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\markboth{\hfil Asymptotic behavior of solutions \hfil EJDE--2001/73}
{EJDE--2001/73\hfil Mauro L. Santos \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No.~73, pp. 1--11. \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} \\
 %
  Asymptotic behavior of solutions to wave equations with
  a memory condition at the boundary
 %
\thanks{ {\em Mathematics Subject Classifications:} 39A11.
\hfil\break\indent
{\em Key words:} Wave equations, asymptotic behavior.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted September 20, 2001. Published November 26, 2001. \hfil\break\indent
Supported by grant 150116/00-4 from the  CNPq/LNCC Brazil} }
\date{}
%
\author{Mauro de Lima Santos}
\maketitle

\begin{abstract}
 In this paper, we study the stability of solutions
 for wave equations whose boundary condition includes a integral
 that represents the memory effect.
 We show  that the dissipation is strong enough to produce
 exponential decay of the solution, provided the relaxation
 function also decays exponentially. When the relaxation function
 decays polynomially, we show that the solution decays polynomially
 and with the same rate.
\end{abstract}

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\section{Introduction}

 The main purpose of this work is study the asymptotic behavior of
 solution of the wave equation with a boundary condition of memory type.
 For this, we consider the following initial boundary-value problem
 \begin{eqnarray}\label{1eq1-1}
 &u_{tt} - \mu (t) u_{xx} = 0  \quad \mbox {in }(0,1) \times (0,\infty),& \\
\label{1eq1-2}
 &u(0,t) = 0, \quad u(1,t) + \int^{t}_{0}g(t-s) \mu (s)u_{x}(1,s)ds = 0,
  \quad \forall t>0 &\\
\label{1eq1-3}
&u(x,0) = u_{0}(x),\quad u_{t}(x,0) = u_{1}(x) \quad \mbox{in } (0,1).&
\end{eqnarray}
The integral in (\ref{1eq1-2}) is a boundary condition which includes the
memory effect. Here, by $u$ we denote the displacement and by $g$
the relaxation function. By $\mu=\mu (t)$ we represent a function of
$W^{1,\infty}_{loc}(0,\infty:\mathbb{R})$, such that $\mu (t)\ge \mu_{0}>0$
and $\mu'(t) \leq 0$   for all $t\ge 0$.
We refer to \cite{H-G} for the physical motivation of this model.

Frictional dissipative boundary condition for the wave equation was
studied by several authors, see for example
\cite{H-G, J-P, K-K, K-Z, L-I, M-L, C-M, M-T, Z-Z} among others.
In these works  existence of solutions and exponential stabilization
were proved for linear and for nonlinear equations.
In contrast with the large literature for frictional dissipative,
for boundary condition with memory, we have only a few works as for
example \cite{C-C, F-M, J-D,  Q1, Q2}.
Let us explain briefly each of the above works.
In \cite{C-C} Ciarletta established theorems of existence, uniqueness
and asymptotic stability for a linear model of heat conduction.
In this case the memory condition describes a boundary that can absorb
heat and due to the hereditary term, can retain part of it.
In \cite{F-M} Fabrizio \& Morro considered a linear electromagnetic model
with boundary condition of memory type and proved the existence,
uniqueness and asymptotic stability of solutions.
While in \cite{Q1} showed  the existence of global smooth solution for the one
dimensional nonlinear wave equation, provided the initial data
$(u_{0},u_{1})$ is small in the $H^{3}\times H^{2}$-norm,
moreover he showed that the solution tends to zero as time goes to infinity.
In all the above works was left open the rate of decay.
 Rivera \& Andrade \cite{J-D} consider a nonlinear one dimensional wave
equation with a viscoelastic boundary condition. They proved the existence,
uniqueness of global smooth solution, provided the initial data
$(u_{0},u_{1})$ is small in the $H^{2}\times H^{1}$-norm and also that
the solution decays uniformly in time (exponentially and algebraically).
Finally, in \cite{Q2} Qin proved a blow up result for the nonlinear one
dimensional wave equation with memory boundary condition.
Our main result is to show that the solution of system
(\ref{1eq1-1})-(\ref{1eq1-3}) decays uniformly in time, with rates depending
on the rate of decay of the relaxation function.
More precisely, denoting by $k$ the resolvent kernel of $g'$
(the derivative of the relaxation function) we show that the solution decays
exponentially to zero provided $k$ decays exponentially to zero.
When $k$ decays polynomially, we show that the corresponding solution also
decays polynomially to zero with the same rate of decay.

The method used here is based on the construction of a suitable Lyapunov
functional $\mathcal{L}$ satisfying
$$
\frac{d}{dt}\mathcal{L}(t)\leq - c_{1} \mathcal{L}(t)+c_{2} e^{-\gamma t} \quad
\mbox {or}\quad \frac{d}{dt}\mathcal{L}(t)
\leq -c_{1} \mathcal{L}(t)^{1+ \frac{1}{\alpha}}+ \frac{c_{2}}{(1+t)^{\alpha +1}}
$$
for some positive constants $c_1$, $c_2$, $\alpha$ and $\gamma$.
To study the existence of solution of (\ref{1eq1-1})-(\ref{1eq1-3}),
we introduce the space
$V:=\{v \in H^{1}(0,1) ; v(0) = 0 \}$.
The notation used in this paper is standard and can be found in
Lions's book \cite{JLL}. In the sequel by $c$ (sometime $c_{1}, c_{2},\dots$)
we denote various positive constants independent of $t$ and on the
initial data.
The organization of this paper is as follows.
In section 2 we establish a existence and regularity result.
In section 3 prove the uniform rate of exponential decay.
Finally in section 4 we prove the uniform rate of polynomial decay.

\section{Existence and Regularity}
In this section we shall study the existence and regularity of solutions
for the (\ref{1eq1-1})-(\ref{1eq1-3}). We assume that the kernel $g$ is
positive and $k$ satisfies:
\begin{eqnarray}\label{2eq2-1}
&0<k(t)\leq b_{0}e^{-\gamma_{0}t}, &\nonumber \\
&-b_{1}k(t)\leq k'(t)\leq -b_{2}k(t), & \\
&-b_{3}k'(t)\leq k''(t)\leq -b_{4}k'(t)& \nonumber
\end{eqnarray}
for some positive constants $b_{i}$, $i=0,1,\dots,4$, and $\gamma_{0}$.
To facilitate our analysis, we introduce the following binary operators:
\begin{eqnarray*}
(f \Box \varphi)(t) &=& \int^{t}_{0}f(t-s)|\varphi(t)-\varphi(s)|^{2} ds,\\
(f \ast \varphi)(t) &=& \int^{t}_{0}f(t-s)\varphi(s) ds,
\end{eqnarray*}
where $\ast$ is the convolution product. Differentiating 
 (\ref{1eq1-2}) we arrive to the Volterra integral equation
$$
\mu(t) u_{x}(1,t) + \frac{1}{g(0)}g'\ast \mu(t) u_{x}(1,t)
= -\frac{1}{g(0)}u_{t}(1,t).
$$
Using the Volterra inverse operator, we obtain
$$
 \mu(t) u_{x}(1,t) = -\frac{1}{g(0)}\{u_{t}(1,t) + k \ast u_{t}(1,t)\}.
$$
With $\tau=\frac{1}{g(0)}$ and using the above identity, we write
\begin{eqnarray}\label{2eq2-2}
\mu(t) u_{x}(1,t) = -\tau \{u_{t}(1,t) + k(0)u(1,t) - k(t)u_{0}(1) + k' \ast u(1,t) \}.
\end{eqnarray}
The following lemma state an  important property of the convolution operator.

\begin{lemma}\label{Lem2.1} For $f,\varphi \in C^{1}([0,\infty[:\mathbb{R})$ we have
\begin{eqnarray*}
\int^{t}_{0} f(t-s) \varphi (s)ds \varphi_{t} & = & -\frac{1}{2} f(t)| \varphi(t) |^{2} + \frac{1}{2} f' \Box \varphi \\
&& - \frac{1}{2} \frac{d}{dt}\left[f \Box \varphi - (\int^{t}_{0}f(s) ds) |\varphi|^{2}\right].
\end{eqnarray*}
\end{lemma}

The proof of this lemma follows by differentiating the term $f \Box \varphi$.

We summarize the well-posedness of (\ref{1eq1-1})-(\ref{1eq1-3}) in the
following theorem.

\begin{theorem}\label{Teo2.1}
Let $(u_{0},u_{1})\in V \times L^{2}(0,1)$, then there exists only one
solution to the (\ref{1eq1-1})-(\ref{1eq1-3}) satisfying
$$
u \in C([0,T]:V)\cap C^{1}([0,T]:L^{2}(0,1)).
$$
Moreover, if $(u_{0},u_{1})\in H^{2}(0,1)\cap V \times V$ satisfies
the compatibility condition
\begin{eqnarray}\label{2eq2-3}
\mu (0)u_{0,x}(1)=-\tau u_{1}(1)\,,
\end{eqnarray}
 then $u \in C([0,T]:H^{2}(0,1)\cap V)\cap C^{1}([0,T]:V)$.
\end{theorem}
This theorem can be showed using the standard Galerkin method,
for this reason we omit it here.

\section{Exponential Decay}

 In this section we show that the solution of (\ref{1eq1-1})-(\ref{1eq1-3})
decays exponentially. Our point of departure  will be to establish some
inequalities for the solution of system (\ref{1eq1-1})-(\ref{1eq1-3}).
For this end, we introduce the functional
$$
F(t)  =  \frac{1}{2}\int^{1}_{0}|u_{t}|^{2} + \mu (t) |u_{x}|^{2} dx
+ \frac{\tau}{2}(k(t)|u(1,t)|^{2} - k'(t)\Box u(1,t)).
$$

\begin{lemma}\label{Lem3.2}
For a strong solution of the system (\ref{1eq1-1})-(\ref{1eq1-3}),
 \begin{eqnarray*}
\frac{d}{dt}F(t)
& \leq &
- \frac{\tau}{2}|u_{t}(1,t)|^{2} + \frac{\tau}{2}k^{2}(t)|u_{0}(1)|^{2} + \frac{\tau}{2}k'(t)|u(1,t)|^{2} \\
&& - \frac{\tau}{2} k''(t) \Box u(1,t) + \frac{1}{2} \int^{1}_{0}\mu' |u_{x}|^{2} dx.
\end{eqnarray*}
\end{lemma}
\paragraph{Proof.}
Multiplying (\ref{1eq1-1}) by $u_{t}$ and  integrating over $[0,1]$  we obtain
\begin{equation}\label{3eq3-1}
\frac{1}{2}\frac{d}{dt}\int^{1}_{0}|u_{t}|^{2} + \mu (t) |u_{x}|^{2} dx
= \mu (t) u_{x}(1,t) u_{t}(1,t) + \frac{1}{2} \int^{1}_{0}\mu' |u_{x}|^{2} dx.
\end{equation}
Using (\ref{2eq2-2}) and Lemma \ref{Lem2.1} the conclusion follows.
Q.E.D.

The following Lemma plays an important role in the construction of
the Lyapunov functional. Let us define the functional
$$
\psi (t) = \int^{1}_{0} x u_{x}u_{t} dx\,.
$$

\begin{lemma}\label{Lem3.3}
The strong solution of (\ref{1eq1-1})-(\ref{1eq1-3}) satisfies
\begin{eqnarray*}
\frac{d}{dt}\psi (t)
& \leq &- \frac{1}{2}\int ^{1}_{0}(|u_{t}|^{2} + \mu (t) |u_{x}|^{2}) dx
+ c |u_{t}(1,t)|^{2} + c k^{2}(t)|u_{0}(1)|^{2} \\
&& + ck(0)k(t)|u(1,t)|^{2} + c k(0) |k'|\Box u(1,t).
\end{eqnarray*}
\end{lemma}

\paragraph{Proof.}
From (\ref{1eq1-1}) it follows that
\begin{eqnarray}
\frac{d}{dt}\psi (t)
&=& \int^{1}_{0}xu_{xt}u_{t} dx + \int^{1}_{0}x u_{x}u_{tt} dx \nonumber\\
&=& \frac{1}{2} \int^{1}_{0}x \frac{d}{dx} |u_{t}|^{2} dx +\frac{1}{2}
\int^{1}_{0}x \mu (t) \frac{d}{dx}|u_{x}|^{2} dx  \label{3eq3-2}\\
&=&- \frac{1}{2}\int^{1}_{0}(|u_{t}|^{2} + \mu (t) |u_{x}|^{2}) dx
+\frac{1}{2}|u_{t}(1,t)|^{2} + \frac{1}{2} \mu (t) |u_{x}(1,t)|^{2}. \nonumber
\end{eqnarray}
Note that
\begin{eqnarray}
\lefteqn{-k(0)u(1,t) - k' \ast u(1,t)} \nonumber\\
&=& - \int^{t}_{0}k'(t-s)[u(1,s) - u(1,t)]ds - k(t) u(1,t) \nonumber \\
& \leq &\Big(\int^{t}_{0}|k'(s)|ds\Big)^{1/2}[|k'| \Box u(1,t)]^{1/2}
+ k(t) |u(1,t)|  \label{3eq3-3}\\
& \leq &|k(t)-k(0)|^{1/2}[|k'| \Box u(1,t)]^{1/2}
+ k(t) |u(1,t)|. \nonumber
\end{eqnarray}
Using (\ref{2eq2-2}) and (\ref{3eq3-3}), it follows that
\begin{eqnarray}\label{3eq3-4}
\lefteqn{\mu (t) |u_{x}(1,t)|^{2} }\\
&\leq& c\{|u_{t}(1,t)|^{2} + k^{2}(t) |u_{0}(1)|^{2} + k(0) |k'|\Box u(1,t)
+ k(0)k(t) |u(1,t)|^{2} \}. \nonumber
\end{eqnarray}
Substituting (\ref{3eq3-4}) into (\ref{3eq3-2}), the conclusion of the
lemma follows. \quad Q.E.D.

Let us introduce the functional
\begin{equation}\label{3eq3-5}
\mathcal{L} (t) = N F(t) +  \psi (t),
\end{equation}
with $N>0$. It is not difficult to see that $\mathcal{L}(t)$ satisfies
\begin{equation}\label{3eq3-6}
q_{0}F(t) \leq \mathcal{L} (t) \leq q_{1}F(t),
\end{equation}
for some positive constants $q_{0}$ and $q_{1}$. Finally, we shall show
the main result of this Section.

\begin{theorem}\label{teo3.1}  Assume that the initial data
$(u_{0},u_{1})\in V \times L^{2}(0,1)$ and that the resolvent $k$ satisfies
(\ref{2eq2-1}). Then there exist positive constants $\alpha_{1}$ and
$\gamma_{2}$ such that
$$
F(t) \leq \alpha_{1} e^{- \gamma_{2}t} F(0),\quad \forall t \ge 0\,.
$$
\end{theorem}

\paragraph{Proof.}
We will suppose that $(u_{0},u_{1})\in H^{2}(0,1)\cap V \times V$ and
satisfies (\ref{2eq2-3}); our conclusion will follow by standard density
arguments. Using Lemmas \ref{Lem3.2} and \ref{Lem3.3} we get
\begin{eqnarray}\label{3eq3-7}
\frac{d}{dt} \mathcal{L} (t) & \leq & - \frac{\tau}{2}N|u_{t}(1,t)|^{2}
+ \frac{\tau}{2}Nk^{2}(t)|u_{0}(1)|^{2} + \frac{\tau}{2}Nk'(t)|u(1,t)|^{2}
\nonumber \\
&&- \frac{\tau}{2}Nk'' \Box u(1,t) - \frac{1}{2}\int^{1}_{0}(|u_{t}|^{2}
+ \mu (t)|u_{x}|^{2})dx + c|u_{t}(1,t)|^{2} \nonumber \\
&& + ck^{2}(t)|u_{0}(1)|^{2} + ck(0)k(t)|u(1,t)|^{2} + ck(0) |k'|\Box u(1,t).
\end{eqnarray}
Then, choosing  $N$ large enough we obtain
$$
\frac{d}{dt}\mathcal{L} (t) \leq - q_{2} F(t) + c k^{2}(t) F(0),
$$
where $q_{2}>0$ is a small constant. Here we use (\ref{2eq2-1}) to conclude
the following estimates for the corresponding two terms appearing in
Lemma \ref{Lem3.2}.
\begin{eqnarray*}
&-\frac{\tau}{2}k'' \Box u(1,t) \leq c_{1}k' \Box u(1,t),& \\
&\frac{\tau}{2}k' |u(1,t)|^{2} \leq -c_{1}k|u(1,t)|^{2}\,.&
\end{eqnarray*}
 Finally, in view of (\ref{3eq3-6}) we conclude that
$$
\frac{d}{dt}\mathcal{L} (t) \leq - \gamma_{1}\mathcal{L} (t) + c k^{2}(t) F(0).
$$
Using the exponential decay of the resolvent kernel $k$ we conclude
$$
\mathcal{L} (t) \leq \{\mathcal{L}(0) + c\} e^{- {\gamma_{2}}t}
$$
for all $t \ge 0$, where $\gamma_{2}=min(\gamma_{0},\gamma_{1})$.
 From (\ref{3eq3-6}) the conclusion follows.  \quad Q.E.D.


\section{Polynomial rate of decay}

The proof of the existence of global solutions for
(\ref{1eq1-1})-(\ref{1eq1-3}) with resolvent kernel $k$ decaying
polynomially is essentially the same as in Section 2. Here our
attention will be focused on the uniform rate of decay when the
resolvent $k$ decays polynomially such as  $(1+t)^{-p}$. In this case,
we will show that the solution also decays polynomially with the
same rate. We shal use the following hypotheses:
\begin{eqnarray}\label{4eq4-1}
& 0 < k(t) \leq b_{0} (1+t)^{-p}, &\nonumber\\
& -b_{1}k^\frac{p+1}{p} \leq  k'(t) \leq -b_{2}k^\frac{p+1}{p}, \\
& b_{3}(-k'(t))^\frac{p+2}{p+1} \leq k''(t) \leq b_{4}(-k'(t))^\frac{p+2}{p+1}
&\nonumber
\end{eqnarray}
where $p>1$ and $b_{i}$, $i=0,1\ldots,4$, are positive constants.
Also we assume that
\begin{equation}\label{4eq4-2}
\int^{\infty}_{0}|k'(t)|^{r}dt < \infty \quad \mbox {if} \quad r> \frac{1}{p+1}.
\end{equation}
The following lemmas will play an important role in the sequel.

\begin{lemma}\label{Lem4.1} Let $m$ and $h$ be integrable functions,
$0 \leq r < 1$ and  $q>0$. Then, for $t \ge 0$,
\begin{eqnarray*}
\lefteqn{\int^{t}_{0}|m(t-s)h(s)|ds }\\
&\leq& \Big(\int^{t}_{0}|m(t-s)|^{1+\frac{1-r}{q}}|h(s)|ds\Big)^{\frac{q}{q+1}}
\Big(\int^{t}_{0}|m(t-s)|^{r}|h(s)|ds\Big)^ \frac{1}{q+1}.
\end{eqnarray*}
\end{lemma}

\paragraph{Proof.} Let
\begin{eqnarray*}
v(s):=|m(t-s)|^{1- \frac{r}{q+1}} |h(s)|^{\frac{q}{q+1}}, \quad
w(s):=|m(t-s)|^{\frac{r}{q+1}} |h(s)|^{\frac{1}{q+1}}\,.
\end{eqnarray*}
Then using H\"{o}lder's inequality with $\delta= \frac{q}{q+1}$ for $v$
and $\delta^{*}=q+1$ for $w$, we arrive to the conclusion. \quad Q.E.D.

\begin{lemma}\label{Lem4.2} Let $p>1$, $0 \leq r < 1$ and $t\ge0$.
Then  for $r>0$,
\begin{eqnarray*}
\lefteqn{ (|k'| \Box u(1,t) )^{\frac{1+(1-r)(p+1)}{(1-r)(p+1)}} }\\
 & \leq & 2\Big( \int^{t}_{0}|k'(s)|^{r}ds ||u||^{2}_{L^{\infty}
 ((0,T),H^{1}(0,1))}\Big)^ {\frac{1}{(1-r)(p+1)}}
 (|k'|^{1+ \frac{1}{p+1}} \Box u(1,t))\,,
\end{eqnarray*}
and for $r=0$,
\begin{eqnarray*}
\lefteqn{ (|k'| \Box u(1,t))^{\frac{p+2}{p+1}} }\\
&\leq& 2 \Big(\int^{t}_{0}||u(.,s)||^{2}_{H^{1}(0,1)} ds
 t ||u(.,s)||^{2}_{H^{1}(0,1)}\Big)^{\frac{1}{p+1}}(|k'|^{1+ \frac{1}{p+1}}
 \Box u(1,t))
\end{eqnarray*}
\end{lemma}

\paragraph{Proof.} The above inequality is a immediate consequence of Lemma
\ref{Lem4.1} with
$$
m(s) :  = |k'(s)|, \quad h(s) :  =  |u(x,t) - u(x,s)|^{2},
\quad q:  =  (1-r)(p+1),
$$
and $t$ fixed. \quad Q.E.D.

\begin{lemma}\label{Lem4.3}
Let  $\alpha>0$, $\beta \ge \alpha + 1$, and $f \ge 0$ be differentiable
function satisfying
$$
f'(t) \leq \frac{- \bar{c}_{1}}{f(0)^\frac{1}{\alpha}}f(t)^{1+\frac{1}{\alpha}}
+\frac{\bar{c}_{2}}{(1+t)^{\beta}}f(0)
$$
for $t \ge 0$ and some positive constants $\bar{c}_{1}$, $\bar{c}_{2}$.
Then there exists a constant $\bar{c}_{3} > 0$ such that for $t\ge0$,
$$
f(t) \leq \frac {\bar{c}_{3}}{(1+t)^{\alpha}}f(0).
$$
\end{lemma}

\paragraph{Proof.} Let $t\ge 0$ and
$$
F(t) = f(t) + \frac{2 \bar{c}_{2}}{\alpha} (1+t)^{-\alpha} f(0).
$$
Then
\begin{eqnarray*}
F' &=& f' - 2 \bar{c}_{2} (1+t)^{-(\alpha +1)}f(0) \\
&\leq& \frac{- \bar{c}_{1}}{f(0)^{\frac{1}{\alpha}}}f^{1+\frac{1}{\alpha}} - \bar{c}_{2} (1+t)^{-(\alpha+1)}f(0),
\end{eqnarray*}
where we used $\beta \ge \alpha + 1$. Hence
$$
F'  \leq  \frac{-c}{f(0)^\frac{1}{\alpha}}(f^{1+\frac{1}{\alpha}} + (1+t)^{-(\alpha + 1)} f(0)^{1 + \frac{1}{\alpha}})  \leq  \frac{-c}{F(0)^ \frac{1}{\alpha}} F^{1 + \frac{1}{\alpha}}.
$$
Integration yields
$$
F(t) \leq \frac{F(0)}{(1+ct)^{\alpha}} \leq \frac{c}{(1 + t)^{\alpha}} f(0)
\quad \mbox{hence}\quad
f(t) \leq \frac{\bar{c}_{3}}{(1 + t)^{\alpha}} f(0)
$$
for some $\bar{c}_{3}$, which proves Lemma \ref{Lem4.3}. \quad Q.E.D.

\begin{theorem}\label{Teo4.1}  Assume that $(u_{0},u_{1})\in V \times L^{2}(0,1)$
and that the resolvent $k$ satisfies (\ref{4eq4-1}).
Then there exists a positive constant $c$ for which
$$
F(t) \leq \frac{c}{(1+t)^{p+1}}F(0).
$$
\end{theorem}

\paragraph{Proof.}
 We will suppose that $(u_{0},u_{1})\in H^{2}(0,1)\cap V \times V$ and
satisfies (\ref{2eq2-3}); our conclusion will follow by standard density
arguments. We define the functional $\mathcal L$ as (\ref{3eq3-5}) and 
we have the equivalence to the energy term $F$ as given in (\ref{3eq3-6})
 again. The negative term
$$
-c k(t) |u(1,t)|^{2}
$$
can be obtained from Lemma \ref{Lem3.3} and estimate
$$
k(t)|u(1,t)|^{2} \leq c \int^{1}_{0} \mu (t) |u_{x}|^{2} dx.
$$
From  Lemmas \ref{Lem3.2} and \ref{Lem3.3}, using the properties 
of $k''$ from the assumption (\ref{4eq4-1}) for the term
$$
- \frac{\tau}{2}k'' \Box u(1,t),
$$
we obtain 
\begin{eqnarray}\label{4eq4-3}
\frac{d}{dt}\mathcal{L} (t) & \leq &
 - c_{1}\Big( \int^{1}_{0}|u_{t}|^{2} + \mu (t) |u_{x}|^{2}dx
 +  k(t)|u(1,t)|^{2} \nonumber\\
 && +N (-k')^{1 +\frac{1}{p+1}}\Box u(1,t)\Big)
+ c_{2} k^{2}(t)F(0).
\end{eqnarray}
Applying Lemma \ref{Lem4.2} with $r>0$ we obtain
\begin{eqnarray*}
\lefteqn{(-k')^{1+\frac{1}{p+1}}\Box u(1,t) } \\
& \ge &\frac{c}{(\int^{t}_{0}|k'|^{r}ds)^{\frac{1}{(1-r)(p+1)}}
F(0)^\frac{1}{(1-r)(p+1)}}((-k') \Box u(1,t))^{1+\frac{1}{(1-r)(p+1)}}
\end{eqnarray*}
(with $c=c(r)$). On the other hand, we have
\begin{eqnarray*}
\lefteqn{ (k(t) |u(1,t)|^{2} + \int^{1}_{0}|u_{t}|^{2}
+\mu (t) |u_{x}|^{2}dx )^{1+ \frac{1}{(1-r)(p+1)}} } \\
 & \leq& c F(0)^{\frac{1}{(1-r)(p+1)}}\Big( k(t) |u(1,t)|^{2}
 + \int^{1}_{0}|u_{t}|^{2} +\mu (t) |u_{x}|^{2}dx \Big).
\end{eqnarray*}
 From (\ref{4eq4-2}) and (\ref{4eq4-3}) and using the last two inequalities,
 we conclude that
\begin{eqnarray*}
\lefteqn{\frac{d}{dt}\mathcal{L} (t)}\\
  &\leq &  -\frac{c}{F(0)^{\frac{1}{(1-r)(p+1)}}}[( k(t) |u(1,t)|^{2} + \int^{1}_{0}|u_{t}|^{2} +\mu (t) |u_{x}|^{2}dx )^{1+\frac{1}{(1-r)(p+1)}}\\
&&+((-k')\Box u(1,t))^{1+\frac{1}{(1-r)(p+1)}}]+ c k^{2}(t) F(0).
 \end{eqnarray*}
Using (\ref{3eq3-6}) and the above inequality can be written as
\begin{equation}\label{4eq4-4}
\frac{d}{dt}\mathcal{L} (t) \leq -\frac{c}{\mathcal{L} (0)^{\frac{1}{(1-r)(p+1)}}} \mathcal{L} (t)^{1+ \frac{1}{(1-r)(p+1)}} + c k^{2}(t) F(0).
\end{equation}
Applying the Lemma \ref{Lem4.3} with $f=\mathcal{L}$, $\alpha=(1-r)(p+1)$
and $\beta=2p$ we have
\begin{equation}\label{4eq4-5}
\mathcal{L} (t) \leq \frac{c}{(1+t)^{\alpha}}\mathcal{L}(0).
\end{equation}
Choosing $r$ such that
$$
\frac{1}{p+1} < r < \frac{p}{p+1},
$$
and taking into account the inequality (\ref{4eq4-5}), we obtain
\begin{eqnarray}\label{4eq4-6}
&\int^{\infty}_{0}F(s) ds \leq c \int^{\infty}_{0} \mathcal{L} (s) ds \leq c
\mathcal{L} (0),& \\
\label{4eq4-7}
&t||u(.,t)||^{2}_{H^{1}(0,1)} \leq c t \mathcal{L} (t) \leq c \mathcal{L} (0),&\\
\label{4eq4-8}
&\int^{t}_{0}||u(.,s)||^{2}_{H^{1}(0,1)} \leq c \int^{\infty}_{0}
\mathcal{L} (t)dt \leq c \mathcal{L} (0).&
\end{eqnarray}
Estimates (\ref{4eq4-6})-(\ref{4eq4-8}) together with Lemma \ref{Lem4.2}
(case $r=0$) imply that
$$
(-k')^{1+\frac{1}{p+1}}\Box u(1,t) \ge \frac{c}{F(0)^{\frac{1}{p+1}}}((-k')
\Box u(1,t) )^{1+ \frac{1}{p+1}}.
$$
Using the same arguments as in the derivation of (\ref{4eq4-4}), we have
$$
\frac{d}{dt} \mathcal{L} (t)
\leq -\frac{c}{\mathcal{L} (0)^{\frac{1}{p+1}}}\mathcal{L}
(t)^{1+ \frac{1}{p+1}} + c k^{2}(t) F(0).
$$
Applying  Lemma \ref{Lem4.3}, we obtain
$$
\mathcal{L} (t) \leq \frac{c}{(1+t)^{p+1}} \mathcal{L}(0).
$$
Then from (\ref{3eq3-6}), we conclude
$$
F(t) \leq  \frac{c}{(1+t)^{p+1}}F(0),
$$
which completes the present proof. \quad Q.E.D.

\paragraph{Acknowledgements:}
The author expresses the appreciation to the referee for his valued suggestions which improved this paper.

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\noindent
{\sc Mauro L. Santos}\\
Departamento de Matem\'atica, Universidade Federal do Par\'a\\
Campus Universitario do Guam\'a \\
Rua Augusto Corr\^ea 01, Cep 66075-110, Par\'a, Brazil \\
e-mail: ls@ufpa.br

\end{document}