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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 179, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/179\hfil Uniform decay]
{Uniform decay for a local dissipative Klein-Gordon-Schr\"odinger 
 type system}

\author[M. N. Poulou, N. M. Stavrakakis \hfil EJDE-2012/179\hfilneg]
{Marilena N. Poulou, Nikolaos M. Stavrakakis}  % in alphabetical order

\address{Marilena N. Poulou \newline
Department of Mathematics,  National Technical University \\
Zografou Campus 157 80, Athens, Hellas, Greece}
\email{mpoulou@math.ntua.gr}

\address{Nikolaos M. Stavrakakis \newline
Department of Mathematics,  National Technical University \\
Zografou Campus 157 80, Athens, Hellas, Greece}
\email{nikolas@central.ntua.gr}

\thanks{Submitted June 7, 2012. Published October 17, 2012.}
\subjclass[2000]{35L70, 35B40}
\keywords{Klein-Gordon-Schr\"{o}dinger system; localized damping; 
\hfill\break\indent existence and uniqueness; energy decay}

\begin{abstract}
 In this article, we consider a nonlinear Klein-Gordon-Schr\"odinger
 type system in  $\mathbb{R}^n$, where the nonlinear term exists and
 the damping term is effective. We prove the existence and uniqueness of a global
 solution and its exponential decay. The result is achieved by using the
 multiplier technique.
\end{abstract}

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

\newcommand{\D}[2]{\frac{\partial \hash 2}{\partial \hash 1}}


\section{Introduction}

 We consider the nonlinear system of Klein-Gordon-Schr\"{o}dinger type with
locally distributed damping
\begin{gather}
 \label{e1.1d}
i\psi_t +\kappa \Delta \psi +i\alpha\psi =  \phi\psi  \chi_{\omega}, \quad
 x \in \Omega,\;  t> 0, \\
\label{e1.2d}
\phi_{tt}- \Delta \phi+\phi+\lambda(x)  \phi_t  =   -\operatorname{Re}(F(x)\cdot \nabla \psi), \quad
 x \in \Omega,\;  t> 0, \\
\label{e1.3d}
\psi = \phi = 0 \quad  \text{on } \Gamma \times (0, \infty), \\
\label{e1.4d}
\psi(0)  = \psi_0, \quad \phi(0) =\phi_0 , \quad \phi_{t}(0)=\phi_1
\end{gather}
where
$\psi_0 \in H_0^{1}(\Omega) \cap H^{2}(\Omega)$,
$\phi_0 \in H_0^{1}(\Omega) \cap H^{2}(\Omega)$,
$\phi_1  \in H_0^{1}(\Omega)$,
$\Omega$  is a bounded domain of  $\mathbb{R}^n$, $n \leq 2 $,
$ \kappa> 0$, $\alpha>0$,  $ \Gamma$ is the smooth boundary of  $\Omega$,
$\omega$ is an  open subset of  $\Omega$
such that  $\operatorname{meas}(\omega)>0$ and
$\lambda \in W^{1,\infty}(\Omega)$  is a nonnegative function.

In what follows   $\chi_{\omega}$  represents the characteristic function of
$\omega$; that is,   $\chi=1$ in $\omega$ and  $\chi=0$ in $\Omega\setminus \omega$.
So that the nonlinearity  term  $\phi \psi$  exists, where the damping
$\lambda(x) \phi_{t}$  takes palce and reciprocally.

 Systems of Klein-Gordon-Schr\"{o}dinger type have been studied for many years.
For example in \cite{cc00,gl97,lw01,ksx97, ps05,ps09} the authors studied
 problems such as existence and uniqueness of solutions, exponential decay,
 the existence of a global attractor and its finite dimensionality in one
or higher dimensions, in bounded or unbounded domains.

  The majority of works in the literature deals with linear dissipative terms,
acting on both equations. Very few is known about the polynomial decay of
a Klein-Gordon-Schr\"{o}dinger type system. In \cite{ca00} the author
proves the polynomial decay when dealing with a localized dissipation
in the wave equation and in \cite{bccs} the authors prove a similar
result when dealing with a KGS system, idea that inspired this work.

  The rest of this article is divided into four sections. In
Section 2, the basic notation is given and the main assumptions made are quoted.
 In Section 3, the existence and uniqueness of global solutions are proved.
 Finally in Section 4, integral inequalities for the energy of the system
are proved using the multiplier method combined with integral inequalities
that can be found in \cite{k94} (See also \cite{l1}).

\section{Notation and assumptions}

   Let us introduce some notation that will be used throughout this work.
Denote by  $ H^{s} (\Omega)$  both the standard real and complex Sobolev spaces
on $\Omega$. For  simplicity reasons sometimes we use  $ H^{s}$, $L^{s}$
 for $ H^{s}(\Omega)$, $L^{s}(\Omega)$.  Let
 $ \|\cdot\|$,  $(\cdot,\cdot)$  denote the norm and  inner product in
 $L^{2}(\Omega)$  respectively, as well as the symbol  $\cdot$  denotes the
inner product in  $\mathbb{R}^n$.  Finally,  $ C$  is a general symbol
for any positive constant.

  Let  $x^0\in \ \mathbb{R}^n$, $n \leq 2 $  and  $n(x)$ be the unit exterior
 normal vector at  $x  \in \Gamma$,  $m(x)=x-x^0$, $x \in \ \mathbb{R}^n$,
$n \leq 2 $ and
\begin{equation} \label{e3.10}
R(x^0):=\sup_{x\ \in \overline{\Omega}}m(x)= \sup_{x\ \in \overline{\Omega}}|x-x^0|.
\end{equation}
We set the norms
\[
\|u\|_{p}^{p}=\int_{\Omega}|u|^{p}dx, \quad
\|u\|^{p}_{\Gamma, p}=\int_{\Gamma}|u(x)|^{p}d\Gamma, \quad
\|u\|_{\infty}=\operatorname{ess\,sup}_{x\in \Omega}|u(x)|.
\]
  Some of the basic tools used are: the embedding inequality
\[
\|u\|_4\leq c_1\|\nabla u \|_2,  \quad \text{for all }
 u  \in H_0^{1}(\Omega);
\]
  Gagliardo-Nirenberg inequality for  $n=1$,
\[
\|u\|_4\leq c_2\|u\|_2^{3/4}\|\nabla u\|^{1/4}_2, \quad
 \text{for all }  u \ \in H^{1}_0(\Omega);
\]
  Gagliardo-Nirenberg inequality for  $n=2$,
\[
\|u\|_4\leq c_3\|u\|^{1/2}\|\nabla u\|^{1/2},\quad \text{for all }
 u \ \in H^{1}_0(\Omega);
\]
  and Young's inequality
\[
ab\leq \frac{1}{p}a^{p}+\frac{1}{p'}b^{p'},  \quad \text{for all }  a,b\geq 0.
\]

\begin{assumption}\label{assum1.1} \rm
Let $ \lambda \in W^{1,\infty}(\Omega)$  be a nonnegative function such that
$\lambda(x)\geq \lambda_0>0$, a.e. in $\omega$.
If  $\lambda(x)\geq \lambda_0>0$  in $\Omega$, then
 $\chi_{\omega}\equiv 1 $  in $\Omega$  (See \cite{ps05}).
\end{assumption}

\begin{assumption}\label{assum1.2} \rm
Let  $ \omega$  be a neighborhood of  $\overline{\Gamma(x^0)}$,  where
$\Gamma(x^0):=\{x  \in \Gamma: m(x)\cdot n(x)>0\}$.
\end{assumption}

\begin{assumption} \label{assum1.3} \rm
We assume that  $  F \in C^{1}(\Omega)$  and  $F  \in L^{\infty}(\Omega)$
 with  $\|F\|_{\infty}=M<+\infty$.
\end{assumption}

\begin{assumption}\label{assum1.4}\rm
  Let there be a neighborhood  $\hat{\omega}$  of  $\overline{\Gamma{(x^0)}}$
 such that  $\hat{\omega}\cap \Omega \subset \omega$   and a vector field
 $h \in (C^{1}(\overline{\Omega}))^n$,  such that  $h=n$ on
$\Gamma(x^0)$ $h\cdot n \geq 0$ a.e. in $\Gamma$,  $h=0$ on
$ \Omega\setminus \hat{\omega}$.
\end{assumption}

 We conclude this section with the following lemma which will play
 essential role when establishing the asymptotic behavior of solutions in Section 4.

\begin{lemma}[{\cite[Lemma 9.1]{k94}}] \label{lem2.1}
Let $E:\mathbb{R}_0^{+}\to \mathbb{R}_0^{+}$ be a non-increasing
function and assume that there exist two constants $p>0$ and $c>0$
such that
\begin{equation*}
\int_s^{+\infty }E^{(p+1)/2}(t)dt\leq cE(s),\quad 0\leq s<+\infty.
\end{equation*}
Then, for all $t\geq 0$,
\[
E(t)\leq \begin{cases}
cE(0)(1+t)^{-2(p-1)} &\text{if } p>1, \\
cE(0)e^{1-wt} &\text{if } p=1,
\end{cases}
\]
where $c$ and $w$ are positive constants.
\end{lemma}

\section{Existence and uniqueness of solutions}

  In this section we derive a priori estimates for the solutions of
the Klein-Gordon-Schr\"{o}dinger system \eqref{e1.1d} - \eqref{e1.4d}.
Let us represent by  $w_{n}$  a basis in  $H_0^{1}(\Omega)\cap H^{2}(\Omega)$
formed by the eigenfunctions of  $-\Delta$,  by  $V_{m}$  the subspace of
$H_0^{1}(\Omega)\cap H^{2}(\Omega)$\ generated by the first m vectors and by
\[
\psi_{m}(t)=\sum_{i=1}^{m}g_{im}(t)w_{i}, \quad
\phi_{m}(t)=\sum_{i=1}^{m}h_{im}w_{i},
\]
  where $(\psi_{m}(t), \phi_{m}(t))$  is the solution of the Cauchy problem
\begin{gather}
 \label{e3.1d}
i(\psi_{t,m},u) +\kappa (\Delta \psi,u) +i\alpha(\psi_{m},u)
=  (\phi_{m}\psi_{m}\chi_{\omega},u) \quad \forall  u  \in V_{m} , \\
\label{e3.2d}
 (\phi_{tt,m},v)- (\Delta \phi_{m},v)+(\phi_{m},v)+ (\lambda (x) \phi_{t,m},v)
=  -\operatorname{Re}(F(x)\cdot \nabla \psi_{m},v),  \quad \forall  v  \in V_{m},
\end{gather}
  with
\begin{equation}  \label{e3.2a}
\begin{gathered}
\psi_{m}(x,0)=\psi_{0m} \to\psi _0, \quad
\phi(x,0) = \phi_{0m} \to \phi_0 \quad \text{in }
 H_0^{1}(\Omega)\cap H^{2}(\Omega), \\
 \phi_{t,m}(0)=\phi_{1m}\to \phi_1 \quad\text{in } H_0^{1}(\Omega) .
\end{gathered}
\end{equation}
 Next we have the following existence and uniqueness result.

\begin{theorem}\label{them2.1}
 Let  $(\psi_0,\phi_0, \phi_1)  \in  (H^{1}_0(\Omega)\cap
H^{2}(\Omega))^{2}\times H_0^{1}(\Omega)$
and Assumption \ref{assum1.1} and \ref{assum1.3} hold.
Then, there exists a unique solution for the problem
\eqref{e1.1d}-\eqref{e1.4d} such that
\begin{gather*}
\psi \in L^{\infty}(0, \infty; H^{1}_0(\Omega)\cap H^{2}(\Omega)), \quad
 \psi_{t}  \in L^{\infty}(0,\infty;L^{2}(\Omega)), \\
\phi \in L^{\infty}(0,\infty; H^{1}_0(\Omega)\cap H^{2}(\Omega)), \quad
 \phi_{t}  \in L^{\infty}(0,\infty;H^{1}_0(\Omega)), \\
\phi_{tt} \in L^{\infty}(0,\infty;L^{2}(\Omega)), \\
\psi(x,0)=\psi_0(x), \quad \phi(x,0)=\phi_0(x), \quad
\phi_{t,0}(x,0)=\phi_1(x), \quad x \in \Omega.
\end{gather*}
\end{theorem}

 \begin{proof}
 The main idea is to use the Galerkin Method. Setting as  $u=\bar{\psi}_{m}(t)$
 in \eqref{e3.1d} and by integrating and taking the imaginary part of the equation
we obtain
\begin{equation} \label{e3.1}
\frac{1}{2} \frac{d}{dt}\|\psi_{m}(t)\|^{2}+\alpha\|\psi_{m}\|^{2}=0.
\end{equation}
  Applying Gronwall's Lemma produces
\begin{equation} \label{e3.2}
\|\psi_{m}(t)\|\leq \|\psi_{m}(0)\| e^{-2\alpha t}.
\end{equation}
  Therefore,
\begin{equation}\label{e3.3}
\|\psi_{m}\|\leq R \quad \text{for all }  t>0.
\end{equation}
Next, by setting  $u=-\bar{\psi}_{t,m}$  in \eqref{e3.1d} and by integrating
and taking the real part,  \eqref{e3.1d} becomes
 \[
 \frac{\kappa}{2} \frac{d}{dt}\int_{\Omega}|\nabla \psi_{m}|^{2}+\alpha \operatorname{Im}
 \int_{\Omega} \psi_{m} \bar{\psi}_{t,m}
= -\operatorname{Re} \int_{\omega} \phi_{m} \psi_{m}  \bar{\psi}_{t,m}.
 \]
   For the right hand side of the equation above we have
\[
\frac{1}{2}\frac{d}{dt} \int_{\omega} \phi_{m} |\psi_{m}|^{2}=
\frac{1}{2}\int_{\omega}\phi_{t,m}|\psi_{m}|^{2}+\operatorname{Re} \int_{\omega} \phi_{m}
\psi \bar{\psi}_{t,m}.
\]
  But from equation \eqref{e3.1d} we also obtain
\[
\alpha \operatorname{Im}\int_{\Omega}\psi_{m} \bar{\psi}_{t,m}dx= \kappa \alpha
\int_{\Omega}|\nabla \psi_{m}|^{2}dx+\alpha \int_{\omega}\phi_{m} |\psi_{m}|^{2}dx.
\]
  Therefore,
\begin{equation} \label{e3.4}
\begin{split}
&\frac{\kappa}{2} \frac{d}{dt}\int_{\Omega}|\nabla \psi_{m}|^{2}+\kappa \alpha
\int_{\Omega}|\nabla \psi_{m}|^{2}dx+\alpha \int_{\omega}\phi_{m} |\psi_{m}|^{2}dx\\
&= -\frac{1}{2}\frac{d}{dt} \int_{\omega} \phi_{m} |\psi_{m}|^{2}+
\frac{1}{2}\int_{\omega}\phi_{t,m}|\psi_{m}|^{2}.
 \end{split}
\end{equation}
  Next, letting   $v=\phi_{t,m}$, equation \eqref{e3.2d} gives
\[ %\label{e3.5}
\frac{1}{2} \frac{d}{dt}\Big[ \|\phi_{t,m}\|^{2}+\|\nabla \phi_{m}\|^{2}
+\|\phi_{m}\|^{2}\Big]+\lambda_0 \|\phi_{t,m}\|^{2}
\leq -\int_{\Omega}(F(x)\cdot \nabla \psi_{m})\phi_{t,m}dx.
\]
  Now, by adding  \eqref{e3.1}, \eqref{e3.4} and the above inequality, we have
\begin{align*} %\label{e3.6}
&\frac{1}{2}\frac{d}{dt}\Big[ \|\psi_{m}\|^{2}+\kappa \|\nabla \psi_{m}(t)\|^{2}
 +\|\phi_{t,m}\|^{2}+\|\nabla \phi_{m}\|^{2}+\|\phi_{m}\|^{2}
 +\int_{\omega}\phi_{m} |\psi_{m}|^{2}dx \Big] \\
&+\alpha \|\psi_{m}\|^{2}
 +\lambda_0 \|\phi_{t,m}\|^{2}+\kappa \alpha \|\nabla \psi_{m}\|^{2}
 +\alpha \int_{\omega}\phi_{m}|\psi_{m}|^{2}dx\\
&-\frac{1}{2}\int_{\omega}\phi_{t,m}|\psi_{m}|^{2}\\
&\leq -\int_{\Omega}(F(x)\cdot \nabla \psi_{m})\phi_{t,m}dx.
\end{align*}
 Evaluating these integrals  by using Assumption \ref{assum1.3},  Gagliardo-Nirenberg
inequality and Young's inequality, we obtain
\begin{gather*}
\Big|\frac{1}{2}\int_{\omega}\phi_{t,m}|\psi_{m}|^{2}dx\Big|\leq
\frac{\alpha}{4}\int_{\Omega}|\phi_{t,m}|^{2}dx+\frac{C^{2}}{4\alpha}\int_{\Omega}|\nabla \psi_{m}|^{2}dx,\\
\Big|\int_{\Omega}(F(x) \cdot \nabla \psi_{m}) \phi_{t,m}dx\Big|\leq \frac{\alpha}{2}
\int_{\Omega}|\phi_{t,m}|^{2}dx+\frac{M^{2}}{2\alpha}\int_{\Omega}
|\nabla \psi_{m}|^{2}dx,\\
\Big|\alpha \int_{\omega}\phi_{m} |\psi_{m}|^{2}dx\Big| \leq
\alpha\int_{\Omega}|\phi_{m}|^{2}dx+\frac{\alpha C^{2}}{4}\int_{\Omega}
|\nabla \psi_{m}|^{2}dx.
\end{gather*}
  Integrating the above expression over  $(0,t)$  and applying Gronwall's
Lemma we obtain the first estimate
\begin{equation} \label{e3.7}
 \|\psi_{m}\|^{2}+\kappa \|\nabla \psi_{m}\|^{2}+\|\phi_{t,m}\|^{2}
+\|\nabla \phi_{m}\|^{2}+\|\phi_{m}\|^{2}+\int_{\omega}\phi_{m} |\psi_{m}|^{2}dx
\leq  L_1,
\end{equation}
where  $L_1$  is a positive constant independent of  $m \in \mathbb{N}$.
Let
\begin{equation*}
E(t)= \|\psi_{m}\|^{2}+\kappa \|\nabla \psi_{m}\|^{2}+\|\phi_{t,m}\|^{2}
+\|\nabla \phi_{m}\|^{2}+\|\phi_{m}\|^{2}+\int_{\omega}\phi_{m} |\psi_{m}|^{2}dx
\end{equation*}
evaluating the integral
\[
\int_{\omega} |\phi_{m}\|\psi_{m}|^{2}dx
\leq \|\phi_{m}\|\|\psi_{m}\|^{2}_4
\leq C\|\phi_{m}\|\|\nabla \psi_{m}\|  \|\psi_{m}\|
\leq \frac{1}{2}\|\nabla \phi_{m}\|^{2}+\frac{\kappa}{2}\|\nabla \psi_{m}\|^{2}+C
\]
one can deduce that
\[
E(t)\geq \|\psi_{m}\|^{2}+\frac{\kappa}{2} \|\nabla \psi_{m}\|^{2}
+\|\phi_{t,m}\|^{2}+\frac{1}{2}\|\nabla \phi_{m}\|^{2}+\|\phi_{m}\|^{2}+C,
\]
  and
\[
E(t)\leq \|\psi_{m}\|^{2}+\frac{3\kappa}{2} \|\nabla \psi_{m}\|^{2}
+\|\phi_{t,m}\|^{2}+\frac{3}{2}\|\nabla \phi_{m}\|^{2}+\|\phi_{m}\|^{2}+C.
\]
so from \eqref{e3.7} we have
\[
\|\psi_{m}\|^{2}+\|\nabla \psi_{m}\|^{2}+\|\phi_{t,m}\|^{2}
+\|\nabla \phi_{m}\|^{2}+\|\phi_{m}\|^{2}\leq L_1+C.
\]
 Next, let  $  u=\Delta \bar{\psi}_{t,m}+\alpha  \Delta \bar{\psi}_{m}$,
in  \eqref{e3.1d}. Taking the real part and integrating over  $\Omega$  produces
\begin{equation} \label{e3.8}
\begin{aligned}
&  \frac{1}{2} \frac{d}{dt}\kappa \|
\Delta \psi_{m}\|^{2}+\kappa \alpha\|
\Delta \psi_{m}\|^{2}\\
&=\operatorname{Re} \int_{\omega} \phi_{m} \psi_{m} \Delta \bar{\psi}_{t,m} dx
+\alpha \operatorname{Re} \int_{\omega} \phi_{m} \psi_{m} \Delta \bar{\psi}_{m}dx.
\end{aligned}
\end{equation}
Furthermore, by letting  $ v=- \Delta \phi_{t,m}$  in \eqref{e3.2d} and integrating,
 we also obtain
\begin{equation} \label{e3.9}
\begin{aligned}
&\frac{1}{2} \frac{d}{dt} \Big(\|
\nabla \phi_{t,m}\|^{2}+\| \Delta \phi_{m}\|^{2}+\|\nabla \phi_{m}\|^{2} \Big)
+\lambda_0\| \nabla \phi_{t,m}\|^{2} \\
&\leq \operatorname{Re} \int_{\Omega} (F(x)\cdot \nabla \psi_{m}) \Delta
\phi_{t,m}dx.
\end{aligned}
\end{equation}
Analyzing  the right hand side of \eqref{e3.8}  produces the equation
\begin{align*}
&\operatorname{Re} \int_{\omega} \phi_{m} \psi_{m} \Delta \bar{\psi}_{t,m}dx\\
&=\frac{d}{dt} \operatorname{Re} \int_{\omega} \phi_{m} \psi_{m}
 \Delta \bar{\psi}_{m} dx  -\operatorname{Re} \int_{\omega} \phi_{t,m} \psi_{m}
\Delta \bar{\psi}_{m}dx-\operatorname{Re} \int_{\omega} \phi_{m} \psi_{t,m}
\Delta \bar{\psi}_{m}dx,
\end{align*}
while by
$\psi_{t,m}=-i(-\Delta \psi_{m}-i\alpha \psi_{m}-\phi_{m} \psi_{m}\chi_{\omega})$,
\begin{align*}
-\operatorname{Re} \int_{\omega} \phi_{m} \psi_{t,m}\Delta \bar{\psi}_{m}dx
&=\operatorname{Re} \int_{\omega} i\phi_{m}[-\Delta \psi_{m}
 -i\alpha \psi_{m}-\phi_{m}\psi_{m}] \Delta \bar{\psi}_{m}dx\\
&=\alpha \operatorname{Re} \int_{\omega} \phi_{m} \psi_{m} \Delta \bar{\psi}_{m}dx
+\operatorname{Im}\int_{\omega} \phi^{2}_{m}\psi_{m}\Delta \bar{\psi}_{m}dx.
\end{align*}
  Substituting the expressions above into \eqref{e3.8} yields
\begin{equation} \label{e3.10b}
\begin{split}
&\frac{1}{2}\frac{d}{dt} \Big( \kappa \|\Delta \psi_{m}\|^{2}-2\operatorname{Re}
 \int_{\omega} \phi_{m} \psi_{m} \Delta \bar{\psi}_{m}dx \Big)
+\kappa \alpha \|\Delta \psi_{m}\|^{2}\\
&=2\alpha \int_{\omega}\phi_{m} \psi_{m} \Delta \bar{\psi}_{m}dx
+\operatorname{Im}\int_{\omega} \phi^{2}_{m}\psi_{m}\Delta \bar{\psi}_{m}dx
-\operatorname{Re} \int_{\omega} \phi_{t,m} \psi_{m} \Delta \bar{\psi}_{m}dx.
\end{split}
\end{equation}
Next, adding \eqref{e3.9} and \eqref{e3.10} gives
\begin{equation} \label{e3.11}
\begin{split}
&\frac{1}{2}\frac{d}{dt} \Big(\kappa \|\Delta \psi_{m}\|^{2}-2\operatorname{Re}
\int_{\omega} \phi_{m} \psi_{m} \Delta \bar{\psi}_{m}dx
+\| \nabla \phi_{t,m}\|^{2}+\| \Delta \phi_{m}\|^{2}+\|\nabla \phi_{m}\|^{2}\Big) \\
&+\kappa \alpha \|\Delta \psi_{m}\|^{2}+\lambda_0\| \nabla \phi_{t,m}\|^{2}\\
&\leq 2\alpha \int_{\omega}\phi_{m} \psi_{m} \Delta \bar{\psi}_{m}dx
 +\operatorname{Im}\int_{\omega} \phi^{2}_{m}\psi_{m}\Delta \bar{\psi}_{m}dx\\
&\quad -\operatorname{Re} \int_{\omega} \phi_{t,m} \psi_{m}
\Delta \bar{\psi}_{m}dx+\operatorname{Re} \int_{\Omega} (F(x)\cdot
\nabla \psi_{m}) \Delta \phi_{t,m}dx.
\end{split}
\end{equation}
  Estimating the integrals on the right hand side of \eqref{e3.11} using
the Sobolev embedding theorem and Young's Inequality
\begin{gather*}
\Big| \operatorname{Re} \int_{\omega} \phi_{m} \psi_{m} \Delta \bar{\psi}_{m} dx\Big|
 \leq\|\phi_{m}\|_4\|\psi_{m}\|_4\|\Delta \psi_{m}\|
 \leq \frac{1}{4}\|\Delta \psi_{m}\|^{2}+C\|\nabla \phi_{m}\|^{2}\|\nabla \psi_{m}\|^{2},
\\
\Big|\operatorname{Im}\int_{\omega} \phi^{2}_{m}\psi_{m}\Delta \bar{\psi}_{m}dx \Big|
  \leq \|\phi_{m}\|^{2}_{6}\|\psi_{m}\|_{6}\|\Delta \psi_{m}\|
  \leq \frac{1}{4}\|\Delta \psi_{m}\|^{2}+C\|\nabla \phi_{m}\|^{4}\|\nabla \psi_{m}\|^{2},
\\
\Big|-\operatorname{Re}\! \int_{\omega} \phi_{m}' \psi_{m} \Delta \bar{\psi}_{m}dx\Big|
 \leq \|\phi_{m}'\|_4\|\psi_{m}\|_4\|\Delta \psi_{m}\|
 \leq \frac{1}{4}\|\Delta \psi_{m}\|^{2}+C\|\nabla \phi_{m}'\|^{2}\|\nabla \psi_{m}\|^{2}.
\end{gather*}
  Now evaluating the last term of \eqref{e3.9},
\begin{align*}
&\int_{\Omega} (F(x)\cdot \nabla \psi_{m}) \Delta \phi_{t,m}dx\\
&=- \int_{\Omega} \nabla(F(x)\cdot \nabla \psi_{m})\nabla \phi_{t,m}dx\\
&=-\int_{\Omega} (F(x)\cdot \Delta \psi_{m}) \nabla \phi_{t,m}dx
-\int_{\Omega}(\nabla F(x)\cdot \nabla \psi_{m}) \nabla \phi_{t,m}dx\\
&\quad -\int_{\Omega}(\nabla \psi_{m}\times (\nabla \times F(x))) \nabla \phi_{t,m}dx
\end{align*}
and taking into consideration Assumption \ref{assum1.3} produces
\begin{gather*}
\big|-\int_{\Omega} (F(x)\cdot \Delta \psi_{m}) \nabla \phi_{t,m}dx\big|
\leq C \|\Delta \psi_{m}\| \|\nabla \phi_{t,m}\|,\\
\big|-\int_{\Omega}(\nabla F(x)\cdot \nabla \psi_{m}) \nabla \phi_{t,m}dx\big|
 \leq C \|\nabla \psi_{m}\| \|\nabla \phi_{t,m}\|, \\
\big|-\int_{\Omega}(\nabla \psi_{m}\times (\nabla \times F(x)))
\nabla \phi_{t,m}dx \big|
 \leq C \|\nabla \psi_{m}\| \|\nabla \phi_{t,m}\|.
\end{gather*}
  Integrating over  $(0,t)$  and applying Gronwall's Lemma we obtain
 the second estimate
\begin{equation} \label{e3.12}
\|\Delta \psi_{m}\|^{2} +\|
\nabla \phi_{t,m}\|^{2}+\| \Delta \phi_{m}\|^{2}+\|\nabla \phi_{m}\|^{2} \leq L_2,
\end{equation}
  where  $L_2$  is a positive constant independent of  $m  \in \mathbb{N}$.
The rest of the proof follows the same basic steps as the one of
\cite[Theorem 3.1]{ksx97}.
\end{proof}

The energy associated to the problem is defined by
\begin{equation}\label{e2.2}
 E(t):=\frac{1}{2} \Big( \|\psi\|^{2}+\kappa
\|\nabla \psi\|^{2}+\int_{\omega} \phi
|\psi|^{2}dx+\|\phi_{t}\|^{2}+\|\nabla \phi\|^{2}+\|\phi\|^{2}\Big).
\end{equation}

\section{Exponential decay}

  Let $\{\psi(t), \phi(t), \phi_{t}(t)\}$ in
$H_0^{1}(\Omega)\cap H^{2}\times H_0^{1}(\Omega) \cap H^{2}(\Omega)\times H_0^{1}(\Omega) $ \ be a solution of the
 \eqref{e1.1d}- \eqref{e1.4d}.

\begin{lemma}\label{lem4.1}
Let $ \kappa, \alpha, \lambda_0$ be large and  $\epsilon $ be small enough.
Then for  $\beta:=\kappa \alpha -\frac{M^{2}}{2\lambda_0}-\frac{1}{2\epsilon}> 0$
 the first order energy satisfies the inequality
\[
E'(t)\leq -\alpha \int_{\Omega}|\psi|^{2}dx
-\beta \int_{\Omega}|\nabla \psi|^{2}dx+c(\alpha ,c_1,\epsilon)
\int_{\Omega}|\phi|^{2}dx -\frac{3}{8}\int_{\Omega}\lambda(x)|\phi_{t}|^{2}dx.
\]
\end{lemma}
\begin{proof}
  Substituting into \eqref{e3.1d}  $u=-\bar{\psi}_{t}$, taking the real part,
next substituting also  $v=\phi_{t}$,  into \eqref{e3.2d} and integrating both
over  $\Omega$,  we have
 \begin{gather*}
 \frac{\kappa}{2} \frac{d}{dt}\int_{\Omega}|\nabla \psi|^{2}dx+\alpha \operatorname{Im}
 \int_{\Omega} \psi \bar{\psi}_{t}dx
= -\operatorname{Re} \int_{\omega} \phi \psi  \bar{\psi}_{t}dx, \\
\frac{1}{2}\frac{d}{dt} \int_{\Omega}(
|\phi_{t}|^{2}dx+|\nabla \phi|^{2}+|\phi|^{2})dx+\int_{\Omega}\lambda(x)|\phi_{t}|^{2}dx=-\operatorname{Re}
\int_{\Omega} (F(x)\cdot \nabla \psi) \phi_{t}dx.
\end{gather*}
  The right hand side of the first equation becomes
\[
\frac{1}{2}\frac{d}{dt} \int_{\omega} \phi |\psi|^{2}dx=
\frac{1}{2}\int_{\omega}\phi_{t}|\psi|^{2}dx+\operatorname{Re} \int_{\omega} \phi
\psi \bar{\psi}_{t}dx.
\]
Also from \eqref{e3.1d} we have
\[
\alpha \operatorname{Im}\int_{\Omega}\psi \bar{\psi}_{t}dx= \kappa \alpha
\int_{\Omega}|\nabla \psi|^{2}dx+\alpha \int_{\omega}\phi |\psi|^{2}dx.
\]
Taking $u=\bar{\psi}$, in \eqref{e3.1d} integrating over $\Omega$
and taking the imaginary part yields
\[
\frac{1}{2}\frac{d}{dt}\int_{\Omega} |\psi|^{2}dx+\alpha \int_{\Omega}|\psi|^{2}dx=0.
\]
 Adding  the above equations gives
\begin{align*}
&\frac{d}{dt}E(t)+\alpha \int_{\Omega}|\psi|^{2}dx+\kappa \alpha
\int_{\Omega}|\nabla \psi|^{2}dx+\alpha \int_{\omega}\phi
|\psi|^{2}dx+\int_{\Omega}\lambda(x)|\phi_{t}|^{2}dx \\
&=\frac{1}{2}\int_{\omega}\phi_{t}|\psi|^{2}dx-\operatorname{Re} \int_{\Omega}
(F(x) \cdot \nabla \psi) \phi_{t}dx.
\end{align*}
Evaluating the integrals by using Assumption \ref{assum1.3}, Gagliardo-Nirenberg
inequality and Young's inequality produces
\begin{gather*}
\Big|\frac{1}{2}\int_{\omega}\phi_{t}|\psi|^{2}dx\Big|\leq
\frac{1}{8}\int_{\Omega}\lambda(x)|\phi_{t}|^{2}dx+\frac{c_1^{2}}{2\lambda_0}\int_{\Omega}|\nabla \psi|^{2}dx,\\
\Big|\int_{\Omega}(F(x) \cdot \nabla \psi) \phi_{t}dx\Big|\leq \frac{1}{2}
\int_{\Omega}\lambda(x)|\phi_{t}|^{2}dx+\frac{M^{2}}{2\lambda_0}\int_{\Omega}
|\nabla \psi|^{2}dx,\\
\Big|\alpha \int_{\omega}\phi |\psi|^{2}dx\Big| \leq
c(\alpha ,c_1,\epsilon) \int_{\Omega}|\phi|^{2}dx+\frac{1}{4\epsilon}\int_{\Omega}
|\nabla \psi|^{2}dx.
\end{gather*}
  Hence for  $ \kappa, \alpha, \lambda_0$  large, $\epsilon $  small enough and
$\beta$ as above, it holds  that
\begin{equation}\label{e4.1}
E'(t)\leq -\alpha \int_{\Omega}|\psi|^{2}dx
-\beta \int_{\Omega}|\nabla \psi|^{2}dx+c(\alpha ,c_1,\epsilon)\int_{\Omega}|\phi|^{2}dx -\frac{3}{8}\int_{\Omega}\lambda(x)|\phi_{t}|^{2}dx,
\end{equation}
which concludes the proof of the Lemma.
\end{proof}

\begin{lemma}\label{lem4.2}
Let  $\{ \psi, \psi, \phi_{t}\}$  be solutions of \eqref{e1.1d} - \eqref{e1.4d},
 $\beta>0$  and Assumptions \ref{assum1.1} - \ref{assum1.3} hold.
Then there exists  $T_0>0$  such that if $T>T_0$ we have
\[
\frac{2}{12}\int_s^TE^{2}(t)dt\leq
\frac{1}{2}\int_s^TE(t)\int_{\Gamma(x^0)}(m\cdot n) \Big(
\frac{\partial \phi}{\partial n}\Big)^{2}d\Gamma dt
+\frac{1}{\alpha}|\chi|+C_0E(s),
\]
  where
\begin{align*}
\chi&:=\frac{1}{4}\Big[  E(t)\Big (\int_{\Omega}
\big(|\psi|^{2}+\kappa|\nabla \psi|^{2}+4\alpha \phi_{t}(m \cdot
\nabla \phi)+\alpha \delta \phi
\Big(4\phi_{t}+2\lambda(x)\phi\Big)\big)dx\\
&\quad + \int_{\omega}\phi |\psi|^{2}dx \Big) \Big]_s^T ,
\end{align*}
$0<s<T<+\infty$ and   $C_0$  depends on $\alpha, \beta, \lambda, R, M, \kappa, n, c_1$.
\end{lemma}

 \begin{proof}
 Multiplying \eqref{e3.1d} by $ E(t)\bar{\psi}$,  taking the imaginary
part and integrating produces
\[
\frac{1}{2}\Big[  E(t)\int_{\Omega}
|\psi|^{2}dx\Big]_s^T-\frac{1}{2}\int_s^T
E'(t)\int_{\Omega}|\psi|^{2}dx+\alpha
\int_s^TE(t)\int_{\Omega}|\psi|^{2}\,dx\,dt=0.
\]
  Next, multiplying  \eqref{e3.1d} by
$ -\frac{1}{2}E(t)\bar{\psi}_{t}$,  taking the real part and
adding the previous equation produces
\begin{equation} \label{e4.3}
\begin{split}
 &\frac{1}{4}\Big[  E(t)\Big (\int_{\Omega}
(|\psi|^{2}+\kappa|\nabla \psi|^{2})dx+ \int_{\omega} \phi
|\psi|^{2}dx \Big ) \Big]_s^T
 -\frac{\kappa}{4}\int_s^T E'(t)\int_{\Omega} |\nabla \psi|^{2}\,dx\,dt  \\
& \quad +\frac{\alpha}{2}\int_s^TE(t) \int_{\Omega} |\psi|^{2}\,dx\,dt
 +\frac{\kappa \alpha}{2} \int_s^T E(t) \int_{\Omega} |\nabla \psi|^{2}\,dx\,dt\\
&\quad +\frac{\alpha}{2} \int_s^T E(t)\int_{\omega} \phi |\psi|^{2}\,dx\,dt
 -\frac{1}{4}\int_s^T E'(t)\int_{\Omega}|\psi|^{2}\,dx\,dt\\
&=\frac{1}{2} \int_s^T E'(t)\int_{\omega} \phi
|\psi|^{2}\,dx\,dt  +\frac{1}{2}\int_s^T E(t) \int_{\omega}
\phi_{t} |\psi|^{2}\,dx\,dt
\end{split}
\end{equation}
Multiplying the second equation by  $\alpha
E(t)(q\cdot \nabla \phi)$,  where  $q \in (W^{1,\infty}(\Omega))^n$,
 integrating by parts and using Green's identity we obtain
\begin{equation}  \label{e4.4}
\begin{split}
&\Big[ \alpha E(t)\int_{\Omega}\phi_{t}(q\cdot
 \nabla \phi)dx\Big]_s^T -\alpha \int_s^T
 E'(t)\int_{\Omega} \phi_{t}(q\cdot \nabla \phi)\,dx\,dt \\
&\quad + \frac{\alpha}{2}\int_s^T E(t) \int_{\Omega} \operatorname{div} q
 |\phi_{t}|^{2}\,dx\,dt
 +\alpha \int_s^T E(t)\int_{\Omega} \frac{\partial \phi}{\partial x_{i}}
 \frac{\partial q_{k}}{\partial x_{i}}  \frac{\partial \phi}{\partial x_{k}}\,dx\,dt\\
&\quad -  \frac{\alpha}{2} \int_s^T E(t) \int_{\Omega}
 \operatorname{div} q |\nabla \phi|^{2}\,dx\,dt
 - \frac{\alpha}{2} \int_s^T E(t) \int_{\Gamma} (q \cdot n)
  \Big(\frac{\partial \phi}{\partial n}\Big)^{2}d\Gamma dt \\
&\quad +\alpha \int_s^T E(t)\int_{\Omega} \phi (q \cdot \nabla \phi)\,dx\,dt
 +\alpha \int_s^T E(t) \int_{\Omega} \lambda(x)\phi_{t}(q\cdot \nabla \phi)\,dx\,dt\\
&=- \alpha \int_s^T E(t) \int_{\Omega} (F(x)\cdot \nabla \psi)
  (q\cdot \nabla \phi)\,dx\,dt.
\end{split}
\end{equation}
  Adding relations \eqref{e4.3} and \eqref{e4.4}, we obtain
\begin{align}
 &\frac{1}{4}\Big[  E(t)\Big (\int_{\Omega}(|\psi|^{2}
 +\kappa|\nabla \psi|^{2}+4\alpha \phi_{t}(q \cdot \nabla \phi))dx
 + \int_{\omega} \phi |\psi|^{2}dx )\Big) \Big]_s^T
\nonumber \\
& +\frac{\alpha}{2}\int_s^T E(t) \int_{\Omega} \operatorname{div} q
  [|\phi_{t}|^{2}-|\nabla \phi|^{2}]\,dx\,dt
 +\alpha \int_s^T E(t) \int_{\Omega} \frac{\partial \phi}{\partial x_{i}}
 \frac{\partial q_{k}}{\partial x_{i}} \frac{\partial \phi}{\partial x_{k}} \,dx\,dt
\nonumber \\
& -\frac{\kappa}{4}\int_s^T E'(t) \int_{\Omega} |\nabla \psi|^{2}\,dx\,dt
  +\alpha\int_s^T E(t) \int_{\Omega} \phi (q\cdot \nabla \phi)\,dx\,dt
\nonumber \\
&-\alpha \int_s^TE'(t)\int_{\Omega} \phi_{t}(q\cdot \nabla \phi)\,dx\,dt
 +\alpha \int_s^T E(t) \int_{\Omega} \lambda(x) \phi_{t}(q\cdot \nabla \phi)\,dx\,dt
\nonumber  \\
 &+\frac{\alpha}{2}\int_s^T E(t) \int_{\Omega} |\psi|^{2}\,dx\,dt
 +\frac{\kappa \alpha}{2} \int_s^T E(t) \int_{\Omega} |\nabla \psi|^{2}\,dx\,dt
\nonumber \\
&+\frac{\alpha}{2} \int_s^T E(t)\int_{\omega} \phi |\psi|^{2}\,dx\,dt
 -\frac{1}{2}\int_s^T E'(t)\int_{\omega}\phi |\psi|^{2}\,dx\,dt
\nonumber \\
& -\frac{1}{4}\int_s^T E'(t)\int_{\Omega}|\psi|^{2}\,dx\,dt
\nonumber \\
&=\frac{1}{2}\int_s^T E(t) \int_{\omega} \phi_{t}|\psi|^{2} \,dx\,dt
 -\alpha  \int_s^TE(t) \int_{\Omega}(F(x)\cdot \nabla \psi)
(q \cdot \nabla \phi)\,dx\,dt
\nonumber \\
&\quad +\frac{\alpha}{2} \int_s^T E(t) \int_{\Gamma} (q \cdot n)
 \Big(\frac{\partial \phi}{\partial n}\Big)^{2}d\Gamma dt. \label{e4.5}
\end{align}
Considering that $ q(x)=m(x)=x-x_0$, relation
\eqref{e4.5} becomes
\begin{equation}
\begin{aligned}
& \frac{1}{4}\Big[  E(t)\Big (\int_{\Omega}
(|\psi|^{2}+\kappa|\nabla \psi|^{2}+4\alpha \phi_{t}(m \cdot\nabla \phi))dx
+\int_{\omega} \phi |\psi|^{2}dx ) \Big) \Big]_s^T
 \\
&\quad +\alpha \int_s^T E(t)\int_{\Omega} |\nabla \phi|^{2} \,dx\,dt
  +\frac{\alpha n}{2}\int_s^T E(t) \int_{\Omega}
[ |\phi_{t}|^{2}-|\nabla \phi|^{2}]\,dx\,dt
 \\
&-\frac{\kappa}{4} \int_s^T E'(t) \int_{\Omega} |\nabla \psi|^{2}\,dx\,dt
 +\frac{\alpha}{2}\int_s^TE(t)\int_{\Omega} |\psi|^{2}\,dx\,dt
\\
& +\alpha \int_s^T E(t) \int_{\Omega} \phi (m\cdot \nabla \phi)\,dx\,dt
+\alpha \int_s^T E(t) \int_{\Omega} \lambda(x) \phi_{t}(m\cdot \nabla \phi)\,dx\,dt
 \\
& -\alpha \int_s^T E'(t)\int_{\Omega} \phi_{t}(q\cdot \nabla \phi)\,dx\,dt
-\frac{1}{4}\int_s^T E'(t)\int_{\Omega}|\psi|^{2}\,dx\,dt\\
& +\frac{\kappa \alpha}{2} \int_s^T E(t) \int_{\Omega} |\nabla \psi|^{2}\,dx\,dt
 \\
& \leq -\frac{\alpha}{2} \int_s^T E(t)\int_{\omega} \phi |\psi|^{2}\,dx\,dt
 +\frac{1}{2} \int_s^T E(t) \int_{\omega} \phi_{t}|\psi|^{2} \,dx\,dt
 \\
&\quad  +\frac{1}{2}\int_s^T E'(t)\int_{\omega}\phi |\psi|^{2}\,dx\,dt
  +\frac{\alpha}{2} \int_s^T E(t) \int_{\Gamma(x^0)} (m \cdot n)
 \Big(\frac{\partial \phi}{\partial n}\Big)^{2}d\Gamma dt
 \\
&\quad -\alpha \int_s^T E(t) \int_{\Omega} (F(x)\cdot \nabla \psi)
 (m \cdot \nabla \phi)\,dx\,dt.
\end{aligned} \label{e4.6}
\end{equation}
  Now multiplying \eqref{e3.2d} by  $\alpha E(t)\xi\phi$,
where  $ \xi  \in W^{1,\infty}(\Omega)$,  and
integrating we obtain
\begin{align}
 &\Big[ \alpha E(t)\int_{\Omega} \phi \xi
\Big(\phi_{t} + \frac{\lambda(x)\phi}{2}\Big)dx
\Big]_s^T- \alpha \int_s^TE(t)\int_{\Omega} \xi
|\phi_{t}|^{2}\,dx\,dt
\nonumber \\
&+ \alpha \int_s^TE(t)\int_{\Omega} \phi (\nabla \xi\cdot \nabla \phi)\,dx\,dt
 + \alpha \int_s^TE(t)\int_{\Omega}\xi |\nabla \phi|^{2}\,dx\,dt
\nonumber\\
&+ \alpha \int_s^TE(t) \int_{\Omega}\xi |\phi|^{2}\,dx\,dt
 + \frac{\alpha}{2}\int_s^TE'(t)\int_{\Omega}\xi\lambda(x)|\phi|^{2}\,dx\,dt
\nonumber \\
&=- \alpha \int_s^TE(t)\int_{\omega}\xi \phi \nabla \psi \,dx\,dt+ \alpha
\int_s^TE'(t)\int_{\Omega}\xi \phi_{t} \phi \,dx\,dt. \label{e4.7}
\end{align}

 Taking  $\xi=2\delta  \in \mathbb{R}$  and adding
\eqref{e4.6} and \eqref{e4.7} produces the expression
\begin{equation} \label{e4.8}
\begin{split}
&\chi + \alpha(\frac{ n}{2}-2\delta) \int_s^T E(t)
\int_{\Omega} |\phi_{t}|^{2}\,dx\,dt +\alpha (1+2\delta - \frac{
n}{2})\int_s^T E(t) \int_{\Omega} |\nabla \phi|^{2} \,dx\,dt
\\
&-\frac{\kappa}{4} \int_s^T E'(t) \int_{\Omega} |\nabla \psi|^{2}\,dx\,dt
+\alpha \int_s^T E(t) \int_{\Omega} \phi (m\cdot \nabla \phi)\,dx\,dt
\\
&-\frac{1}{4}\int_s^T E'(t)\int_{\Omega}|\psi|^{2}\,dx\,dt
 -\alpha \int_s^T E'(t)\int_{\Omega}\phi_{t}(m\cdot \nabla \phi)\,dx\,dt
\\
&+2\alpha \delta \int_s^T E(t)\int_{\Omega}|\phi|^{2}\,dx\,dt
 +\frac{\alpha}{2}\int_s^TE(t) \int_{\Omega} |\psi|^{2}\,dx\,dt\\
&+\alpha \int_s^T E(t) \int_{\Omega} \lambda(x) \phi_{t}(m\cdot
\nabla \phi)\,dx\,dt+\frac{\kappa \alpha}{2} \int_s^T E(t)
\int_{\Omega} |\nabla \psi|^{2}\,dx\,dt
\\
& +\alpha \delta\int_s^TE'(t)\int_{\Omega}\lambda(x)|\phi|^{2}\,dx\,dt
+2\alpha \delta \int_s^TE(t)\int_{\omega} \phi \nabla \psi \,dx\,dt
\\
&\leq  +\frac{1}{2} \int_s^T E'(t)\int_{\omega}\phi |\psi|^{2}\,dx\,dt
 -\frac{\alpha}{2} \int_s^T E(t)\int_{\omega} \phi |\psi|^{2}\,dx\,dt\\
&\quad  +\frac{1}{2} \int_s^T E(t) \int_{\omega} \phi_{t}|\psi|^{2} \,dx\,dt
 + 2\delta \int_s^TE'(t)\int_{\Omega} \phi_{t} \phi \,dx\,dt\\
&\quad -\alpha \int_s^T E(t) \int_{\Omega} (F(x) \cdot \nabla \psi)
 (m \cdot \nabla \phi)\,dx\,dt
 \\
&\quad+\frac{\alpha}{2} \int_s^T E(t) \int_{\Gamma(x^0)} (m \cdot n)
\Big(\frac{\partial \phi}{\partial n}\Big)^{2}d\Gamma dt.
\end{split}
\end{equation}
  Choosing  $\delta= \frac{n-1}{4}$,  if  $n= 2$  or  $\delta  \in (0,1/2)$,  if
$n= 1$  relation \eqref{e4.8}  becomes
\begin{align}
 & \chi + \alpha \int_s^T E^{2}(t)dt
-\frac{\kappa}{4} \int_s^T E'(t) \int_{\Omega} |\nabla \psi|^{2}\,dx\,dt
+\alpha \int_s^T E(t) \int_{\Omega} \phi (m\cdot \nabla \phi)\,dx\,dt
\nonumber \\
&-\alpha \int_s^T E'(t)\int_{\Omega}
\phi_{t}(m\cdot \nabla \phi)\,dx\,dt+\alpha
\int_s^T E(t) \int_{\Omega} \lambda(x) \phi_{t}(m\cdot \nabla \phi)\,dx\,dt
\nonumber \\
&-\frac{1}{4}\int_s^T E'(t)\int_{\Omega}|\psi|^{2}\,dx\,dt
 +\frac{\alpha(n-1)}{4}\int_s^TE'(t)\int_{\Omega}\lambda(x)|\phi|^{2}\,dx\,dt
\nonumber \\
&-\frac{\alpha}{2}\int_s^T E(t) \int_{\Omega}|\phi|^{2}\,dx\,dt
+\frac{\alpha(n-1)}{2} \int_s^T E(t)\int_{\Omega}|\phi|^{2}\,dx\,dt
\nonumber \\
&+\frac{\alpha(n-1)}{2}\int_s^TE(t)\int_{\omega} \phi \nabla \psi \,dx\,dt
\nonumber \\
&\leq \frac{1}{2}\int_s^T E'(t)\int_{\omega}\phi |\psi|^{2}\,dx\,dt
 +\frac{1}{2}\int_s^T E(t) \int_{\omega} \phi_{t}|\psi|^{2} \,dx\,dt
\nonumber \\
&\quad -\alpha \int_s^T E(t) \int_{\Omega} (F(x)\cdot \nabla \psi)
 (m \cdot \nabla \phi)\,dx\,dt
\label{e4.9} \\
&\quad +\frac{\alpha}{2} \int_s^T E(t)\int_{\Gamma(x^0)} (m \cdot n)
 \Big(\frac{\partial \phi}{\partial n}\Big)^{2}d\Gamma dt
 + \frac{\alpha(n-1)}{2} \int_s^TE'(t)\int_{\Omega} \phi_{t} \phi \,dx\,dt.
\nonumber
\end{align}
In order for the proof to be
completed we need to estimate some of the terms appearing in
\eqref{e4.9}.
Let  $ I_1= - \frac{1}{4} \int_s^T
E'(t) \int_{\Omega} (\kappa|\nabla \psi|^{2}+|\psi|^{2})\,dx\,dt$.
Then taking into consideration \eqref{e2.2},
\[
|I_1|\leq \frac{1}{4}\int_s^T|E'(t)|E(t)dt \leq
-\frac{1}{8}\int_s^T(E^{2}(t))'dt \leq \frac{1}{8} E(0)E(s).
\]
  Let  $I_2=  \frac{\alpha(n-2)}{2}
 \int_s^T E(t) \int_{\Omega} |\phi|^{2}\,dx\,dt$.
Then taking into consideration \eqref{e2.2},
\[
|I_2|\leq -\frac{(n-2)}{\alpha \lambda_0}\int_s^TE(t)E'(t)dt\leq
\frac{(n-2)}{2\alpha \lambda_0} E(0)E(s).
\]
  Let $I_3= \alpha \int_s^T E(t)
\int_{\Omega} \phi (m\cdot \nabla \phi)\,dx\,dt$. Then taking into
consideration \eqref{e2.2}, \eqref{e4.1} and Young's inequality gives
\begin{align*}
|I_3| &\leq \frac{R^{2}\alpha^{2}}{4\epsilon}\int_s^TE(t)\int_{\Omega}|\phi|^{2}\,dx\,dt
+\epsilon\int_s^TE(t)\int_{\Omega}|\nabla \phi|^{2}\,dx\,dt\\
&  \leq \frac{R^{2}}{4\epsilon \lambda_0}  E(0)E(s)+2\epsilon\int_s^TE^{2}(t)dt.
\end{align*}
  Let  $ I_4= -\alpha   \int_s^T
E'(t) \int_{\Omega} \phi_{t}(m\cdot \nabla \phi)\,dx\,dt$.  Then
taking into consideration \eqref{e2.2},
\[
|I_4|\leq \alpha R  \int_s^T|E'(t)|E(t)dt\leq
\frac{\alpha R}{2}E(0)E(s).
\]
  Let  $ I_5=  \alpha \int_s^T E(t)
\int_{\Omega} \lambda(x)\phi_{t}(m\cdot \nabla \phi)\,dx\,dt$.
 Then taking into consideration \eqref{e2.2} and
\eqref{e4.1} and Young's inequality produces
\[
|I_5|\leq \frac{\alpha^{2}R^{2}\|\lambda\|_{\infty}}{3\epsilon}E(0)E(t)
+2\epsilon \int_s^T E^{2}(t)dt.
\]
  Let  $ I_{6}=  \frac{\alpha(n-1)}{4}
\int_s^T E'(t) \int_{\Omega} \lambda(x)|\phi|^{2}\,dx\,dt$.
Then taking into consideration \eqref{e2.2},
\[
|I_{6}|\leq
\frac{\alpha(n-1)\|\lambda\|_{\infty}}{2}\int_s^T|E'(t)|E(t)dt\leq
\frac{\alpha(n-1)\|\lambda\|_{\infty}}{4} E(0)E(s).
\]
  Let  $ I_{7}=  \frac{\alpha(n-1)}{2}
\int_s^T E(t) \int_{\omega} \phi \nabla \psi \,dx\,dt$.
 Then taking into consideration \eqref{e2.2},
\eqref{e4.1} and Young's inequality produces
\begin{align*}
|I_{7}| &\leq \frac{\alpha^{2}(n-1)^{2}}{16\epsilon}
 \int_s^TE(t)\int_{\Omega}|\phi|^{2}\,dx\,dt
+2\epsilon \int_s^T E^{2}(t)dt \\
&\leq \frac{(n-1)^{2}}{8\epsilon \lambda_0} E(0)E(s)+2\epsilon \int_s^T E^{2}(t)dt.
\end{align*}
  Let  $ I_{8}=  \frac{1}{2} \int_s^T
E'(t) \int_{\omega}\phi |\psi|^{2}\,dx\,dt$.
 Then from relation \eqref{e2.2} and Young's inequality we obtain
\[
|I_{8}|\leq \frac{c_1}{2}\int_s^T|E'(t)|\int_{\Omega}|\phi| |\nabla \psi|^{2}\,dx\,dt\leq \frac{c_1(\kappa+1)}{4}
E(0)E(s).
\]
  Let  $ I_{9}=  \frac{1}{2} \int_s^T
E(t) \int_{\omega}\phi_{t} |\psi|^{2}\,dx\,dt$. \ Then taking into
consideration \eqref{e2.2} and Young's inequality we obtain
\begin{align*}
|I_{9}| &\leq \frac{c_1^{2}}{16\epsilon}\int_s^TE(t)\int_{\Omega}|
 \nabla \psi|^{2}\,dx\,dt+2\epsilon \int_s^TE^{2}(t)dt \\
&  \leq \frac{c_1^{2}}{32\epsilon \beta} E(0)E(s)+2\epsilon \int_s^TE^{2}(t)dt.
\end{align*}
  Let  $ I_{10}= -\alpha \int_s^T E(t)
\int_{\Omega}(F(x)\cdot \nabla \psi) (m\cdot \nabla \phi)\,dx\,dt$.
Then using \eqref{e2.2}, \eqref{e4.1} and Young's inequality we obtain
\begin{align*}
|I_{10}| &\leq \frac{\alpha^{2} R^{2}M^{2}}{4\epsilon}
\int_s^TE(t)\int_{\Omega}|\nabla \psi|^{2}\,dx\,dt
+2\epsilon \int_s^TE^{2}(t)dt  \\
&\leq \frac{\alpha^{2}R^{2}M^{2}}{8\epsilon \beta } E(0)E(s)
+2\epsilon \int_s^TE^{2}(t)dt.
\end{align*}
  Let  $ I_{11}=  \frac{\alpha(n-1)}{2}
\int_s^T E'(t) \int_{\omega}\phi_{t}  \phi \,dx\,dt$.
Then relation \eqref{e2.2}  implies
\begin{equation*}
|I_{11}|\leq \frac{\alpha(n-1)}{2}\int_s^T|E'(t)|E(t) dt \leq
\frac{\alpha(n-1)}{4} E(0)E(s).
\end{equation*}
Hence, the following inequality
holds, with  $\epsilon= 1/12$,
\begin{equation}
\label{e4.10}
\frac{2}{12}\int_s^TE^{2}(t)dt\leq
\frac{1}{2}\int_s^TE(t)\int_{\Gamma(x^0)}(m\cdot n) \Big(
\frac{\partial \phi}{\partial n}\Big)^{2}d\Gamma dt
+\frac{1}{\alpha}|\chi|+C_0E(s),
\end{equation}
  where
\begin{align*}
C_0&=\Big[\frac{\alpha(n-1)^{2}+8\alpha^{3}R^{2}\|\lambda\|_{\infty}
 \lambda_0+6\alpha R^{2}+(n-2)}{\lambda_0}
+\frac{3c_1^{2}+12\alpha^{2}R^{2}M^{2}}{8\beta}\\
&\quad +\frac{2\alpha(n-1)
 (1+\|\lambda\|_{\infty})+2c_1(\kappa+1)+1+4\alpha R}{8}\Big]E(0).
\end{align*}
\end{proof}

\begin{lemma}\label{lem4.3}
 Let the assumptions in Lemma \ref{lem4.2} and
Assumption \ref{assum1.4} hold.
Then there exists  $T_0>0$  such that if  $T>T_0$, we have
\begin{align*}
&\frac{1}{24}\int_s^T E^{2}(t) dt \\
&\leq \frac{1}{\alpha}|\chi|+ \frac{R}{\alpha}|Y|+(C_0+\frac{R C_1}{\alpha})E(s)
+\frac{3}{2}\|h|_{W^{1,\infty}}R\int_s^T E(t)
 \int_{\hat{\omega}}|\nabla \phi|^{2}\,dx\,dt
\end{align*}
where
\[
Y:=\frac{1}{4}\Big[  E(t)\Big(\int_{\Omega}
\big(|\psi|^{2}+\kappa|\nabla \psi|^{2}+4\alpha \phi_{t}(h \cdot\nabla \phi)\big)dx
+\int_{\hat{\omega}} \phi |\psi|^{2}dx \Big) \Big]_s^T
\]
and  $C_1$  depends on  $\alpha, \beta, \lambda, R, M, \kappa, h, c_1, c_2$.
\end{lemma}

 \begin{proof}
For  $q=h$  expression \eqref{e4.5} becomes
\begin{equation} \label{e4.11}
\begin{split}
&Y-\alpha \int_s^T
E'(t)\int_{\hat{\omega}} \phi_{t}(h\cdot \nabla \phi)\,dx\,dt
 +\frac{\alpha}{2}\int_s^T E(t) \int_{\hat{\omega}}
\operatorname{div} h [|\phi_{t}|^{2}-|\nabla \phi|^{2}]\,dx\,dt
\\
&+\alpha \int_s^T E(t) \int_{\hat{\omega}} \frac{\partial \phi}{\partial x_{i}}
 \frac{\partial h_{k}}{\partial x_{i}} \frac{\partial \phi}{\partial x_{k}} \,dx\,dt
 -\frac{\kappa}{4}\int_s^T E'(t) \int_{\Omega} |\nabla \psi|^{2}\,dx\,dt
 \\
&+\alpha \int_s^T E(t) \int_{\hat{\omega}} \phi (h\cdot \nabla \phi)\,dx\,dt
 +\alpha \int_s^T E(t) \int_{\hat{\omega}} \lambda(x) \phi_{t}(h\cdot
 \nabla \phi)\,dx\,dt
\\
& +\frac{\alpha}{2}\int_s^TE(t) \int_{\Omega} |\psi|^{2}\,dx\,dt
 +\frac{\kappa \alpha}{2} \int_s^T E(t) \int_{\Omega} |\nabla \psi|^{2}\,dx\,dt
\\
&-\frac{1}{4}\int_s^T E'(t)\int_{\Omega}|\psi|^{2}\,dx\,dt
 +\frac{\alpha}{2} \int_s^T E(t)\int_{\omega} \phi |\psi|^{2}\,dx\,dt
\\
&-\frac{1}{2}\int_s^T E'(t)\int_{\omega}\phi |\psi|^{2}\,dx\,dt- \frac{1}{2}\int_s^T
E(t) \int_{\omega} \phi_{t}|\psi|^{2} \,dx\,dt
\\
&+\alpha \int_s^T E(t) \int_{\hat{\omega}}
(F(x)\cdot \nabla \psi) (h \cdot \nabla \phi) \,dx\,dt
\\
&\geq \frac{\alpha}{2} \int_s^T E(t)\int_{\Gamma(x^0)} (h \cdot n)
\Big(\frac{\partial \phi}{\partial n}\Big)^{2}d\Gamma dt.
\end{split}
\end{equation}
  Evaluating some of the terms in the previous expression, we have
\begin{gather*}
\begin{aligned}
&\Big| \frac{\alpha}{2}\int_s^T E(t) \int_{\hat{\omega}} \operatorname{div}
h[|\phi_{t}|^{2} -|\nabla \phi|^{2}]\,dx\,dt\Big|\\
&\leq \frac{\alpha \|h\|_{W^{1,\infty}}}{2\lambda_0}E(0)E(s)
+\frac{\alpha \|h\|_{W^{1,\infty}}}{2}\int_s^T E(t)
\int_{\hat{\omega}}|\nabla \phi|^{2}\,dx\,dt,
\end{aligned}
\\
\Big|\alpha \int_s^T E(t) \int_{\hat{\omega}} \frac{\partial
\phi}{\partial x_{i}} \frac{\partial h_{k}}{\partial x_{i}}
\frac{\partial \phi}{\partial x_{k}} \,dx\,dt\Big|
\leq \alpha \|h\|_{W^{1,\infty}}\int_s^T E(t) \int_{\hat{\omega}}
|\nabla \phi|^{2}\,dx\,dt,
\\
\Big|\frac{\kappa}{4} \int_s^T E'(t)
\int_{\Omega} |\nabla \psi|^{2}-|\psi|^{2}\,dx\,dt\Big|
\leq \frac{\kappa+1}{4}E(0)E(s),
\\
\Big|\alpha \int_s^T E'(t)\int_{\Omega} \phi_{t}(h\cdot
\nabla \phi)\,dx\,dt\Big|\leq  \frac{\alpha\|h\|_{W^{1,\infty}}}{2}E(0)E(s),
\\
 \Big|\alpha \int_s^T E(t) \int_{\hat{\omega}}  \lambda(x)
\phi_{t}(h\cdot \nabla \phi)\,dx\,dt \Big|
 \leq  \frac{\alpha^{2}\|h\|^{2}_{W^{1,\infty}}
\|\lambda\|_{\infty}}{6\epsilon}E(0)E(s)+2\epsilon \int_s^T E^{2}(t) dt,
\\
\Big| \frac{\kappa \alpha}{2} \int_s^T E(t)
\int_{\Omega} |\nabla \psi|^{2}\,dx\,dt \Big|
\leq \frac{\kappa \alpha}{4\beta}E(0)E(s),
\\
\Big| \frac{\alpha}{2} \int_s^T E(t)\int_{\omega} \phi
|\psi|^{2}\,dx\,dt\Big|
 \leq  \frac{a^{2}c_1^{2}}{32\epsilon \beta}E(0)E(s)+2\epsilon \int_s^T E^{2}(t) dt,
\\
\Big|\frac{1}{4}\int_s^T E'(t)\int_{\Omega}|\psi|^{2}\,dx\,dt\Big|
\leq  \frac{1}{4} E(0)E(s),
\\
\Big|\frac{1}{2}\int_s^T E(t) \int_{\omega}
\phi_{t}|\psi|^{2} \,dx\,dt\Big|
\leq  \frac{c_1^{4}}{8\epsilon \beta}E(0)E(s)+2\epsilon \int_s^T E^{2}(t) dt,
\end{gather*}
 and finally
\begin{gather*}
\begin{aligned}
&\Big|\alpha  \int_s^T E(t) \int_{\Omega} (F(x)\cdot \nabla
\psi) (h \cdot \nabla \phi)\,dx\,dt\Big|\\
&\leq \frac{a^{2}\|h\|^{2}_{W^{1,\infty}}M^{2}}{8\epsilon \beta}E(0)E(s)
+2\epsilon \int_s^T E^{2}(t) dt,
\end{aligned}\\
\Big|\alpha \int_s^T E(t) \int_{\Omega} \phi (h\cdot \nabla
\phi)\,dx\,dt\Big|
\leq  \frac{\|h\|^{2}_{W^{1,\infty}}}{4\epsilon\lambda_0}E(0)E(s)
+2\epsilon \int_s^TE^{2}(t)dt,
\\
\begin{aligned}
&\Big|\frac{1}{2}\int_s^T E'(t)\int_{\omega}\phi |\psi|^{2}\,dx\,dt\Big|
\leq  \frac{1}{4}\int_s^T E'(t)\int_{\Omega}|\phi|^{2} \,dx\,dt
+\int_s^T E'(t)\int_{\Omega}|\psi|^{4}\,dx\,dt\\
&\leq  \frac{1+c_2^{2}}{4} E(0)E(s).
\end{aligned}
\end{gather*}
Taking into consideration the evaluations above and substituting
them into \eqref{e4.11}, we obtain
\begin{equation} \label{e4.13}
\begin{split}
\frac{\alpha R}{2} \int_s^T E(t)
\int_{\Gamma(x^0)} \Big(\frac{\partial \phi}{\partial n}\Big)^{2}d\Gamma dt
&\leq R|Y|+10\epsilon R\int_s^T E^{2}(t) dt+R C_1E(s)\\
&\quad + \frac{3\alpha \|h\|_{W^{1,\infty}}R}{2}\int_s^T E(t) \int_{\hat{\omega}}
|\nabla \phi|^{2}\,dx\,dt,
\end{split}
\end{equation}
 where
\begin{align*}
C_1&:=\Big[ \frac{\kappa +3+c_2^{2}+2\alpha \|h\|_{W^{1,\infty}} }{4}
 +\frac{\alpha^{2}c_1^{2}+4\alpha^{2}\|h\|^{2}_{W^{1,\infty}}M^{2}}{32\epsilon \beta}
+ \frac{\alpha \|h|_{W^{1,\infty}}}{2\lambda_0}\\
&\quad +\frac{\|h\|^{2}_{W^{1,\infty}}}{4\epsilon \lambda_0}
 +\frac{\kappa \alpha}{4\beta}+\frac{\alpha^{2}\|h|^{2}_{W^{1,\infty}}
 \|\lambda\|_{\infty}}{6\epsilon}\Big]E(0).
\end{align*}
Combining \eqref{e4.10} and \eqref{e4.13} and choosing
$\epsilon= \frac{\alpha}{80R}$, we deduce
\begin{equation} \label{e4.14}
\begin{split}
\frac{1}{24}\int_s^T E^{2}(t) dt
&\leq \frac{1}{\alpha}|\chi|+ \frac{R}{\alpha}|Y|+(C_0+\frac{R C_1}{\alpha})E(s)\\
&\quad +\frac{3}{2}\|h|_{W^{1,\infty}}R\int_s^T E(t)
\int_{\hat{\omega}}|\nabla \phi|^{2}\,dx\,dt,
\end{split}
\end{equation}
which completes the proof of the Lemma.
\end{proof}

To evaluate  $ \int_s^T E(t) \int_{\hat{\omega}}|\nabla \phi|^{2}\,dx\,dt$,
 we construct a function  $\eta  \in W^{1,\infty}(\Omega)$  such that
\begin{gather}
\label{e4.15}
0\leq \eta \leq 1,  \quad\text{ a.e. in $\Omega$, with $\eta=1$, a.e.
 in $\hat{\omega}$},\\
\label{e4.16}
\eta=0, \quad \text{a.e. in } \Omega\backslash \omega,\\
\label{e4.17}
 \frac{|\nabla \eta|^{2}}{\eta}  \in L^{\infty}(\omega).
\end{gather}
Finally,  we state and prove the main result of the present work.

\begin{theorem} \label{them1.1}
 Let Assumptions \ref{assum1.1} and \ref{assum1.2} hold and  $\beta> 0$.
Then there exists some positive constant  $C=C(E(0))$  such that the following
decay rate holds for each solution  $(\psi, \phi, \phi_{t})$  of
\eqref{e1.1d}- \eqref{e1.4d},
\begin{equation}
\label{e4.2} E(t)\leq \frac{CE(0)}{1+t}, \quad \text{for all } t\geq 0.
\end{equation}
\end{theorem}

\begin{proof}
To prove the Theorem it is sufficient to prove that
\begin{equation*}
 \int_s^T E^{2}(t) dt\leq C E(S), \quad \text{for all }  0\leq s <T<+\infty,
\end{equation*}
 for some positive constant  $C$   independent of $T$.
Letting $\xi=\eta$ in \eqref{e4.7} we obtain
\begin{equation}  \label{e4.18}
\begin{split}
&\Big[ \alpha E(t)\int_{\omega} \phi \eta
\Big(\phi_{t} + \frac{\lambda(x)\phi}{2}\Big)dx\Big]_s^T
- \alpha \int_s^TE(t)\int_{\omega} \eta |\phi_{t}|^{2}\,dx\,dt
\\
&+ \alpha \int_s^TE(t)\int_{\omega} \phi (\nabla \eta \cdot \nabla \phi)\,dx\,dt
+ \alpha \int_s^TE(t)\int_{\omega}\eta |\nabla \phi|^{2}\,dx\,dt\\
&+ \alpha \int_s^TE(t) \int_{\omega}\eta |\phi|^{2}\,dx\,dt
 + \frac{\alpha}{2}\int_s^TE'(t)\int_{\omega}\eta \lambda(x)|\phi|^{2}\,dx\,dt
\\
&=- \alpha \int_s^TE(t)\int_{\omega}\eta \phi \nabla \psi \,dx\,dt
 + \alpha \int_s^TE'(t)\int_{\omega}\eta \phi_{t} \phi \,dx\,dt.
\end{split}
\end{equation}
  Evaluating the integrals above produces
\begin{equation} \label{e4.19}
\begin{split}
&\alpha \int_s^TE(t)\int_{\omega} \eta |\nabla \phi|^{2}\,dx\,dt\\
&\leq |Z|+C_2E(s) +2\epsilon \int_s^T E^{2}(t) dt
 +\frac{\alpha}{2} \int_s^TE(t)\int_{\omega}\eta |\nabla \phi|^{2}\,dx\,dt,
\end{split}
\end{equation}
where
\begin{gather*}
Z:=\Big[ \alpha E(t)\int_{\omega} \phi \eta
\Big(\phi_{t} + \frac{\lambda(x)\phi}{2}\Big)dx \Big]_s^T,
\\
C_2:=\Big[\frac{\alpha \|\lambda\|_{\infty}}{2}
+\|\frac{|\nabla \eta|^{2}}{\eta}\|_{\infty}
+ \frac{1}{2\alpha \lambda_0}+\frac{1}{\alpha \lambda_0}
+\frac{\alpha^{2}}{8\epsilon \beta}+\frac{4\alpha}{3\lambda_0}
+\frac{\alpha}{2}\Big]E(0).
\end{gather*}
Therefore combining \eqref{e4.14} and \eqref{e4.19}, choosing
 $\epsilon=\frac{1}{288\|h\|_{\infty}R}$  and taking into consideration
\[
\int_s^TE(t)\int_{\hat{\omega}} \eta |\nabla \phi|^{2}\,dx\,dt
= \int_s^TE(t)\int_{\hat{\omega}}  |\nabla \phi|^{2}\,dx\,dt,
\]
  we obtain
\begin{equation} \label{e3.20}
\frac{1}{48}\int_s^T E^{2}(t) dt \leq \frac{1}{\alpha}|\chi|
+ \frac{R}{\alpha}|Y|+3\|h\|_{\infty}R|Z|
+(C_0+\frac{R C_1}{\alpha}+\|h\|_{\infty}RC_2)E(s).
\end{equation}
Note that the following estimate holds
\[
\frac{1}{\alpha}|\chi|+ \frac{R}{\alpha}|Y|+3\|h\|_{\infty}R|Z| \leq C_3E(0)E(s),
\]
 where  $C_3=C_3(R, \alpha, \kappa, \delta, \lambda_0,
 c_1, \|h\|_{\infty}, \|\lambda\|_{\infty})$.  Then
\[
\int_s^T E^{2}(t) dt\leq C E(0)E(s),
\]
where  $C=C(R, \alpha, \kappa, \delta, n,  c_1,\beta, \|h\|_{\infty},
 \|\lambda\|_{\infty})$ \ is independent of $T$.
Then employing Lemma \ref{lem2.1} we deduce the desired decay rate.
\end{proof}

\subsection*{Acknowledgments} \
 This work was financially supported by a grant No 65/1790 from
the Basic Research Committee of the National Technical University
of Athens, Greece.

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 \end{document}