\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 243, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/243\hfil A mixed problem for semilinear wave equations]
{A mixed problem for semilinear wave equations with acoustic
boundary conditions in domains with non-locally reacting boundary}

\author[C. L. Frota, L. A. Medeiros, A. Vicente \hfil EJDE-2014/243\hfilneg]
{C\'icero L. Frota, Luiz A. Medeiros, Andr\'e Vicente}  % in alphabetical order

\address{C\'icero L. Frota \newline
Universidade Estadual de Maring\'a - DMA. Av. Colombo, 5790\\
CEP: 87020-900, Maring\'a, PR, Brazil}
\email{clfrota@uem.br}

\address{Luiz A. Medeiros \newline
Universidade Federal do Rio de Janeiro
- IM. Av. Athos da Silveira Ramos 149, CT, Cidade Universit\'aria\\
CEP: 21941-909, Rio de Janeiro, RJ, Brazil}
\email{luizadauto@gmail.com}

\address{Andr\'e Vicente \newline
Universidade Estadual do Oeste do Paran\'a - CCET,
Rua Universit\'aria, 2069, Jd. Universit\'ario\\
CEP: 85819-110, Cascavel, PR, Brazil}
\email{andre.vicente@unioeste.br}

\thanks{Submitted July 14, 2014. Published November 18, 2014.}
\subjclass[2000]{35B40, 35L05}
\keywords{Nonlinear wave equation; non-locally reacting boundary;
\hfill\break\indent acoustic boundary conditions; nonlinear boundary condition; 
stabilization}

\begin{abstract}
 In this article we study the existence, uniqueness and asymptotic stability
 of solution to the mixed problem for the semilinear wave equation with
 acoustic boundary conditions in domains with non-locally reacting boundary.
 We also prove the existence and uniqueness of solution to a problem
 with nonmonotone dissipative term.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
%\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
%\newtheorem{definition}[theorem]{Definition}
%\newtheorem{corollary}[theorem]{Corollary}
%\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks


\section{Introduction}

This article is devoted to the study of the existence, uniqueness and uniform stabilization of solutions $(u,\delta)$ for the problem
\begin{equation}\label{1}
 \begin{gathered}
 u''-\Delta u+\rho(u')=F\quad  \text{in }\Omega\times(0,\infty),\\
 u=0\quad \text{on }\Gamma_0\times(0,\infty), \\
 \frac{\partial u}{\partial \nu}=\delta'
 \quad \text{on }\Gamma_1\times(0,\infty),\\
 u'+f\delta''-  c^2\Delta_{\Gamma}\delta+g\delta'+h\delta=0
 \quad \text{on }\Gamma_1\times(0,\infty),\\
 \delta=0 \quad \text{on }\partial\Gamma_1\times(0,\infty),  \\
 u(x,0)=\phi(x),\quad u'(x,0)=\psi(x)\quad x\in\Omega,\\
 \delta(x,0)=\theta(x),\quad \delta'(x,0)
 =\frac{\partial \phi}{\partial \nu}(x),\quad  x\in\Gamma_1,
 \end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^n$ is an open, bounded and connected
set with smooth boundary $\Gamma$; $\Gamma_1$ is an open and connected
set of $\Gamma$ with smooth boundary, $\partial\Gamma_1$, and
$\Gamma_0=\Gamma\backslash\Gamma_1$. Here $'=\frac{\partial}{\partial t}$,
$\Delta =\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$ and $\Delta_{\Gamma}$
are the spatial Laplace and Laplace-Beltrami operators, respectively, $\nu$
is the unit outward normal vector to $\Gamma$, $c$ is a positive constant,
$\rho:\mathbb{R}\to\mathbb{R}$,
$F:\Omega\times (0,\infty)\to\mathbb{R}$,
$f,g,h:\overline{\Gamma_1}\to\mathbb{R}$,
$\phi,\psi:\Omega\to\mathbb{R}$ and $\theta:\Gamma_1\to\mathbb{R}$
are given functions.

When $c=0$ the boundary conditions \eqref{1}$_3$--\eqref{1}$_4$
are the classical acoustic boundary conditions which were introduced by
Beale and Rosencrans \cite{Beale-Rosencrans} in wave propagation literature.
Precisely, they derived
\begin{equation} \label{2}
 \begin{gathered}
 u''-\Delta u=0 \quad \text{in }\Omega\times(0,\infty),\\
 \frac{\partial u}{\partial \nu}=\delta' \quad \text{on }\Gamma\times(0,\infty),\\
 \rho_0u'+m\delta''+d\delta'+k\delta=0 \quad \text{on }\Gamma\times(0,\infty),
 \end{gathered}
\end{equation}
as a theoretical model to describe the acoustic wave motion into a fluid
in $\Omega\subset\mathbb{R}^3$. Here $\rho_0, m, d, k$ are physical constants.
The function $u(x,t)$ is the velocity potential of the fluid and $\delta(x,t)$
models the normal displacement of the point $x\in\Gamma$ in the time $t$.
To obtain the model, Beale and Rosencrans assumed that each point of the
surface $\Gamma$ acts like a spring in response to the excess pressure and
that each point of $\Gamma$ does not influence each other. Surfaces with this
characteristic are called {\it locally reacting}, see Morse and Ingard \cite{Morse}.
Acoustic boundary conditions has been studied by several authors
\cite{Frigeri,Frota-Cousin-Larkin2,Frota-Cousin-Larkin,Frota-Goldstein,
Liu-Sun,Park-Park,Park-Park-Kang}.
Frota, Cousin and Larkin \cite{Frota-Cousin-Larkin2} obtained decay results
to a nonlinear wave equation when $n=1$ and Frota, Cousin and
Larkin \cite{Frota-Cousin-Larkin} and Park and Park \cite{Park-Park}
obtained decay results when $m=0$ in \eqref{2}$_3$.
The results of Liu and Sun \cite{Liu-Sun} and Park and Park \cite{Park-Park}
are about general decay rates.

Recently, in \cite{Frota-Medeiros-Vicente}, we introduced a new physical
 formulation to the acoustic conditions and we proved the existence,
uniqueness and uniform stabilization to \eqref{1} with
\begin{equation}\label{3}
 u''-M\Big(\int_{\Omega}u^2\,dx\Big)\Delta u+\alpha u'+\beta|u'|^{p}u'=0
\quad \text{in }\Omega\times(0,\infty)
\end{equation}
instead of \eqref{1}$_1$, where $\alpha,\beta$ are positive constants,
$p>1$ if $n=2$, $1<p\leq 2$ if $n=3$ and $M:(0,\infty)\to\mathbb{R}$
 is a given function. The boundary conditions \eqref{1}$_3$--\eqref{1}$_4$
were called {\it acoustic boundary conditions to non-locally reacting boundary},
because for its formulation we assumed that the surface $\Gamma_1$ acts
like an elastic membrane in response to the excess pressure. The exponential
decay was obtained by Nakao's Lemma \cite[Theorem 1]{Nakao}, and the assumption
\begin{equation}\label{4}
 c>4k_1k_2,
\end{equation}
was necessary. Here $c$ is the same of \eqref{1}$_4$, $k_1$ is the constant
of continuity of the trace map and $k_2$ is the Poincar\'{e}'s constant.
This is a strong assumption and physically it means that the velocity
of wave propogation on the surface $\Gamma_1$ is higher than a known constant.
The case $\beta=0$ in \eqref{3} was considered by Vicente and
Frota \cite{Vicente-Frota1}, where the authors assumed sufficiently
small data to prove the existence of solutions. The assumption \eqref{4}
also was necessary to prove the asymptotic stability.
See also the recent work due Vicente and Frota \cite{Vicente-Frota2}
were the authors considered nonlinear boundary equation.
See also the papers of Coclite,  Goldstein and  Goldstein
\cite{Coclite-GGoldstein-JGoldstein,Coclite-GGoldstein-JGoldstein1,
Coclite-GGoldstein-JGoldstein2,Coclite-GGoldstein-JGoldstein3}
were the authors studied problems with Wentzell boundary conditions.
An interesting work is due Gal, G. Goldstein and J. Goldstein
\cite{Gal-GGoldstein-JGoldstein} were they proved, in some special case,
the acoustic and Wentzell boundary conditions are closely related.

On this direction it is important to mention the works of Graber and
Said-Houari \cite{GraberSaid} and Graber \cite{Graber} were the
authors also studied problems with nonhomogeneous boundary
condition. Precisely in \cite{GraberSaid} the authors studied the
interaction between dissipative and sources terms in the domain as
well as in the boundary, for the strongly damped wave equation with
boundary conditions like General Wentzell Boundary Condition (GWBC)
type, which in some special case agree with Acoustic Boundary
Conditions (ABC), see section 4 of \cite{Gal-GGoldstein-JGoldstein}.
Moreover in contrast with our results on stability and decay rates,
due to the source term the authors did an analysis of blow up of
solutions. On the other hand in \cite{Graber} the author proved the
existence, uniqueness and uniform decay of finite energy solutions
when the boundary is taken to be porous and the coupling between the
interior and boundary dynamics is assumed to be nonlinear,
furthermore was added to the boundary a frictional damping term as a
feedback control. Theorem 3 of \cite{Graber}, which gives decay
rates result, assumed a geometrical condition on a the set
$\Gamma_{0}$.


Our purpose in this article is to study the existence, uniqueness and
asymptotic behaviour of solution to \eqref{1}. The existence is done
by employing Galerkin's procedure and compactness arguments. Our
main goal is to prove the exponential decay of solution to \eqref{1}
without the assumption \eqref{4}. This is possible because we use an
alternative method based on an integral inequality instead of
Nakao's Lemma. This technics are used by many authors
\cite{Komornik,Martinez-Vancostenoble,Patcheu}. We observe that the
estimates we made here can be adapted to the problem treated by
Frota, Medeiros and Vicente \cite{Frota-Medeiros-Vicente} therefore
here we also improve their results. Moreover, as an application of
\eqref{1}, we study the existence and uniqueness of global solution
to \eqref{1} without the assumption of monotonicity on $\rho$
($\rho'\geq0$). In some related papers, when $\rho$ is non-monotone,
the authors prove the existence of solution by semigroups theory
\cite{Martinez-Vancostenoble,ODowd, Slemrod}. Here we solve the
problem building an appropriate sequence of functions which converge
to the solution of the problem with non-monotone dissipative term, it
is an adaptation of the ideas of Li and Tsai \cite{Meng-Tsai}.

Our paper is organized as follows: In Section 2 we present some notations.
In Section 3 we prove the existence, uniqueness and uniform decay of solution
to \eqref{1}. Finally, in Section 4 we give the proof of the existence
and uniqueness of solution to \eqref{1} for the nonmonotone case.

\section{Notation}

The inner product and norm in $L^2(\Omega)$ and $L^2(\Gamma_1)$ are denoted,
respectively, by
\begin{gather*}
 (u,v) = \int_{\Omega}u(x)v(x)\,dx, \quad
|u| = \Big(\int_{\Omega} (u(x))^2\,dx \Big)^{1/2}, \\
(\delta,\theta)_{\Gamma_1} =  \int_{\Gamma_1}\delta (x)\theta (x)\,d\Gamma\,,\quad
|\delta|_{\Gamma_1}
 = \Big(\int_{\Gamma_1}(\delta(x))^2\,d\Gamma\Big)^{1/2}.
\end{gather*}
Let $O\subset\Omega$ and $p\geq 1$, we denote by
$$
 |u|_{L^{p}(O)}=\Big(\int_{O}|u(x)|_{\mathbb{R}}^p\,dx\Big)^{1/p}
$$
the norm in $L^{p}(O)$. We define a closed subspace of the Sobolev space
$H^{1}(\Omega)$ as
$$
 V=\{u\in H^1(\Omega); \gamma_0(u)=0\text{ a.e. on }\Gamma_0\},
$$
where $\gamma_0:H^1(\Omega)\to H^{1/2}(\Gamma)$ is the trace map of order zero and $H^{1/2}(\Gamma)$ is the Sobolev space of order $\frac{1}{2}$ defined over $\Gamma$, as introduced by Lions and Magenes \cite{Lions}. We define the following inner product and norm in $V$:
\[
 ((u,v))=\sum_{i=1}^n\int_{\Omega}\frac{\partial u}{\partial x_i}(x)
\frac{\partial v}{\partial  x_i}(x)\,dx,\quad
\|u\|=\Big(\sum_{i=1}^n\int_{\Omega}\Big(\frac{\partial u}{\partial
 x_i}(x)\Big)^2dx\Big)^{1/2},
\]
 respectively. The Poincar\'{e}'s Inequality holds in $V$, thus there
exists a constant $k_0$ such that
\begin{equation}\label{5}
 |u|\leq k_0 \| u\|, \quad \text{for all } u\in V.
\end{equation}
From \eqref{5} we can prove the equivalence, in $V$, of the norm
$\|\cdot\|$ and the usual norm
$\|\cdot\|_{H^1(\Omega)}=\big(|u|^2+\|u\|^2\big)^{1/2}$ of $H^{1}(\Omega)$.

As $\Gamma_1$ is a compact Riemannian manifold, with boundary, endowed
with the natural metric inherited from $\mathbb{R}^n$, it is possible
to give an intrinsic definition of the space $H^k(\Gamma_1)$ by using the
covariant derivative operator. In fact, the Sobolev space $H^{k}(\Gamma_{1})$
is the completion of $C^{\infty}(\Gamma_1)$ with respect to the norm
$$
 \| u \|_{H^{k}(\Gamma_{1})}
= \Big[ \sum_{l=0}^{k} | \nabla^{l} u |_{\Gamma_{1}}^{2} \Big]^{1/2},
$$
where $\nabla^{l}$ is the covariant derivative operator of order $l$,
see Hebey \cite{Hebey}. We also consider the space $H^{1}_{0}(\Gamma_{1})$
the closure of $C^{\infty}_0(\Gamma_{1})$ in $H^{1}(\Gamma_{1})$.
The Poincar\'{e}'s inequality holds in $H_{0}^{1}(\Gamma_{1})$,
thus there exists a constant $k_{1}$ such that
$$
 |\delta|_{\Gamma_{1}} \leq k_{1} |\nabla_{\tau} \delta|_{\Gamma_{1}}, \quad
\text{for all }\delta \in H_{0}^{1}(\Gamma_{1}),
$$
where $\nabla_{\tau}$ is the tangential gradient on $\Gamma_1$,
see Taylor \cite{Taylor}. This allow us to consider the space
$H_0^1(\Gamma_1)$ equipped with the norm and inner product:
$$
 \| \delta \|_{\Gamma_{1}} = | \nabla_{\tau} \delta |_{\Gamma_{1}} \quad
\text{and} \quad ((\delta,\theta))_{\Gamma_1}
= \int_{\Gamma_{1}} \langle \nabla_{\tau} \delta (x) ,
 \nabla_{\tau} \theta (x) \rangle \, d\Gamma.
$$
We denote by $\Delta_{\Gamma}=\operatorname{div}_{\tau}\nabla_{\tau}$
the Laplace-Beltrami operator. We define the operator
$-\Delta_{\Gamma}:H_{0}^{1} (\Gamma_{1}) \to H^{-1}(\Gamma_1)$ such that
$$
 \langle -\Delta_{\Gamma}\delta, \theta\rangle_{H^{-1}(\Gamma_1)
\times H_0^{1}(\Gamma_1)}=\int_{\Gamma_{1}}
\langle \nabla_{\tau} \delta (x) \,,\, \nabla_{\tau} \theta (x) \rangle \, d\Gamma,
$$
for all $\delta,\theta\in H_{0}^1(\Gamma_1)$. We consider
$(H_{0}^{1} (\Gamma_{1}) \cap H^{2}(\Gamma_{1}))$ endowed with the norm
$$
|\delta|_{H_{0}^{1} (\Gamma_{1}) \cap H^{2}(\Gamma_{1})}
= |\Delta_{\Gamma} \delta |_{\Gamma_{1}} \quad
\text{for all } \delta \in (H_{0}^{1} (\Gamma_{1}) \cap H^{2}(\Gamma_{1})),
$$
 which is equivalent to $\| \cdot \|_{H^{2}(\Gamma_{1})}$, see
 Biezuner \cite{Biezuner}.

We define the subspace $W$ of $V$ as
$$
 W=\{u\in (V\cap H^3(\Omega)); \: (\gamma_1(u))_{|_{\Gamma_1}}\in H_0^1(\Gamma_1)\},
$$
here $\gamma_1:H(\Delta , \Omega)\to H^{-\frac{1}{2}}(\Gamma)$
is the Neumann trace map and
$H(\Delta, \Omega)=\{u\in H^1(\Omega); \Delta u\in L^2(\Omega)\}$
is equipped with the norm
\[
\|u\|_{H(\Delta,\Omega)}=\Big(\|u\|_{H^1(\Omega)}^2+|\Delta u|^2\Big)^{1/2}.
\]
 We equipped $W$ with the norm
$$
\|u\|_{W}=\left(\|u\|^2+\|u\|_{H^3(\Omega)}^2\right)^{1/2}.
$$
The space $W$ is necessary to estimate $u_m''(0)$ in the estimate 2.
 We observe that $W$ is dense in $V$. As $\gamma_0$ and $\gamma_1$ are
continuous there exist positive constants $k_2$ and $k_{3}$ such that
\begin{equation}
 |\gamma_0(u)|_{\Gamma_1}\leq k_2 \|u\|_{W}\quad\text{and}\quad
|\gamma_{1} (u) |_{\Gamma_{1}}\leq k_{3} \|u\|_{W},\quad \text{for all }
u \in W.
\end{equation}
Results about Sobolev's Spaces on manifolds can be found in
 Aubin \cite{Aubin}, Hebey \cite{Hebey}, Lions and Magenes \cite{Lions}
and Taylor \cite{Taylor}.

\section{Main Results}

We assume
\begin{equation}\label{7}
 f,g,h\in C(\overline{\Gamma_1}) \text{ with } f,g>0 \text{ and }h\geq 0.
\end{equation}
We suppose that $\rho\in C^1(\mathbb{R})$ and there exist constants
$\kappa_1,\kappa_2>0$ such that
\begin{equation}\label{8}
 |\rho(x)|\leq \kappa_1|x|^q \quad\text{if $|x|>1$, and
$|\rho(x)|\leq \kappa_2|x|$ if $|x|\leq 1$},
\end{equation}
where $q$ satisfies $1\leq q\leq \frac{n+2}{n-2}$ if $n\geq 3$,
or $q\geq 1$ if $n=2$.
We also assume
\begin{equation}\label{9}
 \rho'(x)\geq \kappa_3>0, \quad \text{for all }x\in\mathbb{R}.
\end{equation}

\begin{theorem}\label{thm1}
 Suppose that \eqref{7}--\eqref{9} hold. Let
$(\phi,\psi,\theta)\in W\times (V\cap L^{2q}(\Omega))\times
(H_0^1(\Gamma_1)\cap H^2(\Gamma_1))$ and
$F,F'\in L^2_{\rm loc}(0,\infty;L^2(\Omega))$.
Then, there exists a unique pair $(u,\delta)$ such that
\begin{gather*}
 u,u'\in L^{\infty}_{\rm loc}(0,\infty;V),\quad
 u''\in L^{\infty}_{\rm loc}(0,\infty;L^2(\Omega)),\quad
 u(t)\in H(\Delta,\Omega)\text{ a.e. in } [0,\infty), \\
 \delta\in L^{\infty}_{\rm loc}(0,\infty;H_0^1(\Gamma_1)\cap H^2(\Gamma_1)),\quad
\delta'\in L^{\infty}_{\rm loc}(0,\infty;H_0^1(\Gamma_1)), \\
\delta''\in L^{\infty}_{\rm loc}(0,\infty;L^2(\Gamma_1))
\end{gather*}
and, for all $T>0$,
\begin{equation}\label{10}
 \begin{gathered}
 u''-\Delta u+\rho(u')=F\quad\text{a.e. in }\Omega\times (0,T),\\
 \langle \gamma_1(u(t)),\gamma_0(w)\rangle
=\big(\delta'(t),\gamma_0(w)\big)_{\Gamma_1}\quad\text{a.e. in } [0,T],
\text{ for all }w\in V,\\
 u'+f\delta''-c^2\Delta_{\Gamma}\delta+g\delta'+h\delta=0\quad
\text{a.e. in }\Gamma_1\times (0,T),\\
 u(0)=\phi,\quad u'(0)=\psi,\quad \delta(0)=\theta.
 \end{gathered}
\end{equation}
\end{theorem}

\begin{proof}
Let $(w_j)_{j\in\mathbb{N}}$ be an orthonormal basis in $W$ and let
$(z_j)_{j\in\mathbb{N}}$ be the orthonormal basis in $L^2(\Gamma_1)$
given by eigenfunctions of the operator $-\Delta_{\Gamma}$.
Since the embedding $H_0^1(\Gamma_1)\hookrightarrow L^2(\Gamma_1)$
is compact the existence of the special basis $(z_j)_{j\in\mathbb{N}}$
is a consequence of the Spectral Theory and it is an orthogonal basis
of $(H_0^1(\Gamma_1)\cap H^2(\Gamma_1))$ and $H_0^1(\Gamma_1)$.
For each $m\in\mathbb{N}$, we consider $u_m:\Omega\times[0,T_m]\to\mathbb{R}$
and $\delta_m:\Gamma_1\times[0,T_m]\to\mathbb{R}$ of the form
$$
 u_m(x,t)=\sum_{j=1}^m\alpha_{jm}(t)w_j(x)\quad\text{and}\quad
\delta_m(x,t)=\sum_{j=1}^m\beta_{jm}(t)z_j(x)
$$
the local solution of the approximate problem
\begin{gather}
\left(u_m''(t)+\rho(u_m'(t))-F(t),w_j\right)
+((u_m(t),w_j))-\left(\delta_m'(t),\gamma_0(w_j)\right)_{\Gamma_1}=0, \label{11}
\\
\left(\gamma_0(u_m'(t))+f\delta_m''(t)+g\delta_m'(t)+h\delta_m(t),
 z_j\right)_{\Gamma_1}
 +c^2\left(\left(\delta_m(t),z_j\right)\right)_{\Gamma_1}=0, \label{12}
\\
u_m(0)=\sum_{i=1}^m\phi^iw_i\to \phi\text{ in }W;\:u_m'(0)
=\sum_{i=1}^m\psi^iw_i\to \psi\text{ in }V\cap L^{2q}(\Omega), \label{13}
\\
\delta_m(0)=\sum_{i=1}^m\theta^iz_i\to \theta\text{ in }H_0^1(\Gamma_1)
\cap H^2(\Gamma_1), \delta_m'(0)=(\gamma_1(u_{0m}))_{|_{\Gamma_1}}.\label{14}
\end{gather}
 Here $1 \leq j \leq m$ and $\phi^i,\psi^i,\theta^i$, $i=1,\ldots, m$,
are known scalars. The estimate 1 will allow us to extend the local solution
to the whole interval $[0,T]$, for all $T>0$.
 From \eqref{11} and \eqref{12} we have the following approximate equations
\begin{gather}
\left(u_m''(t)+\rho(u_m'(t))-F(t),w\right)
+\left(\left(u_m(t),w\right)\right)-\left(\delta_m'(t),
\gamma_0(w)\right)_{\Gamma_1}=0,
\label{15}\\
\left(\gamma_0(u_m'(t))+f\delta_m''(t)+g\delta_m'(t)+h\delta_m(t),z\right)_{\Gamma_1}
+c^2\left(\left(\delta_m(t),z\right)\right)_{\Gamma_1}=0,\label{16}
\end{gather}
which hold for all $w\in$ span$\{w_1,\ldots,w_m\}$ and
$z\in\operatorname{span}\{z_1,\ldots,z_m\}$, respectively.
\smallskip

\noindent\textbf{Estimate 1:}
Setting $w=u_m'(t)$ and $z=\delta_m'(t)$ in \eqref{15} and \eqref{16},
respectively, we deduce
\begin{align*}
&\frac{1}{2}\frac{d}{dt}\Big[|u_m'(t)|^2+\|u_m(t)\|^2
+|f^{1/2}\delta_m'(t)|_{\Gamma_1}^2+|h^{1/2}\delta_m(t)|_{\Gamma_1}^2
+c^2\|\delta_m(t)\|_{\Gamma_1}^2\Big]\\
&+\left(\rho(u_m'(t)),u_m'(t)\right)+|g^{1/2}\delta_m'(t)|_{\Gamma_1}^2 \\
&=\left(F(t),u_m'(t)\right).
\end{align*}
Employing the inequality $ab\leq \frac{a^2}{2}+\frac{b^2}{2}$ and
integrating over $(0,t)$, we have
\begin{align*}
&|u_m'(t)|^2+\|u_m(t)\|^2+|f^{1/2}\delta_m'(t)|_{\Gamma_1}^2
+|h^{1/2}\delta_m(t)|_{\Gamma_1}^2+c^2\|\delta_m(t)\|_{\Gamma_1}^2\\
& +\int_0^t\left(\rho(u_m'(\xi)),u_m'(\xi)\right)
 +|g^{1/2}\delta_m'(\xi)|_{\Gamma_1}^2d\xi\\
&\leq |\psi|^2+\|\phi\|^2
 +k_2^2\max_{x\in\overline{\Gamma_1}}|f|\|\phi\|_{W}^2
+\max_{x\in\overline{\Gamma_1}}|h||\theta|_{\Gamma_1}^2\\
&\quad +c^2\|\theta\|_{\Gamma_1}^2
 +\int_0^t|F(\xi)|^2 +|u_m'(\xi)|^2\,d\xi.
\end{align*}
Now, taking into account \eqref{8}, \eqref{9} and employing Gronwall's
lemma we conclude that there exists a positive constant $C_1$, which does not
depend on $m$ and $t$, such that
\begin{equation}\label{19}
|u_m'(t)|^2+\|u_m(t)\|^2+|\delta_m'(t)|_{\Gamma_1}^2
 +|\delta_m(t)|_{\Gamma_1}^2+\|\delta_m(t)\|_{\Gamma_1}^2\leq C_1,
\end{equation}
for all $t\in[0,T]$, where $T>0$ is arbitrary.
\smallskip

\noindent\textbf{Estimate 2:}
 Substituting $t=0$ in \eqref{15}--\eqref{16} and using Green's Formula we obtain
\begin{gather}\label{20}
 \left(u_m''(0)+\Delta u_m(0)+\rho(u_m'(0))-F(0),w\right)=0, \\
\label{21}
 \left(\gamma_0(u_m'(0))+f\delta_m''(0)+c^2\Delta_{\Gamma}\delta_m(0)+g\delta_m'(0)+h\delta_m(0),z\right)_{\Gamma_1}=0.
\end{gather}
Considering
$$
 \Omega_1=\{x\in\Omega;\:|u_m'(x,0)|\leq 1\}\quad\text{and}\quad
\Omega_2=\{x\in\Omega;\:|u_m'(x,0)|> 1\}
$$
from the assumption \eqref{8} and the inclusion
$L^{2q}(\Omega)\hookrightarrow L^{2}(\Omega)$ we obtain
\begin{equation}\label{22}
\begin{aligned}
\Big(\int_{\Omega_1}|\rho(u_m'(x,0))|^2dx\Big)^{1/2}
&\leq C_2\Big(\int_{\Omega_1}|u_m'(x,0)|^{2}dx\Big)^{1/2}\\
&\leq C_2 |u_m'(0)| \leq C_3 |u_m'(0)|_{L^{2q}(\Omega)}\\
&\leq C_3(|u_m'(0)|_{L^{2q}(\Omega)}^q+|u_m'(0)|_{L^{2q}(\Omega)})
\end{aligned}
\end{equation}
and
\begin{equation}\label{23}
\begin{aligned}
\Big(\int_{\Omega_2}|\rho(u_m'(x,0))|^2dx\Big)^{1/2}
&\leq C_4\Big(\int_{\Omega_2}|u_m'(x,0)|^{2q}dx\Big)^{1/2} \\
&\leq C_5|u_m'(0)|_{L^{2q}(\Omega)}^q\\
&\leq C_5(|u_m'(0)|_{L^{2q}(\Omega)}^q+|u_m'(0)|_{L^{2q}(\Omega)}).
\end{aligned}
\end{equation}
Thanks to \eqref{13}, \eqref{22} and \eqref{23} we deduce
\begin{equation}\label{24}
 \left(\rho(u_m'(0)),w\right)\leq |\rho(u_m'(0))||w|\leq C_6 |w|.
\end{equation}
Taking $w=u_m''(0)$ and $z=\delta_m''(0)$ in \eqref{20} and \eqref{21},
respectively, using H\"{o}lder's inequality, noting \eqref{13}, \eqref{14}
and \eqref{24} we obtain
\begin{equation}\label{25}
 |u_m''(0)|+|\delta_m''(0)|_{\Gamma_1}\leq C_7.
\end{equation}
Differentiating \eqref{15} and \eqref{16} with respect to $t$ and taking
$w=u_m''(t)$ and $z=\delta_m''(t)$ we obtain
\begin{equation}\label{26}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\Big[|u_m''(t)|^2+\|u_m'(t)\|^2+|f^{1/2}
\delta_m''(t)|_{\Gamma_1}^2+|h^{1/2}\delta_m'(t)|_{\Gamma_1}^2
+c^2\|\delta_m'(t)\|_{\Gamma_1}^2\Big] \\
&+\left(\rho'(u_m'(t))u_m''(t),u_m''(t)\right)
 +|g^{1/2}\delta_m''(t)|_{\Gamma_1}^2 =\left(F'(t),u_m''(t)\right).
\end{aligned}
\end{equation}
Integrating from $0$ to $t$, taking into account \eqref{9} and \eqref{25} we have
\begin{equation}\label{27}
\begin{aligned}
&|u_m''(t)|^2+\|u_m'(t)\|^2+|f^{1/2}\delta_m''(t)|_{\Gamma_1}^2
+|h^{1/2}\delta_m'(t)|_{\Gamma_1}^2+c^2\|\delta_m'(t)\|_{\Gamma_1}^2
 +|g^{1/2}\delta_m''(\xi)|_{\Gamma_1}^2\\
&\leq C_8  +\int_0^t|F'(\xi)|^2 +|u_m''(\xi)|^2\,d\xi.
\end{aligned}
\end{equation}
This inequality and Gronwall's lemma yield
\begin{equation}\label{28}
 |u_m''(t)|^2+\|u_m'(t)\|^2+|\delta_m''(t)|_{\Gamma_1}^2
+|\delta_m'(t)|_{\Gamma_1}^2+\|\delta_m'(t)\|_{\Gamma_1}^2\leq C_{9},
\end{equation}
for all $t\in[0,T]$, where $T>0$ is arbitrary. Here $C_{9}$ is independent
 of $m$ and $t$.
\smallskip

\noindent\textbf{Estimate 3:}
Substituting $z=\Delta_{\Gamma}\delta_m(t)$ in \eqref{16} we find
$$
 |\Delta_{\Gamma}\delta_m(t)|_{\Gamma_1}
 \leq  C_{10} \Big[
 |f^{1/2}\delta_m''(t)|_{\Gamma_1}+|g^{1/2}\delta_m'(t)|_{\Gamma_1}
+|h^{1/2}\delta_m(t)|_{\Gamma_1} +|\gamma_0(u_m'(t))|_{\Gamma_1}
 \Big],
$$
for all $t\in[0,T]$. As $f,g,h\in C(\overline{\Gamma_1})$ and the map 
$\gamma_0$ is continuous we obtain
$$
 |\Delta_{\Gamma}\delta_m(t)|_{\Gamma_1} 
\leq C_{11}\Big[|\delta_m''(t)|_{\Gamma_1}+|\delta_m'(t)|_{\Gamma_1}
+|\delta_m(t)|_{\Gamma_1}+\|u_m'(t)\|\Big].
$$
Taking into account \eqref{19} and \eqref{28} we conclude that
\begin{equation}\label{29}
 |\Delta_{\Gamma}\delta_m(t)|_{\Gamma_1} \leq C_{12},\quad\text{for all }t\in[0,T],
\end{equation}
where $T>0$ is arbitrary.
\smallskip

\noindent\textbf{Passage to the Limit:} 
Using the estimates \eqref{19}, \eqref{28}--\eqref{29} and compactness argument, 
we can see that there exist a subsequence of $(u_m)_{m\in\mathbb{N}}$ and a 
subsequence of $(\delta_m)_{m\in\mathbb{N}}$, which we will denote by the 
same notations, and functions $u$ and $\delta$, such that
\begin{equation}\label{30}
 \begin{alignedat}{2}
&u_m\stackrel{*}{\rightharpoonup} u \text{ in } L^{\infty}(0,T;V)
 &&  \delta_m\stackrel{*}{\rightharpoonup} \delta
 \text{ in } L^{\infty}(0,T;H_0^1(\Gamma_1)\cap H^2(\Gamma_1))\\
&u_m'\stackrel{*}{\rightharpoonup} u' \text{ in } L^{\infty}(0,T;V)
 && \delta_m'\stackrel{*}{\rightharpoonup} \delta'
 \text{ in } L^{\infty}(0,T;H_0^1(\Gamma_1))\\
&u_m''\stackrel{*}{\rightharpoonup} u'' \text{ in }
 L^{\infty}(0,T;L^2(\Omega))\quad
 &&  \delta_m''\stackrel{*}{\rightharpoonup} \delta''
 \text{ in }  L^{\infty}(0,T;L^2(\Gamma_1)).
 \end{alignedat}
\end{equation}
The convergences above and the Aubin-Lions Theorem lead us to
\begin{equation}\label{31}
u_m\to u \text{ in } L^{2}(0,T;L^2(\Omega)),\quad 
u_m'\to u' \text{ in } L^{2}(0,T;L^2(\Omega)).
\end{equation}
Convergences \eqref{30} and \eqref{31} are sufficient to pass to the 
limit in approximate problem \eqref{11}--\eqref{14} and we can 
conclude $(u,\delta)$ satisfies \eqref{10} 
(see Frota, Medeiros and Vicente \cite{Frota-Medeiros-Vicente}). 
By the regularity \eqref{30} we apply the energy method to obtain 
the uniqueness.
\end{proof}

Now, we  prove the exponential decay of the energy associated to \eqref{1} 
when $F\equiv 0$. Let $(u,\delta)$ be the solution of \eqref{1} 
given by Theorem \ref{thm1}. We define the energy by
\begin{equation}\label{32}
 E(t)=\frac{1}{2}\Big[|u'(t)|^2+\|u(t)\|^2+|f^{1/2}
\delta'(t)|_{\Gamma_1}^2+c^2\|\delta(t)\|_{\Gamma_1}^2
+|h^{1/2}\delta(t)|_{\Gamma_1}^2\Big].
\end{equation}

\begin{lemma}\label{lemma1}
 Let $\varphi:\mathbb{R}^+\to\mathbb{R}^+$ be a non-increasing function and 
assume that there exists a constant $A>0$ such that
 $$
 \int_{t}^{\infty}\varphi(\xi)\,d\xi\leq A\varphi(t),\quad
 \text{for all }t\in\mathbb{R}^+.
 $$
 Then
 $$
 \varphi(t)\leq \varphi(0)\exp\big(1-\frac{t}{A}\big),\quad \text{for all }t\geq 0.
 $$
\end{lemma}

For a proof of the above lemma, see Komornik \cite{Komornik}.

Suppose $F\equiv 0$. Multiplying \eqref{10}$_1$ by $u'$, \eqref{10}$_3$ 
by $\delta'$ and integrating over $\Omega$ and $\Gamma_1$, respectively, we see
\begin{equation}\label{33}
 E'(t)=-\left(\rho(u'(t)),u'(t)\right)-|g^{1/2}\delta'(t)|_{\Gamma_1}^2\leq 0.
\end{equation}

\begin{lemma}\label{lemma2}
 If all assumptions of Theorem \ref{thm1} hold and moreover $F\equiv 0$, 
then we have
 \begin{equation}\label{34}
 2\int_{S}^TE(t)\,dt\leq C_{14}E(S)-\int_{S}^{T}\left(u(t),\rho(u'(t))\right)dt
+\int_{S}^{T}\left(2u(t)-g\delta(t),\delta'(t)\right)_{\Gamma_1}dt,
 \end{equation}
for all $0\leq S<T<\infty$.
\end{lemma}

\begin{proof}
Multiplying \eqref{10}$_1$ by $u$, \eqref{10}$_3$ by $\delta$, integrating 
over $\Omega\times (S,T)$ and $\Gamma_1\times (S,T)$, respectively, and 
integrating by parts we obtain
\begin{equation}\label{35}
\begin{aligned}
&\int_S^T(|u'(t)|^2-\|u(t)\|^2)dt+\int_S^T(|f^{1/2}
\delta'(t)|_{\Gamma_1}^2-c^2\|\delta(t)\|_{\Gamma_1}^2)dt
-\int_{\Omega}uu'dx\Big|_S^T \\
& -\int_{\Gamma_1}(f\delta \delta'+u \delta)d\Gamma\Big|_S^T
 +\int_S^T[2\left(\delta'(t),u(t)\right)_{\Gamma_1}-|h^{1/2}
 \delta(t)|_{\Gamma_1}^2] dt \\
& -\int_S^T\left(u(t),\rho(u'(t))\right)dt
 -\int_S^T\left(g\delta(t),\delta'(t)\right)_{\Gamma_1} dt=0.
\end{aligned}
\end{equation}
We note that
\begin{equation}\label{36}
\begin{aligned}
&-\int_S^T(\|u(t)\|^2+c^2\|\delta(t)\|_{\Gamma_1}^2+|h^{1/2}
 \delta(t)|_{\Gamma_1}^2)dt \\
&=-2\int_S^TE(t)dt+\int_S^T(|u'(t)|^2+|f^{1/2}\delta'(t)|_{\Gamma_1}^2)dt.
\end{aligned}
\end{equation}
Combining \eqref{35} and \eqref{36}, we infer that
\begin{equation}\label{37}
\begin{aligned}
 2\int_S^TE(t)dt
&= -\int_{\Omega}uu'dx\Big|_S^T-\int_{\Gamma_1}(f\delta \delta'+\delta u) 
d\Gamma\Big|_S^T \\
&\quad  +2\int_S^T[|u'(t)|^2+|f^{1/2}\delta'(t)|_{\Gamma_1}^2
+\left(\delta'(t),u(t)\right)_{\Gamma_1}]dt \\
&\quad  -\int_S^T\left(u(t),\rho(u'(t))\right)dt-\int_S^T
\left(g\delta(t),\delta'(t)\right)_{\Gamma_1}dt.
\end{aligned}
\end{equation}
By H\"{o}lder and Young's inequalities and \eqref{32}, we have
$$
 \big|\int_{\Omega}uu'\,dx\big|\leq \frac{1}{2}\int_{\Omega}|u|^2dx
+\frac{1}{2}\int_{\Omega}|u'|^2dx\leq C_{15} E(t),
$$
thus
\begin{equation}\label{38}
 \int_{\Omega}uu'\,dx\Big|_S^T\leq C_{16}E(S).
\end{equation}
Analogously,
\begin{equation}\label{39}
 \int_{\Gamma_1}(f\delta\delta'+\delta u)\,d\Gamma\Big|_S^T\leq C_{17}E(S).
\end{equation}
From \eqref{9} and \eqref{33}, we obtain
\begin{equation}\label{40}
\begin{aligned}
2\int_S^T[|u'(t)|^2+|f^{1/2}\delta'(t)|_{\Gamma_1}^2]dt
&\leq \frac{2}{\kappa_3} \int_S^T\int_{\Omega}\rho(u')u'dx
 +C_{18}\int_{\Gamma_1}g(\delta')^2d\Gamma\,dt \\
& \leq C_{19}\int_S^T\left(\rho(u'(t)),u'(t)\right)
 +|g^{1/2}\delta'(t)|_{\Gamma_1}^2dt \\
& =-C_{19}\int_S^TE'(t) dt\leq C_{20}E(S).
\end{aligned}
\end{equation}
Combining \eqref{37}--\eqref{40}, we obtain \eqref{34}.
\end{proof}

\begin{theorem}\label{theorem2}
 If all assumptions of Theorem {\ref{thm1}} hold and  $F\equiv 0$, then 
there exist positive constants $a$ and $b$ such that $E(t)\leq a\exp(-bt)$, 
for all $t\geq 0$.
\end{theorem}

\begin{proof}
For each $t>0$ fixed, we set
$$
 \Omega_1^t=\{x\in\Omega;\:|u'(x,t)|\leq 1\}\quad\text{and}\quad 
\Omega_2^t=\{x\in\Omega;\:|u'(x,t)|> 1\}.
$$
From \eqref{8} and \eqref{33}, we have
\begin{align*}
|\rho(u'(t))|_{L^2(\Omega_1^t)}^2
&=\int_{\Omega_1^t}|\rho(u')||\rho(u')|dx
 \leq C_{21}\int_{\Omega_1^t}|u'||\rho(u')|dx \\
& \leq C_{21}\left(\rho(u'(t)),u'(t)\right)\\
&=-C_{21}[E'(t)+|g^{1/2}\delta'(t)|_{\Gamma_1}^2]\leq C_{22} |E'(t)|;
\end{align*}
therefore,
\begin{equation}\label{41}
 \big|\int_{\Omega_1^t}u\rho(u')dx\big|
\leq |u(t)|_{L^2(\Omega_1^t)}|\rho(u'(t))|_{L^2(\Omega_1^t)}
\leq C_{23}E^{1/2}(t)|E'(t)|^{1/2}.
\end{equation}
Using the Sobolev's imbedding $V\hookrightarrow L^{q+1}(\Omega)$, 
\eqref{8}, \eqref{32} and \eqref{33}, we obtain
\begin{equation}\label{42}
\begin{aligned}
\Big|\int_{\Omega_2^t}u\rho(u')dx\Big|
&\leq |u(t)|_{L^{q+1}(\Omega_2^t)}|\rho(u'(t))|_{L^{\frac{q+1}{q}}(\Omega_2^t)}\\
& \leq C_{24}\|u(t)\|\Big[\int_{\Omega_2^t}|\rho(u')|^{\frac{1}{q}}
 |\rho(u')|\,dx\Big]^{\frac{q}{q+1}} \\
&\leq C_{25}\|u(t)\|\Big[\int_{\Omega_2^t}u'\rho(u')\,dx\Big]^{\frac{q}{q+1}}
 \leq C_{26}E^{1/2}(t)|E'(t)|^{\frac{q}{q+1}}.
\end{aligned}
\end{equation}
We also observe that
\begin{equation}\label{43}
\begin{aligned}
|\left(2u(t)-g\delta(t),\delta'(t)\right)_{\Gamma_1}|
&\leq C_{27}[\|u(t)\|+|\delta(t)|_{\Gamma_1}]|\delta'(t)|_{\Gamma_1}\\
&\leq C_{28}E^{1/2}(t)|g^{1/2}\delta'(t)|_{\Gamma_1}
 \leq C_{28}E^{1/2}(t)|E'(t)|^{1/2}.
\end{aligned}
\end{equation}
Using Lemma \ref{lemma2}, \eqref{41}--\eqref{43}, we obtain
\begin{equation}\label{44}
 2\int_{S}^TE(t)\,dt\leq C_{29}\Big[E(S)+\int_{S}^{T}E^{1/2}(t)|E'(t)|^{1/2} 
+E^{1/2}(t)|E'(t)|^{\frac{q}{q+1}} dt\Big].
\end{equation}
Now, we  estimate the last term of \eqref{44}. Applying Young's inequality 
and using \eqref{33}, we have
\begin{equation}\label{45}
\begin{aligned}
& E^{1/2}(t)|E'(t)|^{1/2}+E^{1/2}(t)|E'(t)|^{\frac{q}{q+1}}\\
&\leq C_{30}\Big[\frac{1}{2}E(t)+|E'(t)|+\Big(E^{\frac{1}{q+1}}
 E^{\frac{q-1}{2(q+1)}}(t)|E'(t)|^{\frac{q}{q+1}}\Big)^{\frac{q+1}{q}}\Big]\\
&\leq C_{31}\big[E(t)+|E'(t)| +E^{\frac{q-1}{2q}}(0)|E'(t)|\big].
\end{aligned}
\end{equation}
Taking into account \eqref{44}, \eqref{45} and letting $T\to\infty$, 
we conclude that
\begin{equation}\label{46}
 \int_{S}^{\infty}E(t)\,dt\leq C_{32} E(S),\quad \text{for all }S\geq 0.
\end{equation}
Hence, Lemma \ref{lemma1} yields $E(t)\leq C_{33}\exp(-C_{34}t)$, for all $t\geq 0$.
\end{proof}

\begin{remark} \label{rmk3.5} \rm
Suppose $p>1$ if $n=2$ or $1<p\leq 2$ if $n=3$. Setting 
$u''-M(|u(t)|^2)\Delta u+\alpha u'+\beta|u'|^{p}u'=0$,
 in $\Omega\times(0,\infty)$, instead of \eqref{1}$_1$, using the method above 
we can prove the exponential decay without the assumption $c>4k_1k_2$ made
 by Frota, Medeiros and Vicente \cite{Frota-Medeiros-Vicente}.
\end{remark}

\section{Non-monotone dissipative term}

Let $\lambda\in C^1(\mathbb{R})$ be such that
\begin{equation}\label{47}
 |\lambda(x)|\leq \kappa_4|x|^q, \text{ if }|x|>1\quad\text{and}\quad
 |\lambda(x)|\leq \kappa_5|x|, \text{ if }|x|\leq 1,
\end{equation}
where $q$ satisfies $1\leq q\leq \frac{n+2}{n-2}$ if $n\geq 3$ or 
$q\geq 1$ if $n=2$. We also assume
\begin{equation}\label{48}
 \lambda (x)x \geq 0\text{ and }\lambda'(x)>-\alpha^2,\quad
\text{for all }x\in\mathbb{R},
\end{equation}
where $\alpha$ is a constant. We consider
\begin{equation}\label{49}
 \begin{gathered}
 u''-\Delta u+\lambda(u')=0 \quad  \text{in }\Omega\times(0,\infty),\\
 u=0\quad  \text{on }\Gamma_0\times(0,\infty),\\
 \frac{\partial u}{\partial \nu}=\delta' \quad \text{on }\Gamma_1\times(0,\infty),\\
 u'+f\delta''-  c^2\Delta_{\Gamma}\delta+g\delta'+h\delta=0
 \quad\text{on }\Gamma_1\times(0,\infty),\\
 \delta=0 \quad  \text{on }\partial\Gamma_1\times(0,\infty),  \\
 u(x,0)=\phi(x),\quad u'(x,0)=\psi(x)\quad x\in\Omega,\\
 \delta(x,0)=\theta(x),\quad \delta'(x,0)=\frac{\partial \phi}{\partial \nu}(x)\quad
 x\in\Gamma_1,
 \end{gathered}
\end{equation}


\begin{theorem}\label{thm3}
 Suppose that \eqref{47}--\eqref{48} hold. Let 
$(\phi,\psi,\theta)\in W\times (V\cap L^{2q}(\Omega))\times 
(H_0^1(\Gamma_1)\cap H^2(\Gamma_1))$. Then there exists a unique pair 
$(u,\delta)$ which is solution of \eqref{49} and satisfies
\begin{gather*}
 u,u'\in L^{\infty}_{\rm loc}(0,\infty;V),\quad
 u''\in L^{\infty}_{\rm loc}(0,\infty;L^2(\Omega))\,,\quad
 u(t)\in H(\Delta,\Omega)\text{ a.e. in }  [0,\infty),\\
 \delta\in L^{\infty}_{\rm loc}(0,\infty;H_0^1(\Gamma_1)\cap H^2(\Gamma_1)),\quad
\delta'\in L^{\infty}_{\rm loc}(0,\infty;H_0^1(\Gamma_1)), \\
 \delta''\in L^{\infty}_{\rm loc}(0,\infty;L^2(\Gamma_1)).
\end{gather*}
\end{theorem}

\begin{proof}
Let $(u_k)_{k\geq 0}$ and $(\delta_k)_{k\geq 0}$ be the sequences defined by 
$u_0(x,t)=\phi(x)$, for all $(x,t)\in\Omega\times [0,T]$, 
$\delta_0(x,t)=\theta(x)$, for all $(x,t)\in\Gamma_1\times [0,T]$, $T>0$, 
and $(u_k,\delta_k)$, $k=1,2,\ldots$, the solution of
\begin{equation}\label{50}
 \begin{gathered}
 u_k''-\Delta u_k+\lambda(u_k')+\alpha^2u_k'=F_{k-1}=\alpha^2u_{k-1}' \quad
 \text{in }\Omega\times(0,T),\\
 u_k=0\quad \text{on }\Gamma_0\times(0,T),\\
 \frac{\partial u_k}{\partial \nu}=\delta_k' \quad \text{on }\Gamma_1\times(0,T),\\
 u_k'+f\delta_k''-  c^2\Delta_{\Gamma}\delta_k+g\delta_k'+h\delta_k=0\quad
 \text{on }\Gamma_1\times(0,T),\\
 \delta_k=0\quad  \text{on }\partial\Gamma_1\times(0,T),  \\
 u_k(x,0)=\phi(x),\quad u_k'(x,0)=\psi(x)\quad  x\in\Omega;\\
 \delta_k(x,0)=\theta(x),\quad
 \delta_k'(x,0)=\frac{\partial \phi}{\partial \nu}(x)\quad x\in\Gamma_1,
 \end{gathered}
\end{equation}
for $k=1,2,\ldots$, here $u_1'(x,t)=\psi(x)$ for all $(x,t)\in\Omega\times [0,T]$. 
Since $\rho(\xi)=\lambda(\xi)+\alpha^2 \xi$ satisfies \eqref{8}--\eqref{9} 
and $F_{k-1},F_{k-1}'\in L^2(0,T;L^2(\Omega))$, for $k=1,2,\ldots$, 
the existence and uniqueness of solution $(u_k,\delta_k)$ for \eqref{50} 
is given by Theorem \ref{thm1}.

Multiplying \eqref{50}$_1$ by $u_k'$, \eqref{50}$_4$ by $\delta_k'$, 
integrating over $\Omega$ and $\Gamma_1$, respectively, and substituting the 
second equation into the first, we obtain
\begin{equation}\label{51}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\Big[|u_k'(t)|^2+\|u_k(t)\|^2+|f^{1/2}
\delta_k'(t)|_{\Gamma_1}^2+|h^{1/2}\delta_k(t)|_{\Gamma_1}^2
+c^2\|\delta_k(t)\|_{\Gamma_1}^2\Big] \\
&\leq \alpha^2\left(u_{k-1}'(t),u_k'(t)\right).
\end{aligned}
\end{equation}
Denoting by
\begin{gather*}
\begin{aligned}
 e_k(t)&=\frac{1}{2}\operatorname{ess\,sup}_{0<s<t}
\Big\{|u_k'(s)|^2
+\|u_k(s)\|^2+|f^{1/2}\delta_k'(s)|_{\Gamma_1}^2+|h^{1/2}
\delta_k(s)|_{\Gamma_1}^2\\
&\quad +c^2\|\delta_k(s)\|_{\Gamma_1}^2\Big\},
\end{aligned}\\
 e(0)=\frac{1}{2}\Big[|\psi|^2+\|\phi\|^2
+k_2\max_{x\in\overline{\Gamma_1}}|f|\|\phi\|_{W}
+\max_{x\in\overline{\Gamma_1}}|h||\theta|_{\Gamma_1}^2
+c^2\|\theta\|_{\Gamma_1}^2\Big]
\end{gather*}
and integrating \eqref{51}, we have
$$
 e_k(t)\leq e(0)+\alpha^4t\int_0^t|u_{k-1}'(\xi)|^2d\xi
+\frac{1}{4t}\int_0^t|u_k'(\xi)|^2d\xi.
$$
However,
$$
 \frac{1}{4t}\int_0^t|u_k'(\xi)|^2d\xi\leq \frac{1}{2}e_k(t),\quad
\text{for all }k=1,2,\ldots.
$$
Therefore,
\begin{equation}\label{52}
 \frac{1}{2}e_k(t)\leq e(0)+2\alpha^4t^2e_{k-1}(t), 
\quad\text{for all }k=1,2,\ldots.
\end{equation}
Let $M$ be a constant such that $M > 2e(0)$. By induction, it is easy to prove
\begin{equation}\label{53}
 e_k(t)\leq 2e(0)+4\alpha^4t^2M<M,\quad 
\text{for all }t\leq \tau_1=\Big(\frac{M-2e(0)}{4\alpha^4 M}\Big)^{1/2},
\end{equation}
$k=1,2,\ldots$.

Differentiating the equations \eqref{50}$_1$ and \eqref{50}$_4$ with respect 
to $t$, proceeding as in \eqref{22}, \eqref{23}, \eqref{26}, \eqref{27} 
and above we obtain
\begin{equation}\label{54}
 |u_k''(t)|^2+\|u_k'(t)\|^2+|f^{1/2}\delta_k''(t)|_{\Gamma_1}^2
+|h^{1/2}\delta_k'(t)|_{\Gamma_1}^2+c^2\|\delta_k'(t)\|_{\Gamma_1}^2
\leq \overline{M},
\end{equation}
for all $t\in[0,\tau_2]$, where 
$\tau_2=\Big(\frac{\overline{M}-2\overline{e}(0)}{4\overline{M}}\Big)^{1/2}$ 
with $\overline{M}>2\overline{e}(0)$ and
\begin{align*}
 \overline{e}(0)
&=\frac{1}{2}\Big\{[|\Delta\phi|+C_v(|\psi|_{L^{2q}(\Omega)}^{q}
+|\psi|_{L^{2q}(\Omega)})+\alpha^2|\psi|)^{1/2}+\|\psi\|^2+(k_0\|\psi\|_{W}\\
&\quad +c^2|\Delta_{\Gamma}\theta|_{\Gamma_1}
 +k_2\max_{x\in\overline{\Gamma_1}}|g|\|\phi\|_{W}
 +\max_{x\in\overline{\Gamma_1}}|h||\theta|_{\Gamma_1})^{1/2}\\
&\quad +k_2\max_{x\in\overline{\Gamma_1}}|h|\|\phi\|_{W}+c^2k_1\|\phi\|_{W}^2
\Big\}.
\end{align*}
Employing the same argument to obtain \eqref{29}, we have
\begin{equation}\label{55}
 |\Delta_{\Gamma}\delta_k(t)|_{\Gamma_1} \leq C_{35},
\quad\text{for all }t\in[0,\tau],
\end{equation}
where $\tau=\min\{\tau_1,\tau_2\}$. Using the estimates 
\eqref{53}--\eqref{55} and compactness argument, we can see that there exist 
a subsequence of $(u_k)_{k\geq 0}$ and a subsequence of $(\delta_k)_{k\geq 0}$, 
which we will denote by the same notations, and functions $u$ and $\delta$, 
such that
\begin{equation}\label{56}
 \begin{alignedat}{2}
 &u_k\stackrel{*}{\rightharpoonup} u  \text{ in }  L^{\infty}(0,\tau;V),
  && \delta_k\stackrel{*}{\rightharpoonup} \delta
  \text{ in }  L^{\infty}(0,\tau;H_0^1(\Gamma_1)\cap H^2(\Gamma_1)),\\
& u_k'\stackrel{*}{\rightharpoonup} u'
 \text{ in }  L^{\infty}(0,\tau;V),
 && \delta_k'\stackrel{*}{\rightharpoonup} \delta'
 \text{ in } L^{\infty}(0,\tau;H_0^1(\Gamma_1)),\\
& u_k''\stackrel{*}{\rightharpoonup} u''
 \text{ in } L^{\infty}(0,\tau;L^2(\Omega)), \quad 
 &&  \delta_k''\stackrel{*}{\rightharpoonup} \delta''
 \text{ in }  L^{\infty}(0,\tau;L^2(\Gamma_1)).
 \end{alignedat}
\end{equation}
These convergences are sufficient to pass to the limit in \eqref{50}. 
Therefore $(u,\delta)$ is a local solution of \eqref{49}. 
The proof of the uniqueness is standard. Now, we will extend the local 
solution to whole interval $[0,T]$, for all $T>0$. In fact, let $(u,\delta)$ 
the local solution of \eqref{49}, we consider the problem
\begin{equation}\label{57}
 \begin{gathered}
 v''-\Delta v+\lambda(v')=0 \quad \text{in }\Omega\times(0,\infty),\\
 v=0\quad \text{on }\Gamma_0\times(0,\infty),\\
 \frac{\partial v}{\partial \nu}=\sigma' \quad  
 \text{on }\Gamma_1\times(0,\infty),\\
 v'+f\sigma''-c^2\Delta_{\Gamma}\sigma+g\sigma'+h\sigma=0
 \quad \text{on }\Gamma_1\times(0,\infty);\\
 \sigma=0 \quad \text{on }\partial\Gamma_1\times(0,\infty),  \\
 v(x,0)=u(x,\frac{\tau}{2}),\quad v'(x,0)=u'(x,\frac{\tau}{2})\quad
 x\in\Omega;\\
 \sigma(x,0)=\delta(x,\frac{\tau}{2}),\quad
 \sigma'(x,0)=\frac{\partial u}{\partial \nu}(x,\frac{\tau}{2})\quad
 x\in\Gamma_1,
 \end{gathered}
\end{equation}
then we have a local solution $(v,\sigma)$ of \eqref{57} on 
$[\frac{\tau}{2},\frac{3\tau}{2}]$. By uniqueness, we obtain 
$(u,\delta)=(v,\sigma)$ on $[\frac{\tau}{2},\tau]$. Therefore we can extend 
the solution $(u,\delta)$ to whole interval $[0,\frac{3\tau}{2}]$. 
Then we have a global solution of \eqref{49}.
\end{proof}

\subsection*{Acknowledgments}
Andr\'e Vicente was partially supported by the Funda\c{c}\~ao \\
Arauc\'aria.

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\end{document}