(t)),z(t))$.
In view of assumption (\ref{H.3}),
\begin{equation}
I_5\leq -\theta\|u(t)\|_{\gamma +2}^{\gamma
+2}+\theta\int_0^tg(t-\tau )(|u(t)|^{\gamma}u(t),
u(\tau ))d\tau .\label{e3.24}
\end{equation}
Then, combining (3.19)-(3.24) we arrive at
\begin{align}
\rho'(t)\leq& -2\sum_{i,k=1}^n\int_{\Omega}\frac{\partial z}{\partial
x_k}\frac{\partial h_i}{\partial x_k}\frac{\partial z}{\partial
x_i}dx+\int_{\Omega}(\mathop{\rm div}h)|\nabla z|^2dx \nonumber\\
&-\int_{\Omega}(\mathop{\rm div}h)|u'|^2dx-2g(0)(u'(t),h\cdot
\nabla u(t)) \nonumber\\
&-2\int_0^tg'(t-\tau )(u'(t),h\cdot\nabla u(\tau ))d\tau
-\theta\|\nabla z(t)\|_2^2 \nonumber\\
&+(\frac{2n}{\gamma +2}-\theta)\|u(t)\|_{\gamma +2}^{\gamma +2}
+\frac{2}{\gamma +2}\int_{\Omega}
(\nabla\psi\cdot m)|u|^{\gamma +2}dx \label{e3.25}\\
&+2\int_0^tg(t-\tau )\int_{\Omega}|u(x,t)|^{\gamma +1}\psi\sum_{
k=1}^n|m_k||\frac{\partial u}{\partial x_k}(x,\tau )\|dx\,d\tau \nonumber\\
&-2(au'(t),h\cdot\nabla z(t))+\theta\int_0^tg(t-\tau
)(|u(t)|^{\gamma}u(t),u(\tau ))d\tau \nonumber\\
&-\theta(au'(t),z(t))+\theta\|u'(t)\|_2^2 \nonumber\\
&-\theta g(0) ( u'(t), u(t) ) - \theta \int_0^t g'(t-\tau) (u'(t), u(\tau) )
d\tau \nonumber\\
&-\int_{\Gamma}(h\cdot\nu )|\nabla z|^2d\Gamma +\int_{
\Gamma}\frac{\partial z}{\partial\nu}(2h\cdot\nabla z)d\Gamma. \nonumber
\end{align}
Next we handle the boundary terms.  Observe that since $z=0$ on $\Gamma$
we have $\frac{\partial z}{\partial x_k}=\frac{\partial z}{\partial\nu}\nu_k$
which implies
$$h\cdot\nabla z=(h\cdot\nu )\frac{\partial z}{\partial\nu}\quad
\hbox{and}\quad|\nabla z|^2=(\frac{\partial z}{\partial
\nu})^2\quad \hbox{on }\Gamma .$$
From the above expressions and taking (\ref{e1.1}), (\ref{e1.3})
 and (\ref{e3.7}) into account,
\begin{equation}
\begin{aligned}
-\int_{\Gamma}&(h\cdot\nu )|\nabla z|^2d\Gamma +\int_{
\Gamma}\frac{\partial z}{\partial\nu}(2h\cdot\nabla z)d\Gamma\\
=&\int_{\Gamma}(h\cdot\nu )(\frac{\partial z}{\partial\nu})^2d\Gamma \\\
=&\int_{\Gamma (x^0)}(m\cdot\nu )\psi(\frac{\partial z} {\partial
\nu})^2d\Gamma +\int_{\Gamma\backslash\Gamma (x^0)}(m\cdot
\nu )\psi(\frac{\partial z}{\partial\nu})^2d\Gamma\leq 0.
\end{aligned}\label{e3.26}
\end{equation}
Then, from (\ref{e3.25})-(\ref{e3.26}) and after some computations,
 we conclude that
\begin{equation}
\begin{aligned}
\rho'(t)\leq &(n-2-\theta )\|\nabla z(t)\|_
2^2+(\theta -n)\|u'(t)\|_2^2+(\frac{2n} {
\gamma +2}-\theta)\|u(t)\|_{\gamma +2}^{\gamma +2}\\
&+n\int_{Q_1}[1-\psi ]|u'|^2dx+(n-2)\int_{Q_1}[\psi -1]|\nabla z|^2dx\\
&-2\sum_{i,k=1}^n\int_{\Omega}\frac{\partial z}{\partial x_k}
m_i\frac{\partial\psi_i}{\partial x_k}\frac{\partial z}{\partial
x_i}dx+\int_{Q_1\backslash Q_0}(\nabla\psi\cdot m)|\nabla z|^2dx\\
&-\int_{Q_1\backslash Q_0}(\nabla\psi\cdot m)|u'|^2dx
+\frac{2}{\gamma +2}\int_{Q_1\backslash Q_0}(\nabla\psi
\cdot m)|u|^{\gamma +2}dx\\
&+2\int_0^tg(t-\tau )\int_{\Omega}|u(x,t)|^{\gamma +1}\psi\sum_{
k=1}^n|m_k||\frac{\partial u}{\partial x_k}(x,\tau )|dx\,d\tau \\
&-2g(0)(u'(t),h\cdot\nabla u(t))-2\int_0^tg'(t-\tau )(u'(t),h\cdot
\nabla u(\tau ))d\tau \\
&-2(au'(t),h\cdot\nabla z(t))+\theta\int_0^tg(t-\tau
)(|u(t)|^{\gamma}u(t),u(\tau ))d\tau\\
&-\theta(au'(t),z(t))-\theta g(0) ( u'(t), u(t) )
- \theta \int_0^t g'(t-\tau) (u'(t), u(\tau) ) d\tau.
\end{aligned}\label{e3.27}
\end{equation}
Having in mind (\ref{e3.12}), (\ref{e3.16}) and adding and subtracting
suitable terms in order to obtain $-k_1e(t)$ from (\ref{e3.27}),
and supposing without loss of generality that $g(0)\leq 1$, we infer
\begin{equation}
\begin{aligned}
\rho'(t)\leq& -k_1e(t)-\frac{k_1}{2}(\int^t_0 g(s)ds)\|\nabla u(t)\|_2^2
+\frac{k_1}{2}(g\diamond \nabla u)(t) \\
&+2\int_0^tg(t-\tau )\|\nabla u(\tau )\|_2\|\nabla u(t)\|_2d\tau +2(\int_0^tg
(t-\tau )\|\nabla u(\tau )\|_2d\tau)^2\\
&+2n\int_{Q_1}|u'|^2dx+3R(x^0)\max_{x\in\overline {\Omega}}
|\nabla\psi (x)|\int_{Q_1\backslash Q_0}|\nabla z|^2dx \\
&-\theta g(0) ( u'(t), u(t) ) - \theta \int_0^t g'(t-\tau) (u'(t), u(\tau) )
d\tau.\\
&+R(x^0)\max_{x\in\overline {\Omega}}|\nabla\psi (x)|
\int_{Q_1\backslash Q_0}|u'|^2dx\\
&+\frac{2}{\gamma +2}R
(x^0)\max_{x\in\overline {\Omega}}|\nabla\psi (x)|\int_{
Q_1\backslash Q_0}|u|^{\gamma +2}dx \\
&+2R(x^0)\int_0^tg(t-\tau )\|u(t)\|_{2(\gamma
+1)}^{\gamma +1}\|\nabla u(\tau )\|_2d\tau
+2|(u'(t),h\cdot\nabla u(t))| \\
&+2\int_0^t|g'(t-\tau )|\|u'(t)\|_2\|h\cdot\nabla u(\tau)\|_2d\tau
+2|(au'(t),h\cdot\nabla z(t))| \\
&+\theta\int_0^tg(t-\tau )\|u(t)\|_{2(\gamma
+1)}^{\gamma +1}\|u(\tau )\|_2d\tau +\theta|(au'(t),z(t))| \\
&+\theta| ( u'(t), u(t) )| + \theta \int_0^t |g'(t-\tau)| \|u'(t)\|_2
 \| u(\tau) \|_2 d\tau.
\end{aligned}\label{e3.28}
\end{equation}
Next, we analyze some terms on the right hand side in the above
the inequality.\\
Estimate for $I_5:=2\int_0^tg(t-\tau )\|\nabla u(\tau
)\|_2\|\nabla u(t)\|_2 \,d\tau$.
Considering Cauchy-Schwarz inequality and also employing the
inequality $ab\leq\frac{1}{4\eta}a^2+\eta b^2$, for an arbitrary $
\eta >0$, we obtain
$$
|I_5|\leq\frac{1}{2\eta}\|\nabla u(t)\|_
2^2+\eta\|g\|_{L^1(0,\infty )}\int_0^tg(t-\tau
)\|\nabla u(\tau )\|_2^2 d\tau .
$$
Having in mind that $\|g\|_{L^1(0,\infty )}<1$, the last inequality yields
\begin{equation}
\begin{aligned}
|I_5|\leq &(2\eta )^{-1}\|\nabla u(t)\|_2^
2+2\eta(g\diamond\nabla u)(t)+2\eta(\int_0^tg(s)d
s)\|\nabla u(t)\|_2^2 \\
\leq& (2\eta )^{-1}\|\nabla u(t)\|_2^2+4\eta
e(t)+2\eta(\int_0^tg(s)ds)\|\nabla u(t)\|_2^2.
\end{aligned}\label{e3.29}
\end{equation}
Estimate for $I_6:=2(\int_0^tg(t-\tau )\|\nabla u(\tau )\|_2d\tau)^2$.
Considering the Cauchy-Schwarz inequality it holds that
\begin{equation}
%\begin{aligned}
|I_6|\leq\|g\|_{L^1(0,\infty )}\int_0^tg(
t-\tau )\|\nabla u(\tau )\|_2^2d\tau
\leq 4(g\diamond\nabla u)(t)+4\|\nabla u(t)\|_2^2
%\end{aligned}
\label{e3.30}
\end{equation}
where in the last inequality we used $\|g\|_{L^1(0,\infty )}<1$.\\
Estimate for $I_7:=3R(x^0)\max_{x\in\overline {\Omega}}
\|\nabla\psi (x)|\int_{Q_1\backslash Q_0}|\nabla z|^2dx$.

Setting
\begin{equation}
A=3R(x^0)\max_{x\in\overline {\Omega}}|\nabla\psi (x)|\label{e3.31}
\end{equation}
and taking (\ref{e3.16}) into account, we obtain, as in (\ref{e3.29})
and (\ref{e3.30}), the estimate
\begin{equation}
\begin{aligned}
|I_7|\leq&(3A+A(4\eta )^{-1})\|\nabla u(t)\|_2^2+
4\eta Ae(t)+2A(g\diamond\nabla u)(t)\\
&+2A\eta(\int_o^tg(s)ds)\|\nabla u(t)\|_2^2,
\end{aligned}\label{e3.32}
\end{equation}
where $\eta >0$ is an arbitrary number.\\
Estimate for $I_8:=2A(\gamma +2)^{-1}\int_{Q_1\backslash Q_0}|u|^{\gamma +2}dx$.
Considering $k_0>0$ such that $\|v\|_{\gamma+2}\leq k_0\|\nabla v\|_2$
for all $v\in H_0^1(\Omega )$,
and taking (\ref{e3.4}) into account, we deduce
\begin{equation}
|I_8|\leq 2^{(\gamma +2)/2}Ak_0^{\gamma +2}(\gamma +2)^{-1}l^{-\gamma/2}
[E(0)]^{\gamma /2}\|\nabla u(t)\|_2^2.\label{e3.33}
\end{equation}
Estimate for
$I_9:=2R(x^0)\int_0^tg(t-\tau )\|u(t)\|_{2(\gamma +1)}^{\gamma +1}\|\nabla
u(\tau )\|_2d\tau$.
Considering $k_1>0$ such that $\|v\|_{2(\gamma+1)}\leq k_1\|\nabla v\|_2$
for all $v\in H_0^1(\Omega )$, making use of the inequality
$ab\leq\eta a^2+\frac{1}{4\eta}b^2$ for an arbitrary $\eta >0$ and also
the Cauchy-Schwarz one, we obtain
\begin{equation}
\begin{aligned}
|I_9|\leq &k_1^{2(\gamma +1)}R^2(x^0)2^{\gamma}\eta^{-1}l^{-\gamma}
[E(0)]^{\gamma}\|\nabla u(t)\|_2^2+2\eta l^{-1}e(t) \\
&+2\eta(\int_o^tg(s)ds)\|\nabla u(t)\|_2^2.
\end{aligned} \label{e3.34}
\end{equation}
Estimate for $I_{10}:=2|(u'(t),h\cdot\nabla u(t))|$. We have
\begin{equation}
|I_{10}|\leq 4\eta e(t)+R^2(x^0)\eta^{-1}\|\nabla u(t)\|_2^2. \label{e3.35}
\end{equation}
Estimate for $I_{11}:=2\int_0^t|g'(t-\tau )|\|u'(t)\|_2\|h\cdot\nabla
u(\tau)\|_2d\tau$.
Analogously we have done before and taking the assumption (\ref{H.8}) into
account, we arrive at
\begin{equation}
|I_{11}|\leq 2\eta e(t)+2\xi_1R^2(x^0)\eta^{-1}(g\diamond\nabla
u)(t)+2\xi_1R^2(x^0)\eta^{-1}\|\nabla u(t)\|_2^2.\label{e3.36}
\end{equation}
Estimate for $I_{12}:=2|(au'(t),h\cdot\nabla z(t))|$.
From Cauchy-Schwarz inequality, making use of the inequality
 $ab\leq\eta a^2+\frac{1}{4\eta}b^2$ $(\eta >0$ arbitrary) and considering
 (\ref{e3.16}), (\ref{e3.17}) and (\ref{e3.18}),
we deduce
\begin{equation}
|I_{12}|\leq (2\eta )^{-1}\|a\|_{L^{\infty}
(\Omega )}\|u'(t)\|_{L^2(Q_1\backslash Q_0)}^
2+4\eta R(x^0)(6l^{-1}+3)e(t).\label{e3.37}
\end{equation}
Estimate for $I_{13}:=\theta\int_0^tg(t-\tau )\|u(
t)\|_{2(\gamma +1)}^{\gamma +1}\|u(\tau )\|_2d\tau$.
Analogously, we have
\begin{equation}
%\begin{aligned}
I_{13}\leq\theta^2k_1^{2(\gamma +1)}2^{\gamma}l^{-\gamma}(4\eta
)^{-1}\|\nabla u(t)\|_2^2+4\eta\lambda^2e(t)
+2\eta\lambda^2\int_o^tg(s)ds\|\nabla u(t)\|_2^2,
%\end{aligned}
\label{e3.38}
\end{equation}
where $\lambda >0$ comes from the Poincar\'e inequality
$\|v\|_2\leq\lambda\|\nabla v\|_2$ for all $v\in H_0^1(\Omega )$.\\
Estimate for $I_{14}:=\theta|(au'(t),z(t))|$.
Similarly, we deduce
\begin{equation}
I_{14}\leq 4\eta\theta^2\lambda^2\|a\|_{L^{
\infty}(\Omega )}e(t)+3(4\eta )^{-1}\|\nabla u(t)\|_2^2.\label{e3.39}
\end{equation}
Estimate for $I_{15}:= \theta | ( u'(t), u(t) ) |$.
We have
\begin{equation}
 |I_{15}|\leq 2\eta e(t) + \theta^2 \lambda^2 (4\eta)^{-1}
\| \nabla u(t) \|_2^2. \label{e3.40}
\end{equation}
Estimate for
$I_{16}:= \theta \int_0^t |g'(t-\tau)| \|u'(t)\|_2 \| u(\tau) \|_2 d\tau$.
It holds that
\begin{equation}
|I_{16}| \leq 2\eta e(t) + \theta^2 \lambda^2 \xi_1(2\eta)^{-1}
( g\diamond \nabla u)(t)+ \theta^2 \lambda^2 \xi^2(2 \eta )^{-1} \|
\nabla u (t) \|_2^2. \label{e3.41}
\end{equation}
Combining (\ref{e3.28})-(\ref{e3.41}),
\begin{equation}
\begin{aligned}
\rho'(t)\leq &-[k_1-\eta L]e(t) \\
&-[2^{-1}k_1-4\eta (1+A)](\int_o^tg(s)ds)\|\nabla u(t)\|_2^2
+M(\eta )(g\diamond\nabla u)(t) \\
&+N(\eta )\|\nabla u(t)\|_2^2+K(\eta )\|u'(t)\|_{L^2(\omega )}^2,
\end{aligned}\label{e3.42}
\end{equation}
where
\begin{gather*}
L=10+2l^{-1}+4R(x^0)(6l^{-1}+3)+4\lambda^2+4\theta\lambda^2
\|a\|_{L^{\infty}(\Omega )},\\
M(\eta )=2^{-1}k_1+4+2A+2\eta^{-1}\xi_1^2R^2(x^0)+ \theta^2
\lambda^2 \xi_1^2(2 \eta )^{-1},\\
\begin{aligned}
N(\eta )=&3A+4+(\gamma +2)^{-1}2^{(\gamma +2)/2}k_0^{\gamma +2}l^{
-\gamma /2}[E(0)]^{l/2}+\theta^2 \lambda^2 (4 \eta )^{-1}\\
&+ \theta^2\lambda^2\xi_1^2 (2\eta)^{-1}
+(4\eta )^{-1}\Big(4(1+A)\\
&+R^2(x^0)\big(4k_1^{2(\gamma +1)}2^{
\gamma}l^{-\gamma}[E(0)]^{\gamma}+2\xi_1^2+1)\big)+\theta^2k_1^{
2(\gamma +1)}2^{\gamma}l^{-\gamma}\Big), \end{aligned}
\\
K(\eta )=(2\eta )^{-1}+2\eta +3^{-1}A.
\end{gather*}
Choosing $\eta >0$ sufficiently small such that
$$k_2=k_1-\eta L>0\quad \hbox{and}\quad 2^{-1}k_1-4\eta (1+A)\geq 0,
$$
from (\ref{e3.6}), (\ref{e3.8}), (\ref{e3.9}), (\ref{e3.42})
 and considering the assumption (\ref{H.8}), we
obtain
\begin{equation}
\begin{aligned}
e_{\varepsilon}'(t)=&e'(t)+\varepsilon\rho'(t)\\
\leq& -\varepsilon k_2e(t)+\xi_12^{-1}\int_0^tg(s)ds\|\nabla u(t)\|_2^2
-(a_0-\varepsilon K)\|u'(t)\|_{
L^2(\omega )}^2-(2^{-1}\xi_2\\
&-\varepsilon M)(g\diamond\nabla u)(t)
-(2^{-1}g(0)-\varepsilon N)\|\nabla u(t)\|_2^2.
\end{aligned} \label{e3.43}
\end{equation}
Choosing $\varepsilon >0$ small enough such that
$$a_0-\varepsilon K\geq 0,\quad 2^{-1}g(0)-\varepsilon N\geq 0\quad \hbox{and}
\quad 2^{-1}\xi_2-\varepsilon M\geq 0,$$
from (\ref{e3.43}) we deduce
\begin{equation}
e_{\varepsilon}'(t)\leq -\big(\varepsilon k_2-\xi_1\|g\|_{L^1(0,\infty )}
l^{-1}\big)e(t).\label{e3.44}
\end{equation}
For a fixed $\varepsilon >0$ sufficiently small such that the Propositions
\ref{prop3.1} and \ref{prop3.2} hold and considering
$\|g\|_{L^1(0,\infty)}$  small enough, from (\ref{e3.44}) we
conclude that $e_{\varepsilon}'(t)\leq -\delta_1e(t)$ which
completes the proof of Proposition \ref{prop3.2}. \hfill

Combining the Propositions \ref{prop3.1} and \ref{prop3.2}, we deduce
the exponential rate of decay.

\paragraph{Acknowledgments.}
The authors would like to thank the anonymous referees for their important remarks
and comments which allow us to correct, improve, and clarify several
points in this article.


\begin{thebibliography}{00} \frenchspacing

\bibitem{a}M. Aassila, M. M. Cavalcanti and J. A. Soriano, {\em Asymptotic
stability and energy decay rates for solutions of the wave equation
with memory in a star-shaped domain},
SIAM J. Control and Optimization {\bf 38}(5), (2000), 1581-1602.

\bibitem{b} C.  Bardos, G.  Lebeau and J.  Rauch, {\em Sharp sufficient
conditions for the observation, control and stabilization of waves from the
boundary}, SIAM J.  Control and Optimization {\bf 19}, (1992), 1024-1065.

\bibitem{c}M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho, J. A. Soriano,
{\em Existence and uniform decay rates for viscoelastic problems
with nonlinear boundary damping}, Diff. and Integral Equations {\bf 14}(1), (2001), 85-116.

\bibitem{d1}C. M. Dafermos, {\em Energy  methods for non linear
hyperbolic Volterra integro-differential equations}, Comm.
P. D. E. {\bf 3}, (1979),  119-178.

\bibitem{d2} C. M. Dafermos and J. A. Nohel, {\em Energy methods for
nonlinear hyperbolic Volterra integrodifferential equations}, Comm.
P. D. E. {\bf 4}, (1979), 218-278.

\bibitem{d3} C.  M.  Dafermos and J.  A.  Nohel, {\em A nonlinear hyperbolic
Volterra equation in viscoelasticity}, Amer.  J.  Math.  Supplement (1981),
87-116.

\bibitem{j} S. Jiang and J. E. Munoz Rivera, {\em A global existence theorem for
the Dirichlet problem in nonlinear n-dimensional viscoelastic\/}
Diff. and Integral Equations {\bf 9}(4), (1996), 791-810.

\bibitem{k1} I.  Kostin, {\em Long time behaviour of solutions to a semilinear
wave equation with boundary damping}, Dynamics and Control (2002), (to appear).

\bibitem{l1} I.  Lasiecka and D.  Tataru, {\em Uniform boundary stabilization of
semilinear wave equation with nonlinear boundary damping}, Diff.  and Integral
Equations {\bf 6}, (1993), 507-533.

\bibitem{l2} K.  Liu, {\em Locally distributed control and damping for the
conservative systems,} SIAM J.  Control and Optimization {\bf 35}(5), (1997),
1574-1590.

\bibitem{m1} P.  Martinez, {\em A new method to obtain decay rate estimates for
dissipative systems with localized damping}, Rev.  Mat.  Complutense {\bf
12}(1), (1999), 251-283.

\bibitem{m2} J. E. Munoz Rivera, {\em Global solution on a quasilinear wave
equation with memory}, Bolletino U.M.I. {\bf 7}(8-B), (1994), 289-303.

\bibitem{n1} M.  Nakao, {\em Decay of solutions of the wave equation with local
degenerate dissipation}, Israel J.  of Maths {\bf 95}, (1996), 25-42 .

\bibitem{n2} M.  Nakao, {\em Decay of solution of the wave equation with a local
nonlinear dissipation}, Math.  Ann.  {\bf 305}(3), (1996), 403-417.

\bibitem{z1} E. Zuazua, {\em Exponential decay for the semilinear wave
equation with localized damping in unbounded domains}, J. Math.
Pures et Appl. {\bf 70}, (1991), 513-529.

\bibitem{z2} E. Zuazua,
{\em Exponential decay for the semilinear wave equation with locally
distributed damping}, Comm. P. D. E. {\bf 15}, (1990), 205-235.

\end{thebibliography}
\smallskip

\noindent\textsc{Marcelo M. Cavalcanti} (e-amil: marcelo@gauss.dma.uem.br)\\
\textsc{Val\'eria N. Domingos Cavalcanti} (e-mail: valeria@gauss.dma.uem.br)\\ 
\textsc{Juan A. Soriano} (e-mail: soriano@gauss.dma.uem.br)\\[2pt]
Departamento de Matem\'atica\\
Universidade Estadual de Maring\'a\\
87020-900 Maring\'a - PR, Brasil


\end{document}