\documentclass[twoside]{article} \usepackage{amsfonts,latexsym} % font used for R in Real numbers \pagestyle{myheadings} \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} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode@=11 \@addtoreset{equation}{section} \catcode@=12 %\newcommand{\mathbb{R}}{\mathbb{R}} \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} &00$. It is not difficult to see that$\mathcal{L}(t)$satisfies $$\label{3eq3-6} q_{0}F(t) \leq \mathcal{L} (t) \leq q_{1}F(t),$$ 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 $$\label{4eq4-2} \int^{\infty}_{0}|k'(t)|^{r}dt < \infty \quad \mbox {if} \quad r> \frac{1}{p+1}.$$ 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 $$\label{4eq4-4} \frac{d}{dt}\mathcal{L} (t) \leq -\frac{c}{\mathcal{L} (0)^{\frac{1}{(1-r)(p+1)}}} \mathcal{L} (t)^{1+ \frac{1}{(1-r)(p+1)}} + c k^{2}(t) F(0).$$ Applying the Lemma \ref{Lem4.3} with$f=\mathcal{L}$,$\alpha=(1-r)(p+1)$and$\beta=2p$we have $$\label{4eq4-5} \mathcal{L} (t) \leq \frac{c}{(1+t)^{\alpha}}\mathcal{L}(0).$$ 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. \begin{thebibliography}{99} \frenchspacing \bibitem{C-D} C. 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