\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 53, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/53\hfil Nonlinear transmission problem] {Nonlinear transmission problem with a dissipative boundary condition of memory type} \author[D. Andrade, L. Harue Fatori, J. E. Mu\~noz Rivera \hfil EJDE-2006/53\hfilneg] {Doherty Andrade, Luci Harue Fatori, Jaime E. Mu\~noz Rivera} % in alphabetical order \address{Doherty Andrade \newline Maring\'a State University, Departament of Mathematics, 87020-900 - Maring\'a, Brazil} \email{doherty@uem.br} \address{Luci H. Fatori \newline Londrina State University, Departament of Mathematics, 86051-990 - Londrina, Brazil} \email{lucifatori@uel.br} \address{Jaime E. Mun\~oz Rivera \newline Laborat\'orio Nacional de Computa\c c\~ao Cient\'\i fica, Av. Get\'ulio Vargas, 333, 25651-070 - Petr\'opolis, Brazil} \email{rivera@lncc.br} \date{} \thanks{Submitted November 10, 2005. Published April 28, 2006.} \subjclass[2000]{35B40, 35L70} \keywords{Wave equation; asymptotic behavior; memory} \begin{abstract} We consider a differential equation that models a material consisting of two elastic components. One component is clamped while the other is in a viscoelastic fluid producing a dissipative mechanism on the boundary. So, we have a transmission problem with boundary damping condition of memory type. We prove the existence of a global solution and its uniformly decay to zero as time approaches infinity. More specifically, the solution decays exponentially provided the relaxation function decays exponentially. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \newcommand{\der}[2]{\frac{\partial #1}{\partial #2}} \section{Introduction} In this paper, we model the oscillation of a solid consisting of two elastic materials. We suppose that a part of the boundary is inside a viscoelastic fluid producing a dissipative mechanism of memory type while the other part of the boundary is clamped. The corresponding mathematical equations which model this situation is called a transmission problem with boundary dissipation. Boundary dissipation was studied for several authors, see for example, \cite{Gr 1, Sh 2, La10, Tu12, Co14, On15, Wy18, Ca28} and the references therein, all of them dealing with frictional damping. Models with memory dissipation are physically and mathematically more interesting, physically because our model follows the constitutive equations for materials with memory and Mathematically because the estimates we need to show the exponential decay are more delicate and depends on the relaxation function, see for example \cite{2} and the references therein. Memory dissipation is produced by the interaction of materials with memory. Such types of dissipation are subtle and their analysis are more delicate than the frictional damping, because introduce another type of technical difficulties. So, we have only a few works in this direction. In this work we show the existence of solutions of a nonlinear transmission problem with boundary dissipation of memory type. Moreover we will prove that under suitable conditions on the relaxation functions the solution will decay uniformly as time goes to infinity. The transmission problem considered here is \begin{gather} \rho_1 u_{tt}-\gamma_1 \Delta u + f(u)=0, \quad \mbox{in } \Omega_{1}\times]0,T[, \label{eq1.1} \\ \rho_2 v_{tt}-\gamma_2 \Delta v + g(v)=0, \quad \mbox{in } \Omega_{2}\times]0,T[,\label{eq1.2} \end{gather} with boundary condition \begin{equation} \label{eq1.3} u(x,t)+\int_0^t k(t-\tau) \frac{\partial u}{\partial \nu}d\tau =0 \quad\mbox{ on }\quad \Gamma \end{equation} and satisfying the transmission condition \begin{equation} \label{eq1.4} u=v,\quad\mbox{ and }\quad \gamma_1 \frac{\partial u}{\partial \nu} = \gamma_2 \frac{\partial v}{\partial \nu} \quad\mbox{on } \quad\Gamma_1. \end{equation} Additionally we assume that $v$ satisfies Dirichlet boundary condition over $\Gamma_2$, \begin{equation}\label{eq1.5} v(x,t)=0,\quad \mbox{on } \Gamma_2\times]0,T[, \end{equation} and verifies the initial conditions \begin{gather*} u(x,0)= u_0(x), \quad \mbox{and}\quad u_t(x,0)= u_1(x) \quad \mbox{in } \Omega_1\\ v(x,0)= v_0(x), \quad\mbox{and}\quad v_t(x,0)= v_1(x) \quad \mbox{in } \Omega_2. \end{gather*} \begin{figure}[htb] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(62,40)(0,10) \put(30,30){\circle{16}} \put(29,29){$\bullet$} \put(31,31){$x_0$} \put(37.5,29){$\Gamma_2$} \qbezier(12,30)(13,39)(30,40) \qbezier(12,30)(13,21)(30,20) \qbezier(30,40)(47,39)(48,30) \qbezier(30,20)(47,21)(48,30) \put(16,29){$\Omega_2$} \put(42.5,37.5){$\Gamma_1$} \qbezier(55.98,45)(48.61,55.76)(22.5,43) \qbezier(22.5,43)(-1.61,26.76)(4.02,15) \qbezier(4.02,15)(11.39,4.24)(37.5,17) \qbezier(37.5,17)(61.62,33.24)(55.98,45) \put(11,17){$\Omega_1$} \put(54,28){$\Gamma$} \end{picture} \end{center} \caption{The configuration\label{fig1}} \end{figure} The transmission problem \eqref{eq1.1}-\eqref{eq1.2} can be consider as a semilinear wave equation with discontinuous coefficients and discontinuous and non linear terms; that is, denoting \begin{gather*} U= \begin{cases} u(x), &\mbox{if } x \in \Omega_1\\ v(x), &\mbox{if } x \in \Omega_2,\end{cases} \quad \rho(x)= \begin{cases} \rho_1, &\mbox{if } x \in \Omega_1\\ \rho_2,&\mbox{if } x \in \Omega_2,\end{cases} \\ a(x)= \begin{cases} \gamma_1, &\mbox{if } x \in \Omega_1\\ \gamma_2, &\mbox{if } x \in \Omega_2,\end{cases} \quad F(x)=\begin{cases} f(x), &\mbox{if } x \in \Omega_1\\ g(x), &\mbox{if } x \in \Omega_2.\end{cases} \end{gather*} Note that \eqref{eq1.1}-\eqref{eq1.2} is equivalent to $$ \rho(x) U_{tt}- a(x) \Delta U+F(U)=0,\quad \mbox{ in } \Omega \times (0,T) $$ where $\Omega = \Omega_1 \times \Omega_2$. \section{Existence of solutions} \begin{lemma}\label{lemma2.1} For each function $\alpha\in C^{1}$ and each $\varphi \in W^{1,2}(0,T)$, we have \begin{equation} \int_{0}^{t}\alpha(t-\tau )\varphi (\tau )d\tau \varphi _{t} =-\frac{1}{2} \alpha(t)|\varphi (t)|^{2}+\frac{1}{2}\alpha'\Box \varphi -\frac{1}{2}\frac{d}{dt}\big\{ \alpha\Box \varphi -\Big( \int_{0}^{t}\alpha\Big) |\varphi |^{2}\big\}.\label{eq2.1} \end{equation} \end{lemma} Let $a$ be a function that satisfies \begin{equation}\label{eq2.2} k(0) a+ k' * a= -\frac{k'}{k(0)}. \end{equation} By $*$ we denote the convolution product; that is, $k*g(\cdot,t)=\int_{0}^{t}k(t-\tau )g(\cdot ,\tau )\,d\tau $. The function $a$ is called the resolvent kernel of $k$. Using the Volterra's resolvent, we have $$ \frac{\partial u}{\partial \nu}= -\frac{1}{k(0)}u_t-a*u_t $$ after performing an integration by parts, the above identity is equivalent to \begin{equation}\label{eq2.3} \frac{\partial u}{\partial \nu}= -\frac{1}{k(0)}u_t-a(0)u-a'*u+a(t)u_0. \end{equation} We assume the following hypotheses on $a$: \begin{gather} a(t)>0,\quad a'(t) <0,\quad a''(t)>0,\quad \forall t \geq 0 \label{eq2.4}\\ -c_0 a'(t) \leq a''(t)\leq -c_1 a'(t),\quad \forall t \geq 0,\label{eq2.5} \end{gather} where $c_i$ are positive constants. To facilitate our calculation we introduce the following notation \begin{gather} (\alpha \Box f) (t)= \int_0^t \alpha(t-\tau) \left| f(t) - f(\tau)\right|^2d\tau, \label{eq2.6}\\ (\alpha \diamondsuit f) (t)= \int_0^t g(t-\tau) \left[ f(t) - f(\tau)\right] d\tau .\label{eq2.7} \end{gather} It follows that \begin{equation} (\alpha *f) (t)= \Big(\int_0^t \alpha(s) ds\Big)f(t)-(\alpha \diamondsuit f) (t). \label{eq2.8} \end{equation} From hypothesis \eqref{eq2.2}, we know that the behavior of $a$ is similar to the behavior of $k$. We can find the following Lemma in \cite{jaime+racke}. \begin{lemma}\label{lemma2.2} If $b$ and $\alpha$ satisfy $b+\alpha=-b*\alpha$, then \begin{itemize} \item[(i)] Suppose that $|\alpha(t)|\leq c_\alpha{\rm e}^{-\gamma t}$, for all $t>0$, for some $\gamma >0$, and $c_\alpha>0$, then for any $0 <\varepsilon <\gamma$ and $c_\alpha< \gamma - \varepsilon$, we have $$ |b(t)| \leq \frac{c_\alpha(\gamma- \varepsilon)}{\gamma-\varepsilon-c_\alpha}{\rm e}^{-\varepsilon t}, \quad \forall t >0. $$ \item[(ii)] If $\alpha$ satisfies $ |\alpha(t)|\leq c_\alpha(1+t)^{-p}$, for some $p>1$, $c_\alpha>0$ and $$ \frac{1}{c_\alpha} > c_p:= \sup_{0\leq t<\infty}\int_0^t(1+t)^p (1+t-\tau)^{-p} (1+\tau)^{-p} d\tau, $$ then $$ |b(t)| \leq \frac{c_\alpha}{1-c_\alpha c_p}(1+t)^{-p}, \forall\, t> 0. $$ \end{itemize} \end{lemma} Let us introduce the following two vector spaces \begin{gather*} W= \{w\in H^1(\Omega_2): w(x)=0 \mbox{ on } \Gamma_2\},\\ V=\{ (u,v) \in H^1(\Omega_{1}) \times W : u=v \hbox{ on } \Gamma_{1} \}. \end{gather*} Let us consider $f, g \in C^1(\mathbb{R})$ satisfying \begin{gather} |f(s)| \leq C_1 |s|^\rho + C_2 \quad \hbox{and} \quad |g(s)|\leq C_1 |s|^\rho + C_2,\label{eq2.9}\\ |f'(s)| \leq C_1 |s|^{\rho -1} + C_2 \quad \hbox{and} \quad |g'(s)|\leq C_1 |s|^{\rho -1} + C_2 \label{eq2.10} , \end{gather} where $C_1$ and $C_2$ are positive constants. When the space dimension is $n\leq 2$, we use $1 \le \rho < \infty $, and when $n\geq 3$, we use $1\leq \rho\leq \frac{n}{n-2}$. We also assume that for $s\in \mathbb{R}$, \begin{equation} F(s)=\int_0^sf(\sigma)\,d\sigma \geq 0 \quad \mbox{and} \quad G(s)=\int_0^sg(\sigma)\,d\sigma \geq 0. \label{eq2.11} \end{equation} Let us introduce the definition of weak solution to system \eqref{eq1.1}--\eqref{eq1.5}. \begin{definition} \rm We say that the couple $(u,v)$ is a weak solution of \eqref{eq1.1}--\eqref{eq1.5} when $$ (u,v)\in L^\infty(0,T;V), \quad (u_t,v_t)\in L^\infty(0,T;L^2(\Omega_1)\times L^2(\Omega_2)), $$ and satisfies \begin{align*} &\int_0^T \int_{\Omega_{1}} [\rho_1 u \phi_{tt} + \gamma_1\nabla u \nabla \phi + f(u) \phi ]\, dx\,dt \\ & + \int_0^T \int_{\Omega_{2}} [ \rho_2 v \psi_{tt} +\gamma_2\nabla v \nabla \psi + g(v) \psi ]\, dx\,dt \\ &= \int_{\Omega_{1}} u_1 \phi(0) dx - \int_{\Omega_{1}} u_0 \phi_t(0) dx + \int_{\Omega_{2}} v_1 \psi(0) dx - \int_{\Omega_{2}} v_0 \psi_t(0) dx \\ &\quad -\int_\Gamma \left( \frac{1}{k(0)} u_t+ a(0)u+a'* u - a(t) u_0\right) \phi\,d\Gamma, \end{align*} for any $(\phi , \psi )\in C^2 (0,T; V)$ such that $$ \phi(T)=\phi_t (T) = \psi(T)= \psi_t(T)=0. $$ \end{definition} To show the existence of strong solutions we need a regularity result for the elliptic system associated with the problem \eqref{eq1.1}--\eqref{eq1.5}. For the reader's convenience we recall the following result whose proof can be found in the book by O. A. Ladyzhenskaya and N. N. Ural'tseva \cite[Theorem 16.2]{Lad}. \begin{lemma}\label{lemma2.4} For any given functions $F\in L^2 (\Omega_{1} )$, $G \in L^2 (\Omega_{2})$, $g \in H^{1/2}(\Gamma)$, $\gamma_1,\gamma_2 \in \mathbb{R}^{+}$, then there exists only one solution $(u,v)$, with $u \in H^2(\Omega_{1})$ and $v \in H^2(\Omega_{2})$, to the system \begin{gather*} - \gamma_1\Delta u = F \quad \mbox{in}\quad \Omega_{1},\\ - \gamma_2\Delta v = G \quad \mbox{in}\quad\Omega_{2},\\ v(x)= 0 \quad \mbox{on}\quad \Gamma_2\\ \frac{\partial u}{\partial \nu}= g, \mbox{ on } \Gamma, \\ u(x)=v(x) \quad \mbox{on } \Gamma_{1} \\ \gamma_1 \frac{\partial u}{\partial \nu} = \gamma_2\frac{\partial v}{\partial \nu} \quad \mbox{on } \Gamma_{1} \,. \end{gather*} \end{lemma} The existence result is summarized in the following theorem. \begin{theorem}\label{theorem2.5} Suppose that $f$ and $g$ are $C^1$-functions satisfying \eqref{eq2.9}--\eqref{eq2.11} and let us take initial data such that $$ (u_0,v_0)\in V , \quad (u_1,v_1) \in L^2(\Omega_{1})\times L^2(\Omega_{2}),\quad u_0=0 \mbox{on } \Gamma. $$ Then, there exists a solution $(u,v)$ of system \eqref{eq1.1}--\eqref{eq1.5}, such that $$ (u,v)\in C(0,T;V)\cap C^1(0,T;L^2(\Omega_{1})\times L^2(\Omega_{2})). $$ In addition, if the second-order regularity holds, that is, $(u_0,v_0)\in H^2(\Omega_{1})\times H^2(\Omega_{2})$ and $(u_1,v_1) \in V$, and \begin{gather*} u_2:=\frac{\gamma_1}{\sigma_1} \Delta u_0- f(u_0) \in L^2(\Omega_1)\\ v_2:=\frac{\gamma_2}{\sigma_2} \Delta u_0- g(v_0) \in L^2(\Omega_2), \end{gather*} satisfying the compatibility conditions \begin{gather*} \frac{\partial u_0}{\partial \nu}= - \frac{1}{k(0)}u_1- au_0\, \mbox{ on}\, \Gamma \\ u_0=v_0\quad\mbox{and}\quad \gamma_1\displaystyle\frac{\partial u}{\partial \nu}= \gamma_2\displaystyle\frac{\partial v}{\partial \nu}, \quad \mbox{on } \Gamma_1 \end{gather*} then there exists a strong solution satisfying $(u,v)$ in the space $$ C(0,T;H^2(\Omega_{1})\times H^2(\Omega_{2}))\cap C^1(0,T;V) \cap C^2(0,T;L^2(\Omega_{1})\times L^2(\Omega_{2})). $$ \end{theorem} \begin{proof} To show the existence of solutions we use the Galerkin method. Let $(\varphi_i, \omega_i)$, $i=1, \dots, \infty$ be a basis of $V$ and let us write $$ (u^m(t),v^m(t))= \sum_{i=1}^m h_i (t) (\varphi_i, \omega_i) , $$ where $u^m$ and $v^m$ satisfy \begin{equation} \label{eq2.12} \begin{aligned} &\int_{\Omega_{1}} \{\rho_1 u_{tt}^m \varphi_i + \gamma_1\nabla u^m \nabla \varphi_i + f(u^m) \varphi_i \} dx \\ &+ \int_{\Omega_{2}} \{\rho_2 v_{tt}^m \omega_i + \gamma_2\nabla v^m \nabla \omega_i + g(v^m) \omega_i \} dx \\ &= -\int_\Gamma \left(\frac{1}{k(0)} u_t^m+a(0)u^m+a'\ast u^m- a(t)u_0^m\right) \phi_i \,d\Gamma, \quad i=1,2,\dots,m. \end{aligned} \end{equation} This is a $m$-dimensional system of ODEs in $h_i(t)$ and has a local solution in $t$. With the estimates obtained below, we can extend $u^m$ and $v^m$ to the whole interval $[0,T]$. \subsection*{Weak Solutions} Multiplying the above equation by $h_i'(t)$ and summing up from $i=1$ to $m$, we have $$ \frac{d}{dt}E^m(t)=-\frac{1}{2k(0)}\int_\Gamma | u_t^m|^2 d \Gamma + \frac12a'(t)\int_\Gamma | u^m |^2 d\Gamma - \frac 12\int_\Gamma a'' \Box u^m d\Gamma, $$ where \begin{align*} E^m(t) &= \frac{1}{2} \int_{\Omega_{1} } \{ \rho_1| u_t^m |^2 + \gamma_1| \nabla u^m |^2 + 2 F(u^m) \} dx +a(t)\int_\Gamma | u|^2 d\Gamma -\int_\Gamma a' \Box u d\Gamma \\ &\quad + \frac{ 1}{2} \int_{\Omega_{2} } \{\rho_2| v_t^m |^2 + \gamma_2| \nabla v^m |^2 + 2 G(v^m) \} dx. \end{align*} Then we deduce that \begin{gather} (u^m ,v^m)\quad \hbox{is bounded in}\quad L^{\infty} (0,T;H^1(\Omega_{1})\times H^1(\Omega_{2})), \label{eq2.13} \\ (u_t^m ,v_t^m)\quad \hbox{is bounded in}\quad L^{\infty} (0,T;L^2(\Omega_{1})\times L^2(\Omega_{2})), \label{eq2.14} \end{gather} which imply that \begin{gather*} (u^m, v^m) \rightharpoonup (u,v) \quad \hbox{weakly $\star$ in } L^{\infty} (0,T;H^1(\Omega_{1})\times H^1(\Omega_2)),\\ (u_t^m,v_t^m) \rightharpoonup (u_t,v_t) \quad \hbox{weakly $\star$ in } L^{\infty} (0,T;L^2(\Omega_{1})\times L^2(\Omega_{2})). \end{gather*} Application of the Lions-Aubin's Lemma \cite[Theorem 5.1]{Lions2}, we have \[ (u^m,v^m) \rightarrow (u,v) \quad \hbox{strongly in } L^{2} (0,T;L^2(\Omega_{1})\times L^2(\Omega_{2})), \] and consequently \begin{gather*} u^m \rightarrow u \quad \hbox{a.e. in} \; \Omega_{1} \quad \hbox{and} \quad f(u^m) \rightarrow f(u) \quad \hbox{a.e. in} \; \Omega_{1}, \\ v^m \rightarrow v \quad \hbox{a.e. in} \; \Omega_{2} \quad \hbox{and} \quad g(v^m) \rightarrow g(v) \quad \hbox{a.e. in} \; \Omega_{2}. \end{gather*} From the growth condition \eqref{eq2.9}, we have \begin{gather*} f(u^m) \quad \hbox{is bounded in } L^{\infty} (0,T;L^{2}(\Omega_{1})),\\ g(v^m) \quad \hbox{is bounded in } L^{\infty} (0,T;L^{2}(\Omega_{2})); \end{gather*} therefore, \begin{gather*} f(u^m) \rightharpoonup f(u) \quad \mbox{weakly in } L^{2} (0,T;L^{2}(\Omega_{1})), \\ g(v^m) \rightharpoonup g(v)\quad \mbox{weakly in } L^{2} (0,T;L^{2}(\Omega_{2})). \end{gather*} The rest of the proof of the existence of weak solution is a matter of routine. \subsection*{Strong Solutions} To show the regularity we take a basis of such that $(u_0,v_0)$ and $(u_1,v_1)$ are in $B=\{(\phi_i,w_i), i \in \mathbb{N}\}$. Therefore, $$ u_0^m=u_0,\quad v_0^m=v_0,\quad u_1^m=u_1,\quad v_1^m=v_1, \quad \forall m. $$ Differentiate the approximate equation and multiply by $h_{i}''(t)$. Using a similar argument as before, we obtain \begin{equation}\label{eq2.15} \frac{d}{dt}E_2^m(t) \leq \int_{\Omega_{1} } | f'(u^m)| u_t^m u_{tt}^m dx+ \int_{\Omega_{2} } | g'(v^m)| v_t^m v_{tt}^m dx , \end{equation} where \begin{equation}\label{eq2.16} \begin{aligned} E_2^m(t)&=\frac{1}{2}\int_{\Omega_{1} } \rho_1| u_{tt}^m |^2 + \gamma_1| \nabla u^m_t |^2 \,dx + \frac{1}{2} \int_{\Omega_{2} } \rho_2| v_{tt}^m |^2 + \gamma_2| \nabla v^m_t |^2 \,dx \\ &\quad +\frac{1}{2} a(t)\int_\Gamma | u_t^m |^2 \,d\Gamma + \frac{1}{2}\int_\Gamma a' \Box u_t^m \,d\Gamma. \end{aligned} \end{equation} Note that $E^m_2(0)$ is bounded, in fact is constant, because of our choice of the basis. Let us estimate the right hand side of \eqref{eq2.16}. From \eqref{eq2.10} we have \begin{align*} &\int_{\Omega_1} | f'(u^m)u^m_tu_{tt}^m | \, dx \\ &\leq \frac{C_1}{2} \int_{\Omega_1} | u^m |^{2(\rho-1)}| u^m_t |^2 \,dx + \frac{C_2}{2} \int_{\Omega_1} | u^m_t |^{2}\,dx + \frac{C_1+C_2}{2} \int_{\Omega_1} | u^m_{tt} |^{2}\,dx . \end{align*} But since $(\rho-1) \leq 2/(n-2)$ and $ \frac{1}{r} + \frac{1}{s} =1$ with $ r= n/2 $ and $s=n/(n-2)$, we see that $$ \int_{\Omega_1} | u^m |^{2(\rho -1)}| u^m_t |^2 \,dx \leq \Big( \int_{\Omega_1} | u^m |^{2^{\ast}} dx \Big)^{1/r} \Big( \int_{\Omega_1} | u^m_t |^{2^{\ast}} dx \Big)^{1/s}, $$ where $2^{\ast} = 2n/(n-2)$. Then from Sobolev imbeddings and \eqref{eq2.13} there exists a constant $C>0$ such that $$ \int_{\Omega_1} | u^m |^{2(\rho-1)}| u^m_t |^2 \,dx \leq C + C \int_{\Omega_1} | \nabla u^m_t |^{2} \, dx. $$ It follows that $$ \int_{\Omega_1} | f'(u^m)u^m_tu_{tt}^m | \, dx \leq C + C \int_{\Omega_1} \{ | u^m_{tt} |^{2} + | \nabla u^m_t |^2 \}\,dx , $$ and similarly $$ \int_{\Omega_2} | g'(v^m) v^m_t v_{tt}^m | \, dx \leq C + C \int_{\Omega_2} \{ | v^m_{tt} |^{2} + | \nabla v^m_t |^2 \}\,dx . $$ Hence, from \eqref{eq2.15} and the Gronwall inequality we conclude that \begin{gather*} (u_t^m ,v_t^m)\quad \hbox{is bounded in}\quad L^{\infty} (0,T;H^1(\Omega_{1})\times H^1(\Omega_{2})), \\ (u_{tt}^m ,v_{tt}^m)\quad \hbox{is bounded in}\quad L^{\infty} (0,T;L^2(\Omega_{1})\times L^2(\Omega_{2})), \end{gather*} which imply that \begin{gather*} (u^m_t,v^m_t) \rightharpoonup (u_t,v_t) \quad \hbox{weakly $\ast$ in} \quad L^{\infty} (0,T;H^1(\Omega_{1})\times H^1(\Omega_{2})),\\ (u_{tt}^m, v_{tt}^m) \rightharpoonup (u_{tt},v_{tt}) \quad \hbox{weakly $\ast$ in} \quad L^{\infty} (0,T;L^2(\Omega_{1})\times L^2(\Omega_{2})). \end{gather*} Therefore, $(u,v)$ satisfies \eqref{eq1.1}-\eqref{eq1.5}. Moreover $$ \frac{\partial u}{\partial \nu}=- \frac{1}{k(0)} u_t - a(0)u - a' \ast u + a(t) u_0. $$ Integrating by parts, $$ \frac{\partial u}{\partial \nu}= -\frac{1}{k(0)}u_t-a\ast u_t. $$ Since $u_t$ is bounded in $H^1(\Omega_1)$, $\frac{\partial u}{\partial \nu} \in H^{\frac{1}{2}}(\Gamma)$. So we have \begin{gather*} - \gamma_1\Delta u = u_{tt}- f(u) \in L^2(\Omega_1),\\ - \gamma_2\Delta v = v_{tt}- g(v) \in L^2(\Omega_2)\\ u=v \quad\mbox{and} \quad \gamma_1 \frac{\partial u}{\partial \nu}= \gamma_2\frac{\partial v}{\partial \nu}\quad \mbox{in } \Gamma_1\\ v=0 \quad \mbox{in } \Gamma_2,\quad \frac{\partial u}{\partial \nu}\in H^{1/2}(\Gamma). \end{gather*} Then using Lemma \ref{lemma2.4} we have the required regularity to $(u,v)$. \end{proof} \section{Asymptotic behavior} In this section we prove that the solution decay exponentially as time approaches infinity. First, we need some preliminaries results. \begin{lemma}\label{lemma3.1} Suppose that the initial data satisfies the second order regularity as in Theorem \ref{theorem2.5}, then \[ \frac{d}{dt}E(t)=-\frac{1}{k(0)}\int_{\Gamma }|u_{t}|^{2}\,d\Gamma +\frac{a'(t)}{2}\int_{\Gamma }|u|^{2}\,d\Gamma -\frac{1}{2}\int_{\Gamma }a^{\prime \prime }\Box u\,d\Gamma, \] where \begin{equation} \begin{aligned} E(t)&=\frac{1}{2}\int_{\Omega _{1}}\rho _{1}|u_{t}|^{2}+\gamma _{1}|\nabla u|^{2}+2F(u)dx+\gamma_{1}\int_{\Gamma }a(t)|u|^{2}\, -a'\Box u \, d\Gamma \\ &\quad +\frac{1}{2}\int_{\Omega _{2}}\rho _{2}|v_{t}|^{2}+\gamma _{2}|\nabla v|^{2}+2G(v)dx . \end{aligned}\label{eq3.1} \end{equation} \end{lemma} \begin{proof} Multiply by $u_t$ equation \eqref{eq1.1} and by $v_t$ equation \eqref{eq1.2}, summing up and using identity \eqref{eq2.3} and Lemma \ref{lemma2.1} we get the result. \end{proof} Let $f$ and $g$ be such that \begin{gather}%\label{hipFG} 0\leq F(s):=\int_0^s f(t)dt \leq \frac{1}{m+1} s f(s),\label{eq3.2}\\ 0\leq G(s):=\int_0^s g(t)dt\leq \frac{1}{l+1} s g(s),\label{eq3.3}\\ F(s) \leq G(s)\label{eq3.4} \end{gather} where $l,m>1$. Note that odd polynomials satisfy \eqref{eq3.2}-\eqref{eq3.3}. Let \begin{equation}\label{eq3.5} \delta < \min\{\frac{l-1}{l+1}n, \frac{m-1}{m+1}n,1\} \end{equation} and $$ J_0(t)= \int_{\Omega_1} \rho_1 u_t q \cdot \nabla u \,dx + \int_{\Omega_2}\rho_2 v_t q \cdot \nabla v \,dx. $$ \begin{lemma}\label{lemma3.2} Under the hypothesis of Lemma \ref{lemma3.1}, consider $q(x)=x-x_0\in C^{1}(\overline{\Omega })$, $\gamma_1> \gamma_2$ and $\rho_1>\rho_2$. Then any strong solution of \eqref{eq1.1}--\eqref{eq1.5} satisfies \begin{align*} \frac{d}{dt}J_0(t) &\leq \gamma_1 \int_\Gamma \frac{\partial u}{\partial \nu} q \cdot \nabla u \,dx- \frac{\gamma_1}{2}\int_\Gamma q \cdot \nu | \nabla u |^2\,dx + \frac{\rho_1}{2} \int_\Gamma q \cdot \nu | u_t |^2 \,d\Gamma\\ &\quad - \frac n2\int_{\Omega_1}\rho_1 | u_t|^2 - \gamma_1 | u |^2 dx+ n \int_{\Omega_1}F(u)dx-\gamma_1\int_{\Omega_1} | \nabla u |^2\,dx\\ &\quad -\frac n2\int_{\Omega_2}\rho_2 | v_t|^2 - \gamma_2 | \nabla v |^2 \,dx+ n \int_{\Omega_2} G(v)\,dx- \gamma_2\int_{\Omega_2}| \nabla v|^2 dx. \end{align*} \end{lemma} \begin{proof} Using equation \eqref{eq1.1}, \begin{align*} &\frac{d}{dt}\int_{\Omega_1} \rho_1 u_t q_k \frac{\partial u}{\partial x_k} dx \\ &= \int_{\Omega_1}\rho_1 u_{tt}q_k \frac{\partial u}{\partial x_k}dx+ \int_{\Omega_1}\rho_1 u_t q_k \frac{\partial u_t }{\partial x_k} dx \\ &=\int_{\Omega_1} \gamma_1 \Delta u q_k \frac{\partial u}{\partial x_k}dx- \int_{\Omega_1}f(u) q_k \frac{\partial u}{\partial x_k}dx+ \frac{\rho_1}{2} \int_{\Omega_1} q_k \frac{\partial | u_t|^2}{\partial x_k}dx \\ &= \gamma_1\int_{\partial \Omega_{1}}\frac{\partial u}{\partial \nu}q_k \frac{ \partial u}{\partial x_k}d\Gamma- \frac{\gamma_1}{2}\int_{\partial {\Omega_{1}}} q_k \nu_k | \nabla u |^2 d\Gamma + \frac{\gamma_1}{2} \int_{ \Omega_1} \frac{\partial q_k}{\partial x_k} | \nabla u |^2 dx\\ &\quad - \int_{\partial \Omega_{1}}F(u) q_k\nu_k d\Gamma+ \int_{\Omega_1}F(u)\frac{ \partial q_k}{\partial x_k} dx + \frac{\rho_1}{2} \int_{\partial \Omega_{1}} q_k\nu_k| u_t|^2 d\Gamma \\ &\quad - \frac{\rho_1}{2} \int_{\Omega_{1}} \frac{\partial q_k}{\partial x_k} | u_t|^2 dx - \gamma_1 \int_{\Omega_{1}} \nabla u \cdot \nabla q_k \frac{\partial u}{\partial x_k} dx. \end{align*} So we have, \begin{equation}\label{eq3.6} \begin{aligned} \frac{d}{dt}\int_{\Omega _{1}}\rho _{1}u_{t}q_{k}\frac{\partial u}{\partial x_{k}}dx &=\gamma _{1}\int_{\partial \Omega_1 }\frac{\partial u}{\partial \nu }q_{k} \frac{\partial u}{\partial x_{k}}d\Gamma-\frac{\gamma_1}{2}\int_{\partial \Omega_1 }q_{k} \nu_{k}|\nabla u|^{2}d\Gamma \\ &\quad -\int_{\partial \Omega_1 }q_k \nu_k F(u)d\Gamma +\frac{\rho _{1}}{2}\int_{\partial \Omega_1 }q_k \nu_k |u_{t}|^{2}d\Gamma \\ &\quad -\frac{1}{2}\int_{\Omega _{1}}\frac{\partial q_{k}}{\partial x_{k}}\left\{ \rho _{1}|u_{t}|^{2}-\gamma _{1}|\nabla u|^{2}\right\} dx\\ &\quad +\int_{\Omega _{1}}F(u)\frac{\partial q_k}{\partial x_k}dx -\gamma _{1}\int_{\Omega _{1}}\nabla u\cdot\nabla q_{k}\frac{\partial u}{\partial x_{k}}dx. \end{aligned} \end{equation} Similarly using equation \eqref{eq1.2}, we obtain \begin{equation}\label{eq3.7} \begin{aligned} \frac{d}{dt}\int_{\Omega _{2}}\rho _{2}v_{t}q_{k} \frac{\partial v}{\partial x_{k}} dx &=-\gamma _{2}\int_{\partial\Omega_2}\frac{\partial v}{\partial \nu }q_{k} \frac{\partial v}{\partial x_{k}}d\Gamma+\frac{\gamma_{2}}{2}\int_{\partial\Omega_2}q_k \nu_k |\nabla v|^{2}d\Gamma \\ &\quad +\int_{\Gamma _{1}}q_k \nu_k G(v)d\Gamma -\frac{\rho_2}{2}\int_{\Gamma_{1}}q_k \nu_k|v_{t}|^{2}d\Gamma \\ &\quad -\frac{1}{2}\int_{\Omega _{2}}\frac{\partial q_{k}}{\partial x_{k}}\left\{ \rho _{2}|v_{t}|^{2}-\gamma _{2}|\nabla v|^{2}\right\} dx \\ &\quad +\int_{\Omega _{2}}\frac{\partial q_{k}}{\partial x_{k}}G(v)dx-\gamma _{2}\int_{\Omega _{2}}\nabla v\cdot \nabla q_{k}\frac{\partial v}{\partial x_{k}}dx. \end{aligned} \end{equation} Using that $\nabla u = \frac{\partial u}{\partial \nu}\nu + \nabla_\tau u$ and $v=0$ on $\Gamma_2$ we have from \eqref{eq3.6} and \eqref{eq3.7} that \begin{align*} &\frac{d}{dt}J_0(t)\\ &= \gamma_1 \int_\Gamma \frac{\partial u}{\partial \nu}q \cdot \nabla u \, d\Gamma - \frac{\gamma_1}{2}\int_{\Gamma}q \cdot \nu | \nabla u |^2\,d \Gamma + \frac{\rho_1}{2} \int_{\Gamma_1} q \cdot \nu | u_t |^2 \, d \Gamma \\ &\quad + \frac{\gamma_1}{2}\int_{\Gamma_1}\frac{\partial u}{\partial \nu } q \cdot \nabla_\tau u\, d\Gamma - \frac{\gamma_1}{2}\int_{\Gamma_1} q\cdot \nu | \nabla_\tau u |^2 \, d \Gamma+ \frac{\rho_1}{2}\int_\Gamma q \cdot \nu | u_t |^2\, d\Gamma\\ &\quad - \int_{\partial \Omega_1} q \cdot \nu F(u) \, d\Gamma- \frac{\gamma_2}{2}\int_{\Gamma_1} q \cdot \nu | \frac{\partial v}{\partial \nu}|^2 \, d\Gamma\\ &\quad - \gamma_2 \int_{\Gamma_1}\frac{\partial v}{\partial \nu} q\cdot \nabla_\tau v \, dx + \frac{\gamma_2}{2}\int_{\Gamma_1} q \cdot \nu | \nabla_\tau v |^2 \,d\Gamma\\ &\quad +\int_{\Gamma_1} q \cdot \nu G(v) \, d\Gamma +\frac{\gamma_1}{2}\int_{\Gamma_1}q\cdot \nu | \frac{\partial u}{\partial \nu}|^2 \,d\Gamma\\ &\quad -\frac{\rho_2}{2}\int_{\Gamma_1} | v_t |^2 q \cdot \nu \,d\Gamma+ \frac{\gamma_2}{2}\int_{\Gamma_2} q\cdot \nu | \frac{\partial v}{\partial \nu}|^2 d\Gamma -\frac{1}{2}\int_{\Omega_1}\frac{\partial q_k}{\partial x_k}(\rho_1 | u_t |^2 - \gamma_1 | \nabla u |^2)\, dx\\ &\quad + \int_{\Omega_1} F(u) \frac{\partial q_k}{\partial x_k} dx-\gamma_1\int_{\Omega_1} \nabla u \cdot \nabla q_k \frac{\partial u}{\partial x_k} dx - \frac12\int_{\Omega_2}\frac{\partial q_k}{\partial x_k}(\gamma_2 | v_t|^2 - \rho_2| \nabla v |^2) dx\\ &\quad +\int_{\Omega_2}G(v) \frac{\partial q_k}{\partial x_k} dx -\gamma_2\int_{\Omega_2}\nabla v \cdot \nabla q_k \frac{\partial v}{\partial x_k}\,dx. \end{align*} Since $u=v$ in $\Gamma_1$ then $\nabla_\tau u= \nabla_\tau v$ in $\Gamma_1$; therefore, \begin{align*} &\frac{d}{dt}J_0(t)\\ &= \gamma_1 \int_\Gamma \frac{\partial u}{\partial \nu}q \cdot \nabla u \, d\Gamma - \frac{\gamma_1}{2}\int_{\Gamma}q \cdot \nu | \nabla u |^2\,d \Gamma + \frac{\rho_1}{2} \int_\Gamma q \cdot \nu | u_t |^2 \, d \Gamma \\ &\quad + \frac{\gamma_1}{2}\big(\frac{\gamma_1-\gamma_2}{\gamma_2}\big) \int_{\Gamma_1}|\frac{\partial u}{\partial \nu }|^2 q \cdot \nu\, d\Gamma - \big(\frac{\gamma_1-\gamma_2}{2}\big)\int_{\Gamma_1} q\cdot \nu | \nabla_\tau u |^2 \, d \Gamma\\ &\quad + \big(\frac{\rho_1-\rho_2}{2}\big)\int_{\Gamma_1} q \cdot \nu | u_t |^2\, d\Gamma- \int_{\Gamma} q \cdot \nu F(u) \, d\Gamma\\ &\quad - \int_{\Gamma_1}q\cdot \nu [ F(u)-G(u) ]\,d\Gamma +\frac{\gamma_2}{2}\int_{\Gamma_2} q \cdot \nu | \frac{\partial v}{\partial \nu}|^2 d\Gamma\\ &\quad -\frac{n}{2}\int_{\Omega_1}\rho_1 | u_t |^2 - \gamma_1 | \nabla u |^2\, dx+ n\int_{\Omega_1} F(u) dx\\ &\quad -\gamma_1\int_{\Omega_1} |\nabla u|^2\,dx- \frac{n}{2}\int_{\Omega_2}\rho_2 | v_t |^2 - \gamma_2 | \nabla v|^2dx +n\int_{\Omega_2} G(v) dx- \gamma_2\int_{\Omega_2} | \nabla v |^2 dx. \end{align*} Using that $(x-x_0)\cdot \nu >0$ in $\Gamma$ then we conclude our proof. \end{proof} \begin{lemma}\label{lemma3.3} Under the hypothesis in Lemma \ref{lemma3.1}, \begin{align*} &\frac{d}{dt}\big\{ \int_{\Omega_{1}}\rho _{1}uu_{t}dx +\int_{\Omega _{2}}\rho _{2}v_{t}vdx\big\}\\ &=\int_{\Omega _{1}}\rho _{1}| u_{t}| ^{2}-\gamma _{1}|\nabla u|^{2}dx +\gamma_1\int_{\Gamma}\frac{\partial u}{\partial \nu}ud\Gamma\\ &\quad -\int_{\Omega_1}f(u)u dx +\int_{\Omega_ 2}\rho _{2}| v_{t}| ^{2}-\gamma_{2}|\nabla v|^{2}dx -\int_{\Omega_2}g(v)v dx . \end{align*} \end{lemma} \begin{proof} Multiply \eqref{eq1.1} by $u$ and \eqref{eq1.2} by $v$ and summing up the product the our result follows. \end{proof} Let us define the functional \[ \Phi (t)=J_0(t)+\big( \frac{n-\delta}{2}\big) \big[\int_{\Omega _{1}}\rho _{1}uu_{t}dx+\int_{\Omega _{2}}\rho _{2}v_{t}vdx\big] \] where we consider $q(x)=x-x_0$ as before. \begin{lemma}\label{lemma3.4} Under the hypotheses of Lemmas \ref{lemma3.1} and \ref{lemma3.2}, there exists a positive constant $\delta_0$ such that \[ \frac{d}{dt} \Phi(t) \leq C\int_\Gamma \big|\frac{ \partial u} {\partial \nu}\big|^2 d \Gamma + \big(\frac{n-\delta}{2}\big)\gamma_1\int_\Gamma u \frac{\partial u}{\partial \nu }d\Gamma - \delta_{0} E_{0}(t) +\frac{\rho_1}{2}\int_{\Gamma}q\cdot\nu |u_t|^2d\Gamma, \] where $$ E_0(t)= \frac12\int_{\Omega_1} \rho_1 | u_t |^2 + \gamma_1 | \nabla u |^2 + F(u) dx + \frac12\int_{\Omega_2} \rho_2 | v_t |^2 + \gamma_2 | \nabla v |^2 +G(v) dx. $$ \end{lemma} \begin{proof} From Lemma \ref{lemma3.2} and Lemma \ref{lemma3.3} we have, \begin{align*} &\frac{d}{dt}\Phi(t) \\ &\leq C\int_\Gamma \big| \frac{\partial u}{\partial \nu}\big|^2 d\Gamma -\frac{\delta}{2} \Big[\int_{\Omega_1} \rho_1 | u_t |^2 + \gamma_1 | \nabla u |^2 dx + \int_{\Omega_2} \rho_2 | v_t |^2 + \gamma_2 | \nabla v |^2 dx\Big]\\ &\quad +n\int_{\Omega_1}F(u) dx\ + n\int_{\Omega_2}G(v) dx -\big(\frac{n-\delta}{2}\big)\int_{\Omega_1} f(u) u dx \\ &\quad -\big(\frac{n-\delta}{2}\big)\int_{\Omega_2} g(v) v dx +\big(\frac{n-\delta}{2}\big)\gamma_{1}\int_{\Gamma}\frac{\partial u}{\partial \nu}u d\Gamma +\frac{\rho_1}{2}\int_{\Gamma}q\cdot\nu|u_t|^2d\Gamma. \end{align*} Using the hypotheses on $F$ and $G$, we obtain \begin{align*} n \int_{\Omega_1} F(u) dx - \frac{n-\delta}{2}\int_{\Omega_1}f(u)u dx&\leq \Big( \frac{n}{m+1}- \frac{n-\delta}{2}\Big)\int_{\Omega_1}f(u)u dx\\ &\leq -\alpha \int_{\Omega_1}f(u)u dx, \end{align*} where by our assumption on $\delta$ we have that $\alpha>0$. Similarly \[ n \int_{\Omega_2} G(v) dx - \frac{n-\delta}{2}\int_{\Omega_2}g(v)v dx \leq -\beta \int_{\Omega_2}g(v)v dx. \] From where it follows that \begin{align*} \frac{d}{dt}\Phi(t) &\leq \gamma_1\int_{\Gamma} |\frac{\partial u}{\partial \nu}|^2 d\Gamma+ \big(\frac{n-\delta}{2}\big)\gamma_1\int_{\Gamma} u \frac{\partial u }{\partial \nu}d\Gamma\\ &\quad - \frac{\delta}{2} \Big[\int_{\Omega_1} \rho_1 | u |^2 + \gamma_1 | \nabla u |^2 dx + \int_{\Omega_2} \rho_2 | v_t |^2 + \gamma_2 | \nabla v |^2 dx\Big]\\ &\quad -\alpha\int_{\Omega_1} u f(u) dx - \beta \int_{\Omega_2} v g(v) dx +\frac{\rho_1}{2}\int_{\Gamma}q\cdot\nu|u_t|^2d\Gamma, \end{align*} which implies that for $\delta_0=\min \left\{\frac{\delta}{2}, \alpha(m+1), \beta(l+1)\right\}$, we have \[ \frac{d}{dt}\Phi(t) \leq \gamma_1\int_{\Gamma}|\frac{\partial u}{\partial \nu}|^2 d\Gamma + \left(\frac{n-\delta}{2}\right)\gamma_1\int_{\Gamma} u \frac{\partial u }{\partial \nu}d\Gamma - \delta_0 E_0(t) +\frac{\rho_1}{2}\int_{\Gamma}q\cdot\nu|u_t|^2d\Gamma . \] \end{proof} \begin{theorem}\label{theorem3.5} With hypotheses in Lemma \ref{lemma3.2}, there exists a positive constants such that any strong solution satisfies $$ E(t)\leq C E(0) \exp(-\delta_1 t), $$ provided \eqref{eq2.4}-\eqref{eq2.5} holds. \end{theorem} \begin{proof} Note that from \eqref{eq1.3} and \eqref{eq2.8} we have $$ \frac{\partial u}{\partial \nu}= - \frac{1}{k(0)}u_t- a(t) u - a'\diamondsuit u $$ from where it follows $$ | \frac{\partial u}{\partial \nu}|^2\leq 2\big\{\frac{1}{k^2(0)} | u_t|^2 + a^2(t)| u |^2 + | a' \diamondsuit u |^2 \big\}. $$ Since $$ | a' \diamondsuit u |^2 = \Big| \int_0^t a' (t-s) \{ u(s)-u(t) \} ds \Big|^2 \leq \Big(\int_0^t |a'(t-s)| ds\Big)| a'| \Box u. $$ From this inequality and \eqref{eq2.5} it follows that \begin{equation}\label{eq3.8} \left | \frac{\partial u }{\partial \nu}\right|^2 \leq k_0 \{| u_t|^2 + a(t)| u |^2 + a' \Box u\}. \end{equation} On the other hand, \begin{equation} \label{eq3.9} \begin{aligned} \big| \int_\Gamma u \frac{\partial u }{\partial \nu }d\Gamma\big| &\leq \Big( \int_\Gamma | u|^2d\Gamma\Big)^{1/2}\Big( \int_\Gamma \big| \frac{\partial u}{\partial \nu }\big|^2 d\Gamma \Big)^{1/2}\\ &\leq \delta_1\int_\Gamma |u|^2 d\Gamma + C_{\delta_1}\int_{\Gamma}\{|u_t|^2 + a(t)| u |^2 + a' \Box u\}d\Gamma\\ &\leq \delta_1\int_\Gamma | u |^2 d\Gamma+ C\int_{\Gamma}\{| u_t|^2 + | u |^2 + a' \Box u\}d\Gamma. \end{aligned} \end{equation} Since $$\int_\Gamma | u |^2 \,d\Gamma \leq C\int_\Omega | \nabla u |^2 + | \nabla v |^2 \,dx,$$ we have that $\mathcal{L}(t) = NE(t) + \Phi(t)$ satisfies \begin{align*} \frac{d}{dt}\mathcal{L}(t) &\leq -\frac{N\gamma_1 }{k(0)}\int_\Gamma | u_t |^2d\Gamma +\frac{ N\gamma_{1}a'(t)}{2}\int_\Gamma | u |^2 d\Gamma -\frac{N\gamma_1 }{2}\int_\Gamma a''\Box u d\Gamma\\ &\quad + C\int_\Gamma \big| \frac{\partial u }{\partial \nu}\big|^2 d\Gamma + \big(\frac{n-\delta}{2}\big)\gamma_1 \int_\Gamma u \frac{\partial u}{\partial \nu }d\Gamma \\ &\quad - \frac{\delta_0}{2}E_{0}(t) +\rho_1\int_\Gamma q\cdot\nu |u_t|^2 d\Gamma. \end{align*} Using \eqref{eq4.2} and \eqref{eq4.3} we conclude that \begin{equation} \begin{gathered} \frac{d}{dt}\mathcal{L}(t)\leq -\big(\frac{N\gamma_1 }{k(0)}-C_{2}\big)\int_\Gamma | u_t |^2d\Gamma -\big(\frac{N\gamma_1 }{2}-C_{2}\big) \int_\Gamma a''\Box u d\Gamma- \frac{\delta_0}{2}E_{0}(t) \\ \frac{d}{dt}\mathcal{L}(t)\leq -\frac{\delta_0}{2}E(t) \leq -c \mathcal{L}(t). \end{gathered}\label{eq3.10} \end{equation} from where our conclusion follows. \end{proof} We remark that standard density arguments, the above result is also valid for weak solutions. \section{Polynomial rate of decay} Here our attention turns to the uniform rate of decay when $k$ decays polynomially as $(1+t)^{-p}$. In this case we will show that the solution also decays polynomially with the same rate. Let us consider the following hypotheses: \begin{equation}\label{eq4.1} \begin{gathered} 0 < a(t) \leq b_{0} (1+t)^{-p}, \\ -b_{1}a^{1+\frac{1}{p}}(t)\leq a'(t) \leq -b_{2}a^{1+\frac{1}{p}}(t), \\ b_{3}[-a'(t)]^{1+\frac{1}{p+1}}\leq a''(t) \leq b_{4}[-a'(t)]^{1+\frac{1}{p+1}}, \end{gathered} \end{equation} where $p>1$ and $b_{i}>0$ for $i=0,\dots,4$. The following lemmas will play an important role in the sequel. \begin{lemma}\label{lemma4.1} Let $m$ and $h$ be integrable functions, $0 \leq r < 1$ and $q>0$. Then, for $t \ge 0$, \begin{align*} &\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)^{q/(q+1)} \Big(\int^{t}_{0}|m(t-s)|^{r}|h(s)|ds\Big)^{1/(q+1)}. \end{align*} \end{lemma} \begin{proof} Let \[ v(s):=|m(t-s)|^{1- \frac{r}{q+1}} |h(s)|^{\frac{q}{q+1}}, \quad w(s):=|m(t-s)|^{\frac{r}{q+1}} |h(s)|^{\frac{1}{q+1}}. \] Applying H\"{o}lder's inequality to $|m(s)h(s)|=v(s)w(s)$ with exponents $\delta= q/(q+1)$ for $v$ and $\delta^{*}=q+1$ for $w$ our conclusion follows. \end{proof} \begin{lemma}\label{lemma4.2} Let $\phi\in L^\infty(0,T;L^2(\Gamma))$. Then, for $p>1$, $0 \leq r < 1$ and $t\ge0$, we have \begin{align*} &\big(\int_{\Gamma }|a'| \Box \phi d \Gamma \Big)^{\frac{1+(1-r)(p+1)}{(1-r)(p+1)}}\\ &\leq 2 \Big(\int^{t}_{0}|a'(s)|^{r}ds \|\phi \|^{2}_{L^{\infty}(0,t; L^{2}(\Gamma ))}\Big)^ {\frac{1}{(1-r)(p+1)}} \int_{\Gamma }|a'|^{1+ \frac{1}{p+1}} \Box \phi \,d \Gamma , \end{align*} while for $r=0$ we get \begin{align*} & \Big(\int_{\Gamma{1}} |a'| \Box \phi d \Gamma \Big)^{\frac{p+2}{p+1}}\\ & \leq 2 \Big(\int^{t}_{0}\|\phi (s,.)\|^{2}_{L^{2}(\Gamma )} ds + t \|\phi (s,.)\|^{2}_{L^{2}(\Gamma )}\Big)^{p+1} \int_{\Gamma}|a'|^{1+ \frac{1}{p+1}} \Box \phi \,d \Gamma. \end{align*} \end{lemma} \begin{proof} The above inequalities are a immediate consequence of Lemma \ref{lemma4.1} taking \[ m(s) : = |a'(s)| , \quad h(s) : = \int_{\Gamma }|\phi(t,x) - \phi_(s,x)|^{2} d \Gamma , \quad q: = (1-r)(p+1). \] This concludes our assertion. \end{proof} \begin{theorem}\label{theorem4.4} With the hypotheses in Lemma \ref{lemma3.1} and Lemma \ref{lemma3.2}, if the resolvent kernel $a(t)$ satisfies condition \eqref{eq4.1}, then there is a positive constant $c$ such that \begin{eqnarray*} E(t) \leq \frac{c}{(1+t)^{p+1}}E(0). \end{eqnarray*} \end{theorem} \begin{proof} Note that from \eqref{eq1.3} and \eqref{eq2.8} we have $$ \frac{\partial u}{\partial \nu}= - \frac{1}{k(0)}u_t- a(t) u - a'\diamondsuit u $$ from which it follows $$ | \frac{\partial u}{\partial \nu}|^2\leq 2\big\{\frac{1}{a^2(0)} | u_t|^2 + a^2(t)| u |^2 + | a' \diamondsuit u |^2 \big\}. $$ Since $$ | a' \diamondsuit u |^2 = \Big| \int_0^t a' (t-s) \{ u(s)-u(t) \} ds \Big|^2 \leq \Big(\int_0^t |a'(t-s)|^{1-\frac1p} ds\Big) [-a']^{1+\frac1p} \Box u, $$ and \eqref{eq2.5}, it follows that \begin{equation} \label{eq4.2} \left | \frac{\partial u }{\partial \nu}\right|^2 \leq k_0 \{| u_t|^2 + [-a']^{1+\frac1p} (t)| u |^2 + [-a']^{1+\frac1p} \Box u\}. \end{equation} On the other hand, \begin{equation}\label{eq4.3} \begin{aligned} \Big| \int_\Gamma u \frac{\partial u }{\partial \nu }d\Gamma\Big| &\leq \Big( \int_\Gamma |u|^2d\Gamma\Big)^{1/2} \Big( \int_\Gamma \big| \frac{\partial u}{\partial \nu }\big|^2 d\Gamma \Big)^{1/2}\\ &\leq \delta_1\int_\Gamma |u|^2 d\Gamma + \delta_1\int_{\Gamma} \big\{|u_t|^2 + [-a']^{1+\frac1p} (t)| u |^2 + [-a']^{1+\frac1p} \Box u\big\}d\Gamma\\ &\leq C\int_{\Gamma}\big\{| u_t|^2 + | u |^2 + [-a']^{1+\frac1p} \Box u\big\}d\Gamma. \end{aligned} \end{equation} Since $$ \int_\Gamma | u |^2 \,d\Gamma \leq C\int_\Omega | \nabla u |^2 + | \nabla v |^2 \,dx, $$ we have $\mathcal{L}(t) = NE(t) + \Phi(t)$ which satisfies \begin{align*} \frac{d}{dt}\mathcal{L}(t) &\leq -\frac{N\gamma_1}{k(0)}\int_\Gamma | u_t |^2d\Gamma +\frac{N\gamma_{1}a'(t)}{2}\int_\Gamma | u |^2 d\Gamma -\frac{N\gamma_1 }{2}\int_\Gamma [-a']^{1+\frac1p} \Box u d\Gamma\\ &\quad +C\int_\Gamma [-a']^{1+\frac1p} \Box ud\Gamma + C\int_\Gamma \big| \frac{\partial u }{\partial \nu}\big|^2 d\Gamma + \big(\frac{n-\delta}{2}\big)\gamma_1 \int_\Gamma u \frac{\partial u}{\partial \nu }d\Gamma \\ &\quad - \frac{\delta_0}{2}E_{0}(t) +\rho_1\int_\Gamma q\cdot\nu |u_t|^2 d\Gamma. \end{align*} Using \eqref{eq4.2} and \eqref{eq4.3}, we conclude that \begin{equation} \begin{aligned} \frac{d}{dt}\mathcal{L}(t) &\leq -\big(\frac{N\gamma_1}{k(0)}-C_{2}\big) \int_\Gamma | u_t |^2d\Gamma -\big(\frac{N\gamma_1 }{2}-C_{2}\big) \int_\Gamma [-a']^{1+\frac{1}{p}}\Box u d\Gamma\\ &\quad - \frac{\delta_0}{2}E_{0}(t), \end{aligned}\label{eq4.4} \end{equation} from where we have that for $N$ large enough we get \begin{equation}\label{eq4.5} \frac{d}{dt}\mathcal{L}(t) \leq -\frac{N\gamma_1}{2k(0)}\int_\Gamma | u_t |^2d\Gamma -\frac{N\gamma_1}{4}\int_\Gamma [-a']^{1+\frac{1}{p}}\Box u d\Gamma- \frac{\delta_0}{2}E_{0}(t). \end{equation} Let us fix $01$ we get, for $t\geq 0$, the following bounds \begin{gather*} t\|u(t,.)\|^{2}_{L^{2}(\Gamma )} \leq c t \mathcal{L} (t) \leq \infty,\\ \int^{t}_{0}\|u(s,.)\|^{2}_{L^{2}(\Gamma )} \leq c \int^{\infty}_{0} \mathcal{L} (t) \leq \infty. \end{gather*} Using the above estimates in Lemma \ref{lemma4.2} with $r=0$, we get \[ \int_{\Gamma }[-a']^{1+\frac{1}{p+1}}\Box u d \Gamma \ge \frac{c}{E(0)^{\frac{1}{p+1}}}\Big(\int_{\Gamma }[- a']\Box u d \Gamma \Big)^{1+ \frac{1}{p+1}}. \] Using these inequalities and the same arguments as in the derivation of \eqref{eq4.9}, we have \[ \frac{d}{dt} \mathcal{L} (t) \leq -\frac{c}{\mathcal{L} (0)^{\frac{1}{p+1}}}\mathcal{L} (t)^{1+ \frac{1}{p+1}}. \] From where we obtain $ \mathcal{L} (t) \leq \displaystyle\frac{c}{(1+t)^{p+1}} \mathcal{L}(0)$. 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