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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 161, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/161\hfil wave equations with nonlinear boundary conditions]
{Observability inequality and decay rate for wave equations with
 nonlinear boundary conditions}

\author[Yuan Gao, Jin Liang, Ti-Jun Xiao \hfil EJDE-2017/161\hfilneg]
{Yuan Gao, Jin Liang, Ti-Jun Xiao}

\address{Yuan Gao \newline
Shanghai Key Laboratory for Contemporary Applied Mathematics,
School of Mathematical Sciences, Fudan University,
Shanghai 200433, China}
\email{gaoyuan12@fudan.edu.cn}

\address{Jin Liang \newline
 School of Mathematical Sciences,
Shanghai Jiao Tong University,
Shanghai 200240, China}
\email{jinliang@sjtu.edu.cn}

\address{Ti-Jun Xiao (corresponding author)\newline
Shanghai Key Laboratory for Contemporary Applied Mathematics,
School of Mathematical Sciences, Fudan University,
Shanghai 200433, China}
\email{tjxiao@fudan.edu.cn}

\dedicatory{Communicated by Jerome A Goldstein}

\thanks{Submitted September 8, 2016. Published July 4, 2017.}
\subjclass[2010]{35L70, 35B35, 76Q05, 35L20}
\keywords{Nonlinear wave; coupled boundary equations; uniform decay rates;
\hfill\break\indent observability inequality}

\begin{abstract}
 We study a class of wave propagation problems concerning the nonlinearity
 of dynamic evolution for boundary material. We establish an observability
 inequality for the related linear system, and make a connection between
 the linear system and the original nonlinear coupled system.
 Also, we obtain the desired energy decay rate for the original nonlinear
 boundary value problem.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

We are concerned with the nonlinear boundary value problem
\begin{gather}
u_{tt}(x,t)=\Delta u(x,t) ,\quad x\in\Omega,\, t>0;\label{be} \\
u(x,t)=0, \quad x\in\Gamma_{0},\, t>0;\label{bc0} \\
u_{t}(x,t)+f(z_{t})+g(z)=0 \quad x\in\Gamma_{1},\, t>0;\label{abc1}\\
\frac{\partial u} {\partial \nu}=z_t \quad x\in\Gamma_{1},\; t>0;\label{abc2}\\
u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x),\quad x \in\Omega,\quad
z(x,0)=z_0(x),\quad x\in\Gamma_1;\label{1.6}
\end{gather}
where $\Delta$ is the Laplacian operator, $\Omega$ is a bounded domain
in $\mathbb{R}^{n}$ with a boundary $\Gamma=\Gamma_{0}\cup \Gamma_{1}$
(disjoint, closed, and nonempty) of class $C^{2}$, and $f,g$ are given
functions on $\mathbb{R}$.

For some similar systems with or without source terms in \eqref{be},
there exist several results about uniform decay rate of the solutions to
these systems. For instance, \cite{Frota2006,Graber2010,Graber2011,Graber2012}
study the porous boundary condition with
the interface described by
\begin{gather*}
u_{t}+f(x) z_{t}+g(x) z=0, \quad x\in\Gamma_{1},\, t>0;\\
\frac{\partial u} {\partial \nu}+\rho(u_t)=z_t, \quad x\in\Gamma_{1},\, t>0,
\end{gather*}
where $ \rho$ is a given function. In this paper, we focus on the investigation
of the problem above concerning the nonlinearity of dynamic evolution for
 boundary material, which is always described by boundary displacement $z$.
We allow for nonlinear damping $f(z_t)$ and
nonlinear potential $g(z)$ ($f$ and $g$ may depend on $x$ also, which can be handled
similarly) in the boundary displacement equation (1.3). Such nonlinearity enables
our results to possess wide applicability.

Since our system is a coupled system and we hope to control the whole coupled
system by only using a single boundary damping, which is different from and
much more complex than the case of single equations, we will make efforts
to establish the observability of the related linear system, to find a useful
connection between the linear system and the original nonlinear system,
and finally to obtain the decay rate of the energy. We also would like
to state that our idea is stimulated by the significant papers
\cite{Alabau2011, Lasiecka2003, Beale1974, Frota2006, 3G03, GLX,Graber2011,XL0,XL}.

We first present some notation, basic definitions and assumptions
(cf., e.g., \cite{Alabau2011, GLX}).
Throughout this paper, $c,c_i$ are as generic constants whose values may
change from line to line. We make the following assumptions:
\begin{itemize}
 \item[(H1)] there exists $x_0\in\mathbb{R}^n$ such that
$m(x)\cdot\nu(x)\leq0$ for $x\in\Gamma_0$, where $m(x)=x-x_0$ and
$\nu(x)$ is the unit normal vector pointing to the exterior of $\Omega$.

 \item[(H2)] The function $g\in C(\mathbb{R})$ is monotone nondecreasing such
that $g(0)=0$; the function $f\in C^1(\mathbb{R})$ satisfies $f(0)=0$ and
$\inf_{s\in\mathbb{R}}f'(s)>0,$ and there exists a continuous strictly
increasing odd function $\rho\in C([-1,1];\mathbb{R}),$ which is continuously
differentiable in a neighbourhood of 0 with $\rho(0)=\rho'(0)=0$, such that
\begin{equation}\label{A1_f}
\begin{gathered}
 c_1\rho(|v|)\leq|f(v)|\leq c_2 \rho^{-1}(|v|),\quad
|v|\leq 1,\text{ a.e. on }\Gamma_1, \\
 c_1|v|\leq |f(v)|\leq c_2|v|,\quad |v|\geq 1,\text{ a.e. on }\Gamma_1.
 \end{gathered}
\end{equation}
Moreover, $g(s)$ is locally Lipschitz continuous such that
\begin{equation}\label{A1_g}
c_1|v|\leq |g(v)|\leq c_2|v|,\quad |v|\geq 1,\text{ a.e. }\Gamma_1.
\end{equation}
\end{itemize}

Also we define
 \begin{equation}\label{H}
 H(x):=\sqrt{x}\rho(\sqrt{x}),\quad x\in [0,r_0^2],
 \end{equation}
$r_0>0$ being small enough such that $H$ is strictly convex on
$[0,r_0^2]$. We define
\begin{equation}\label{L}
L(y):= \begin{cases}
 \hat{H}^{\star}(y)/y, & \text{if $y\in(0,\infty)$,} \\
 0, & \text{if $y=0$.}
 \end{cases}
\end{equation}
Here
$$
\hat{H}^{\star}: =\sup_{x\in\mathbb{R}}\{xy-\hat{H}(x)\}
$$
stands for the convex conjugate function of $\hat{H}$ (the extension of $H$
to $\mathbb{R}$ in which $\hat{H}(x)=+\infty$ for
$x\in\mathbb{R}\setminus [0,r_0^2]$).
Moreover, we define a function $\Lambda_H$ on $(0,r_0^2]$ by
$$
\Lambda_H(x):=\frac{H(x)}{xH'(x)},
$$
and write
\begin{equation*}
\psi(x):=\frac{1}{H'(r_0^2)}+\int_{1/x}^{H'(r_0^2)}
\frac{1}{v^2(1-\Lambda_H((H')^{-1}(v)))}dv,\quad x\geq \frac{1}{H'(r_0^2)}.
\end{equation*}
Then, there exists $\delta > 0$ such that $\psi$ is strictly increasing on
$[0,\delta]$.

Let
$$
V(\Omega)=\{u(x)\in H^1(\Omega),u|_{\Gamma_0}=0\},
$$
and define the inner products and norms on $V(\Omega),$ $L^2(\Omega)$, and
$L^2(\Gamma_1)$ respectively as follows
\begin{gather*}
((u,v))_V=\int_{\Omega}\nabla u(x)\cdot\nabla v(x)dx, \quad
\| u\|_V =\Big( \int _{\Omega }| \nabla u\left( x\right) | ^{2}dx
\Big) ^{1/2} , \\
 (u,v)=\int_{\Omega}u(x)v(x)dx,\quad |u|=(\int_{\Omega}(u(x))^2dx)^{1/2},\\
\langle\phi,\psi\rangle=\int_{\Gamma_1}\phi(x)\psi(x)d\Gamma,\quad
 |\phi|_{\Gamma_1}=(\int_{\Gamma_1}(\phi(x))^2dx)^{1/2}.
\end{gather*}
Clearly, the $\|\cdot\|_V$ is equivalent to the usual $H^1$ norm.

Define the ``finite energy space'' by
\[
\mathcal{H}:=V(\Omega)\times L^2(\Omega)\times L^2(\Gamma_1),
\]
where the norm on $\mathcal{H}$ is given by
\[
|(u,v,z)|_{\mathcal{H}}^2=\| u\\| _{V}^{2}+| v| ^{2}
+| z| _{\Gamma_1}^{2}.
\]
Define the energy of system \eqref{be}-\eqref{1.6} by
$$
E(t):=\frac{1}{2}\int_{\Omega}|\nabla u|^2+u_t^2 dx
+\frac{1}{2}\int_{\Gamma_1}z_t^2 d\Gamma+\int_{\Gamma_1}G(z) d\Gamma,
$$
where $G(x) = \int_0^x g(s)ds$ is the anti-derivative of $g$.

\section{Main results and proofs}

Rewrite the system \eqref{be}-\eqref{1.6} as
\begin{equation}\label{11}
 \frac{\partial}{\partial t}
\begin{pmatrix}
 u \\
 u_t \\
 z
 \end{pmatrix}
=\begin{pmatrix}
 u_t \\
 \Delta u \\
 f^{-1}(-u_t|_{\Gamma_1}-g(z)) \\
 \end{pmatrix}
 =\mathcal{A}\begin{pmatrix}
 u \\
 u_t \\
 z
 \end{pmatrix} .
\end{equation}
The action of the operator $\mathcal{A}$ is given by the matrix of operators 
that appears in \eqref{11}. The remaining boundary conditions are encoded 
in the domain of $\mathcal{A}$, given by
\begin{equation*}
 D(\mathcal{A})= \Big\{\begin{pmatrix}
 u \\
 v \\
 z
 \end{pmatrix}\in\mathcal{H};v\in V(\Omega), 
\Delta u\in L^2(\Omega),\frac{\partial u}{\partial \nu}\Big|_{\Gamma_1}
=f^{-1}(-v|_{\Gamma_1}-g(z))\Big\} .
\end{equation*}
From (H2), one knows that $f$ is strictly increasing, and its inverse function 
$f^{-1}$ is Lipschitz continuous.
Thus, using the standard method of nonlinear monotone operators and the semigroup
 theory (cf. \cite{Barbu2010}), we can obtain wellposedness of the system.


To study the energy decay rates of \eqref{be}-\eqref{1.6}, we first give 
an observability inequality of the following linear system, which has the 
same initial values as the original nonlinear system:
\begin{gather}
P_{tt}(x,t)=\Delta P(x,t) ,\quad x\in\Omega,\, t>0; \label{con1} \\
 P(x,t)=0, \quad x\in\Gamma_{0},\, t>0; \\
P_{t}(x,t)+ M_{t}(x,t)+ M(x,t)=0, \quad x\in\Gamma_{1},\, t>0;\label{conM} \\
\frac{\partial P(x,t)} {\partial \nu}=M_t, \quad x\in\Gamma_{1},\, t>0; \\
P(x,0)=u_0(x),\quad P_t(x,0)=u_1(x),\quad x \in\Omega; \\
\label{con6}
M(x,0)=z_0(x),\quad x\in\Gamma_1.
\end{gather}

Using the multiplier method (cf., e.g., \cite{Lasiecka2003, XL}), 
we can prove the following observability inequality.

\begin{theorem}[Observability inequality]\label{obs}
There is a constant $T_0>0 $, depending only on $\Omega$, such that for 
$T\geq T_0$, there corresponds a positive constant $C_T$ satisfying
\begin{equation}\label{ob1}
C_T E_p(0)\leq \int_0^T\int_{\Gamma_1}M_t^2\,dx\,dt,
\end{equation}
 where 
$$
E_p(t):=\frac{1}{2}\int_{\Omega}P_t^2+|\nabla P|^2 dx
+\frac{1}{2}\int_{\Gamma_1}M^2 d\Gamma
$$
 is the energy of \eqref{con1}-\eqref{con6}.
\end{theorem}

\begin{proof} The proof is divided into the following 5 steps.
\smallskip

\noindent\textbf{Step 1:} Let $\xi(t)\in C_0^{\infty}(\mathbb{R})$ 
be the cutoff function defined by
$$
\xi(t)=\begin{cases}
 1, & t\in[\epsilon_0,T-\epsilon_0] \\
 \text{a $C^{\infty}$ function with range in $(0,1)$,} 
& t\in(0,\epsilon_0)\cup(T-\epsilon_0,T) \\
 0, & t\in(-\infty,0)\cup(T,\infty),
 \end{cases}
$$
 for $\epsilon_0\in (0, T/2)$.

 Let $h$ be a $[C^2(\bar{\Omega})]^n$-vector field, which will be specified later. 
Then, multiplying \eqref{con1} by $h\cdot\nabla P$, integrating in time and 
space and using the boundary condition, we obtain
\begin{align*}
 0 &= \int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega}h\cdot\nabla P(P_{tt}
 -\Delta P) \,dx\,dt \\
 &= (h\cdot\nabla P,P_t)_{L^2(\Omega)}\Big|_{\epsilon_0}^{T-\epsilon_0}
 -\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega}
 \Big[ \nabla\cdot(\frac{h}{2}(P_t^2))-\frac{\nabla\cdot h}{2}P_t^2 \\
 &\quad -\nabla\cdot(\frac{h}{2}|\nabla P|^2)\Big]\,dx\,dt
  -\int_{\Gamma_1}\int_{\epsilon_0}^{T-\epsilon_0}h\cdot\nabla PM_t d\Gamma dt \\
&\quad  +\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega} J|\nabla P|^2 \,dx\,dt
 -\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega}\frac{\nabla\cdot h}{2}|\nabla P|^2 \,dx\,dt \\
 &= (h\cdot\nabla P,P_t)_{L^2(\Omega)}\Big|_{\epsilon_0}^{T-\epsilon_0}
 -\int_{\Gamma}\int_{\epsilon_0}^{T-\epsilon_0}\frac{h\cdot\nu}{2}
 (P_t^2-|\nabla P|^2)d\Gamma dt\\
 &\quad +\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega}
 \frac{\nabla\cdot h}{2}(P_t^2-|\nabla P|^2)\,dx\,dt
 +\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega}J|\nabla P|^2 \,dx\,dt\\
 &\quad -\int_{\Gamma_1}\int_{\epsilon_0}^{T-\epsilon_0}h
 \cdot\nabla PM_t d\Gamma dt,
\end{align*}
where $J:=\frac{\partial h_i(x)}{\partial x_j}$.

By (H1) we can take $h$ such that
\begin{gather*}
 h\cdot\nu= 0\quad \text{on }\Gamma_0, \\
 J=\frac{\partial h_i(x)}{\partial x_j}\geq \rho_0 I\quad \text{on }\Omega,
\end{gather*}
for some constant $\rho_0>0$. Hence,
\begin{align*}
&\rho_0\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega}|\nabla P|^2 \,dx\,dt \\
&\leq \int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega}
 J|\nabla P|^2 \,dx\,dt \\
&\leq \int_{\epsilon_0}^{T-\epsilon_0}\int_{\Gamma_1}h\cdot\nabla PM_t d\Gamma dt
 + \int_{\epsilon_0}^{T-\epsilon_0}\int_{\Gamma_1}
 \frac{h\cdot\nu}{2}(P_t^2-|\nabla P|^2)d\Gamma dt\\
&\quad -(h\cdot\nabla P,P_t)_{L^2(\Omega)}\Big|_{\epsilon_0}^{T-\epsilon_0}
 -\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega}\frac{\nabla\cdot h}{2}
 (P_t^2-|\nabla P|^2)\,dx\,dt.
\end{align*}
Since 
$$
|\nabla P|^2=(M_t^2+|\frac{\partial P}{\partial\tau}|^2), \quad 
E'_p=-\int_{\Gamma_1}M_t^2 d\Gamma\leq 0,
$$ 
we have
\begin{equation}\label{n14}
\begin{aligned}
&\rho_0\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega}|\nabla P|^2 \,dx\,dt \\
&\leq \Big|\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega}
 \frac{\nabla\cdot h}{2}(P_t^2-|\nabla P|^2) \,dx\,dt \Big|\\
&\quad +C_h\Big[\int_{\Sigma_1}M_t^2 d\Gamma dt
 +\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Gamma_1}P_t^2
 +|\frac{\partial P}{\partial\tau}|^2 d\Gamma dt\Big]+C_h E_p(0),
\end{aligned}
\end{equation}
where $\Sigma_1:=(0,T)\times\Gamma_1$, and $C_h$ is a positive constant 
depending on $h$.
Write
$$
{\rm l.o.t}(P,M):=\|(P,P_t,M)\|_{C([0,T];H^{1-\epsilon}(\Omega)
\times H^{-\epsilon}(\Omega)\times H^{-\epsilon}(\Gamma_1))},
$$
for $\epsilon>0$.

Multiplying \eqref{con1} by $P\nabla\cdot h$, integrating in time and space, 
and using the boundary condition and Sobolev Trace Theory, we obtain
\begin{equation}\label{n12}
\begin{aligned}
&\Big| \int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega}\nabla\cdot 
h(P_t^2-|\nabla P|^2)\,dx\,dt\Big| \\
&= \Big|\langle P_t,P\nabla\cdot h\rangle _{H^{-\epsilon}(\Omega)
 \times H^{\epsilon}(\Omega)}\Big|_{\epsilon_0}^{T-\epsilon_0}
 +\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega} P\nabla P\cdot
 \nabla(\nabla\cdot h)\,dx\,dt\\
&\quad -\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Gamma_1}P\nabla\cdot 
 h M_td\Gamma dt \Big|\\
&\leq C_{\epsilon}\int_{\Sigma_1}M_t^2 d\Gamma dt
 +\epsilon\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega}|\nabla P|^2\,dx\,dt
+{\rm l.o.t}(P,M).
\end{aligned}
\end{equation}
Let $\min\{\nabla h\}=d_0>0$. Combining \eqref{n12} and \eqref{n14} gives
\begin{equation}\label{n16}
\begin{aligned}
 &\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega} |\nabla P|^2+P_t^2 \,dx\,dt \\
 &\leq C_{\epsilon,h}\Big\{\int_{\Sigma_1}M_t^2 d\Gamma dt
+\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Gamma_1}(P_t^2
+|\frac{\partial P}{\partial\tau}|^2)d\Gamma dt\Big\} \\
&\quad +C_hE_p(0)+{\rm l.o.t}(P,M).
 \end{aligned}
\end{equation}
Using \cite[Lemma 4]{Lasiecka2003} to estimate 
$\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Gamma_1}|
\frac{\partial P}{\partial\tau}|^2 d\Gamma dt$ in \eqref{n16}, we obtain
\begin{equation}\label{n18}
\begin{aligned}
&\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Omega} |\nabla P|^2+P_t^2 \,dx\,dt \\
 &\leq C_{T,\epsilon_0,h}\Big\{\int_{\Sigma_1}M_t^2+\xi^2P_t^2 d\Gamma dt
+\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Gamma_1}P_t^2 d\Gamma dt\Big\}\\
&\quad + C_hE_p(0)+{\rm l.o.t}(P,M).
\end{aligned}
\end{equation}
\smallskip

\noindent\textbf{Step 2:} We estimate
 $\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Gamma_1}P_t^2 d\Gamma dt
+\int_{\Sigma_1}\xi^2P_t^2 d\Gamma dt$.
The boundary condition on $\Gamma_1$ shows that
\[
\int_{\epsilon_0}^{T-\epsilon_0}\int_{\Gamma_1}P_t^2 d\Gamma dt
\leq \int_{\Sigma_1}\xi^2P_t^2 d\Gamma dt 
\leq 2\int_{\Sigma_1}M_t^2+M^2 d\Gamma dt.
\]
By \eqref{n18}, we have
\begin{equation}\label{n20}
 \int_{\epsilon_0}^{T-\epsilon_0}E_p(t)dt
\leq C_{T,\epsilon_0,h,f}\int_{\Sigma_1}(M_t^2+M^2) d\Gamma dt+C_hE_p(0)
+{\rm l.o.t}(P,M).
\end{equation}
From $E'_p=-\int_{\Gamma_1}M_t^2 d\Gamma $, it follows that
\begin{equation}\label{n30}
\begin{aligned}
& (T-2\epsilon_0)\Big[E_p(0)-\int_{\Sigma_1}M_t^2 d\Gamma dt\Big] \\
 &\leq  (T-2\epsilon_0)E_p(T) \\
 &\leq \int_{\epsilon_0}^{T-\epsilon_0}E_p dt \\
 &\leq C_{T,\epsilon_0,h,f}\int_{\Sigma_1}(M_t^2+M^2)d\Gamma dt+C_hE_p(0)
+{\rm l.o.t}(P,M).
\end{aligned}
\end{equation}
\smallskip

\noindent\textbf{Step 3:} We estimate $\int_{\Sigma_1}M^2d\Gamma dt$.
Multiplying \eqref{conM} by $M$ and integrating in time and space, we obtain
\begin{align*}
 0&=\int_{\Sigma_1}M(P_t-M_t+M)d\Gamma dt \\
 &=\int_{\Gamma_1}MP d\Gamma\Big|_{t=0}^{t=T} -\int_{\Sigma_1}(M_tP+MM_t-M^2)d\Gamma dt.
\end{align*}
Hence
\begin{equation}\label{n40}
\begin{aligned}
\int_{\Sigma_1}M^2d\Gamma dt 
&= \Big|\int_{\Sigma_1}MM_td\Gamma dt +\int_{\Sigma_1}M_tP d\Gamma dt
-\int_{\Gamma_1}MP d\Gamma \Big|_{t=0}^{t=T} \big| \\
& \leq  \epsilon_1\int_{\Sigma_1}M^2d\Gamma dt
+C_{\epsilon_1}\int_{\Sigma_1}M_t^2d\Gamma dt+{\rm l.o.t}.(P,M),
\end{aligned}
\end{equation}
where $\epsilon_1$ is arbitrarily small.
Combining this with \eqref{n30}, we obtain
\begin{equation}\label{n34}
 (T-2\epsilon_0-C_h)E_p(0)\leq C_{T,\epsilon_0,h}\int_{\Sigma_1}M_t^2d\Gamma dt
+{\rm l.o.t}(P,M).
\end{equation}
Therefore, for $T>T_0:=2\epsilon_0-C_h$, we almost get \eqref{ob1} except 
for the lower-order terms ${\rm l.o.t}(P,M)$.
\smallskip

\noindent\textbf{Step 5:} We claim that for 
$$
T>T_1=\max\{T_0,2{\rm diam}(\Omega)\},
$$ 
there exists a constant $C_T>0$ such that the solution of 
\eqref{con1}-\eqref{con6} satisfies the  inequality
\begin{equation}\label{l.o.t}
 {\rm l.o.t} (P,M)\leq C_T\|M_t\|_{L^2(\Sigma_1)}^2.
\end{equation}
Suppose this is false. Then there exists a sequence 
$$
{(P(0)^n,P_t(0)^n,M(0)^n)}\subset \mathcal{H},
$$ 
and a corresponding solution sequence ${(P^n,P_t^n,M^n)}$ 
of \eqref{con1}-\eqref{con6} such that
\begin{gather*}
 {\rm l.o.t} (P^n,M^n) = 1\quad \forall n, \\
 \|M^n_t\|_{L^2(\Sigma_1)}^2 \to 0 \quad n\to\infty.
\end{gather*}
Thus, by \eqref{n34}, we see that $\|(P(0)^n,P_t(0)^n,M(0)^n)\|_{\mathcal{H}}$ 
is bounded for $T$ large enough. Hence there is a subsequence, 
still denoted by 
$$
{(P(0)^n,P_t(0)^n,M(0)^n)},\quad 
 (P(0)^{\ast},P_t(0)^{\ast},M(0)^{\ast}),
$$ 
such that
\begin{equation}\label{n51}
(P(0)^n,P_t(0)^n,M(0)^n)\to (P(0)^{\ast},P_t(0)^{\ast},M(0)^{\ast}),\quad 
\text{in $\mathcal{H}$ weakly}.
\end{equation}
Let $(P^{\ast},P_t^{\ast},M^{\ast})$ be the solution corresponding to 
$(P(0)^{\ast},P_t(0)^{\ast},M(0)^{\ast})$. Then from
$$
E'_p=-\int_{\Gamma_1}M_t^2 d\Gamma<0,
$$ 
it follows that
 \begin{equation}\label{n52}
 (P^n,P_t^n,M^n)\to (P^{\ast},P_t^{\ast},M^{\ast}),\quad 
\text{weak star in $L^{\infty}(0,T;\mathcal{H})$.}
 \end{equation}
 Clearly, $\|(P^n,P_t^n,M^n)\|_{C(0,T;\mathcal{H})}$ is bounded by the 
wellposedness of the system. Let
 \begin{gather*}
 X := H^1(\Omega)\times L^2(\Omega) \times L^2(\Gamma_1), \\
 B := H^{1-\epsilon}(\Omega)\times H^{-\epsilon}(\Omega)
 \times H^{-\epsilon}(\Gamma_1), \\
 Y := H^{-\epsilon}(\Omega)\times ({H^1(\Omega)})'
 \times H^{-\epsilon}(\Gamma_1) .
 \end{gather*}
We claim that $X\hookrightarrow B$ compactly. Indeed,
 for all  $s,t\in\mathbb{R}$ with $s>t$, for an arbitrary bounded set 
$\{\psi_n\}\subset H^s(\Omega)$, we can extend the domain of $\psi_n$ 
to $\hat{\Omega}$, such that $\psi_n|_{\partial\hat{\Omega}}=0$. 
It is known that $H_0^s(\hat{\Omega})$ is compactly embedded in 
$H_0^t(\hat{\Omega})$. Hence, there exists a $\psi\in H_0^t(\hat{\Omega})$ 
such that $\|\psi_{n_i}-\psi\|_{H_0^t(\hat{\Omega})}\to 0$.
 Hence $\|\psi_{n_i}-\psi\|_{H^t(\Omega)}\to 0$.

We also claim that 
$$
\|(P_t^n,P_{tt}^n,M_t^n)\|_{L^2(0,T;Y)}\leq C\quad \text{uniformly}.
$$
Indeed, it suffices to estimate $\|P_{tt}^n\|_{L^2(0,T;({H^1(\Omega)})')}$.
By \eqref{con1} and the boundary condition, we see that for all $t\in(0,T)$ 
and $u\in H^1(\Omega)$,
\begin{equation}\label{Ptt}
 \langle P_{tt},u\rangle=\int_{\Omega}\Delta P udx
=\int_{\Gamma_1}M_tud\Gamma -\int_{\Omega}\nabla P\cdot\nabla u dx
\end{equation}
is meaningful. Hence $P_{tt}\in L^{\infty}(0,T;{(H^1(\Omega)})')$.

We deduce then by a classic compactness result (see \cite{Simon1987}) that
\begin{equation*}
 (P^n,P_t^n,M^n)\to (P^{\ast},P_t^{\ast},M^{\ast})\quad {\rm in}~~L^{\infty}(0,T;B)~~{\rm strongly}.
\end{equation*}
Therefore,
\begin{equation}\label{n55}
 \|(P^{\ast},P^{\ast}_t,M^{\ast})\|_{C([0,T];H^{1-\epsilon}(\Omega)
\times H^{-\epsilon}(\Omega)\times H^{-\epsilon}(\Gamma_1))} = 1.
\end{equation}

On the other hand, by \eqref{n51}, we have $M_t^{\ast}=0$. 
Differentiating \eqref{conM} in time, we obtain $P_{tt}^{\ast}|_{\Gamma_1}=0$. 
Let $a(t,x)=P_{tt}^{\ast}(t,x)$ such that
\begin{gather*}
 a_{tt}=\Delta a, \quad \text{in } \Omega\times(0,T), \\
 \frac{\partial a}{\partial \nu} =(\frac{\partial P}{\partial\nu})_{tt}=0, \quad 
\text{on } \Gamma\times(0,T), \\
 a = 0, \quad \text{on }\Gamma_1.
\end{gather*}
Using Holmgren's Uniqueness Theorem \cite{Graber2011}, with $T>2{\rm diam}(\Omega)$,
$$
a(t,x)=P_{tt}^{\ast}(t,x)=0, \quad\text{in } \Omega\times(0,T).
$$
Then from
\begin{gather*}
\Delta P^{\ast} = 0,\quad \text{in } \Omega, \\
P^{\ast}|_{\Gamma_0}=0, \quad \frac{\partial P^{\ast}}{\partial\nu}|_{\Gamma_1} =0,
\end{gather*}
we know that $P^{\ast}=0$. So we obtain $M^{\ast}=0$ due to \eqref{conM}. 
Thus $(P^{\ast},M^{\ast})=(0,0)$ contradicts \eqref{n55}.
A combination of Steps 1-5 completes the proof.
\end{proof}

Next we show a connection between linear and nonlinear systems.

\begin{theorem}\label{lem2}
Assume that $(u,u_t,z)$ and $(P,P_t,M) $ are solutions of system 
\eqref{be}-\eqref{1.6} and \eqref{con1}-\eqref{con6} respectively. Then
\begin{equation}\label{compare}
\int_{\Sigma_1}M_t^2 d\Gamma dt\leq C\int_{\Sigma_1}z_t^2+f(z_t)^2 d\Gamma dt.
\end{equation}
\end{theorem}

\begin{proof}
Set $\xi=u-P$, $\eta=z-M$. Then $(\xi,\xi_t,\eta)$ is the solution of
\begin{equation} \label{com1}
\begin{gathered}
\xi_{tt}(x,t)=\Delta \xi(x,t) ,\quad x\in\Omega,\, t>0; \\
\frac{\partial \xi(x,t)} {\partial \nu}=0 \quad x\in\Gamma_{0},\, t>0; \\
\xi_{t}(x,t)+f(z_{t})-M_t+g( z)-M=0 \quad x\in\Gamma_{1},\, t>0; \\
\frac{\partial \xi} {\partial \nu}(x,t)=\eta_t(x,t) \quad x\in\Gamma_{1},\, t>0; \\
\xi(x,0)=0,\quad \xi_t(x,0)=0,\quad x \in\Omega; \\
\eta(x,0)=0,\quad x\in\Gamma_1.
\end{gathered}
\end{equation}
Multiplying \eqref{com1} by $\xi_t$, integrating in time and space, we obtain
\begin{equation}\label{*1}
\begin{aligned}
& \int_0^t\int_{\Omega}(\frac{\xi_t^2}{2}+\frac{|\nabla \xi|^2}{2})_t \,dx\,dt\\
&= \int_0^t\int_{\Gamma_1}\frac{\partial \xi}{\partial \nu}\xi_t d\Gamma dt\\
&=\int\int_{\Gamma_1}\eta(M_t-f(z_t)+M-g(z))d\Gamma dt \\
&= \int\int_{\Gamma_1}(z_t-M_t)[M_t-f(z_t)+M-g(z)]d\Gamma dt.
\end{aligned}
\end{equation}
Take $\epsilon>0$ small enough. \eqref{A1_g} implies that there exist 
$c_1>0,\,c_2>0$ such that
\begin{equation*}
c_1|v|\leq |g(v)|\leq c_2|v|,\quad |v|\geq \epsilon,\text{a.e. }\Gamma_1.
\end{equation*}
Assuming $z>\epsilon$, we have, by \eqref{*1} and 
$\int_{\Gamma_1}-z_tf(z_t) d\Gamma\leq 0$,
\begin{align*}
\int_0^t\int_{\Omega}(\frac{\xi_t^2}{2}+\frac{|\nabla \xi|^2}{2})_t \,dx\,dt 
&\leq \int_0^t\int_{\Gamma_1}-M_t^2+M_tf(z_t)+z_tM_t d\Gamma dt \\
&\quad +\int_0^t\int_{\Gamma_1\cap\{\eta_t\geq0\}}\max\{-\eta\eta_t,-c_1\eta\eta_t\}
 d\Gamma dt \\
&\quad +\int_0^t\int_{\Gamma_1\cap\{\eta_t<0\}}\max\{-\eta\eta_t,-c_2\eta\eta_t\}d\Gamma dt.
\end{align*}
Therefore
\begin{align*}
&\int_0^t\int_{\Omega}(\frac{\xi_t^2}{2}
 +\frac{|\nabla \xi|^2}{2})_t \,dx\,dt+\int_0^t\int_{\Gamma_1}M_t^2 d\Gamma dt\\
&\leq\int_0^t\int_{\Gamma_1} M_tf(z_t)+z_tM_t d\Gamma dt \\
&\quad +\int_0^t\int_{\Gamma_1\cap\{\eta_t\geq0\}}
 \max\{-(\frac{\eta^2}{2})_t,-c_1(\frac{\eta^2}{2})_t\}d\Gamma dt\\
&\quad +\int_0^t\int_{\Gamma_1\cap\{\eta_t<0\}}
 \max\{-(\frac{\eta^2}{2})_t,-c_2(\frac{\eta^2}{2})_t\}d\Gamma dt.
\end{align*}
Noting the initial values and using Young's inequality, we obtain
\begin{equation}
\begin{aligned}
&\int_0^t\int_{\Omega}(\frac{\xi_t^2}{2}+\frac{|\nabla \xi|^2}{2})_t \,dx\,dt
+\int_0^t\int_{\Gamma_1}M_t^2 d\Gamma dt\\
&\leq c\int_0^t\int_{\Gamma_1} f(z_t)^2+z_t^2 d\Gamma dt 
\end{aligned} \label{05t1},
\end{equation}
giving \eqref{compare}.
Similarly, we obtain \eqref{compare} for $z<-\epsilon$.

Finally, choose $\epsilon $ small enough such that $|g(z)|\leq c\epsilon$ and 
$|z|\leq\epsilon$. By \eqref{*1} we have
$$
\int_0^t\int_{\Omega}(\frac{\xi_t^2}{2}+\frac{|\nabla \xi|^2}{2})_t \,dx\,dt 
=\int_0^t\int_{\Gamma_1}(z_t-M_t)[M_t-f(z_t)+M-z+z-g(z)]d\Gamma dt,
$$
and
\begin{align*}
&\int_0^t\int_{\Omega}(\frac{\xi_t^2}{2}+\frac{|\nabla \xi|^2}{2}
 +\frac{\eta^2}{2})_t \,dx\,dt\\
&\leq  \int_0^t\int_{\Gamma_1}[-M_t^2+z_tM_t+M_tf(z_t)+zz_t-M_tz-z_tg(z)
 +M_tg(z)]d\Gamma dt.
\end{align*}
By Young's inequality and H\"older's inequality, we obtain
\begin{equation}
\begin{aligned}
&\int_0^t\int_{\Omega}(\frac{\xi_t^2}{2}+\frac{|\nabla \xi|^2}{2}
 +\frac{\eta^2}{2})_t \,dx\,dt+\int_0^t\int_{\Gamma_1}M_t^2d\Gamma dt\\
&\leq  \int_0^t\int_{\Gamma_1}[\epsilon_0 M_t^2
 +C(\epsilon_0)(z_t^2+f(z_t)^2+\epsilon^2)]d\Gamma dt.
\end{aligned} \label{05t2}
\end{equation}
Since the constant in \eqref{05t1} dose not depend on $\epsilon$, we can 
let $\epsilon\to 0$ in \eqref{05t2}.
Noticing the initial values, we then obtain \eqref{compare}.
\end{proof}


\begin{theorem}[Decay rate]\label{th2}
Suppose that 
$$
\lim_{x\to 0^+}\frac{H'(x)}{\Lambda_H(x)}=0,
$$
and $T$ is a time such that \eqref{ob1} holds.
Then the energy of system \eqref{be}-\eqref{1.6} satisfies
$$
E(t)\leq C(T, E(0)) L\Big(\frac{1}{\psi^{-1}(\frac{t-T}{T_\star})}\Big),
$$
for $t$ large enough; moreover, if 
$$
\limsup_{x\to 0^+}\Lambda_H(x)<1,
$$ 
then we have
$$
E(t)\leq C(T, E(0))(H')^{-1}\left(\frac{c_0}{t-T}\right),\quad
\text{for $t$ large enough.}
$$
Here, $C(T, E(0))$ is a positive constant depending on $T$ and $E(0)$, 
and $T_{\star}>0$ depends on $T$.
\end{theorem}

\begin{proof}
Clearly, we see that
\begin{align*}
\int_{\Gamma_1}G(z) d\Gamma \
&=\int_{\Gamma_1}\int_0^zg(s)ds \\
&\leq \int_{\Gamma_1\cap\{z\geq 1\}}\int_0^z c_2 s dsd\Gamma
+ \int_{\Gamma_1\cap\{z\leq -1\}}\int_0^z c_1 s dsd\Gamma \\
&\quad +\int_{\Gamma_1\cap\{|z|\leq 1\}}\int_0^z g(s) dsd\Gamma\\
&\leq \frac{c}{2}\int_{\Gamma_1}z^2d\Gamma.
\end{align*}
Setting $c_0=\max(c,1)$, we have
\begin{equation}\label{E0}
E(0)\leq c_0E_p(0).
\end{equation}
Let $w$ satisfy
$$
H^{\star}(w(s))=\frac{sw(s)}{\beta},\quad s\in[0,\beta r_0^2),
$$
where
\begin{equation}\label{beta}
\beta >\max\Big\{\frac{E(0)}{c_0 L(H'(r_0^2))},\frac{E(0)}{c_0\delta}\Big\},
\end{equation}
$ r_0$ is as in \eqref{H}, and $\delta>0$ is a constant such that $\psi$ 
is strictly increasing on $[0,\delta]$.
Then the definition of $L$ implies 
\begin{equation}\label{w}
 w(s)=L^{-1}\left(\frac{s}{\beta}\right), \quad \forall s\in [0,\beta r_0^2).
\end{equation}
From the property of $L$, it follows that $w$ is a strictly increasing function from $[0,\beta r_0^2)$ onto $[0,+\infty)$.
Thus, by using the optimal-weight convexity method (cf. \cite[Lemma 2.1]{Alabau2011}), we deduce that
\begin{align*}
& w(E_p(0))\int_{\Sigma_1}z_t^2+f(z_t)^2d\Gamma dt\\
&\leq c_3 TH^{\star}(w(E_p(0)))+c_4(w(E_p(0))+1)\int_{\Sigma_1}z_tf(z_t)d\Gamma dt.
\end{align*}
This and Theorems \ref{obs} and \ref{lem2}  yield
\begin{align*}
&C_T E_p(0)w(E_p(0)) \\
&\leq w(E_p(0))\int_0^T\int_{\Gamma_1}M_t^2 d\Gamma dt 
 \leq Cw(E_p(0))\int_0^T\int_{\Gamma_1}z_t^2+f(z_t)^2 d\Gamma dt \\
&\leq T\tilde{c}_3 H^{\star}(w(E_p(0)))+c_6(w(E_p(0))+1)
 \int_{\Sigma_1}z_tf(z_t)d\Gamma dt \\
&\leq T c_5\frac{E_p(0)w(E_p(0))}{\beta}+c_6(H'(r_0^2)+1)
 \int_{\Sigma_1}z_tf(z_t)d\Gamma dt ,
\end{align*}
where we used \eqref{w} and $\beta>\frac{E(0)}{c_0 L(H'(r_0^2))}$ 
in the last inequality.
From this and \eqref{E0}, we have
$$
\Big(\tilde{C}_T-\frac{\tilde{c}_5T}{\beta}\Big)
\frac{E(0)}{c_0}w\Big(\frac{E(0)}{c_0}\Big)\leq E(0)-E(T).
$$
Thanks to $\beta>\frac{T\tilde{c}_5}{\tilde{C}_T}$, we set
\begin{equation}\label{rhoT}
 \rho_T:=\frac{1}{c_0}\Big(\tilde{C}_T-\frac{T\tilde{c}_5}{\beta}\Big)>0
\end{equation}
and deduce that
$$
E(T)\leq E(0)\Big[1-\rho_T w(\frac{E(0)}{c_0})\Big]
=E(0)\Big[1-\rho_T L^{-1}(\frac{E(0)}{c_0\beta})\Big].
$$
Denoting $E_k:=\frac{E(kT)}{c_0\beta}$, we obtain
$$
E_1\leq E_0[1-\rho_T L^{-1}(E_0)].
$$
From  the invariance by time translation $t-kT$ for system 
\eqref{be}-\eqref{1.6} and \eqref{con1}-\eqref{con6}, we have
$$
E_{k+1}\leq E_k[1-\rho_T L^{-1}(E_k)].
$$
Because $\beta> \frac{E(0)}{c_0\delta}$, we can apply \cite[Theorem 1.5]{Alabau2011} 
to complete the proof.
\end{proof}

\begin{remark}\rm 
Under the assumptions of Theorem \ref{th2}, we have
$$
L\Big(\frac{1}{\psi^{-1}(\frac{t-T}{T_{\star}})}\Big)\to 0, \text{ as }t\to 0.
$$
Moreover, by taking special $f$ and $g$, we can see clearly the meaning of 
the decay rate (please see the examples in \cite[Section 4]{Alabau2011}).
\end{remark}

\subsection*{Acknowledgments}
This work was supported partly by the National Natural Science Foundation of China
(11371095, 11571229), and the Shanghai Key
Laboratory for Contemporary Applied Mathematics.

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\end{document}