\input amstex \documentstyle{amsppt} \magnification=\magstephalf \hcorrection{1cm} \vcorrection{-6mm} \nologo \TagsOnRight \NoBlackBoxes \headline={\ifnum\pageno=1 \hfill\else% {\tenrm\ifodd\pageno\rightheadline \else \leftheadline\fi}\fi} \def\rightheadline{EJDE--1998/28\hfil A hyperbolic problem \hfil\folio} \def\leftheadline{\folio\hfil G. G. Doronin, N. A. Lar'kin, \& A. J. Souza \hfil EJDE--1998/28} \def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt % Electronic Journal of Differential Equations, Vol. {\eightbf 1998}(1998), No.~28, pp.~1--10.\hfil\break ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \hfill\break ftp 147.26.103.110 or 129.120.3.113 (login: ftp)\bigskip} } \topmatter \title A hyperbolic problem with nonlinear second-order boundary damping \endtitle \thanks {\it 1991 Mathematics Subject Classifications:} 35A05, 35L20, 35A35, 49M15.\hfil\break\indent {\it Key words and phrases:} Initial Boundary-Value Problem, Faedo-Galerkin Method. \hfil\break\indent \copyright 1998 Southwest Texas State University and University of North Texas.\hfil\break\indent Submitted May 22, 1998. Published October 30, 1998.\hfil\break\indent Partially supported by the Conselho Nacional de Desenvolvimento Tecnol\'ogico e Cient\'{\i}fico, \hfil\break\indent Brazil, under Grants 523258/95-0, 450042/97-9 and 452722/97-8. \endthanks \author G. G. Doronin, N. A. Lar'kin, \& A. J. Souza \endauthor \address G. G. Doronin \hfill\break Current address: DME-CCT-UFPB, CEP 58109-970, Campina Grande, PB, Brazil. Permanent address: 630090, Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia \endaddress \email gleb\@dme.ufpb.br \endemail \address N. A. Lar'kin \hfill\break Departamento de Matem\'atica, Funda\c{c}\~ao Universidade Estadual de Maring\'a, CEP 97020-900, Maring\'a, PR, Brazil \endaddress \email nalarkin\@gauss.dma.uem.br \endemail \address A. J. Souza \hfill\break Departamento de Matem\'atica e Estat\'{\i}stica, Universidade Federal da Para\'{\i}ba, CEP 58109-970, Campina Grande, PB, Brazil \endaddress \email cido\@dme.ufpb.br \endemail \abstract The initial boundary value problem for the wave equation with nonlinear second-order dissipative boundary conditions is considered. Existence and uniqueness of global generalized solutions are proved. \endabstract \endtopmatter \document \head{ 1. Introduction}\endhead In [1], J.L. Lions considers nonlinear problems on manifolds in which the unknown $\omega$ satisfies the Laplace equation in a cylinder $Q$ and a nonlinear evolution equation of the form $$ \frac{\partial \omega}{\partial \nu}+\omega_{tt}+|\omega_t|^{\rho}\omega_t=0 \tag{1.1} $$ on the lateral boundary $\Sigma$ of $Q$. Here $\nu$ is an outward normal vector on $\Sigma$. This problem models water waves with free boundaries ([2], [3]). The boundary condition $$ \frac{\partial \omega}{\partial \nu}+|\omega_t|^{\rho}\omega_t=0 \tag{1.2} $$ arises when one studies flows of a gas in channels with porous walls [4,\,5]. The presence of the second derivative with respect to $t$ in the boundary condition is due to internal forces acting on particles of the medium at the outward boundary. Motivated by this, we study in the present paper the wave equation $$ u_{tt}-\Delta u=f\quad \text{in}\quad Q \tag{1.3} $$ with the nonlinear boundary condition $$ \frac{\partial u}{\partial \nu}+K(u)u_{tt}+|u_t|^{\rho} u_t=0\quad\text{on} \quad \Sigma \tag{1.4} $$ and with the initial data $$ u(x,0)=u_t(x,0)=0. \tag{1.5} $$ The term $K(u)u_{tt}$ models internal forces when the density of the medium depends on the displacement. In [1] it is shown that (1.1) can be replaced by the evolution equation $$ u_{tt}+A(u)+|u_t|^{\rho}u_t=0\quad \text{on}\quad\Sigma, $$ where $A$ is a linear positive self-adjoint operator. In that sense, the expression (1.2) looks like a semilinear hyperbolic equation on the manifold $\Sigma$. Equation (1.4) also behaves as a hyperbolic equation with nonlinear principal operator. Generally speaking, quasilinear hyperbolic equations do not have global regular solutions. There are examples of ``blow-up'' at a finite time. (See, for instance, [6].) Nevertheless, the presence of linear damping allows proof of the existence of global solutions for small initial data ([7]). Moreover, a nonlinear damping makes it possible to prove global existence theorems for some quasilinear wave equations without restrictions on a size of the initial conditions ([8], [9]). Here we use the ideas from [8] to prove the existence of global generalized solutions to the problem (1.3)-(1.5). We exploit the Faedo-Galerkin method, a priori estimates and compactness arguments. Uniqueness is proved in the one-dimensional case. We consider the classical wave equation only to simplify calculations. Similar results hold for a second-order evolution equation of the form $$ u_{tt}+A(t)u+F(u,u_t)=f, $$ where $A(t)$ is a linear, strictly elliptic operator, and $F(u,u_t)$ is a suitable function of $u$ and $u_t$. Moreover, hyperbolic-parabolic or elliptic equations also may be considered. \head { 2. The Main Result}\endhead For $T>0$, let $\Omega$ be a bounded open set of $R^n$ with sufficiently smooth boundary $\Gamma$ and $Q=\Omega\times(0,T)$. We consider the hyperbolic problem $$ u_{tt}-\Delta u=f(x,t)\;,\quad(x,t)\in Q; \tag{2.1} $$ $$ \left. \left( \frac{\partial u}{\partial \nu }+K(u)u_{tt}+|u_t|^{\rho}u_t\right) \right|_{ \Sigma_1}=0;\quad \left. u\right|_{\Sigma_0}=0; \tag{2.2} $$ $$ u(x,0)=u_t(x,0)=0. \tag{2.3} $$ Here $K(u)$ is a continuously differentiable positive function; $\nu $ is the outward unit normal vector on $\Gamma$; $\Gamma =\Gamma _0 \cup \Gamma _1$; $\Gamma _0\cap \Gamma _1=\varnothing$ ; $\Sigma_i=\Gamma _i\times (0,T)\ \ (i=0,1)$;\ \ $\rho \in (1,\infty)$. We denote by $H_1(\Omega)$ the Sobolev space $H^1(\Omega )$ with the condition $u|_{\Gamma_0}=0$; $(u,v)(t)=\int_\Omega u(x,t)v(x,t)\,dx$; $||u||$ is the norm in $L^2(\Omega )$: $||u||^2(t)=(u,u)(t)$; $\Delta u=\sum_{i=1}^n\partial ^2 u/\partial x_i^2$. \subheading{Definition} A function $u(x,t)$ such that $$ \align &u\in L^\infty (0,T;H_1(\Omega )),\\ & u_t\in L^\infty (0,T;H_1(\Omega ))\cap L^{\rho +2}(\Sigma_1), \\ & u_{tt}\in L^\infty (0,T;L^2(\Omega )\cap L^2(\Gamma_1)), \\ & u(x,0)=u_t(x,0)=0 \endalign $$ is a generalized solution to (2.1)-(2.3) if for any functions $v\in H_1(\Omega)\cap L^{\rho +2}(\Gamma)$ and $\varphi\in C^1(0,T)$ with $\varphi(T)=0$ the following identity holds: $$\gathered \int\limits_0^T\left\{ (u_{tt},v)(t)+(\nabla u,\nabla v)(t)+\int\limits_{\Gamma_1}\left[ |u_t|^\rho u_t-K^{\prime}(u)u_t^2 \right] v\,d\Gamma \right\}\varphi(t)\,dt \\ - \, \int\limits_0^T \varphi^{\prime}(t)\int\limits_{\Gamma_1}K(u)u_t v\,d{\Gamma}\,dt= \int\limits_0^T\,(f,v)\varphi(t)\,dt \,. \endgathered \tag{2.4} $$ We consider functions $K(u)$ satisfying the assumptions $$ \gather 00$ there exists at least one generalized solution to the problem (2.1)-(2.3). If $n=1$, this solution is unique. \endproclaim \demo{Proof} We prove the existence part of the Theorem by the Faedo-Galerkin method. First, we construct approximations of the generalized solution. Then we obtain a priori estimates necessary to guarantee convergence of approximations. Finally, we prove the uniqueness in the one-dimensional case. \enddemo \head {3. Approximate solutions} \endhead Let $ \{w_j(x)\}$ be a basis in $H_1(\Omega )\cap L^{\rho +2}(\Gamma_1)$. We define the approximations $$ u^N(x,t)=\sum_{i=1}^Ng_i(t)w_i(x), \tag{3.1} $$ where $g_i(t)$ are solutions to the Cauchy problem $$ \gather (f,w_j)(t)= (u_{tt}^N,w_j)(t)+(\nabla u^N,\nabla w_j)(t) +\int_{\Gamma _1}\left\{ K(u^N)u_{tt}^N+|u_t^N|^\rho u_{t}^N\right\} w_j\,d\Gamma\,; \tag{3.2} \\ g_j(0)=g_j^{\prime }(0)=0;\;\;j=1,...,N\text{.} \tag{3.3} \endgather $$ It can be seen that (3.2) is not a normal system of ODE; therefore, we can not apply the Caratheodory theorem directly. To overcome this difficulty, we have to prove that the matrix $A$ defined by $$ (Ag^{\prime \prime })_j=g_j^{\prime \prime }(t)+\int\limits_{\Gamma _1}\left\{ K(u^N)\sum_{i=1}^Ng_i^{\prime \prime }(t)w_i(x)\right\} w_j(x)\,d\Gamma \;;\;\ j=1,...,N \tag{3.4} $$ has an inverse. Multiplying (3.2) by $g_j^{\prime \prime }(t)$ and summing over $j$, we obtain the quadratic form $$ q\,(g_1^{\prime \prime },...,g_N^{\prime \prime })=\sum_{j=1}^N\left[ (g_j^{\prime \prime })^2+\sum_{i=1}^N\int\limits_{\Gamma _1}K(u^N)w_iw_jd\Gamma \;g_i^{\prime \prime }g_j^{\prime \prime }\right]. $$ The condition $K(u)\geq K_0>0$ implies that for any $g^{\prime \prime} (t)\neq 0$ $$ q\,=\sum_{j=1}^N(g_j^{\prime \prime })^2+\int\limits_{\Gamma _1}K(u^N)\left( \sum_{j=1}^Ng_j^{\prime \prime }w_j\right) ^2d\Gamma \geq \sum_{j=1}^N(g_j^{\prime \prime })^2+K_0||u^N_{tt}||_{L^2(\Gamma _1)}^2>0. $$ Hence, the quadratic form $q$ is positive definite and all eigenvalues of the symmetric matrix $A$ in (3.4) are positive. Thus, (3.2) can be reduced to normal form and, by the Caratheodory theorem, the problem (3.2),(3.4) has solutions $g_j(t)\in H^3(0,t_N)$ and all the approximations (3.1) are defined in $ (0,t_N)$. \head{ 4. A priori estimates} \endhead Next, we need a priori estimates to show that $t_N=T$ and to pass to the limit as $N\rightarrow\infty$. To simplify the exposition, we omit the index $N$ whenever it is unambiguous to do so. Multiplying (3.2) by $2g_j^{\prime }$ and summing from $j=1$ to $j=N$, we obtain $$ \align 2(f,u_t)(t) =&\frac d{dt}\left( ||u_t||^2+||\nabla u||^2\right)(t) +\,2\int\limits_{\Gamma_1}|u_t|^{\rho +2}d\Gamma \\ &+\int\limits_{\Gamma _1}\left\{ \frac d{dt}\left( K(u)u_t^2\right) -K^{\prime }(u)(u_t)^3\right\}\, d\Gamma\,. \endalign $$ Integrating with respect to $\tau $ from $0$ to $t$, we get $$ \align 2\int\limits_0^t(f,u_\tau )\,d\tau=& \left( ||u_t||^2+||\nabla u||^2\right)(t) \\ &+\,2\int\limits_0^t\int\limits_{\Gamma _1}\left\{ |u_\tau |^{\rho +2}-\frac 12K^{\prime }(u)(u_\tau )^3\right\} d\Gamma \,d\tau +\int\limits_{\Gamma _1}K(u)u_t^2\,d\Gamma\,. \endalign $$ Notice that $$\gather 2\int\limits_0^t\int\limits_{\Gamma _1}|u_\tau |^2\left\{ |u_\tau |^\rho -\frac 12K^{\prime }(u)u_\tau \right\}\, d\Gamma \,d\tau \\ \geq 2\int\limits_0^t\int\limits_{\Gamma _1}|u_\tau |^2\left\{ |u_\tau |^\rho -\varepsilon |u_\tau |^\rho -C(\varepsilon )|K^{\prime }(u)|^{\frac \rho {\rho -1}}\right\}\, d\Gamma \,d\tau\,, \endgather $$ where $\varepsilon$ is an arbitrary positive number. From now on, we denote by ``$C$'' all constants independent of $N$. Fixing $\varepsilon =1/2$, taking into account (2.6), and applying the Cauchy-Schwarz inequality, we get $$ \gathered \left( ||u_t||^2+||\nabla u||^2\right) (t)+\int\limits_0^t\int\limits_{\Gamma _1}|u_\tau |^{\rho +2}\,d\Gamma\,d\tau +\int\limits_{\Gamma _1}K(u)u_t^2\,d\Gamma \\ \leq \int\limits_0^t\left( ||f||^2+||u_\tau ||^2\right) (\tau )\,d\tau +C\int\limits_0^t\int\limits_{\Gamma _1}|u_\tau |^2(1+K(u))\,d\Gamma\,d\tau\,. \endgathered \tag{4.1} $$ Note that $K(u)\geq C_0(1+K(u))$ where $2 C_0=\min \{1,K_0\}$. Therefore, for the function $$ E_1(t)=\left( ||u_t||^2+||\nabla u||^2\right) (t)+C_0\int\limits_{\Gamma _1}(1+K(u))|u_t|^2d\Gamma $$ we have from (4.1) the inequality $$ E_1(t)\leq C\left(1+\int\limits_0^tE_1(\tau )\,d\tau\right). $$ By Gronwall's lemma, we conclude that, for all $t\in (0,T)$ and for all $N\geq 1$, $$ E_1(t)\leq C. $$ This and (4.1) give that for all $t\in(0,T)$, $$ \gather \int\limits_0^t\int\limits_{\Gamma _1}|u_\tau |^{\rho +2}\,d\Gamma\,d\tau \leq C, \\ \int_{\Gamma_1}K(u^N)(u_t^N)^2\,d\Gamma \leq C, \endgather $$ where $C$ does not depend on $N$. In order to obtain the second a priori estimate, we observe that $$ ||u_{tt}||(0)\leq ||f||(0); \tag{4.2} $$ $$ \int\limits_{\Gamma _1}u_{tt}^2(x,0)d\Gamma \leq ||f||^2/K(0). \tag{4.3} $$ Indeed, multiplying (3.2) by $g_j^{\prime \prime}(0)$, summing over $j$, and setting $t=0$, we obtain $$ (u_{tt},u_{tt})(0)+\int\limits_{\Gamma _1}K(0)u_{tt}^2(x,0)\,d\Gamma =(f,u_{tt})(0) $$ which implies (4.2). Consequently, $$ \int\limits_{\Gamma _1}K(0)u_{tt}^2(x,0)\,d\Gamma \leq ||f||(0)\cdot ||u_{tt}||(0)\leq ||f||^2(0), $$ which gives (4.3). Differentiating (3.2) with respect to $t$, multiplying by $g_j''$, and summing over $j$, we obtain the identity $$ \align (f_t,u_{tt})(t)=&\dfrac 12\dfrac d{dt}\left( ||u_{tt}||^2+||\nabla u_t||^2\right) (t) \\ &+\int\limits_{\Gamma _1}\left\{ K(u)u_{tt}u_{ttt}+K^{\prime}(u)u_tu_{tt}^2+ (\rho +1)|u_t|^\rho u_{tt}^2\right\}\, d\Gamma\,. \endalign $$ Notice that $$ K(u)u_{tt}u_{ttt}=\dfrac 12\dfrac d{dt}\left( K\left( u\right) u_{tt}^2\right) -\dfrac 12K^{\prime }(u)u_tu_{tt}^2\ $$ and $$ \align \left|\, \int\limits_{\Gamma _1}K^{\prime }(u)u_tu_{tt}^2d\Gamma \right| \leq & \varepsilon \int\limits_{\Gamma _1}|u_t|^\rho |u_{tt}|^2d\Gamma +C(\varepsilon )\int\limits_{\Gamma _1}|K^{\prime }(u)|^{\frac \rho {\rho -1}}\cdot |u_{tt}|^2\,d\Gamma \\ \leq & \varepsilon \int\limits_{\Gamma _1}|u_t|^\rho |u_{tt}|^2d\Gamma +C(\varepsilon )\int\limits_{\Gamma _1}(1+K(u))|u_{tt}|^2\,d\Gamma \,. \endalign $$ Setting $\varepsilon =\rho$, we have $$ \gathered \dfrac 12\dfrac d{dt}\left( ||u_{tt}||^2+||\nabla u_t||^2+\int\limits_{\Gamma _1}K(u)u_{tt}^2d\Gamma \right) (t)+\int\limits_{\Gamma _1}|u_t|^\rho |u_{tt}|^2d\Gamma \\ \leq \left(||f_t||^2+||u_{tt}||^2\right)(t)+ C\int\limits_{\Gamma_1}(1+K(u))|u_{tt}|^2d\Gamma \,. \endgathered \tag{4.4} $$ Defining $E_2(t)$ as $$ E_2(t)=\left(||u_{tt}||^2+||\nabla u_t||^2+C_0\int\limits_{\Gamma _1}(1+K(u))|u_{tt}|^2\,d\Gamma\right)(t) $$ and taking into account (4.2), (4.3), we reduce (4.4) to the form $$ E_2(t)\leq C\left( 1+\int\limits_0^tE_2(\tau )d\tau \right) . $$ By Gronwall's lemma, for all $t\in (0,T),\;N\geq 1$ we obtain $$ E_2(t)\leq C \,. $$ Taking into consideration that $u|_{\Sigma_0}=0$, we obtain the following statements $$\align &u^N\in L^\infty (0,T;H^1(\Omega )); \\ &u_t^N\in L^\infty (0,T;H^1(\Omega ))\cap L^{\rho +2}(\Sigma )\cap L^\infty (0,T;L^2(\Gamma )); \\ &u_{tt}^N\in L^\infty (0,T;L^2(\Omega )\cap L^2(\Gamma )); \tag{4.5}\\ &\dfrac \partial {\partial t}|u_t^N|^{1+\rho /2}\in L^2(\Sigma ); \\ &K^{1/2}(u^N)u_{tt}^N\in L^\infty (0,T;L^2(\Gamma )). \endalign $$ \head {5. Passage to the limit} \endhead Multiply (3.2) by $\varphi \in C^1(0,T)$ with $\varphi (T)=0$ and integrate with respect to $t$ from $0$ to $T$. After integration by parts, we obtain $$ \gathered \int\limits_0^T \left\{ (u_{tt}^N,w_j) +(\nabla u^N,\nabla w_j)+\int\limits_{\Gamma_1}|u_t^N|^{\rho}u_t^N w_j\,d\Gamma \right\} \varphi(t)\,dt \\ -\int\limits_0^T \varphi^{\prime}(t)\int\limits_{\Gamma_1} K(u^N)u_t^Nw_j(x)\,d\Gamma \,dt+\varphi(t)K(u^N)u_t^N|_0^T \\ -\int\limits_0^T\varphi(t)\int\limits_{\Gamma_1} K^{\prime}(u^N)(u_t^N)^2w_j\,d\Gamma\,dt= \int\limits_0^T(f,w_j)\varphi(t)\,dt. \endgathered \tag{5.1} $$ Because of (4.5) we can extract a subsequence ${u^\mu}$ from ${u^N}$ such that: $$ \align &u^{\mu} \rightarrow u \text{ weakly star in } L^{\infty}(0,T;H_1(\Omega));\\ &u_t^{\mu} \rightarrow u_t \text{ weakly star in } L^{\infty}(0,T;H_1(\Omega))\cap L^{\rho +2}(\Sigma);\\ &u_{tt}^{\mu} \rightarrow u_{tt}\text{ weakly star in } L^{\infty}(0,T;L^2(\Omega) \cap L^2(\Gamma));\\ &u^{\mu}, u_t^{\mu}\rightarrow u, u_t \quad \text{ a.e. on }\Sigma. \endalign $$ Therefore, $$ \gather |u_t^{\mu}|^{\rho}u_t^{\mu} \in L^q(\Sigma),\ \ q=(\rho+2)/(\rho+1)>1 , \text{ and converges a.e. on }\Sigma; \\ K(u^{\mu})u_t^{\mu}\in L^q(\Sigma), \ \ \text{ and converges a.e. on }\Sigma; \\ K^{\prime}(u^{\mu})(u_t^{\mu})^2 \in L^q(\Sigma), \ \ \text{ and converges a.e. on }\Sigma. \endgather $$ Thus, we are able to pass to the limit in (5.1) to obtain $$\gathered \int\limits_0^T \left\{ (u_{tt},w_j)+(\nabla u,\nabla w_j)+\int\limits_{\Gamma_1} \left( |u_t|^{\rho}u_t-K^{\prime}(u)u_t^2 \right) w_j\,d\Gamma \right\} \varphi(t)\,dt \\ -\int\limits_0^T\varphi^{\prime}(t)\int\limits_{\Gamma_1}K(u)u_tw_j\,d \Gamma\,dt=\int\limits_0^T(f,w_j)\varphi(t)\,dt\,. \endgathered \tag{5.2} $$ It can be seen that all the integrals in (5.2) are defined for any function $\varphi(t)\in C^1(0,T),\ \varphi(T)=0$. Taking into account that $\{w_j(x)\}$ is dense in $H^1(\Omega) \cap L^{\rho+2} (\Gamma)$, we conclude that (2.4) holds. If $n=1,2$, one can get more regular solutions. In this case $u\in L^{\infty}(0,T;L^q(\Gamma))$ for any $q\in [1,\infty)$. Hence, $K(u)u_{tt} \in L^{\infty}(0,T;L^p(\Gamma))$, where $p$ is an arbitrary number from the interval $[1,2)$. This allows us to rewrite (5.2) in the form $$ \int\limits_0^T(f,w_j)\,dt= \int\limits_0^T \left\{ (u_{tt},w_j)+(\nabla u,\nabla w_j)+\int\limits_{\Gamma_1} \left( K(u)u_{tt}+|u_t|^{\rho}u_t \right) w_j\,d\Gamma \right\} \varphi(t)\,dt. $$ Taking into account that almost every point $t\in (0,T)$ is a Lebesgue point and that $w_j(x)$ are dense in $H^1(\Omega)$ and therefore in $L^q(\Gamma)$, we obtain $$ (u_{tt},v)(t)+(\nabla u, \nabla v)(t)+\int_{\Gamma_1}\{K(u)u_{tt} +|u_t|^{\rho}u_t\}v\,d\Gamma=(f,v)(t), $$ where $v$ is an arbitrary function from $H^1(\Omega)$. \head {6. Uniqueness}\endhead Let $n=1$. Let $u$ and $v$ be two solutions to (2.1)-(2.3), and set $z(x,t)=u(x,t)-v(x,t)$. Then for fixed $t$, for every function $\phi \in H_1(\Omega) $, we have $$\gather (z_{tt},\phi)(t)+(\nabla z,\nabla \phi)(t) \\ +\int\limits_{\Gamma_1}\left\{ K(u)z_{tt}+v_{tt}(K(u)-K(v))+|u_t|^{\rho}u_t -|v_t|^{\rho}v_t \right\}\,\phi\,d\Gamma =0\,. \endgather $$ Since $z_t(x,t)\in L^{\infty}(0,T;H_1(\Omega))$, we may take $\phi=z_t$, and this equation can be reduced to the inequality $$ \gather \frac 12 \frac {d}{dt} \left[E(t)+\int\limits_{\Gamma_1}K(u)(z_t)^2\,d\Gamma \right] \\ +\int\limits_{\Gamma_1}\left\{v_{tt}z_t(K(u)-K(v))-\frac 12 K^{\prime}(u)u_t (z_t)^2\right\}\,d\Gamma\leq 0\,. \endgather $$ Here we set $E(t)=\|z_t\|^2(t)+\|\nabla z\|^2(t)$ and use the monotonicity of $|u_t|^{\rho}u_t$, the differentiability of $K$, and the regularity of $K(u)u_{tt}$ (see the end of previous section). Condition (2.6) then implies that $$ \align \frac 12 \frac {d}{dt}& \left[E(t)+\int\limits_{\Gamma_1}K(u)(z_t)^2\,d\Gamma \right] \\ \leq &C\max_{\Gamma_1}(1+K(u))^{\frac{\rho-1}{\rho}}|u_t|\int\limits_{\Gamma_1} (z_t)^2\,d\Gamma+\frac 12 \int\limits_{\Gamma_1}\left\{|z_t|^2+|v_{tt}|^2 |K(u)-K(v)|^2\right\}\,d\Gamma \\ \leq &C\int\limits_{\Gamma_1}|z_t|^2\,d\Gamma+\max_{\Gamma_1}|K(u)-K(v)|^2 \int\limits_{\Gamma_1}|v_{tt}|^2\,d\Gamma \\ \leq& C_1\|z_t\|_{L^2(\Gamma_1)}^2+C_2\|v_{tt}\|_{L^2(\Gamma_1)}^2\cdot \|z\|_{C(\Gamma_1)}^2\,. \endalign $$ Integrating from $0$ to $t$, using (2.5) and the Sobolev embedding theorem ([10]), we obtain $$ \|z_t\|^2(t)+\|\nabla z\|^2(t)+\|z_t\|_{L^2(\Gamma_1)}^2(t)\leq C\int\limits_{0}^{t}\left[ \|z_t\|_{L^2(\Gamma_1)}^2(\tau)+ \|\nabla z\|^2(\tau)\right]\,d\tau\,. $$ This implies that $\|z\|=0$ and $u=v$ a.e. in $Q$. The proof of the Theorem is completed. \subheading{Remark} We use homogeneous initial conditions (2.3) for technical reasons. Non-homogeneous initial data also can be considered without any restrictions on their size ([10]). In fact, suppose that initial conditions are imposed as follows $$ u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x),\quad x\in \Omega. $$ Using the transformation $v(x,t)=u(x,t)-u_0(x)-u_1(x)\cdot t$, we obtain the problem $$ \gather v_{tt}-\Delta v=F(x,t)\quad \ \text{in}\ Q; \tag{6.1} \\ \frac {\partial v}{\partial \nu}+ \frac {\partial \phi}{\partial \nu}+ K(v+\phi)v_{tt}+|v_t+u_1|^{\rho}(v_t+u_1)=0\quad \text{on } \Sigma_1; \tag{6.2} \\ v+\phi =0\quad \quad\text{on } \Sigma_0; \tag{6.3}\\ v(x,0)=v_t(x,0)=0\quad \text{in } \Omega.\tag{6.4} \endgather $$ Here $\phi (x,t)=u_0(x)+u_1(x)\cdot t$ and $\ F(x,t)=(f+\Delta \phi)(x,t)$ are given functions. It is clear that for regular solutions the compatibility conditions $$ \frac {\partial u_0}{\partial \nu}+K(u_0)(f+\Delta u_0)+ |u_1|^{\rho}u_1\ |_{\Gamma_1}=0;\quad u_0\ |_{\Gamma_0}=0 $$ need to be satisfied. This implies that conditions (6.2)-(6.4) are also compatible. If $(u_0,\,u_1)(x)\in H^2(\Omega)$, than $F(x,t)\in H^1(0,T;L^2(\Omega))$. Moreover, if $u_1\in L^{\rho +2}(\Gamma_1)$, then we are able to obtain necessary a priori estimates and to pass to the limit by the method of Sections 4 and 5. Of course, the use of conditions (6.2), (6.3) in place of (2.2) complicates calculations, but does not affect the final result. \subheading{Acknowledgments} We thank the referee for his/her valuable suggestions. \head{ REFERENCES}\endhead \frenchspacing \item{[1]} Lions, J.-L., {\it Quelques m\'{e}thodes de r\'{e}solution des probl\`{e}mas aux limites non lin\'{e}aires}, Paris, Dunod, 1969. \item{[2]} Garipov, R.M., {\it On the linear theory of gravity waves: the theorem of existence and uniqueness}, Archive Rat. Mech. Anal., 24 (1967), 352-367. \item{[3]} Friedman A., Shinbrot M., {\it The initial value problem for the linearized equations of water waves}, J. Math. 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