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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 129, pp. 1--14.\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-2017/129\hfil 
 Memory boundary feedback stabilization]
{Memory boundary feedback stabilization for Schr\"odinger
equations with variable coefficients}

\author[A. Nawel, K. Melkemi \hfil EJDE-2017/129\hfilneg]
{Abdesselam Nawel, Khaled Melkemi}

\address{Abdesselam Nawel \newline
Department of Mathematics,
University of Laghouat, Algeria}
\email{nawelabedess@gmail.com}

\address{Khaled Melkemi \newline
Department of Mathematics,
University of Batna II, Algeria}
\email{k.melkemi@univ-batna2.dz}

\dedicatory{Communicated by Jesus Ildefonso Diaz}

\thanks{Submitted February 20, 2017. Published May 11, 2017.}
\subjclass[2010]{93D15, 35J10, 35B40, 53C17}
\keywords{Schr\"odinger equation; exponential stabilization;
\hfill\break\indent boundary condition of memory type; Riemannian geometry}

\begin{abstract}
 First we consider the boundary stabilization of Schr\"odinger
 equations with constant coefficient memory feedback.
 This is done by using Riemannian geometry methods and the
 multipliers technique.
 Then we explore the stabilization limits of  Schr\"odinger equations
 whose elliptical part has a variable coefficient. We  established
 the exponential decay of solutions using the multipliers techniques.
 The introduction of dissipative boundary conditions of memory type allowed
 us to obtain an accurate estimate on the uniform rate of decay of the energy
 for Schr\"odinger equations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

Let $\Omega $ be an open bounded domain in
 $\mathbb{R}^n$ with boundary $\Gamma :=\partial \Omega $.
It is assumed that $\Gamma $ consists of two parts $\Gamma _0$ and $\Gamma _1$
such that $\Gamma _0,\Gamma _1\neq \emptyset $,
$\overline{\Gamma _0}\cap\overline{\Gamma _1}=\emptyset $.
We consider the mixed problem for Schr\"odinger equation
\begin{gather}
y_{t}-\mathbf{i}Ay =0 \quad \text{in } \Omega \times \mathbb{R} _{+},  \label{1.1a} \\
y( 0,x) =y_0( x) \quad \text{in }  \Omega ,  \label{1.1b} \\
y =0 \quad \text{on }\Gamma _0\times \mathbb{R}_{+},  \label{1.1c} \\
\frac{\partial y}{\partial \nu _{A}} =u \quad \text{on }
 \Gamma _1\times \mathbb{R}_{+},  \label{1.1d}
\end{gather}
where
\begin{equation*}
Ay=\sum_{i,j=1}^n\frac{\partial }{\partial x_i}
\Big( a_{ij}(x) \frac{\partial y}{\partial x_{j}}\Big),
\end{equation*}
the functions $a_{ij}=a_{ji}$ are $C^{\infty }$ functions in $\mathbb{R}^n$,
\[
\frac{\partial y}{\partial \nu _{A}}
=\sum_{i,j=1}^na_{ij}(x,t) \frac{\partial y}{\partial x_i}\nu _i,
\]
$\nu =( \nu_1,\nu _2,\dots,\nu _n) $ is the unit normal of
$\Gamma $ pointing toward the exterior of $\Omega$, $\nu _{A}=A\nu$, and
$A=(a_{ij}) $ is a matrix function. We assume that
\begin{equation}
\sum_{i,j=1}^na_{ij}( x) \xi _i\overline{\xi _{j}}>\alpha
_1\sum_{i=1}^n\xi _i^2\quad \forall x\in \Omega ,\; \zeta
\in \mathbb{C} ^n,\;\zeta \neq 0,  \label{1.2}
\end{equation}
for some positive constant $\alpha _1$, and
\begin{equation*}
u=-\int_0^{t}k( t-s) y_{s}( s) ds-by_{t}\,.
\end{equation*}
Then we transform the boundary condition in $\Gamma _1$.
Suppose that $y_0=0$ on $\Gamma _1$, by integration by parts
\begin{align*}
\int_0^{t}k( t-s) y_{t}( s) ds
&= [ k(t-s) y( s) ] _0^{t}+\int_0^{t}k'(t-s) y( s) ds \\
&= \int_0^{t}k'( t-s) y( s) ds+k(0) y( t) -k( t) y_0 \\
&= \int_0^{t}k'( t-s) y( s) ds+k(0) y( t) .
\end{align*}

Throughout the paper we assume that
\begin{equation}
u=-\int_0^{t}k'( t-s) y( s) ds-k( o)
y( t) -by_{t}.\text{\ }  \label{1.3}
\end{equation}
where\ $k:\Gamma _1\times \mathbb{R}_{+}\to\mathbb{R}_{+}\in C^2
(\mathbb{R}^{+},L^{\infty }( \Omega ) ) $ and
 $b:\Gamma_1\to\mathbb{R}_{+}\in L^{\infty }( \Omega ) $.
We define the corresponding energy functional by
\begin{equation}
2E( t) :=\int_{\Omega }| \nabla _{g}y|_{g}^2\,d\Omega
+\int_{\Gamma _1}k| y| ^2\,d\Gamma_1-\int_0^{t}\int_{\Gamma _1}k'( t-s) | y(
t) -y( s) | \,d\Gamma _1ds.  \label{1.4}
\end{equation}

Our goal  is to stabilize the system \eqref{1.1a}--\eqref{1.1d} and
\eqref{1.3}; to find a suitable feedback $u=F( x,y_{t}) $ such
that the energy \eqref{1.4} decays to zero exponentially as
$t\to +\infty $ for every solution $y$ of which $E( 0)<+\infty$.
The approach adopted uses Riemannian geometry. This method was
first introduced to boundary-control problems by Yao [10] for the
exactly controllability of wave equations.

The stabilization of partial differential equations has been considered by
many authors.
The asymptotic behaviour of the wave equation with memory and linear
feedbacks with constant coefficients has been studied by Guesmia [4], and by
Aassila et al [1] in the nonlinear case. This study has been generalised by
Chai \& Guo [2] for variable coefficients by using a very different
method, namely, the Riemannian geometry method.

On the other hand, the stabilization of the Schr\"odinger equations has
been studied by Machtyngier \& Zuazua [9] in the Neumann boundary
conditions, and by Cipolatti et al [3] with nonlinear feedbacks. This study
has been considered by Lasiecka \& Triggani [8] with constant
coefficients acting in the Dirichlet boundary conditions.

The objective of this work, we consider the boundary stabilization for
system \eqref{1.1a}--\eqref{1.1d} and \eqref{1.3} with variable coefficients and
memory feedbacks by using multipliers techniques.

Our paper is organized as follows.
In subsection 1.1, we introduce some notation and results on Riemannian geometry.
 Our main results are studied in section 2.
Section 3 is devoted to the proof of the main results.

\subsection{Notation}

The definitions here are standard and classical in the literature, see Hebey [5].
Let $A( x) $ and $G( x) $, respectively, be the
coefficient matrix and its inverse
\begin{equation}
A( x) = (a_{ij}( x) ), \quad
G( x) = [ A( x) ] ^{-1}=(g_{ij}( x))  \label{1.6}
\end{equation}
for $i,j = 1,\dots,n$; $x\in\mathbb{R}^n$.

\subsection*{Euclidean metric on $\mathbb{R}^n$}
Let $( x_1,\dots,x_n) $ be the natural coordinate system in
$\mathbb{R}^n$. For each $x\in\mathbb{R}^n$, denote
\begin{equation*}
X\cdot Y=\sum_{i=1}^n\alpha _i\beta _i,\quad
| X|_0^2=X\cdot X, \quad \text{for }
 X=\sum_{i=1}^n\alpha _i\frac{\partial }{\partial x_i},\quad
 Y=\sum_{i=1}^n\beta _i\frac{\partial }{\partial x_i}\in T_x
\mathbb{R}^n\,.
\end{equation*}
For $f\in C^{1}( \overline{\Omega }) $ and
$X=\sum_{i=1}^n \alpha _i\frac{\partial }{\partial x_i}$, we denote by
\begin{equation}
\nabla _0f=\sum_{i=1}^n\frac{\partial f}{\partial x_i}\frac{\partial }{
\partial x_i}\quad \text{and}\quad
\operatorname{div}_0( X) =\sum_{i=1}^n\frac{\partial \alpha _i( x) }{\partial x_i}
 \label{1.7}
\end{equation}
the gradient of $f$ and the divergence of $X$ in the Euclidean metric.

\subsection*{Riemannian metric}
For each $x\in\mathbb{R}^n$, define the inner product and the corresponding
norm on the tangent space $T_x\mathbb{R}^n$ by
\begin{gather*}
g( X,Y) = \langle X,Y\rangle _{g}=X\cdot G( x)
Y=\sum_{i,j=1}^ng_{ij}( x) \alpha _i\beta _{j}   \\
| X| _{g}^2 = \langle X,Y\rangle _{g}\quad \text{for }
X=\sum_{i=1}^n\alpha _i\frac{\partial }{\partial x_i},\quad
Y=\sum_{i=1}^n\beta _i\frac{\partial }{\partial x_i}
\in T_x\mathbb{R}^n  %\label{1.8}
\end{gather*}
Then $(\mathbb{R}^n,g) $ is a Riemannian manifold with a Riemannian metric $g$.
Denote the Levi-Cevita connection in metric $g$ by $D$. Let $H$ be a vector
field on $(\mathbb{R}^n,g) $. The covariant differential $DH$ of $H$ determines a
bilinear form on $T_x\mathbb{R}^n\times T_x\mathbb{R}^n$.
For each $x\in\mathbb{R}^n$, by
\begin{equation*}
DH( X,Y) =\langle D_{X}H,Y\rangle _{g}, \forall  X,Y\in T_x\mathbb{R}^n
\end{equation*}
where $D_{X}H$ is the covariante derivative of $H$ with respect to $X$.
The following lemma provides some useful equalities.

\begin{lemma}[{\cite[lemma 2.1]{y1}}]  \label{lem1.1}
Let $f$, $h\in C^{1}( \overline{\Omega }) $ and
let $H,X$ be  a vector field on $\mathbb{R}^n$.
Then using the above notation, we have
\begin{itemize}
\item[(i)]
\begin{equation}
\langle H(x),A(x)X(x)\rangle _{g}=H( x) X( x) ,\quad \forall x\in\mathbb{R}^n
 \label{1.9} \\
\end{equation}

\item[(ii)] The gradient $\nabla _{g}f$  of $f$
 in the Riemannian metric $g$ is given by
\begin{equation}
\nabla _{g}f( x) =\sum_{i=1}^n\Big( \sum_{j=1}^na_{ij}(
x) \frac{\partial f}{\partial x_{j}}\Big) \frac{\partial }{\partial
x_i}=A( x) \nabla _0f.  \label{1.10}
\end{equation}

\item[(iii)]
\begin{equation}
\frac{\partial y}{\partial \nu _{A}}=( A( x) \nabla
_0y) .\nu =\nabla _{g}y.\nu .  \label{1.11}
\end{equation}

\item[(iv)]
\begin{equation}
\langle \nabla _{g}f,\nabla _{g}H\rangle _{g}=\nabla _{g}f( h)
=\nabla _0f\cdot A( x) \nabla _0h.  \label{1.12}
\end{equation}

\item[(v)]
\begin{equation}
\begin{aligned}
&\langle \nabla _{g}f,\nabla _{g}H( f) \rangle  \\
&=DH( \nabla_{g}f,\nabla _{g}f) +\frac{1}{2}\operatorname{div}_0( | \nabla
_{g}f| _{g}^2H) ( x) -\frac{1}{2}|
\nabla _{g}f| _{g}^2\operatorname{div}_0(H)
\quad x\in \mathbb{R}^n.
\end{aligned} \label{1.13}
\end{equation}

\item[(vi)]
\begin{equation}
\begin{aligned}
Ay &= -\sum_{i,j=1}^n\frac{\partial }{\partial x_i}(a_{ij}(
x) \frac{\partial y}{\partial x_{j}})   \\
&= -\operatorname{div}_0( A( x) \nabla _0y) =-\operatorname{div}_0(
\nabla _{g}y) ,\quad y\in C^2( \Omega) .
\end{aligned}  \label{1.14}
\end{equation}
\end{itemize}
\end{lemma}

\section{Statement of main result}

To obtain the boundary stabilization of problem \eqref{1.1a}--\eqref{1.1d}
 and \eqref{1.3}, we  assumed that there exists a vector
field $H$ on the Riemannian manifold $(\mathbb{R}^n,g) $ such that:
\begin{gather}
\forall X\in T_x\mathbb{R}^n,\;\exists a>0,\quad
\langle D_{X}H,X\rangle _{g}\geq a| X| _{g}^2,  \label{2.1} \\
H( x) \cdot \nu <0\quad \text{on }\Gamma _0,  \label{2.2} \\
H( x) \cdot \nu \geq 0\quad \text{on }\Gamma _1.  \label{2.3}
\end{gather}
$k\geq 0$ and $k'\leq 0$, on $\Gamma _0\times\mathbb{R}_{+}$. Moreover,
\begin{gather}
\varphi =\inf_{( x,t) \in \Gamma _0\times \Re ^{+}}(-k')\neq 0,  \label{2.4}\\
k''\geq 0\,.  \label{2.5}
\end{gather}

We have the following result of existence and uniqueness of weak solution to
\eqref{1.1a}--\eqref{1.1d} and \eqref{1.3}.

\begin{theorem} \label{thm2.1}
For all initial data
$y_0\in V=H_{\Gamma _0}^{1}(\Omega ) =\{y\in H^{1}( \Omega ) $;
$y=0$ on $\Gamma _0\}$, problem \eqref{1.1a}--\eqref{1.1d} and
\eqref{1.3}  admits a unique global weak solution
$y\in C(\mathbb{R}^{+},V)$. Furthermore, if
$y_0\in H^{3}( \Omega) \cap H_{\Gamma _0}^{1}( \Omega ) $
and $\frac{\partial y_0}{\partial \nu _{A}}=-\frac{1}{2}ky( 0) $
on $\Gamma _1$, then the solution has the regularity
\begin{equation*}
\mathit{y\in C}^{1}(\mathbb{R}^{+},V) .
\end{equation*}
\end{theorem}

\begin{proof}
Existence of a solution is proved using Galerkin method [6].

Suppose $y$ is the unique global weak solution of problem
\eqref{1.1a}--\eqref{1.1d} and \eqref{1.3}.
The variational formulation of the problem is
\begin{align*}
\int_{\Omega}y_{t}\overline{v}\,d\Omega
&= \mathbf{i}\int_{\Omega }Ay \overline{v}\,d\Omega \\
&= \mathbf{i}\sum_{i,j=1}^n\int_{\Omega }\frac{\partial }{\partial x_i}
\Big( a_{ij}( x) \frac{\partial y}{\partial x_{j}}\Big)
\overline{v}\,d\Omega  \\
&= \mathbf{i}\sum_{i,j=1}^n\int_{\Gamma _1}a_{ij}( x) \frac{
\partial y}{\partial x_i}\nu _i\overline{v}\,d\Gamma _1
-\mathbf{i} \sum_{i,j=1}^n\int_{\Omega }a_{ij}( x) \frac{\partial y}{
\partial x_i}\frac{\partial \overline{v}}{\partial xj}\,d\Omega \\
&= \mathbf{i}\sum_{i,j=1}^n\int_{\Gamma _1}\frac{\partial y}{\partial
\nu _{A}}\overline{v}\,d\Gamma _1
-\mathbf{i}\sum_{i,j=1}^n\int_{\Omega}a_{ij}( x)
 \frac{\partial y}{\partial x_i}\frac{\partial
\overline{v}}{\partial xj}\,d\Omega
\end{align*}
for $v\in V$,  and
\begin{align*}
\int_{\Omega}y_{t}\overline{v}\,d\Omega
& =\mathbf{i}\int_{\Gamma_1}(-\int_0^{t}k'( t-s) y( s) ds
 -k(0) y( t) -by_{t})\overline{v}\,\,d\Gamma _1 \\
& -\mathbf{i}\sum_{i,j=1}^n\int_{\Omega }a_{ij}( x)
\frac{ \partial y}{\partial x_i}\frac{\partial \overline{v}}{\partial xj}\,d\Omega .
\end{align*}

We introduce the following notation:
$\langle\cdot,\cdot\rangle$ is the scalar product in $L^2(\Omega )$,
\begin{gather*}
a(y,v) = \sum_{i,j=1}^n\int_{\Omega }a_{ij}( x) \frac{\partial
y}{\partial x_i}\frac{\partial \overline{v}}{\partial xj}\,d\Omega , \\
\beta (t,y,v) = \int_{\Gamma _1}(-\int_0^{t}k'(t-s) y( s) ds
 -k( 0) y( t) )\overline{v}\,d\Gamma _1 \quad \beta (0,y,v)=0.
\end{gather*}
We have
\[
\langle y_{t},v\rangle +\mathbf{i}a( y,v) +\mathbf{i}\beta (t,y,v)
=-\mathbf{i}\langle by_{t},v\rangle
\]


\subsection*{Uniqueness}
Let $y$ and $z$ be two solutions. then $w=y-z$ satisfies
\begin{gather}
 w_{t}-\mathbf{i}Aw=0\quad  \text{in } ]0,+\infty )\times \Omega \label{1.a} \\
 w( 0,x) =0\quad  \text{in } \Omega  \label{1.b} \\
 w=0\quad\text{on } ]0,+\infty )\times \Gamma _0  \label{1.c} \\
 \frac{\partial w}{\partial \nu _{A}}
=-\int_0^{t}k'(t-s) w( s) ds-k( 0) w( t) -bw_{t},\quad
\text{on } ]0,+\infty [ \times \Gamma _1\,.  \label{1.d}
\end{gather}
Multiplying \eqref{1.a} by $\overline{w}_{t}$, integrating over $\Omega $.
Applying the Green formula, we obtain
\begin{equation*}
\int_{\Omega }| w_{t}| ^2\,d\Omega -\mathbf{i}\sum_{i,j=1}^n
\int_{\Gamma _1}a_{ij}( x) \frac{\partial w}{\partial x_i}\nu
_i\overline{w_{t}}\,d\Gamma +\mathbf{i}\sum_{i,j=1}^n\int_{\Omega
}a_{ij}( x) \frac{\partial w}{\partial x_i}\frac{\partial
\overline{w_{t}}}{\partial xj}\,d\Omega =0\,.
\end{equation*}
Taking the imaginary part of above equation,
\begin{align*}
\operatorname{Re}\int_{\Gamma _1}\frac{\partial y}{\partial \nu _{A}}
\overline{w_{t}}\,d\Gamma _{_1}
&= \operatorname{Re}\int_{\Omega }\nabla _{g}w\nabla _{g}
\overline{w}_{t}\,d\Omega \\
&= \frac{1}{2}\int_{\Omega }\big[ \frac{d}{dt}| \nabla _{g}w|_{g}^2\big] \,d\Omega \\
&= \frac{1}{2}\frac{d}{dt}\int_{\Omega }| \nabla _{g}w( t)
| _{g}^2\,d\Omega\,.
\end{align*}
For $t=0$, we have
\begin{equation*}
\int_{\Omega }| \nabla _{g}w| _{g}^2\,d\Omega +\int_{\Gamma
_1}k| w| ^2\,d\Gamma _1-\int_0^{t}\int_{\Gamma
_1}k'( t-s) | w( t) -w( s)
| ^2\,d\Gamma _{_1}ds=0
\end{equation*}
so
\begin{equation*}
\int_{\Omega }| \nabla _{g}w| _{g}^2\,d\Omega
=-\int_{\Gamma _1}k| w| ^2\,d\Gamma
_1+\int_0^{t}\int_{\Gamma _1}k'( t-s) |
w( t) -w( s) | ^2\,d\Gamma _1ds.
\end{equation*}
We deduce
\begin{equation*}
\int_{\Omega }| \nabla _{g}w| _{g}^2\,d\Omega \leq 0\,
\end{equation*}
and so $\nabla _{g}w=0$,
this with the condition at limit,
$w=0$
\end{proof}

\subsection{Existence of solutions}

Let $(e_n)_{n\in \aleph }$ be a set of functions in $V$ that form an orthonormal
basis for $L^2(\Omega )$.
Let $V_{m}$ be the space generated by $(e_1,\dots,e_{m})$, and
\begin{equation*}
y^{m}(t)=\sum_{i=1}^{m}\alpha _{im}(t)e_{i.}
\end{equation*}
be a solution of the Cauchy problem
\[
\langle y_{t}^{m},v\rangle +\mathbf{i}a( y^{m},v) +\mathbf{i}
\beta ( t,y^{m},v) =-\mathbf{i}\langle by_{t}^{m},v\rangle \,. 
\]
With $v=y_{t}^{m}$, we have
\begin{equation*}
\langle y_{t}^{m},y_{t}^{m}\rangle +\mathbf{i}a( y^{m},y_{t}^{m})
+\mathbf{i}\beta ( t,y^{m},y_{t}^{m}) =-\mathbf{i}\langle
by_{t}^{m},y_{t}^{m}\rangle\,.
\end{equation*}
Taking the imaginaire part, it results
\begin{align*}
&\operatorname{Re}\Big( \sum_{i,j=1}^n\int_{\Omega }a_{ij}( x) \frac{
\partial y^{m}}{\partial x_i}\frac{\partial \overline{y_{t}^{m}}}{\partial
xj}\,d\Omega \Big) \\
&=-\int_{\Gamma _1}b| y_{t}^{m}|^2\,d\Gamma _1 
 -\operatorname{Re}\Big( \int_{\Gamma _1}\int_0^{t}k'(
t-s) y^{m}(s)\overline{y_{t}^{m}}( t) ds\,d\Gamma _1\Big)\\
&\quad -\operatorname{Re}\Big( \int_{\Gamma _1}\int_0^{t}k( 0) y^{m}(t)
\overline{y_{t}^{m}}( t) ds\,d\Gamma _1\Big)\,.
\end{align*}
However,
\begin{align*}
&-\int b| y_{t}^{m}| ^2\,d\Gamma _1-\operatorname{Re}\Big(
\int_{\Gamma _1}\int_0^{t}k'( t-s) y^{m}(s)\overline{
y_{t}^{m}}( t) ds\,d\Gamma _1\Big) \\
&-\operatorname{Re}\Big( \int_{\Gamma _1}\int_0^{t}k( 0) y^{m}(t)
\overline{y_{t}^{m}}( t) \,d\Gamma _1\Big) \\
& =-\int_{\Gamma_1}b| y_{t}^{m}| ^2\,d\Gamma _1 
 +\frac{1}{2}\int_{\Gamma _1}k'| y^{m}|^2\,d\Gamma _1 \\
&\quad -\frac{1}{2}\int_{\Gamma _1}\int_0^{t}k''( t-s) | y^{m}
( t) -y^{m}( s) | ^2\,ds\,d\Gamma _1 \\
&\quad +\frac{1}{2}\frac{d}{dt}\big[ \int_{\Gamma _1}k|
y^{m}| ^2\,d\Gamma _1\big] 
+\frac{1}{2}\frac{d}{dt}\Big[\int_{\Gamma _1}\int_0^{t}k'( t-s) |
y^{m}( t) -y^{m}( s) | ^2\,ds\,d\Gamma _1\Big],
\end{align*}
so
\begin{align*}
&\operatorname{Re}\Big( \sum_{i,j=1}^n\int_{\Omega }a_{ij}( x) \frac{
\partial y^{m}}{\partial x_i}\frac{\partial \overline{y_{t}^{m}}}{\partial
xj}\,d\Omega \Big) \\
&= -\int b| y_{t}^{m}| ^2\,d\Gamma_1-\frac{1}{2}\int_{\Gamma _1}k'| y^{m}|
^2\,d\Gamma _1
-\frac{1}{2}\int_{\Gamma _1}\int_0^{t}k''(
t-s) | y^{m}( t) -y^{m}( s) | ^2\,ds\,d\Gamma _1 \\
&\quad -\frac{1}{2}\frac{d}{dt}\Big[ \int_{\Gamma _1}k|
y^{m}| ^2\,d\Gamma _1\Big] 
+\frac{1}{2}\frac{d}{dt}\Big[ \int_{\Gamma _1}\int_0^{t}k'( t-s) | y^{m}( t) 
-y^{m}( s) | ^2\,ds\,d\Gamma _1\Big]\,.
\end{align*}
We have
\[
-\int_{\Gamma _1}b| y_{t}^{m}| ^2\,d\Gamma _1+\frac{1
}{2}\int_{\Gamma _1}k'| y^{m}| ^2\,d\Gamma
_1-\frac{1}{2}\int_{\Gamma _1}\int_0^{t}k''(
t-s) | y^{m}( t) -y^{m}( s) |
^2\,ds\,d\Gamma _1<0, 
\]
thus
\begin{align*}
& \operatorname{Re}\Big( \sum_{i,j=1}^n\int_{\Omega }a_{ij}( x) \frac{
\partial y^{m}}{\partial x_i}\frac{\partial \overline{y_{t}^{m}}}{\partial
xj}\,d\Omega \Big) \\
&\leq -\frac{1}{2}\frac{d}{dt}\Big[ \int_{\Gamma
_1}k| y^{m}| ^2\,d\Gamma _1\Big] 
 +\frac{1}{2}\frac{d}{dt}\Big[ \int_{\Gamma _1}\int_0^{t}k'( t-s) 
| y^{m}( t) -y^{m}( s) | ^2\,ds\,d\Gamma _1\Big]\,.
\end{align*}
We deduce that
\begin{equation*}
\operatorname{Re}\Big( \sum_{i,j=1}^n\int_{\Omega }a_{ij}( x) \frac{
\partial y^{m}}{\partial x_i}\frac{\partial \overline{y_{t}^{m}}}{\partial
xj}\,d\Omega \Big) \leq 0\,.
\end{equation*}
On the other hand, 
\begin{align*}
\operatorname{Re}\Big( \sum_{i,j=1}^n\int_{\Omega }a_{ij}( x) \frac{
\partial y^{m}}{\partial x_i}\frac{\partial \overline{y_{t}^{m}}}{\partial
xj}\,d\Omega \Big) 
&= \frac{1}{2}\frac{d}{dt}\Big(
\sum_{i,j=1}^n\int_{\Omega }a_{ij}( x) \frac{\partial y^{m}}{
\partial x_i}\frac{\partial \overline{y^{m}}}{\partial xj}\,d\Omega \Big)
\\
&= \frac{1}{2}\frac{d}{dt}\Big( \int_{\Omega }| \nabla
_{g}y^{m}| _{g}^2\,d\Omega \Big).
\end{align*}
Therefore,
\begin{equation*}
\frac{1}{2}\int_{\Omega }| \nabla _{g}y^{m}( T)
| _{g}^2\,d\Omega \leq \frac{1}{2}\int_{\Omega }| \nabla
_{g}y^{m}( 0) | _{g}^2\,d\Omega,
\end{equation*}
from where
\begin{equation*}
| y^{m}| _{V}^2\leq | y^{0m}|_{V}^2
\end{equation*}
and
$v^{m}\to v$ in $V$, or 
\begin{equation}
\langle y_{t}^{m},v^{m}\rangle +\mathbf{i}a( y^{m},v^{m}) +
\mathbf{i}\beta ( t,y^{m},v^{m}) =-\mathbf{i}\int_{\Gamma
_1}by_{t}^{m}\overline{v^{m}}\,d\Gamma _1.  \label{e**}
\end{equation}
For $\zeta \in D( ]0,T[) $, we put
\begin{equation*}
\psi ^{m}=\zeta v^{m}\quad\text{and}\quad \psi =\zeta v\,;
\end{equation*}
so we have
\begin{equation*}
\psi ^{m}\to \psi \quad \text{in }L^2( 0,T,V)\,.
\end{equation*}
Multiplying \eqref{e**} by $\zeta $ and integrating over $]0,T[$, we find
\begin{align*}
&\int_0^{T}\langle y_{t}^{m},\psi ^{m}\rangle dt
+\mathbf{i}\int_0^{T}a( y^{m},\psi ^{m}) dt
+\mathbf{i}\int_0^{T}\beta( t,y^{m},\psi ^{m}) dt\\
&=-\mathbf{i}\int_0^{T}\int_{\Gamma_1}by_{t}^{m}\overline{\psi ^{m}}
 \,d\Gamma _1dt\,.
\end{align*}
Passing to the limit, 
\[
\int_0^{T}\langle y,\psi _{t}\rangle dt+\mathbf{i}\int_0^{T}a(
y,\psi ) dt+\mathbf{i}\int_0^{T}\beta ( t,y,\psi ) dt
=- \mathbf{i}\int_0^{T}\int_{\Gamma _1}by\overline{\psi _{t}}\,d\Gamma _1dt
\]
for all $v\in V$ and all $\zeta \in D(]0,T[)$.
$y\in C] 0,T[ ,V)$.

\subsection*{Regularity}
We have
\begin{align*}
&\langle y_{tt}^{m},v\rangle +\mathbf{i}\int_{\Gamma_1}
\Big(-\int_0^{t}k''( t-s) y^{m}(
s) ds-k( 0) y_{t}^{m}(t)\Big)\overline{v}\,\,d\Gamma _1\\
&+\mathbf{i}\sum_{i,j=1}^n\int_{\Omega }a_{ij}( x) \frac{
\partial y_{t}^{m}}{\partial x_i}\frac{\partial \overline{v}}{\partial xj}
\,d\Omega 
-\mathbf{i}\int_{\Gamma _1}by_{tt}^{m}v\,d\Gamma _1=0\,.
\end{align*}
Putting $v=y_{tt}^{m}$,
\begin{align*}
&\langle y_{tt}^{m},y_{tt}^{m}\rangle +\mathbf{i}\int_{\Gamma
_1}(-\int_0^{t}k''( t-s) y^{m}(
s) ds-k( 0) y_{t}^{m}(t))\overline{y_{tt}^{m}}\,\,d\Gamma _1 \\
&+\mathbf{i}\sum_{i,j=1}^n\int_{\Omega }a_{ij}( x) \frac{
\partial y_{t}^{m}}{\partial x_i}\frac{\partial \overline{y_{tt}^{m}}}{
\partial xj}\,d\Omega \text{ }-\mathbf{i}\int_{\Gamma _1}by_{tt}^{m}
\overline{y_{tt}^{m}}\,d\Gamma _1=0\,.
\end{align*}
Taking the imaginary part,
\begin{align*}
&\operatorname{Re}\Big( \sum_{i,j=1}^n\int_{\Omega }a_{ij}( x) \frac{
\partial y_{t}^{m}}{\partial x_i}\frac{\partial \overline{y_{tt}^{m}}}{
\partial xj}\,d\Omega \Big) \\
&=-\int_{\Gamma _1}b|y_{tt}^{m}| ^2\,d\Gamma _1 
 -\operatorname{Re}\Big( \int_{\Gamma _1}\int_0^{t}k'(
t-s) y_{t}^{m}(s)\overline{y_{tt}^{m}}( t) ds\,d\Gamma_1\Big) \\
&\quad -\operatorname{Re}\Big(\int_{\Gamma _1}\int_0^{t}k( 0)
y_{t}^{m}(t)\overline{y_{tt}^{m}}( t) ds\,d\Gamma _1\Big),
\end{align*}
we have
\begin{align*}
& -\int_{\Gamma _1}b| y_{tt}^{m}| ^2\,d\Gamma _1-
\operatorname{Re}( \int_{\Gamma _1}\int_0^{t}k''(
t-s) y^{m}(s)\overline{y_{tt}^{m}}( t) ds\,d\Gamma _1)
\\
& -\operatorname{Re}( \int_{\Gamma _1}\int_0^{t}k( 0)
y_{t}^{m}(t)\overline{y_{tt}^{m}}( t) \,d\Gamma _1)  \\
&=-\int b| y_{tt}^{m}| ^2\,d\Gamma _1 
 -\frac{1}{2}\int_{\Gamma _1}k''|y_{t}^{m}| ^2\,d\Gamma _1 \\
&\quad +\frac{1}{2}\int_{\Gamma_1}\int_0^{t}k'''( t-s) |y_{t}^{m}( t) 
 +y^{m}( s) | ^2\,ds\,d\Gamma_1 \\
&\quad  -\frac{1}{2}\frac{d}{dt}\Big[ -\int_{\Gamma _1}k'|
y_{t}^{m}| ^2\,d\Gamma _1-\frac{1}{2}\int_{\Gamma
_1}\int_0^{t}k''( t-s) |
y_{t}^{m}( t) +y^{m}( s) | ^2\,ds\,d\Gamma_1\Big]\,.
\end{align*}
We deduce that
\begin{equation*}
-\int_{\Gamma _1}b| y_{tt}^{m}| ^2\,d\Gamma _1-\frac{1
}{2}\int_{\Gamma _1}k''| y_{t}^{m}|
^2\,d\Gamma _1+\frac{1}{2}\int_{\Gamma _1}\int_0^{t}k'''( t-s) 
| y_{t}^{m}( t) +y^{m}(s) | ^2\,ds\,d\Gamma _1\leq 0;
\end{equation*}
so
\begin{align*}
&\operatorname{Re}( \sum_{i,j=1}^n\int_{\Omega }a_{ij}( x) \frac{
\partial y_{t}^{m}}{\partial x_i}\frac{\partial \overline{y_{tt}^{m}}}{
\partial xj}\,d\Omega ) \\
&\leq -\frac{1}{2}\frac{d}{dt}(-\int_{\Gamma_1}k'| y_{t}^{m}| ^2\,d\Gamma _1 
 +\frac{1}{2}\int_{\Gamma _1}\int_0^{t}k''(t-s) | y_{t}^{m}( t) +y^{m}( s)
| ^2\,ds\,d\Gamma _1)\,.
\end{align*}
On the other hand,
\begin{equation*}
\operatorname{Re}\Big( \sum_{i,j=1}^n\int_{\Omega }a_{ij}( x) \frac{
\partial y_{t}^{m}}{\partial x_i}\frac{\partial \overline{y_{tt}^{m}}}{
\partial xj}\,d\Omega \Big) 
=\frac{1}{2}\frac{d}{dt}\Big( \int_{\Omega
}| \nabla _{g}y_{t}^{m}| _{g}^2\,d\Omega \Big)
\end{equation*}
and
\begin{align*}
&\frac{1}{2}\frac{d}{dt}( \int_{\Omega }| \nabla
_{g}y_{t}^{m}| _{g}^2\,d\Omega ) \\
&\leq -\frac{1}{2}\frac{d}{dt}\Big(-\int_{\Gamma _1}k'| y_{t}^{m}|
^2\,d\Gamma _1 
+\frac{1}{2}\int_{\Gamma _1}\int_0^{t}k''(
t-s) | y_{t}^{m}( t) +y^{m}( s)| ^2ds\,d\Gamma _1\Big)\,.
\end{align*}
Therefore,
\begin{gather*}
-\frac{1}{2}\frac{d}{dt}\Big[ -\int_{\Gamma _1}k'|
y_{t}^{m}| ^2\,d\Gamma _1+\frac{1}{2}\int_{\Gamma
_1}\int_0^{t}k''( t-s) |
y_{t}^{m}( t) +y^{m}( s) | ^2ds\,d\Gamma _1
\Big] \leq 0, \\
\frac{1}{2}\frac{d}{dt}\Big( \int_{\Omega }| \nabla
_{g}y_{t}^{m}| _{g}^2\,d\Omega \Big) \leq 0\,.
\end{gather*}
We have
\begin{gather*}
\frac{1}{2}\int_{\Omega }| \nabla _{g}y_{t}^{m}( T)| _{g}^2\,d\Omega 
\leq \frac{1}{2}\int_{\Omega }| \nabla_{g}y_{t}^{m}( 0) | _{g}^2\,d\Omega,\\
| y_0^{m}| _{V}^2\leq |y_0^{0m}| _{V}^2
\end{gather*}

Our main result is the following theorem.

\begin{theorem} \label{thm2.2}
Assume \eqref{1.2}--\eqref{1.4} and \eqref{2.1}--\eqref{2.5},
and that the problem
\begin{align*}
y_{t}-\mathbf{i}Ay = 0\quad \text{on } \Omega \\
y = 0\quad \text{in } \Sigma _0 \\
\frac{\partial y}{\partial \nu _{A}} = 0 \quad \text{in }\Sigma
\end{align*}
has $y=0$ as the unique solution.
Then for all given initial data $y_0\in H_{\Gamma _0}^{1}(\Omega )$,
there exist two positive constants $M$
 and $\omega $ such that
\begin{equation*}
E(t)\leq Me^{-\omega t}E(0),\quad \text{for }t>0.
\end{equation*}
\end{theorem}

\section{Proof of main result}

  For simplicity, we assume that $y$ is a weak solution. By a
classical density argument, Theorem \ref{thm2.2} still holds for a weak solution.

\begin{lemma} \label{lem3.1}
The energy defined by \eqref{1.4} is decreasing and satisfies
\begin{align*}
E( T) -E( 0) 
&= -\int_0^{T}\int_{\Gamma_1}| y_{t}| ^2\,d\Gamma _1dt
 +\frac{1}{2}\int_0^{T}\int_{\Gamma _1}k'| y| ^2\,d\Gamma_1\,dt \\
&\quad-\frac{1}{2}\int_0^{T}\int_0^{t}\int_{\Gamma _1}k''( t-s) | y( t) -y( s) |
^2\,d\Gamma _1\,ds\,dt,
\end{align*}
whenever $0<T<\infty$.
\end{lemma}

\begin{proof}
Differentiating the energy $E(\cdot) $ defined by \eqref{1.4} and using
Green's second theorem, we have
\begin{align*}
2E'( t) 
&= \int_{\Omega }\nabla _{g}\overline{y_{t}}\nabla _{g}y\,d\Omega 
+\int_{\Omega }\nabla _{g}\overline{y}\nabla_{g}y_{t}\,d\Omega
 +\int_{\Gamma _1}k'| y|^2\,d\Gamma _1 \\
&\quad +2\operatorname{Re}\int_{\Gamma _1}ky_{t}y\,d\Gamma _1
-\int_0^{t}\int_{\Gamma_1}k''( t-s) | y( t) -y( s) | ^2\,d\Gamma _1\,ds \\
&\quad -\int_0^{t}\int_{\Gamma _1}k'( t-s) \frac{d}{dt}
(y( t) -y( s) )( \overline{y( t)
-y( s) }) ds\,d\Gamma _1.
\end{align*}
Then
\begin{align*}
E'( t) &= \operatorname{Re}\int_{\Omega }\langle \nabla
_{g}y_{t},\nabla _{g}\overline{y}\rangle _{g}\,d\Omega +\frac{1}{2}
\int_{\Gamma _1}k'| y| ^2\,d\Gamma _1+ \\
&\quad +\operatorname{Re}\int_{\Gamma _1}k\overline{y_{t}}y\,d\Gamma _1-\frac{1}{2}
\int_0^{t}\int_{\Gamma _1}k''( t-s) |
y( t) -y( s) | ^2\,d\Gamma _1ds \\
&\quad -\operatorname{Re}\int_0^{t}\int_{\Gamma _1}k'( t-s)
\overline{y_{t}}( t) ( y( t) -y( s)
) \,d\Gamma _1ds.
\end{align*}
Then
\begin{align*}
E'( t) &= \operatorname{Re}\int_{\Gamma _1}\frac{\partial y}{
\partial \nu _{A}}\overline{y_{t}}\,d\Gamma _1+\frac{1}{2}\int_{\Gamma
_1}k'| y| ^2\,d\Gamma _1 \\
&\quad +\operatorname{Re}\int_{\Gamma _1}k\overline{y_{t}}y\,d\Gamma _1-\frac{1}{2}
\int_0^{t}\int_{\Gamma _1}k''( t-s) |
y( t) -y( s) | ^2\,d\Gamma _1ds \\
&\quad -\operatorname{Re}\int_0^{t}\int_{\Gamma _1}k'( t-s)
\overline{y_{t}}( t) ( y( t) -y( s)
) \,d\Gamma _1ds
\end{align*}
By the boundary condition,  we have $\operatorname{Re}\int_{\Gamma _1}y
\overline{y_{t}}( k-k( 0) -\int_0^{t}k'(
t-s) ds) \,d\Gamma _1=0$. Then  we find that
\begin{align*}
E'( t) 
&= -\int_{\Gamma _1}b| y_{t}| ^2\,d\Gamma _1
 +\frac{1}{2}\int_{\Gamma _1}k'| y| ^2\,d\Gamma _1 \\
&\quad -\frac{1}{2}\int_0^{t}\int_{\Gamma _1}k''(t-s) | y( t) -y( s) |
^2\,d\Gamma _1ds.
\end{align*}
This completes the proof.
\end{proof}

\begin{lemma} \label{lem3.2}
For all $0\leq S<T<\infty $ we have
\begin{align*}
&\int_{S}^{T}\int_{\Gamma _1}\big( \frac{\partial y}{\partial \nu _{A}}
\big) ^2\frac{H\cdot\nu }{| \nu _{A}( x) |_{g}^2}\,d\Gamma _1ds
-\int_{S}^{T}\int_{\Gamma _1}| \nabla_{g}y| _{g}^2H\nu \,d\Gamma _1ds \\
&+2\operatorname{Re}\int_{S}^{T}\int_{\Gamma _1}\big( \frac{\partial y}{\partial
\nu _{A}}\big) H( \overline{y}) \,d\Gamma _1ds
+\operatorname{Im}\int_{S}^{T}\int_{\Gamma _1}y\overline{y_{t}}H\upsilon 
\,d\Gamma _1ds\\
&=2\operatorname{Re}\int_{S}^{T}\int_{\Omega }DH( \nabla _{g}y,\nabla
_{g}y) \,d\Omega\,ds-\int_{S}^{T}\int_{\Omega }| \nabla
_{g}y| _{g}^2\operatorname{div}H\,d\Omega\,ds \\
&\quad +\operatorname{Im}\int_{\Omega }y_{t}H( y) \,d\Omega \big|_{S}^{T}
 +\operatorname{Im}\int_{S}^{T}\int_{\Omega }\overline{y_{t}}y \operatorname{div}H\,d\Omega\,ds
\end{align*}
\end{lemma}

\begin{proof}
We multiply \eqref{1.1a} by $H\cdot\nabla \overline{y}$ and integrate
over $]S,T[\times \Omega $, to obtain
\begin{equation}
\int_{S}^{T}\int_{\Omega }y_{t}H\cdot\nabla \overline{y}\,d\Omega\,ds-\mathbf{i}
\int_{S}^{T}\int_{\Omega }AyH\cdot\nabla \overline{y}\,d\Omega\,ds=0 \,. \label{3.1a}
\end{equation}
We have
\begin{equation} \label{3.1b}
\begin{aligned}
&\int_{S}^{T}\int_{\Omega }y_{t}H\cdot\nabla \overline{y}\,d\Omega\,ds\\
&= \int_{\Omega }yH\cdot\nabla \overline{y}\,d\Omega \big|_{S}^{T}
 -\int_{S}^{T}\int_{\Omega }yH\cdot\nabla \overline{y_{t}}\,d\Omega\,ds \\
&= \int_{\Omega }yH\cdot\nabla \overline{y}\,d\Omega \big|_{S}^{T}
 -\int_{S}^{T}\int_{\Gamma }y\overline{y_{t}}H\cdot\nu \,d\Gamma ds
 +\int_{S}^{T}\int_{\Omega }\overline{y_{t}}\operatorname{div}( y.H) \,d\Omega\,ds.
\end{aligned}
\end{equation}
  Substituting \eqref{3.1b}  in \eqref{3.1a}, we obtain
\begin{align*}
&\int_{\Omega }yH\cdot\nabla \overline{y}\,d\Omega \big|_{S}^{T}
-\int_{S}^{T}\int_{\Gamma }y\overline{y_{t}}H\cdot\nu \,d\Gamma ds \\
&-\mathbf{i}\Big[ \int_{S}^{T}\int_{\Omega }( A\overline{y}H\cdot\nabla y
+AyH\cdot\nabla \overline{y}) \,d\Omega\,ds\Big] 
 +\int_{S}^{T}\int_{\Omega }\overline{y_{t}}y \operatorname{div}H\,d\Omega\,ds=0.
\end{align*}
Hence
\begin{equation}
\begin{aligned}
&2\operatorname{Re}\int_{S}^{T}\int_{\Omega }AyH\cdot\nabla
 \overline{y}\,d\Omega\,ds\\
& =\operatorname{Im}\int_{\Omega }yH\cdot\nabla \overline{y}\,d\Omega \big|_{S}^{T}
-\operatorname{Im} \int_{S}^{T}\int_{\Gamma }y\overline{y_{t}}H\cdot\nu \,d\Gamma\,ds\\ 
&\quad +\operatorname{Im}\int_{S}^{T}\int_{\Omega }\overline{y_{t}}y
\operatorname{div}H\,d\Omega\,ds
\end{aligned} \label{3.1c}
\end{equation}
Using \eqref{1.13}, we rewrite the integral
on the left-hand side of \eqref{3.1c} as
\begin{equation} \label{3.1d}
\begin{aligned}
&\int_{S}^{T}\int_{\Omega }AyH\cdot\nabla \overline{y}\,d\Omega\,ds\\
&=\int_{S}^{T}\int_{\Gamma }\frac{\partial y}{\partial \nu _{A}}H(
y) \,d\Gamma ds
-\int_{S}^{T}\int_{\Omega }DH( \nabla _{g}y,\nabla_{g}\overline{y}) \,d\Omega\,ds   \\
&\quad -\frac{1}{2}\int_{S}^{T}\int_{\Omega }\operatorname{div}( | \nabla
_{g}y| ^2.H) \,d\Omega\,ds   \\
&\quad +\frac{1}{2}\int_{S}^{T}\int_{\Omega }| \nabla _{g}y|
_{g}^2\operatorname{div}H\,d\Omega\,ds
\end{aligned}
\end{equation}
Recalling the boundary condition \eqref{1.1b},  on $\Gamma $ we have
\begin{equation}
y=y_{t}=0, \quad | \nabla _{g}y| _{g}^2=\frac{1}{|
\nu _{A}( x) | _{g}^2}( \frac{\partial y}{\partial \nu _{A}}) ^2,\quad
H( y) =\frac{H\cdot\nu }{|
\nu _{A}( x) | _{g}^2}\big( \frac{\partial y}{
\partial \nu _{A}}\big) .  \label{3.2}
\end{equation}

Thus using this equation and \eqref{1.1c}, we find that this simplifies  the
sought-after identity.
\end{proof}
  
\subsection*{Completion of the proof of theorem \ref{thm2.1}}

Set $C_0=\big( \alpha _1\sup_{x\in \overline{\Omega }}\frac{
| H| ^2}{| H\cdot\nu | }\big) ^2$,
 $C_1=\frac{\alpha _1}{2}\sup_{x\in \overline{\Omega }}|
\operatorname{div} H| $,
\begin{gather*} 
C_2=\frac{\alpha _1}{2\varepsilon }\sup_{x\in
\overline{\Omega }}| \nabla _{g}( \operatorname{div}H) | +
\frac{1}{4}\sup_{x\in \overline{\Omega }}| \operatorname{div}H| 
+\frac{\varepsilon \alpha _1}{2}\sup_{x\in \overline{\Omega }}| H(x) |,\\
C_{3}=2\Big[ \frac{| k( 0) | _{L^{\infty
}( \Gamma _1) }}{e\delta f}+| h| _{L^{\infty
}( \Gamma _1) }\Big]
\end{gather*}
 where $h( x) =\frac{k(0) }{\delta ( 1+ek'( 0) ) }$,
 $x\in\Gamma _1$.
From assumptions \eqref{2.1}, \eqref{2.2} \eqref{2.3}, we deduce that
\begin{equation}
\begin{aligned}
&2a\int_{S}^{T}\int_{\Omega }| \nabla _{g}y|_{g}^2\,d\Omega\,ds \\
&\leq -\int_{S}^{T}\int_{\Gamma _1}| \nabla_{g}y| _{g}^2H\cdot\nu \,d\Gamma _1ds
 +2 \operatorname{Re}\int_{S}^{T}\int_{\Gamma _1}
\big( \frac{\partial y}{\partial \nu _{A}}\big) H(
\overline{y}) \,d\Gamma _1ds \\
&\quad +\operatorname{Im}\int_{S}^{T}\int_{\Gamma _1}y\overline{y_{t}}H\cdot
 \nu \,d\Gamma_1ds
+\int_{S}^{T}\int_{\Omega }| \nabla _{g}y|_{g}^2\operatorname{div}H\,d\Omega\,ds\\
&\quad -\operatorname{Im}\int_{\Omega }yH( y) \,d\Omega \big|_{S}^{T}
 -\operatorname{Im} \int_{S}^{T}\int_{\Omega }\overline{y_{t}}y
 \operatorname{div}HdQ
\end{aligned} \label{3.3a}
\end{equation}

Also for an arbitrary  positive constant $\varepsilon $ we have
the estimates
\begin{gather}
\begin{aligned}
&-\int_{S}^{T}\int_{\Gamma _1}| \nabla _{g}y|
_{g}^2H\cdot\nu \,d\Gamma _1ds+2\operatorname{Re}\int_{S}^{T}\int_{\Gamma _1}
\big(\frac{\partial y}{\partial \nu _{A}}\big) H( \overline{y})
\,d\Gamma _1ds\text{ \ }   \\
&\leq C_0\int_{S}^{T}\int_{\Gamma _1}\big( \frac{\partial y}{\partial
\nu _{A}}\big) ^2\,d\Gamma _1ds , 
\end{aligned}\label{3.3b} \\
\begin{aligned}
&\big| \operatorname{Im}\int_{S}^{T}\int_{\Gamma _1}y\overline{y_{t}}H\cdot\nu
\,d\Gamma _1ds
-\operatorname{Im}\int_{\Omega }yH( y) \,d\Omega \big|_{S}^{T}| \\
&\leq \frac{\sup_{x\in \overline{\Omega }}| H| }{2\alpha _1}
 \int_{S}^{T}\int_{\Gamma _1}| y| ^2\,d\Gamma_1ds \\
&\quad +2\alpha _1\sup_{x\in \overline{\Omega }}| H|
\int_{S}^{T}\int_{\Gamma _1}| y_{t}| ^2\,d\Gamma
_1ds+\varepsilon \sup_{x\in \overline{\Omega }}| H( x)| E( S) \\
&\quad +\Big( \frac{\varepsilon \alpha _1}{2}\sup_{x\in \overline{\Omega }
}| H( x) | \Big) | y|_{C( [ S,T] ,L^2( \Omega ) ) }^2,
\end{aligned} \label{3.3c} \\
\begin{aligned}
&| \int_{S}^{T}\int_{\Omega }| \nabla _{g}y|
_{g}^2\operatorname{div}H\,d\Omega\,ds-\operatorname{Im}\int_{S}^{T}
\int_{\Omega }\overline{y_{t}} y\operatorname{div}H\,d\Omega\,ds| \\
&\leq C_1\int_{S}^{T}\int_{\Gamma _1}\big(
\frac{\partial y}{\partial \nu _{A}}\big) ^2\,d\Gamma _1ds 
+\frac{\varepsilon }{2}\int_{S}^{T}\int_{\Omega }| \nabla _{g}
\overline{y}| _{g}^2\,d\Omega\,ds \\
&\quad +\Big( \frac{\alpha _1}{2\varepsilon }\sup_{x\in \overline{\Omega }
}| \nabla _{g}( \operatorname{div}H) | +\frac{1}{4}\sup_{x\in
\overline{\Omega }}| \operatorname{div}H| \Big)
\int_{S}^{T}\int_{\Gamma _1}| y| ^2\,d\Gamma _1ds,
\end{aligned} \label{3.3d}\\
\begin{aligned}
\big| \int_{\Sigma _1}( \frac{\partial y}{\partial \nu _{A}}
) ^2d\Sigma _1\big| 
& \leq 3[\int_{S}^{T}\int_{\Gamma_1}\Big( -\int_0^{t}k'( t-s) y( s)
ds\Big) ^2\,d\Gamma _1ds   \\
&\quad  +\int_{S}^{T}\int_{\Gamma _1}k( 0) | y|
^2\,d\Gamma _1ds+\int_{S}^{T}\int_{\Gamma _1}| y_{t}|
^2\,d\Gamma _1ds]. 
\end{aligned} \label{3.3e}
\end{gather}

Now using a compactness-uniqueness argument in the style of
\cite{l1}, we deduce
\begin{equation}
| y| _{C([ S,T] ,L^2( \Omega) ) }^2\leq \int_{S}^{T}\int_{\Gamma _1}|
y_{t}| ^2\,d\Gamma _1ds  \label{3.3f}
\end{equation}
Inserting  \eqref{3.3b}, \eqref{3.3c}, \eqref{3.3d} and 
\eqref{3.3f} in \eqref{3.3a}, leads to
\begin{equation}
\begin{aligned}
&2a\int_{S}^{T}\int_{\Omega }| \nabla _{g}y|_{g}^2\,d\Omega\,ds\\
&\leq 3( C_0+C_1) \Big[ \int_{S}^{T}\int_{
\Gamma _1}( -\int_0^{t}k'( t-s) y(s) ds) ^2\,d\Gamma _1ds\Big] \\
&\quad + 3( C_0+C_1) \Big[ \int_{S}^{T}\int_{\Gamma _1}k(
0) | y| ^2\,d\Gamma _1ds+\int_{S}^{T}\int_{\Gamma
_1}| by_{t}| ^2\,d\Gamma _1ds\Big] \\
&\quad  +C_2\int_{S}^{T}\int_{\Gamma _1}| y| ^2\,d\Gamma
_1ds+\frac{\varepsilon }{2}\int_{S}^{T}\int_{\Omega }| \nabla _{g}
\overline{y}| _{g}^2\,d\Omega\,ds \\
&\quad +\varepsilon \sup_{x\in \overline{\Omega }}| H( x)| E( S)
+2\alpha _1\sup_{x\in \overline{\Omega }}| H| \int_{S}^{T}\int_{\Gamma _1}|
y_{t}| ^2\,d\Gamma _1ds \\
&\quad +( \frac{\varepsilon \alpha _1}{2}\sup_{x\in \overline{\Omega }
}| H( x) | ) | y|_{C( [S,T] ,L^2( \Omega ) ) }^2
\end{aligned} \label{3.4}
\end{equation}

Choosing $\varepsilon $ so that $4a-\varepsilon >0$, we obtain
\begin{align*}
&( 4a-\varepsilon )\int_{S}^{T}E( t) dt \\
&\leq [3C_{3}( C_0+C_1) +\varepsilon \sup_{x\in \overline{\Omega }
}| H( x) | ]E( S) 
+2\alpha _1\sup_{x\in \overline{\Omega }}| H| \frac{1
}{\sup_{x\in \overline{\Gamma _1}}| b| }E( S)
\\
&\quad +[3( C_0+C_1) | k( 0) |
_{L^{\infty }}^2+C_2]\frac{1}{\varphi }E( S) \\
&\quad +[3( C_0+C_1) +\frac{\varepsilon \alpha _1}{2}\sup_{x\in
\overline{\Omega }}| H( x) | ]\sup_{x\in
\overline{\Gamma _1}}| b| E( S) ,
\end{align*}
where
\begin{gather*}
\int_{S}^{T}E(t)dt\leq CE( S), \\
\begin{aligned}
C &= \frac{1}{( 4a-\varepsilon ) }[3C_{3}(
C_0+C_1) +\varepsilon \sup_{x\in \overline{\Omega }}|
H( x) | +2\alpha _1\sup_{x\in \overline{\Omega }
}| H| \frac{1}{\sup_{x\in \overline{\Gamma _1}}| b| } \\
&\quad +[3( C_0+C_1) +\frac{\varepsilon \alpha _1}{2}\sup_{x\in
\overline{\Omega }}| H( x) | ]\sup_{x\in \overline{\Gamma _1}}| b| \\
&\quad +[3( C_0+C_1) | k( 0) |_{L^{\infty }}^2+C_2]\frac{1}{\varphi }].
\end{aligned}
\end{gather*}
 Letting $T\to +\infty $, we obtain for every $S\in\mathbb{R}_{+}$,
\begin{equation*}
\int_{S}^{+\infty }E(t)dt\leq CE( S) .
\end{equation*}
The desired conclusion follows now from Komornik 
\cite[Theorem 8.1]{k1}.


\subsection*{Acknowledgments}
I would like to thank to Prof.\  S. E. Rebiai for
support received about  the regularity questions for the Schrodinger equation.

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\end{document}