\documentclass[twoside]{article} \usepackage{amssymb} % used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Existence and boundary stabilization \hfil EJDE--1998/08}% {EJDE--1998/08\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 1998}(1998), No.~08, pp. 1--21. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113} \vspace{\bigskipamount} \\ Existence and boundary stabilization of a nonlinear hyperbolic equation with time-dependent coefficients \thanks{ {\em 1991 Mathematics Subject Classifications:} 35B40, 35L80. \hfil\break\indent {\em Key words and phrases:} Boundary stabilization, asymptotic behaviour. \hfil\break\indent \copyright 1998 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted July 6, 1997. Published March 10, 1998.} } \date{} \author{M. M. Cavalcanti, V. N. Domingos Cavalcanti \& J. A. Soriano} \maketitle \begin {abstract} In this article, we study the hyperbolic problem \begin{eqnarray*} &K(x,t)u_{tt} - \sum_{j=1}^n\left(a(x,t)u_{x_j}\right) + F(x,t,u,\nabla u) = 0&\\ &u=0\quad\hbox{on }\Gamma_1,\quad \frac {\partial u}{\partial\nu}+\beta (x) u_t =0\quad\hbox{on }\Gamma_0 &\\ &u(0)=u^0,\quad u_t(0)=u^1\quad\hbox{in }\Omega\,, & \end{eqnarray*} where $\Omega$ is a bounded region in ${\mathbb R}^n$ whose boundary is partitioned into two disjoint sets $\Gamma_0, \Gamma_1$. We prove existence, uniqueness, and uniform stability of strong and weak solutions when the coefficients and the boundary conditions provide a damping effect. \end{abstract} \catcode\@=11 \renewcommand{\theequation}{\thesection.\arabic{equation}} \@addtoreset{equation}{section} \catcode\@=12 \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}{Proposition}[section] \ \section{Introduction} Let $\Omega$ be a bounded domain of ${\mathbb R}^n$ with $C^2$ boundary $\Gamma$. Assume that $\Gamma$ has a partition $\Gamma_0,\Gamma_1$, such that each set has positive measure, and $\overline {\Gamma}_0\cap\overline {\Gamma}_1$ is empty. See the definition of these two sets in (\ref{eq:1.2}) below, and note that this condition excludes domains with connected boundary. Our objective is to study the problem \begin{eqnarray} &K(x,t)\frac {\partial^2u}{\partial t^2}+A(t)u+F(x,t,u,\nabla u)=0 \quad\hbox{in}\quad Q=\Omega\times ]0,\infty [ & \label{*}\\ &u=0\quad\hbox{on}\quad\Sigma_1=\Gamma_1\times ]0,\infty [ & \nonumber\\ &\frac {\partial u}{\partial\nu_A}+\beta (x)\frac {\partial u}{\partial t}=0\quad\hbox{on}\quad\Sigma_0=\Gamma_0\times ]0,\infty [&\nonumber\\ &u(0)=u^0,\quad\frac {\partial u}{\partial t}(0)=u^1\quad\hbox{ in}\quad\Omega\,, & \nonumber \end{eqnarray} where $A(t)=-\sum_{j=1}^n\frac {\partial}{\partial x_j}\left(a(x,t)\frac { \partial}{\partial x_j}\right)$. Stability of solutions for this problem with $K(x,t)=1$, $A(t)=-\Delta$ and $F=0$ has been studied by many authors; see for example J. P. Quinn \& D. L. Russell [10], G. Chen [2,3,4], J. Lagnese [6,7], and V. Komornik \& E. Zuazua [5] who also studied the nonlinear problem with $F=F(x,t,u)$. To the best of our knowledge, this is the first publication on boundary stabilization with time-dependent coefficients and the nonlinear term $F=F(x,t,u,\nabla u)$. Stability of problems with the nonlinear term $F(x,t,u,\nabla u)$ require a careful treatment, because we do not have any information about the influence of integral $\int_\Omega F(x,t,u,\nabla u)u'\,dx$ on the energy $$\label{eq:1.1} e(t)=\frac 12\int_\Omega (K(x,t)|u'(x,t)|^2+a(x,t)|\nabla u(x,t)|^2)\,dx\,,$$ or about the sign of the derivative $e'(t)$. When the coefficients depend on time, there are some technical difficulties that we need to overcome. First, semigroup arguments are not suitable for finding solutions to (\ref{*}); therefore, we make use of a Galerkin approximation. For strong solutions, this approximation requires a change of variables to transform (\ref{*}) into an equivalent problem with initial value equals zero. Secondly, the presence of $\nabla u$ in the nonlinear part brings up serious difficulties when passing to the limit. The goal of this work is to investigate conditions on the coefficients that lead to exponential decay of an energy determined by the solution. To this end, we use the perturbed-energy method developed by V. Komornik \& E. Zuazua in [5]. By establishing adequate hypotheses on $K(x,t)$, $a(x,t)$ and $F(x,t,u,\nabla u)$, the above method allow us to solve (\ref{*}) when $\beta (x)=(x-x^0)\cdot\nu (x)$ with $x^0$ a point in ${\mathbb R}^n$ and $\nu (x)$ the exterior unit normal. Our paper is divided in 4 sections. In \S 2, we establish notation and state our results. In \S 3, we prove solvability of (\ref{*}) using the Galerkin method. In \S 4, we prove exponential decay of solutions. \section{Notation and statement of results} For the rest of this article, let $x^0$ be a fixed point in ${\mathbb R}^n$. Then put $$m=m(x)=x-x^0\,,$$ and partition the boundary $\Gamma$ into two sets: $$\label{eq:1.2} \Gamma_0=\{x\in\Gamma \,:\,m(x)\cdot\nu (x)\geq 0\}\,,\quad \Gamma_1=\{x\in\Gamma \,:\,m(x)\cdot\nu (x)<0\}\,.$$ Consider the Hilbert space $$V=\{v\in H^1(\Omega )\,:\,v=0\quad\hbox{on}\quad\Gamma_1\}\,,$$ and define the following: \begin{eqnarray*} &(u,v)=\int_\Omega u(x)v(x)\,dx,\quad |u|^2=\int_\Omega |u(x)|^2\,dx,& \\ &(u,v)_{\Gamma_0}=\int_{\Gamma_0}u(x)v(x)\,d\Gamma ,\quad|u|_{\Gamma_0}^2= \int_{\Gamma_0}|u(x)|^2\,dx,& \\ & \|u\|_{\infty}=\mbox{ess}\,\sup_{t\geq 0}\|u(t)\|_{L^\infty(\Omega )}, \quad u'=u_t=\frac {\partial u}{\partial t},\quad u_{x_i}=\frac {\partial u}{\partial x_i}& \end{eqnarray*} and $$\label{eq:2.1} R(x^0)=\max_{x\in\overline {\Omega}} \|x-x^0\|$$ Now, we state the general hypotheses. \paragraph{(A.1) Assumptions on $F(x,t,u,\nabla u)$.} Suppose $F:\overline {\Omega}$$\times [0,\infty [\times {\mathbb R}^{n+1} \rightarrow {\mathbb R} is an element of the space C^1(\overline {\Omega}\times [0,\infty [\times {\mathbb R}^{n+1}) and satisfies $$\label{eq:2.3} |F(x,t,\xi ,\zeta )|\leq C_0(1+|\xi |^{\gamma +1}+|\zeta |)$$ where C_0 is a positive constant, and \zeta =(\zeta_1,...,\zeta_n). Let \gamma be a constant such that \gamma>0 for n=1,2, and 0<\gamma\leq 2/(n-2) for n\geq 3. Assume that there is a non-negative function C(t) in the space L^\infty(0,\infty )\cap L^1(0,\infty ), such that \begin{eqnarray} &F(x,t,\xi ,\zeta )\eta\geq |\xi |^{\gamma}\xi\eta -C(t)(1+|\eta \|\zeta |), \quad\forall\eta\in {\mathbb R}\,,& \label{eq:2.5}\\ &F(x,t,\xi ,\zeta )\left(m\cdot\zeta\right )\geq |\xi |^{\gamma}\xi\,\left(m\cdot\zeta\right)-C(t)(1+|\zeta \|m\cdot\zeta |)\,.\label{eq:2.6} \end{eqnarray} Assume that there exist positive constants C_0,\dots,C_n, such that \begin{eqnarray} &|F_t(x,t,\xi ,\zeta )|\leq C_0\left(1+|\xi |^{\gamma +1}+|\zeta |\right), & \label{eq:2.7}\\ &\left|F_{\xi}(x,t,\xi ,\zeta )\right|\leq C_0(1+|\xi |^{\gamma}),&\label{eq:2.8}\\ &\left|F_{\zeta_i}(x,t,\xi ,\zeta )\right|\leq C_i \quad\mbox{for }i=1,2,\dots ,n\,.&\label{eq:2.9} \end{eqnarray} We also assume that there exist positive constants D_1, D_2, such that for all \eta, \hat{\eta} in {\mathbb R} and for all \zeta, \hat{\zeta} in {\mathbb R}^n, $$\label{eq:2.10} (F(x,t,\xi_{},\zeta )-F(x,t,\hat{\xi },\hat{\zeta }))(\eta -\hat{\eta }) \geq -D_1(|\xi |^{\gamma}+|\hat{\xi }|^{\gamma})|\xi -\hat{\xi } \|\eta -\hat{\eta }|-D_2|\eta -\hat{\eta }\|\zeta -\hat{\zeta }|\,.$$ The following is an example of a function F that satisfies the above conditions.$$ F(x,t,u,\nabla u)=|u|^{\gamma}u+\varphi (t) \sum_{i=1}^n\sin\left(\frac {\partial u}{\partial x_i}\right)\,, $$where \varphi is a function sufficiently regular. \paragraph{(A.2) Assumptions on the initial data.}$$u^0,u^1\in V\cap H^2(\Omega )\,\quad \mbox{and}\quad \frac {\partial u^0}{\partial\nu_A}+\beta (x)u^1=0 \hbox{ on }\Gamma_0\,. $$\paragraph{(A.3) Assumptions on the coefficients.} \begin{eqnarray*} &K\in W^{1,\infty}(0,\infty ;C^1(\overline { \Omega})),\quad a\in W^{1,\infty}(0,\infty ;C^1(\overline {\Omega}))\cap W^{2,\infty}(0,\infty ;L^\infty(\Omega ))&\\ &a_t,K_t\in L^1(0,\infty ;L^\infty(\Omega )),\quad \beta\in W^{1,\infty}(\Gamma_0)\,.& \end{eqnarray*} Also assume that there exist positive constants a_0, k_0, such that $$\label{eq:2.13} K\geq k_0, \quad a\geq a_0,\quad\hbox{in }Q,\quad\hbox{and}\quad \beta (x)\geq 0\quad\hbox{a.e. on \Gamma_0}.$$ For short notation, define \begin{eqnarray*} &a(t,u,v)=\sum_{j=1}^n\int_\Omega a(x,t)\frac {\partial u}{\partial x_j}\frac {\partial v}{\partial x_j}\,dx,&\\ &a'(t,u,v)=\sum_{j=1}^n\int_\Omega a_t(x,t)\frac {\partial u}{\partial x_j}\frac {\partial v}{\partial x_j}\,dx,&\\ &a''(t,u,v)=\sum_{j=1}^n\int_\Omega a_{tt}(x,t)\frac {\partial u}{\partial x_j} \frac {\partial v}{\partial x_j}\,dx\,.& \end{eqnarray*} We observe that from the above assumptions on a, there exist positive constants a_1, a_2, and a_3 such that, \begin{eqnarray} &a_0|\nabla u|^2\leq a(t,u,u)\leq a_1|\nabla u|^2\quad\forall u\in V\quad \hbox{and}\quad t\geq 0\,,& \label{eq:2.17}\\ &|a'(t,u,v)|\leq a_2|\nabla u\|\nabla v|\quad\forall u\in V \quad\hbox{and}\quad t\geq 0\,,& \label{eq:2.18}\\ &|a''(t,u,v)|\leq a_3|\nabla u\|\nabla v|\quad\forall u\in V \quad\hbox{and}\quad t\geq 0\,.\label{eq:2.19} \end{eqnarray} Now, we are in a position to state our results. \begin{theorem} Under Assumptions (A1, A2, A3), Problem (\ref{*}) possesses a unique strong solution, u: ]0,\infty [\times\Omega \rightarrow {\mathbb R}, such that$$u\in L^\infty(0,\infty ;V\cap H^2(\Omega )),\,u'\in L^\infty( 0,\infty ;V),\mbox{ and }u''\in L^\infty(0,\infty ;L^2(\Omega )).$$\end{theorem} Now, we present a result on stability of strong solutions, which will be extended to weak solutions. Let \begin{eqnarray*} H(t)&=&\|\nabla a(t)\|_{L^\infty(\Omega )}+\|\nabla K(t)\|_{L^{\infty}(\Omega )}\\ &&+\|a_t(t)\|_{L^\infty(\Omega )}+\|K_t(t)\|_{L^{\infty}(\Omega )}+C(t)\,. \end{eqnarray*} \begin{theorem} Assume that there are positive constants \alpha,r,\epsilon, \theta_0, such that for all t sufficiently large, $$\label{eq:2.20} \int_0^t\exp(\epsilon\theta_0s)H(s)\,ds\leq\alpha t^r\,.$$ Then the energy (\ref{eq:1.1}) determined by the strong solution u decays exponentially. This is, for some positive constants \delta ,\epsilon, \theta_1, $$\label{eq:2.21} E(t)=e(t)+\frac 1{\gamma +2}\int_\Omega |u(x,t)|^2\,dx\leq \delta \exp(-\epsilon\theta_1t)\,.$$ \end{theorem} Notice that (\ref{eq:2.20}) requires the integral to have polynomial growth. Therefore, each term in H(t) behaves as a function of the form Q(t)\exp(-\beta t) with Q(t) a polynomial and \beta >\epsilon\theta_0. An example of a function that satisfies (\ref{eq:2.20}) is H(t)=t\exp(-\beta t). In fact, \begin{eqnarray*} \lefteqn{ \int_0^t\exp(\epsilon\theta_0s)s\exp(-\beta s)\,ds} &&\\ &=&-\frac t{\beta -\theta_0\epsilon}\exp(-(\beta -\theta_0\epsilon )t) -\frac 1{(\beta -\theta_0\epsilon )^2}\exp(-(\beta -\theta_0\epsilon )t) +\frac 1{(\beta -\theta_0\epsilon )^2}\\ &\leq&\alpha t+\delta\,, \end{eqnarray*} for some positive constants \alpha and \delta. \begin{theorem} Suppose that \{u^0, u^1\} is in V \times L^2(\Omega ), and that assumptions (A1), (A3) hold. Then (\ref{*}) has a unique weak solution, u:\Omega\times ]0,\infty [\rightarrow {\mathbb R}, in the space$$ C([0,\infty );V)\cap C^1([0,\infty );L^2(\Omega ))\,. $$Furthermore, Theorem~2.2 holds for the weak solution u. \end{theorem} \paragraph{Remark} Notice that as t increases, (\ref{*}) converges to an equation of constant coefficients, and F=|u|^\gamma u. Hence, (\ref{*}) can be seen as a disturbance of a much better known problem, which was studied in [5]. Also note that both equations have solutions with the same exponential decay, (\ref{eq:2.21}). \section{Existence of strong and weak solutions} In this section, we prove the existence and uniqueness of strong and weak solutions to (\ref{*}). First we consider strong solutions, and then using a density argument we extend the same result to weak solutions. A variational formulation of Problem (\ref{*}) leads to the equation$$ \int_\Omega Ku''w\,dx+\int_\Omega a(x,t)\nabla u\nabla w\,dx+\int_\Omega F(x,t,u,\nabla u)w\,dx+\int_{\Gamma_0}\beta u'w\,d\Gamma =0\,,$$for all w in the space V. Strong solutions to (\ref{*}) with the boundary condition \int_{\Gamma_0}\beta u'w\,d\Gamma can not be obtained by the method of \lq\lq special basis"; therefore, bases formed with eigenfunctions can not be used for (\ref{*}). Differentiating the above expression with respect to t does not help, because of the technical difficulties when estimating u''(0). To avoid these difficulties, we transform (\ref{*}) into an equivalent problem with initial value equal to zero. In fact, the change of variables \begin{eqnarray} &v(x,t)=u(x,t)-\phi (x,t)&\label{eq:3.1}\\ &\phi (x,t)=u^0(x)+tu^1(x),\quad t\in [0,T] &\label{eq:3.2} \end{eqnarray} leads to \begin{eqnarray} &K(x,t)v''+A(t)v+F(x,t,\phi +v,\nabla\phi +\nabla v)=f\quad\hbox{in }Q=\Omega\times (0,T),&\label{eq:3.3}\\ &v=0\quad\mbox{on }\Sigma_1=\Gamma_1\times (0,T),&\nonumber\\ &\frac {\partial v}{\partial\nu_A}+\beta (x)v'=g\quad \hbox{on }\Sigma_0=\Gamma_0\times (0,T),&\nonumber\\ &v(x,0)=v'(x,0)=0\,,& \end{eqnarray} where f(x,t)=-A(t)u^0(x)-tA(t)u^1(x), (x,t)\in\Omega\times [0,T], and g(x,t)=-t\frac {\partial u^1}{\partial\nu_A}. Note that if v is a solution of (\ref{eq:3.3}) on [0,T], then u=v+\phi is a solution of (\ref{*}) in the same interval. From estimates obtained below, we are able to prove that $$\label{eq:3.6} |A(t)v(t)|^2+|\nabla v'(t)|^2\leq C,\quad \forall t\in [0,T]\,.$$ Thus, from (\ref{eq:3.1}) and (\ref{eq:3.2}) we obtain the same inequality (\ref{eq:3.6}) for the solution u. Then using standard methods, we extend u to the interval (0,\infty ). Hence, it is sufficient to prove that (\ref{eq:3.3}) has a local solution, which shall be done by using the Galerkin method. Let (\omega_{\nu})_{\nu\in {\bf N}} be a set of functions in V\cap H^2(\Omega ), that form and orthonormal basis for L^2(\Omega ). Let V_m be the space generated by \omega_1,\omega_2,\dots ,\omega_m and let $$\label{eq:3.7} v_m(t)=\sum_{i=1}^mg_{jm}(t)\omega_j$$ be the solution to the Cauchy problem \begin{eqnarray} \lefteqn{ (K(t)v_m''(t),w)+a(t,v_m(t),w)+(\beta v_m'(t),w)_{\Gamma_0} } &&\nonumber\\ \lefteqn{ +\int_\Omega F(x,t,v_m+\phi ,\nabla v_m+\nabla\phi )w\,dx }&& \nonumber \\ &=& (f(t),w)+(g(t),w)_{\Gamma_0}, \quad \forall w\in V_m\,, \label{eq:3.8} \\ && v_m(0)=v_m'(0)=0\,.\nonumber \end{eqnarray} Observe that all the terms in the above expression are well defined. In particular, \int_\Omega F(x,t,v_m+\phi ,\nabla v_m+\nabla\phi )w\,dx exists because of (\ref{eq:2.3}). By standard methods in differential equations, we can prove the existence of a solution to (\ref{eq:3.8}) on some interval [0,t_m). Then this solution can be extended to the close interval by the use of the first estimate below. \subsection*{A priori estimates} \paragraph{First Estimate:} Taking w=2v_m'(t) in (\ref{eq:3.8}), we have \begin{eqnarray*} \lefteqn{ \frac d{dt}\{|\sqrt {K(t)}v_m'(t)|^2+a(t,v_m(t),v_m(t))\}+2(\beta ,v_m^{\prime 2}(t))_{\Gamma_0} } &&\\ \lefteqn{ +2\int_\Omega F(x,t,v_m+\phi ,\nabla v_m+\nabla\phi )v_m'\,dx }&& \\ &=&(K_t(t),v_m^{\prime 2}(t))+a'(t,v_m(t),v_m(t)) +2(f(t),v_m'(t)) \\ &&+2\frac d{dt}(g(t),v_m(t))_{\Gamma_0}-2(g'(t),v_m(t))_{\Gamma_0}\,. \end{eqnarray*} Integrating the above expression over [0,t], we obtain \begin{eqnarray} \lefteqn{ |\sqrt {K(t)}v_m'(t)|^2+a(t,v_m(t), v_m(t))+2\int_0^t(\beta ,v_m'{}^2(s))_{\Gamma_0}\,ds }&& \label{eq:3.9} \\ \lefteqn{+2\int_0^t\int_\Omega F(x,s,v_m+\phi ,\nabla v_m+\nabla\phi )v_ m'\,dx\,ds }&& \nonumber\\ &=&\int_0^t(K_s(s),v_m^{\prime 2}(s))\,ds +\int_0^ta'(s,v_m(s),v_m(s))\,ds \nonumber\\ &&+2\int_0^t(f(s),v_m'(s))\,ds+2(g(t),v_m(t))_{\Gamma_0} -2\int_0^t(g'(s),v_m(s))_{\Gamma_0}\,ds\,. \nonumber \end{eqnarray} \paragraph{Estimate for I_1:=2\int_0^t\int_\Omega F(x,s,v_m+\phi ,\nabla v_m+\nabla\phi )v_m'\,dx\,ds.} We have \begin{eqnarray*} I_1&=&2\int_0^t\int_\Omega F(x,s,v_m+\phi ,\nabla v_m+\nabla\phi )(v_m'+\phi')\,dx\,ds\\ &&-2\int_0^t\int_\Omega F(x,s,v_m+\phi ,\nabla v_m+\nabla\phi )\phi' \,dx\,ds\,. \end{eqnarray*} From (\ref{eq:2.3}) and (\ref{eq:2.5}) it follows that \begin{eqnarray} I_1&\geq&\frac 2{\gamma +2}\|v_m(t)+ \phi (t)\|_{L^{\gamma +2}(\Omega )}^{\gamma +2}-\frac 2{\gamma +2} \|\phi (0)\|_{L^{\gamma +2}(\Omega )}^{\gamma +2} \label{eq:3.10}\\ &&-2C\int_0^t\int_\Omega (1+|v_m'+\phi'||\nabla v_m+\nabla\phi | )\,dx\,ds \nonumber \\ &&-2C\int_0^t\int_\Omega (1+|v_m+\phi |^{\gamma +1}+|\nabla v_m+ \nabla\phi |)|\phi'|\,dx\,ds\,. \nonumber \end{eqnarray} Substituting (\ref{eq:3.10}) in (\ref{eq:3.9}), observing that (\ref{eq:2.13}), (\ref{eq:2.17}), (\ref{eq:2.18}) hold, and noting that v_m(0)=v_m'(0)=0, it follows that \begin{eqnarray*} \lefteqn{ k_0|v_m'(t)|^2+a_0|\nabla v_m(t)|^2+\frac 2{\gamma +2}\|v_m(t)+ \phi (t)\|_{L^{\gamma +2}(\Omega )}^{\gamma +2} +2\int_0^t(\beta ,v_m'(s)^2)_{\Gamma_0}\,ds } && \hspace{11cm}\\ &\leq&\frac 2{\gamma +2} \|\phi (0)\|_{L^{\gamma +2}(\Omega )}^{\gamma +2}+\int_0^t(K_s(s) ,v_m^{\prime 2}(s))\,ds+a_2\int_0^t|\nabla v_m(s)|^2\,ds \\ &&+2\int_0^t(f(s),v_m'(s))\,ds+2(g(t),v_m(t))_{\Gamma_0} -2\int_0^t(g'(s),v_m(s))_{\Gamma_0}\,ds \\ &&+2C\int_0^t\int_\Omega (1+|v_m'+\phi'| |\nabla v_m+\nabla\phi |)\,dx\,ds\\ &&+2C\int_0^t\int_\Omega (1+|v_m+\phi |^{\gamma +1}+ |\nabla v_m+\nabla\phi |)|\phi'|\,dx\,ds\,. \end{eqnarray*} Using Young, H\"older and the Schwarz inequalities we obtain \begin{eqnarray*} \lefteqn{ k_0|v_m'(t)|^2+\frac {a_0}2|\nabla v_m(t)|^2 +\frac 2{\gamma +2}\|v_m(t)+\phi (t)\|_{L^{\gamma +2}(\Omega )}^{\gamma +2} +2\int_0^t(\beta ,v_m^{\prime 2}(s))_{\Gamma_0}\,ds } && \hspace{11cm}\\ &\leq& L_0+L_1\int_0^t\left(|v_m'(s)|^2+|\nabla v_m(s)|^2+\|v_m(s)+ \phi (s)\|_{L^{\gamma +2}}^{\gamma +2}\right)\,ds\,. \end{eqnarray*} From this inequality and the Gronwall's inequality, we obtain the first estimate, $$\label{eq:3.13} |v_m'(t)|^2+|\nabla v_m(t)|^2+\|v_m(t)+\phi (t)\|_{L^{\gamma +2} (\Omega )}^{\gamma +2}+\int_0^t(\beta ,v_m'(s)^2)_{\Gamma_0}\,ds\leq L\,,$$ where L is a positive constant independent of m and t\in [0,T]. \paragraph{Second Estimate:} First, we prove that v_m''(0) is bounded in the L^2(\Omega ) norm. Indeed, considering t=0 in (\ref{eq:3.8}) we obtain \begin{eqnarray*} &(K(0)v_m''(0),w)+a(0,v_m(0),w)+(\beta v_m'(0),w)_{\Gamma_0} +\int_\Omega F(x,0,u^0,\nabla u^0)w\,dx &\\ &=(-A(0)u^0,w)\,\quad\forall w\in V_m\,.& \end{eqnarray*} From this inequality and the fact that v_m(0)=v_m'(0)=0, we get$$(K(0)v_m''(0),w)=-\int_{ \Omega}F(x,0,u^0,\nabla u^0)w\,dx-(A(0)u^0,w),\quad\forall w\in V_m\,. $$With w=v_m''(0) in the above equation, we obtain$$(K(0),v_m^{\prime\prime 2}(0))=-\int_{ \Omega}F(x,0,u^0,\nabla u^0)v_m''(0)\,dx-(A(0)u^0,v_m''(0)) $$From this equation, (\ref{eq:2.3}), and (\ref{eq:2.13}), we conclude that \begin{eqnarray*} k_0|v_m''(0)|^2&\leq& C\int_\Omega (1+|u^0|^{\gamma +1}+|\nabla u^0|) |v_m''(0)|\,dx+|A(0)u^0\|v_m''(0)|\\ &\leq& C(\Omega )[1+|\nabla u^0|^{\gamma +1}+|\nabla u^0|+|A(0)u^0|]|v_m''(0|\,. \end{eqnarray*} That is$$|v_m''(0)|\leq C(\Omega ,k_0)[1+|\nabla u^0|^{\gamma +1}+|\nabla u^0|+|A(0)u^0|],\quad\forall m\in {\mathbb N}\,.$$Therefore, $$\label{eq:3.17}\mbox{v_m''(0) is bounded in L^2(\Omega ).}$$ % Taking the derivative of (\ref{eq:3.8}) with respect to t, it follows that \begin{eqnarray*} && (K_t(t)v_m''(t),w)+(K(t)v_m^{\prime\prime\prime}(t ),w)+a'(t,v_m(t),w)+a(t,v_m'(t),w) \\ && (\beta v_m''(t),w)_{\Gamma_0}+\int_\Omega F_t(x, t,v_m+\phi ,\nabla v_m+\nabla\phi )w\,dx \\ &&+\int_\Omega F_{v_m+\phi}(x,t,v_m+\phi ,\nabla v_m+\nabla\phi )(v_m'+\phi')w\,dx \\ && +\sum_{i=1}^n\int_\Omega F_{v_mx_i+\phi x_i}(x,t,v_m+\phi ,\nabla v_m+\nabla\phi )(v_{mx_i}'+\phi_{x_i}')w\,dx \\ && \quad = (f'(t),w)+(g'(t),w)_{\Gamma_0}\,. \end{eqnarray*} % Substituting w by 2v_m''(t) in the above expression it results that \begin{eqnarray*} \lefteqn{ \frac d{dt}\{|\sqrt {K(t)}v_m^{\prime \prime}(t)|^2+a(t,v_m'(t),v_m'(t))+2a'(t,v_m(t),v_m'(t))\} +2(\beta ,v_m^{\prime\prime 2}(t))_{\Gamma_0} } && \hspace{11cm} \\ &=&-(K_t(t),v_m^{\prime\prime 2}(t))+2a'(t,v_m'(t),v_m'(t))+2a''(t,v_m(t),v_ m'(t))\\ &&+a'(t,v_m'(t),v_m'(t))-2\int_\Omega F_t(x,t,v_m+\phi ,\nabla v_ m+\nabla\phi )v_m''\,dx\\ &&-2\int_\Omega F_{v_m+\phi}(x,t,v_m+\phi ,\nabla v_m+\nabla\phi )(v_m'+\phi')v_m''\,dx\\ &&-2\sum_{i=1}^n\int_\Omega F_{v_mx_i+\phi x_i}(x,t,v_m+\phi ,\nabla v_m+\nabla\phi )(v_{mx_i}'+\phi_{x_i}')v_m''\,dx\\ &&+2(f'(t),v_m''(t))+2\frac d{dt}(g'(t),v_m'(t))_{\Gamma_0}\,. \end{eqnarray*} % Integrating both sides of this equation over [0,t] and observing that v_m'(0)=0, we obtain \begin{eqnarray} \lefteqn{ |\sqrt {K(t)}v_m''(t) |^2+a(t,v_m'(t),v_m'(t))+2\int_0^t (\beta ,v_m^{\prime\prime 2}(s))_{\Gamma_0}\,ds } \label{eq:3.19}\\ &=&|\sqrt {K(0)}v_m''(0)|^2-2a'(t,v_m(t),v_m'(t))-\int_ 0^t(K_s(s),v_m^{\prime\prime 2}(s))\,ds \nonumber\\ &&+3\int_0^ta'(s,v_m'(s),v_m'(s))\,ds+2\int_0^ta''(s,v_m(s),v_m'(s))\,ds \nonumber\\ &&-2\int_0^t\int_\Omega F_s(x,s,v_m+\phi ,\nabla v_m+\nabla\phi )v_m''\,dx\,ds\nonumber\\ &&-2\int_0^t\int_\Omega F_{v_m+\phi}(x,s,v_m+\phi ,\nabla v_m+\nabla \phi )(v_m'+\phi')v_m''\,dx\,ds\nonumber\\ &&-2\sum_{i=1}^n\int_0^t\int_\Omega F_{v_mx_i+\phi x_i}(x,s,v_m+ \phi ,\nabla v_m+\nabla\phi )(v_{mx_i}'+\phi_{x_i}')v_m'' \,dx\,ds\nonumber\\ &&+2\int_0^t(f'(s),v_m''(s))\,ds+2(g'(t),v_m'(t))_{\Gamma_0}\,.\nonumber \end{eqnarray} From (\ref{eq:2.7}), (\ref{eq:2.8}), (\ref{eq:2.9}), (\ref{eq:2.13}), (\ref{eq:2.17}), (\ref{eq:2.18}), (\ref{eq:2.19}), (\ref{eq:3.13}), (\ref{eq:3.17}), (\ref{eq:3.19}) and using Young, H\"older and Schwarz inequalities, and the Sobolev injection, we have \begin{eqnarray*} \lefteqn{ k_0|v_m''(t)|^2+\frac { a_0}2|\nabla v_m'(t)|^2+2\int_0^t(\beta ,v_m^{\prime\prime 2}(s)) \,ds }&&\\ &\leq & L_2+L_3\int_0^t(|v_m''(s)|^2+|\nabla v_m'(s)|^2)\,ds\,. \end{eqnarray*} Then using the Gronwall's inequality, we obtain the second estimate,$$ |v_m''(t)|^2+|\nabla v_m'(t)|^2+\int_0^t(\beta ,v_m^{\prime\prime 2}(s))\,ds \leq L\,, $$where L is a positive constant independent of m\in {\mathbb N} and t\in [0,T]. The above estimates, allows us passing to the limit in the linear terms. Next we analyze the nonlinear term. \subsection*{Analysis of the nonlinear term F} From (\ref{eq:2.3}) there is positive constant M such that \begin{eqnarray*} \lefteqn{ \int_\Omega |F(x,t,v_m+\phi ,\nabla v+\nabla\phi )|^2\,dx }\\ &\leq& M\left(1+\|v_m(t)+\phi (t)\|_{L^{2(\gamma +1)}}^{2(\gamma +1)}+ \left|\nabla v_m(t)+\nabla\phi (t)\right|^2\right)\,. \end{eqnarray*} Therefore, from the first estimate it follows that $$\label{eq:3.22} \left\{F(x,t,v_m+\phi ,\nabla v_m+ \nabla\phi )\right\}_{m\in {\bf N}}\hbox{ is bounded in } L^2(0,T;L^2(\Omega ))\,.$$ Consequently, there exists a subsequence of \{v_m\}_{m\in {\bf N}} (which we still denote by the same symbol) and a function \chi in L^2(0,T;L^2(\Omega )) such that $$\label{eq:3.23} F(x,t,v_m+\phi ,\nabla v_m+\nabla\phi )\rightharpoonup\chi\quad \hbox{weak in }L^2(0,T;L^2(\Omega ))\,.$$ % From the above estimates after passing to the limit, we conclude that $$\label{eq:3.24} Kv''+A(t)v+\chi =f\quad\hbox{ in } L^2(0,T;L^2(\Omega )).$$ We observe that$$v\in L^\infty(0,T;V),\quad v'\in L^\infty(0,T;V),\quad v^{\prime \prime}\in L^\infty(0,T;L^2(\Omega ))\,.$$Moreover,$$\frac {\partial v}{\partial\nu_A}+\beta v'=g\quad \hbox{in } L^\infty (0,TH^{1/2}(\Gamma_0))\,.$$% On the other hand, integrating the approximate problem (\ref{eq:3.8}) over [0,T] and considering that w=v_m, we obtain \begin{eqnarray} \lefteqn{ \int_0^T\left(K(t)v_m'' (t),v_m(t)\right)\,dt+\int_0^Ta(t,v_m(t),v_m(t))\,dt } && \hspace{10cm}\label{eq:3.25}\\ \lefteqn{ +\int_0^T\left(\beta v_m'(t),v_m(s)\right)_{\Gamma_0}\,ds +\int_0^T\int_\Omega F(x,t,v_m+\phi ,\nabla v_m+\nabla\phi )v_m\,dx,\,dt } \nonumber\\ &=&\int_0^T(f(t),v_m(t))\,dt+\int_0^T(g(t),v_m(t))_{\Gamma_0}\,dt\,. \nonumber \end{eqnarray} To simplify notation, subsequences will be denoted by the same symbol as the corresponding original sequences. Notice that from the first and second estimates, and the Aubin-Lions Theorem there exists a subsequence of \{v_m\}_{m\in {\mathbb N}}, such that \begin{eqnarray} &v_m\rightarrow v\quad\hbox{strong in } L^2(0,T;L^2(\Omega ))\,,& \label{eq:3.26} \\ &v_m'\rightarrow v'\quad\hbox{strong in } L^2(0,T;L^2(\Omega ))\,. \label{eq:3.27}\end{eqnarray} % Now, the first estimate yields $$\label{eq:3.28} \left|\sqrt {\beta}v_m'(s)\right|_{H^{1/2}(\Gamma_0)}^2\leq C_0 \left|\nabla v_m'(s)\right|^2\leq L;\quad s\in [0,T]\,,$$ and from the second estimate we get $$\label{eq:3.29}\left|\sqrt {\beta}v_m'' (s)\right|_{\Gamma_0}^2\leq L;\quad s\in [0,T]\,.$$ Combining (\ref{eq:3.28}) and (\ref{eq:3.29}), noting that the injection H^{1/2}(\Gamma_0)\hookrightarrow L^2(\Gamma_0) is compact, and considering Aubin-Lions Theorem it follows that $$\label{eq:3.30}\sqrt {\beta}v_m'\rightarrow\sqrt { \beta}v'\quad\hbox{in}\quad L^2(0,T;L^2(\Gamma_0)).$$ In a similar way$$\sqrt {\beta}v_m\rightarrow\sqrt {\beta}v\quad \hbox{in}\quad L^2(0,T; L^2(\Gamma_0)).$$Moreover, because of the second estimate$$v_m''\rightharpoonup v''\quad \hbox{weak\quad in}\quad L^2(0,T;L^2(\Omega )).$$Then, considering the strong convergences given in (\ref{eq:3.26}), (\ref{eq:3.27}) and (\ref{eq:3.30}) and the corresponding weak converges, we are able to pass to the limit in (\ref{eq:3.25}). \begin{eqnarray} \lefteqn{ \lim_{m\rightarrow\infty}\int_0^Ta (t,v_m(t),v_m(t))\,dt} &&\label{eq:3.31} \\ &=&-\int_0^T\left(K(t)v''(t),v(t)\right)\,dt-\int_0^T\left (\beta v'(t),v(t)\right)_{\Gamma_0}\,dt \nonumber\\ &&-\int_0^T\int_\Omega \chi (t)v(t)\,dx\,dt+\int_0^T(f(t),v(t))\,dt+ \int_0^T(g(t),v)_{\Gamma_0}\,dt\,.\nonumber \end{eqnarray} Substituting (\ref{eq:3.24}) in (\ref{eq:3.31}), applying Green formula and noting that$$\frac {\partial v}{\partial\nu_A}=-\beta v'+g\quad\hbox{a.e. on }\Gamma_0$$we deduce that$$\lim_{m\rightarrow\infty}\int_0^Ta(t,v_m(t),v_m(s))\,dt=\int_0^Ta (t,v(t),v(t))\,\,dt$$and that $$\label{eq:3.32}\lim_{m\rightarrow\infty}\int_0^T\left (\nabla v_m(t),\nabla v_m(t)\right)\,dt=\int_0^T\left(\nabla v(t),\nabla v(t)\right)\,\,dt\,.$$ Finally, taking into account that \begin{eqnarray*} \lefteqn{ \int_0^T\left|\nabla v_m(t)-\nabla v(t)\right|^2\,dt }\\ &=&\int_0^T\left|\nabla v_m(t)\right|^2\,dt-2\int_0^T\left(\nabla v_ m(t),\nabla v(t)\right)\,dt+\int_0^T\left|\nabla v(t)\right|^2\,dt\,, \end{eqnarray*} from (\ref{eq:3.32}) and the first estimate we deduce that$$\lim_{m\rightarrow\infty}\int_0^T\left|\nabla v_m(t)-\nabla v(t )\right|^2\,dt=0\,.$$Therefore,$$\nabla v_m\rightarrow\nabla v\quad\hbox{in}\quad L^2(0,T;L^2(\Omega ))\,,$$and consequently$$\nabla v_m\rightarrow\nabla v\quad\hbox{a.e.\quad in}\quad Q_T= \Omega\times (0,T)\,.$$% From (\ref{eq:3.26}) and the above convergence, we obtain$$F(x,t,v_m+\phi,\nabla v_m+\nabla \phi)\rightarrow F(x,t,v+\phi,\nabla v + \nabla \phi)\quad\hbox{a. e. in } Q_T\,.$$Applying Lemma 1.3 in [8, Chant. 1], it follows from the above convergence, (3.13) and (3.14) that$$F(x,t,v_m+\phi,\nabla v_m+\nabla \phi)\rightharpoonup F(x,t,v+\phi,\nabla v +\nabla \phi)\quad\hbox{weak in } L^2(0,T;L^2(\Omega ))\,.$$Note that the function v:\Omega\rightarrow {\mathbb R} is a weak solution to the Dirichlet-Neumann problem \begin{eqnarray*} &A(t)v_{}=f^{*} \quad\mbox{in }\Omega \,,&\\ &v=0\quad \mbox{in }\Gamma_1\,, \qquad \frac {\partial v}{\partial\nu_A}=g^{*}\quad \mbox{ in }\Gamma_0\,,& \end{eqnarray*} where f^{*}=f-Kv''-F(x,t,v+\phi ,\nabla v+\nabla\phi ), f^{*}\in L^2(\Omega ), g^{*}=-\beta v'+g, g^{*}\in H^{1/2}(\Gamma_0), and t is a fixed value in [0,T]. The theory of elliptic problems states that the solution v belongs to the space L^\infty(0,T;H^2(\Omega )); therefore, v\in L^\infty(0,T;V\cap H^2(\Omega )). \hfill \subsection*{Uniqueness of the solution} Let u and \hat {u} be two solutions of (\ref{*}), and put z=u-\hat {u}. From (\ref{eq:2.10}), (\ref{eq:2.13}), (2.11) and (2.12), it follows that \begin{eqnarray*} \lefteqn{ k_0\left|z'(t)\right|^2+a_0\left|\nabla z(t)\right|^2+2\int_0^t\left (\beta ,z^{\prime 2}(s)\right)_{\Gamma_0}\,ds } && \\ &\leq& 2D_1\int_0^t\int_\Omega \left(\left|u\right|^{\gamma}+\left |\hat u\right|^{\gamma}\right)\left|z\right|\left|z'\right|\,dx\,\,dt +2D_2\int_0^t\int_\Omega \left|z'\right|\left|\nabla z\right|\,dx\,ds \\ &&+\|K_1\|_{\infty}\int_0^t\left|z'(s)\right|^2\,ds+a_2\int_0^t\left |\nabla z(s)\right|^2\,ds\,. \end{eqnarray*} Since 0<\gamma\leq 2/(n-2), for n\geq 3, we have the Sobolev immersion H^1(\Omega )\hookrightarrow L^{2(\gamma +1)}(\Omega ). This immersion is also true for all \gamma >0 when n=1,2. Therefore, with$$\frac {\gamma}{2(\gamma +1)}+\frac 1{2(\gamma +1)}+\frac 12=1\,,$$and using the generalized H\"older and the Poincar\'e inequalities, we conclude that$$\left|z'(t)\right|^2+a_0\left|\nabla z(t)\right|^2+2\int_0^t\left (\beta ,z^{\prime 2}(s)\right)_{\Gamma_0}\,ds \leq C\int_0^t\left\{\left|z'(s)\right|^2+\left|\nabla z(s)\right |^2\right\}\,ds\,.$$Applying Gronwall's lemma in the last inequality we obtain z=0 and therefore, u=\hat {u}. This concludes the proof of Theorem (2.1). \hfill \paragraph{Existence of weak solutions.} We have just proved the existence of strong solutions to (\ref{*}) when u^0 and u^1 are smooth. Now by a density argument and a procedure analogous to the one in the third estimate, we prove the existence of a weak solution. The main step in this approach is obtaining a sequence that satisfy the hypothesis of compatibility (A.2). For this purpose, we define the following sequence. Given \{u^0,u^1\} in V\times L^2(\Omega ), consider$$u_{\mu}^1\in H_0^1(\Omega )\cap H^2(\Omega )\quad \mbox{such that}\quad u_{\mu}^1\rightarrow u^1\quad \mbox{in } L^2(\Omega )\,,$$and$$u_{\mu}^0\in D(-\Delta )=\{u\in V\cap H^2(\Omega );\frac {\partial u}{\partial\nu}=0\quad\mbox{on }\Gamma_0\}\,, \quad \mbox{such that } u_{\mu}^0\rightarrow u^0\quad\mbox{in }\,V. $$Uniqueness of a weak solution is guaranteed by the Visik-Ladyshenskaya method. See for example Lions and Magenes [9, section 8]. \section{Asymptotic behaviour} In this section we prove exponential decay for strong solutions of (\ref{*}), and by a density argument we obtain the same results for weak solutions. Let us consider the modified energy$$ E(t)=e(t)+\frac 1{\gamma +2}\int_\Omega |u(x,t)|^{\gamma +2}\,dx\,, $$which by (\ref{eq:2.5}) satisfies \begin{eqnarray} E'(t)&\leq&\frac 12a'(t,u,u)+\frac 12\int_\Omega K_t(x,t)|u'|^2\,dx\label{eq:4.2} \\ &&-\int_{\Gamma_0}(m\cdot\nu )|u'|^2\,d\Gamma +C(t)\int_\Omega (1+|u'\|\nabla u|)\,dx\,.\nonumber \end{eqnarray} % Let \mu and \lambda be positive constants such that \begin{eqnarray} &\int_{\Gamma_0}(m\cdot\nu )v^2\,d\Gamma\leq\mu\int_\Omega |\nabla v|^2\,dx\quad \forall v\in \, V& \label{eq:4.3} \\ &|v|^2\leq\lambda |\nabla v|^2\quad \forall v\in V\,.& \label{eq:4.4} \end{eqnarray} For an arbitrary \epsilon>0 define the perturbed energy $$E_\epsilon(t)=E(t)+\epsilon\psi (t)\,,\label{eq:4.5}$$ where $$\psi (t)=2\int_\Omega K(x,t)u'(m\cdot\nabla u)\,dx+\theta\int_\Omega K(x,t)u'u\,dx\,, \label{eq:4.6}$$ \theta \in ]n-2,n[, and \theta >\frac {2n}{\gamma +2}. For short notation, put $$k_1=\min\left\{2(\theta -n+2),2(n-\theta ),(\gamma +2)(\theta -\frac { 2n}{\gamma +2})\right\}>0\,.\label{eq:4.7}$$ \begin{proposition} There exists \delta_0>0 such that$$|E_\epsilon(t)-E(t)|\leq\epsilon\delta_0E(t),\, \forall t\geq 0\;\forall\epsilon >0\,.$$\end{proposition} \paragraph{Proof:} From (\ref{eq:2.1}), (\ref{eq:2.17}), (\ref{eq:4.4}), and (\ref{eq:4.6}) we obtain \begin{eqnarray*} |\psi (t)|&\leq& 2a_0^{-1/2}\|K\|_{\infty}^{1/2}R(x^0)|\sqrt {K}u' (t)|a^{1/2}(t,u,u)\\ &&+a_0^{-1/2}\lambda^{1/2}\theta \|K\|_{\infty}^{1/2}|\sqrt {K}u' (t)|a^{1/2}(t,u,u)\\ &\leq& a_0^{-1/2}\|K\|_{\infty}^{1/2}(2R(x^0)+\lambda^{1/2}\theta)E(t)\,. \end{eqnarray*} Putting \delta_0=a_0^{-1/2}\|K\|_{\infty}^{1/2}(2R(x^0)+\lambda^{1/2}\theta ), we deduce$$|E_\epsilon(t)-E(t)|=\epsilon |\psi (t)|\leq\epsilon\delta_0E(t)\,. $$Which proves this proposition. \hfill \smallskip For a positive constant M, let \begin{eqnarray*} H(t)&=&M\big(\|\nabla a(t)\|_{L^\infty(\Omega )}+\|\nabla K(t) \|_{L^\infty(\Omega )}\\ &&+\|a_t(t)\|_{L^\infty(\Omega )}+\|K_t(t)\|_{L^\infty( \Omega )}+C(t)\big)\,. \end{eqnarray*} \begin{proposition} There exist positive constants \delta_1,\delta_2,\epsilon_1 such that$$E_\epsilon'(t)\leq -\epsilon\delta_1E(t)+H(t)E(t)+\delta_2C(t)\,,$$for all t\geq 0 and for all \epsilon\in (0,\epsilon_1]. \end{proposition} \paragraph{ Proof:} Differentiating each term in (\ref{eq:4.6}) with respect to t and substituting Ku''=-A(t)u-F(x,t,u,\nabla u) in the expression obtained, \begin{eqnarray*} \lefteqn{\psi'(t)} && \\ &=&2\int_\Omega K_tu'(m\cdot\nabla u)\,dx-2\int_\Omega A(t)u(m\cdot \nabla u)\,dx\\ &&-2\int_\Omega F(x,t,u,\nabla u)(m\cdot\nabla u)\,dx +2\int_\Omega Ku'(m\cdot\nabla u')\,dx+\theta\int_\Omega K_tu'u\,dx\\ &&-\theta\int_\Omega A(t)uu\,dx-\theta\int_\Omega F(x,t,u,\nabla u)u\,dx+ \theta\int_\Omega K|u'|^2\,dx\,. \end{eqnarray*} From (\ref{eq:2.6}) and the above identity we have \begin{eqnarray} \psi'(t) &\leq& 2\int_\Omega K_tu'(m\cdot\nabla u)\,dx-2\int_\Omega A(t)u(m\cdot\nabla u)\,dx \nonumber \\ &&-2\int_\Omega |u|^{\gamma}u(m\cdot\nabla u)\,dx +2C(t)\int_\Omega (1+|\nabla u\|m\cdot\nabla u|)\,dx \nonumber\\ &&+2\int_\Omega Ku'(m\cdot\nabla u')\,dx +\theta\int_\Omega K_tu'u\,dx-\theta\int_\Omega A(t)uu\,dx \label{eq:4.8}\\ &&-\theta\int_\Omega F(x,t,u,\nabla u)u\,dx +\theta\int_\Omega K|u'|^2\,dx\,.\nonumber \end{eqnarray} Now, we estimate one by one the terms on the right-hand side of the above inequality. \paragraph{Estimate for I_1:=-2\int_\Omega A(t)u(m\cdot\nabla u)\,dx.} Using Green and Gauss formula, we obtain \begin{eqnarray} I_1&=&(n-2)\int_\Omega a(x,t)|\nabla u|^2\,dx+\int_\Omega (\nabla a\cdot m)|\nabla u|^2\,dx\nonumber\\ &&-\int_\Gamma a(x,t)(m\cdot\nu )|\nabla u|^2\,d\Gamma +2\int_\Gamma \frac {\partial u}{\partial\nu_A}(m\cdot\nabla u)\,d\Gamma \,. \label{eq:4.9}\end{eqnarray} \paragraph{Estimate for I_2:=-2\int_\Omega |u|^{\gamma}u(m\cdot\nabla u)\,dx.} By the Gauss formula, \begin{eqnarray} I_2&=&-\frac 2{\gamma +2}\int_\Omega \nabla (|u|^{\gamma +2})\cdot m\,dx \label{eq:4.10}\\ &=&\frac {2n}{\gamma +2}\int_\Omega |u|^{\gamma +2}\,dx-\frac 2{\gamma +2}\int_\Gamma (m\cdot\nu )|u|^{\gamma +2}\,d\Gamma\,.\nonumber \end{eqnarray} From (\ref{eq:1.2}) and noting that u|_{\Gamma_1}=0, we have $$-\frac 2{\gamma +2}\int_\Gamma (m\cdot\nu )|u|^{\gamma +2}\,d\Gamma \leq 0\,.\label{eq:4.11}$$ \paragraph{Estimate for I_3:=2\int_\Omega Ku'(m\cdot\nabla u')\,dx.} By Gauss Theorem we get \begin{eqnarray} I_3&=&\int_\Omega K(x,t)\,\,m\cdot\nabla (|u'|^2)\,dx \label{eq:4.12}\\ &=&-\int_\Omega (\nabla K\cdot m)|u'|^2\,dx-n\int_\Omega K(x,t)|u' |^2dx+\int_{\Gamma_0}(m\cdot\nu )K(x,t)|u'|^2d\Gamma . \nonumber \end{eqnarray} \paragraph{Estimate for I_4:=-\theta\int_\Omega A(t)uu\,dx.} By Green's formula and observing that \frac {\partial u}{\partial \nu_A}=-(m\cdot\nu )u' on \Gamma_0, it follows that $$I_4=-\theta\int_\Omega a(x,t)|\nabla u|^2\,dx-\theta\int_{\Gamma_ 0}(m\cdot\nu )u'u\,d\Gamma\, .\label{eq:4.13}$$ \paragraph{Estimate for I_5:=-\theta\int_\Omega F(x,t,u,\nabla u)u\,dx.} From (\ref{eq:2.5}) we deduce that $$I_5\leq -\theta\int_\Omega |u|^{\gamma +2}\,dx+\theta C(t)\int_{ \Omega}(1+|u\|\nabla u|)\,dx\,.\label{eq:4.14}$$ Thus, substituting (\ref{eq:4.9})--(\ref{eq:4.14}) in (\ref{eq:4.8}) we conclude that \begin{eqnarray} \psi'(t) &\leq& (\theta -n)\int_\Omega K(x,t)|u'|^2\,dx+(n-2-\theta ) \int_{\Omega}a(x,t)|\nabla u|^2\,dx \label{eq:4.15}\\ &&+(\frac {2n}{\gamma +2}-\theta )\int_{\Omega}|u|^{\gamma +2}\,dx +\int_\Omega (\nabla a\cdot m)|\nabla u|^2\,dx\nonumber\\ &&-\int_\Omega (\nabla K\cdot m)|u'|^2\,dx +2\int_\Omega K_tu'(m\cdot\nabla u)\,dx +\theta\int_\Omega K_tu'u\,dx \nonumber\\ &&+2C(t)\int_\Omega (1+|\nabla u\|m\cdot\nabla u|)\,dx +\theta C(t)\int_\Omega (1+|u\|\nabla u|)\,dx \nonumber\\ &&-\int_\Gamma (m\cdot\nu )a(x,t)|\nabla u|^2\,d\Gamma +2\int_\Gamma \frac {\partial u}{\partial\nu_A}(m\cdot\nabla u)\,d\Gamma \nonumber\\ &&+\int_{\Gamma_0}(m\cdot\nu )K(x,t)|u'|^2\,d\Gamma -\theta\int_{\Gamma_0}(m\cdot\nu )u'u\,d\Gamma \,.\nonumber \end{eqnarray} % On the other hand, \frac {\partial u}{\partial x_k}=\frac { \partial u}{\partial\nu}\nu_k on \Gamma_1 implies$$m\cdot\nabla u=(m\cdot\nu )\frac {\partial u}{\partial\nu} \quad\mbox{and}\quad |\nabla u|^2=(\frac {\partial u}{\partial\nu})^2 \quad\mbox{on }\quad \Gamma_1\,.$$Consequently, \begin{eqnarray} \lefteqn{-\int_\Gamma (m\cdot\nu )a(x,t)|\nabla u|^2\,d\Gamma }\label{eq:4.16}\\ &=&-\int_{\Gamma_0}(m\cdot\nu )a(x,t)|\nabla u|^2\,d\Gamma - \int_{_{}\Gamma_1}(m\cdot\nu )a(x,t)(\frac {\partial u}{\partial\nu})^2 \,d\Gamma \nonumber \end{eqnarray} and $$2\int_\Gamma \frac {\partial u}{\partial\nu_A}(m\cdot\nabla u) \,d\Gamma =-2\int_{\Gamma_0}(m\cdot\nu )u'(m\cdot\nabla u)\,d\Gamma + 2\int_{\Gamma_1}a(x,t)(m\cdot\nu )(\frac {\partial u}{\partial\nu} )^2\,d\Gamma\,.\label{eq:4.17}$$ In the above equality, we used that$\frac {\partial u}{\partial \nu_A}=-(m\cdot\nu )u'$on$\Gamma_0$. Replacing (\ref{eq:4.16}) and (\ref{eq:4.17}) in (\ref{eq:4.15}), and using that$\int_{\Gamma_1}a(x,t)(m\cdot\nu )(\frac {\partial u}{\partial\nu} )^2\,d\Gamma\leq 0$, we obtain \begin{eqnarray} \psi'(t) &\leq& (\theta -n)\int_\Omega K(x,t)|u'|^2\,dx+(n-2-\theta )\int_{ \Omega}a(x,t)|\nabla u|^2\,dx \label{eq:4.18} \\ &&+(\frac {2n}{\gamma +2}-\theta )\int_{\Omega}|u|^{\gamma +2}\,dx +\int_\Omega (\nabla a\cdot m)|\nabla u|^2\,dx \nonumber \\ &&-\int_\Omega (\nabla K\cdot m)|u'|^2\,dx +2\int_\Omega K_tu'(m\cdot\nabla u)\,dx +\theta\int_\Omega K_tu'u\,dx\nonumber \\ &&+2C(t)\int_\Omega (1+|\nabla u\|m\cdot\nabla u|)\,dx +\theta\int_\Omega C(t)(1+|u\|\nabla u|)\,dx \nonumber \\ &&-\int_{\Gamma_0}(m\cdot\nu )a(x,t)|\nabla u|^2\,d\Gamma -2\int_{\Gamma_ 0}(m\cdot\nu )u'(m\cdot\nabla u)\,d\Gamma \nonumber\\ &&+\int_{\Gamma_0}(m\cdot\nu )K(x,t)|u'|^2\,d\Gamma -\theta\int_{\Gamma_0}(m\cdot\nu )u'u\,d\Gamma \,.\nonumber \end{eqnarray} However, since \begin{eqnarray*} \lefteqn{ -2\int_{\Gamma_0}(m\cdot\nu )u'(m\cdot\nabla u)\,d\Gamma }&& \\ &\leq& a_0^{-1}{\mathbb R}^2(x^0)\int_{\Gamma_0}(m\cdot\nu )|u'|^2\,d\Gamma +\int_{\Gamma_0}(m\cdot\nu )a(x,t)|\nabla u|^2\,d\Gamma \,, \end{eqnarray*} from (\ref{eq:4.18}) it results that \begin{eqnarray} \lefteqn{ \psi'(t) } && \nonumber \\ &\leq& (\theta -n)\int_\Omega K(x,t)|u'|^2\,dx+(n-2-\theta )\int_{ \Omega}|\nabla u|^2\,dx \label{eq:4.19} \\ &&+(\frac {2n}{\gamma +2}-\theta )\int_\Omega |u|^{\gamma +2}\,dx +\int_\Omega (\nabla a\cdot m)|\nabla u|^2\,dx-\int_\Omega (\nabla K\cdot m)|u'|^2\,dx \nonumber \\ &&+2\int_\Omega K_tu'(m\cdot\nabla u)\,dx +\theta\int_\Omega K_tu'u\,dx+2C(t)\int_\Omega (1+|\nabla u\|m\cdot \nabla u|)\,dx \nonumber \\ &&+\theta\int_\Omega C(t)(1+|u\|\nabla u|)\,dx +a_0^{-1}{\mathbb R}^2(x^0)\int_{\Gamma_0}(m\cdot\nu )|u'|^2\,d\Gamma \nonumber \\ &&+\int_{\Gamma_0}(m\cdot\nu )K(x,t)|u'|^2\,d\Gamma -\theta\int_{\Gamma_0}(m\cdot\nu )u'u\,d\Gamma \,. \nonumber \end{eqnarray} % Let$k_2$be a positive real number such that$0