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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 202, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/202\hfil Boundary stabilization]
{General boundary stabilization result of memory-type thermoelasticity
 with second sound}

\author[F. Boulanouar, S. Drabla \hfil EJDE-2014/202\hfilneg]
{Fairouz Boulanouar, Salah Drabla}  % in alphabetical order

\address{Fairouz Boulanouar  \newline
Department of Mathematics, Faculty of Sciences,
University Farhat Abbas of Setif1, Setif 19000, Algeria}
\email{boulanoir\_b@yahoo.com}

\address{Salah Drabla \newline
Department of Mathematics, Faculty of Sciences,
University Farhat Abbas of Setif1, Setif 19000, Algeria}
\email{drabla\_s@univ-setif.dz}

\thanks{Submitted August 1, 2014. Published September 30, 2014.}
\subjclass[2000]{35B37, 35L55, 74D05, 93D15, 93D20}
\keywords{Thermoelasticity with second sound; viscoelastic damping;
\hfill\break\indent  decay; relaxation function; boundary stabilization; convexity}

\begin{abstract}
 In this article we consider an n-dimensional system of visco-ther\-moelasticity
 with second sound, where a viscoelastic dissipation is acting on a part of
 the  boundary. We prove an explicit general decay rate result without
 imposing $u_0=0$ as in \cite{Mus}. This allows a larger class of
 relaxation functions and initial data, hence, generalizes some previous
 results existing in the literature.
\end{abstract}

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


\section{Introduction}

The Classical Fourier law of heat conduction expresses that the heat flux
within a medium is proportional to the local temperature gradient in the
system. A well known consequence of this law is that heat perturbations
propagates with an infinite speed. Experiments showed that heat conduction
in some dielectric crystals at low temperatures is free of this paradox and
disturbances, which are almost entirely thermal, propagate in a finite
speed. This phenomenon in dielectric crystals is called second sound (see
\cite{Coleman}). To overcome this physical paradox, Maxwell \cite{Maxwell},
Cattaneo \cite{Cattenao} adopted a non-classical heat flux Maxwell-Cattaneo
law to get rid of this unphysical results. This Maxwell-Cattaneo relation
contains an extra inertial term with respect to the Fourier law
\[
\tau _0q_{t}+q+\kappa \nabla \theta =0,
\]
where $q=q(x,t) \in \mathbb{R}^n$ is the heat flux vector,
$\tau _0$ is the relaxation time and $\kappa $ is the heat conductivity.
The conservation of energy equation introduces the hyperbolic equation,
which describes heat propagation with finite speed.

Result concerning existence, blow up, and asymptotic behavior of smooth, as
well as weak solutions in thermoelasticity with second sound have been
established over the past two decades by many mathematicians. Tarabek
\cite{Tara} treated problems related to
\begin{equation}
\begin{gathered}
u_{tt}-a(u_x,\theta ,q) u_{xx}+b(u_x,\theta ,q)
\theta _x=\alpha _1(u_x,\theta ) qq_x \\
\theta _{t}+g(u_x,\theta ,q) q_x+d(u_x,\theta
,q) u_{tx}=\alpha _2(u_x,\theta ) qq_{t} \\
\tau (u_x,\theta ) q_{t}+q+k(u_x,\theta ) \theta_x=0,
\end{gathered} \label{1.1}
\end{equation}
in both bounded and unbounded situations and established global existence
results for small initial data. He also showed that these
`` classical'' solutions tend to
equilibrium as $t$ tends to infinity; however, no rate of decay has been
discussed. In his work, Tarabek used the usual energy argument and exploited
some relations from the second law of thermodynamics to overcome the
difficulty arising from the lack of Poincar\'{e}'s inequality in the
unbounded domains. Racke \cite{RacT} discussed \eqref{1.1} and established
exponential decay results for several linear and nonlinear initial boundary
value problems. In particular he studied \eqref{1.1}, with
$\alpha_1=\alpha _2=0$, and for a rigidly clamped medium with temperature hold
constant on the boundary. i.e
\[
u(t,0) =u(t,1) =0,\quad \theta (t,0)
=\theta (t,1) =\bar{\theta },\quad t\geq 0,
\]
and showed that, for small enough initial data and classical solutions decay
exponentially to the equilibrium state. Messaoudi and Said-Houari
 \cite{MSE} extended the decay result of\ \cite{RacT} to the case when
$\alpha_1\neq 0$, $\alpha _2\neq 0$.

For the multi-dimensional case ($n=2, 3$), Racke \cite{RacA} established an
existence result for the following $n$-dimensional problem
\begin{equation}
\begin{gathered}
u_{tt}-\mu \Delta u-(\mu +\lambda ) \nabla (
\operatorname{div}u) +\beta \nabla \theta =0 \quad \text{in }\Omega \times (
0,+\infty ) \\
\theta _{t}+\gamma \operatorname{div}q+\delta \operatorname{div}u_{t}=0 \quad
\text{in }\Omega \times (0,+\infty ) \\
\tau q_{t}+q+\kappa \nabla \theta =0 \quad \text{in }\Omega \times (
0,+\infty ) \\
u(.,0) =u_0,\quad u_{t}(.,0) =u_1,\quad \theta
(.,0) =\theta _0,\quad q(.,0) =q_0 \quad \text{in }\Omega \\
u=\theta =0 \quad \text{on }\Gamma \times [ 0,+\infty ) ,
\end{gathered} \label{1.2}
\end{equation}
where $\Omega $ is a bounded domain of $\mathbb{R}^n$,
with a smooth boundary $\Gamma$,
$u=u(x,t)$, $q=q(x,t) \in \mathbb{R}^n$, and
 $\mu , \lambda , \beta , \gamma ,\delta , \tau , \kappa$,
are positive constants, where $\mu , \lambda $
are Lame moduli and $\tau $ is the relaxation time, a small parameter
compared to the others. In particular if $\tau =0$, \eqref{1.2} reduces to
the system of classical thermoelasticity, in which the heat flux is given by
Fourier's law instead of Cattaneo's law. He also proved, under the
conditions $rotu=rotq=0$, an exponential decay result for \eqref{1.2}. This
result applies automatically to the radially symmetric solution, since it is
only a special case.

Messaoudi \cite{MLE} considered \eqref{1.2}, in the presence
of a source term, and proved a blow up result for solutions with negative
initial energy. This result was extended later to certain solutions with
positive energy by Messaoudi and Said-Houari \cite{MSB}.

In this article, we are concerned with the  system
\begin{equation}
\begin{gathered}
u_{tt}-\mu \Delta u-(\mu +\lambda ) \nabla (
\operatorname{div}u) +\beta \nabla \theta =0 \quad
 \text{in }\Omega \times (0,+\infty ) \\
c\theta _{t}+\kappa \operatorname{div}q+\beta \operatorname{div}u_{t}=0 \quad
 \text{in }\Omega \times (0,+\infty ) \\
\tau _0q_{t}+q+\kappa \nabla \theta =0 \quad
\text{in }\Omega \times (0,+\infty ) \\
u(.,0) =u_0,\quad u_{t}(.,0) =u_1,\quad \theta
(.,0) =\theta _0,\quad q(.,0) =q_0 \quad \text{in }\Omega \\
u=0 \quad \text{on }\Gamma _0\times [ 0,+\infty ) \\
u(x,t) =-\int_0^{t}g(t-s)(\mu \frac{\partial u}{
\partial \nu }+(\mu +\lambda ) (\operatorname{div}u) \nu )
(s) ds \quad \text{on }\Gamma _1\times [ 0,+\infty )
\\
\theta =0 \quad \text{on }\Gamma \times [ 0,+\infty ) ,
\end{gathered}  \label{1.3}
\end{equation}
which models the transverse vibration of a thin elastic body, taking in
account the heat conduction given by Cattaneo's law. Here, $\Omega $ is a
bounded domain of $\mathbb{R}^n$ $(n\geq 2) $ with a smooth boundary
$\Gamma$, such that $
\{ \Gamma _0,\Gamma _1\}$ is a partition of $\Gamma , \nu $ is the outward
normal to $\Gamma $, $u=u(x,t) \in \mathbb{R}^n$ is the displacement vector,
$\ q=q(x,t) \in \mathbb{R}^n$ is the heat flux vector,
$\theta =\theta (x,t)$ is the difference temperature and the relaxation
function $g$ is a positive differentiable function. The coefficients
$c,\kappa ,\beta ,\mu ,\lambda $, $\tau _0$ are positive constants, where
$\mu , \lambda $ are Lame moduli and $\tau _0$ is the relaxation time,
a small parameter compared to the others. The boundary condition on
$\Gamma _1$ is the nonlocal boundary condition responsible for the memory effect.

Messaoudi and Al-Shehri treated system \eqref{1.3} of
thermoelasticity with second sound in \cite{MAZ} subject to boundary
condition of memory type. If $k$ is the resolvent kernel of
$-g'/g(0)$, they showed in \cite{MA} that the energy decays at the same rate
as of $(-k') $, while in \cite{MAZ}, when $(-k') $ decays exponentially,
the energy decays at a polynomial rate. Recently, Mustafa \cite{Mus}
treated system \eqref{1.3} for $k$ satisfying
\begin{gather}
k(0)>0,\quad \lim_{t\to \infty }k(t)=0,\quad k'(t)\leq 0, \label{k1}\\
k''(t)\geq H(-k'(t)),\quad \forall t>0,  \label{k2}
\end{gather}
where $H$ is a positive function, which is linear or strictly increasing,
strictly convex of class $C^2$ on $(0,r]$, $r<1$, and $H(0)=0$ and proved
for $u_0=0$ on $\Gamma _1$, an explicit energy decay formula which is
not necessarily of exponential or polynomial-type decay.

Our aim in this work is to investigate \eqref{1.3}  for
resolvent kernels satisfying \eqref{k1}  and \eqref{k2},
  when $u_0\neq 0$ on $\Gamma _1$ is taken into account.
The proof is based on the multiplier method and makes use of some estimates
of \cite{Mus} with the necessary modification needed to obtain our result.
The paper is organized as follows. In section 2, we present some notations
and material needed for our work. In section 3, we establish some technical
lemmas and state our main theorem, while the proof of our main result will
be given in section 4.

\section{Notation and transformation}

In this section we introduce some notation and prove some lemmas.
To establish our result, we shall make the following assumption:
\begin{itemize}
\item[(A1)] The partition $\Gamma _0$ and $\Gamma _1$ are closed,
disjoint, with $\operatorname{meas}(\Gamma _0) >0$ and satisfying
\[
\Gamma _1=\{ x\in \Gamma : m(x) .\nu \geq \delta>0\},\quad
\Gamma _0=\{ x\in \Gamma : m(x) .\nu \leq 0\} ,
\]
where $m(x) =x-x_0$, for some $x_0\in \mathbb{R}^n$.
\end{itemize}
Similarly to \cite{MS8, MA, MS10}, applying Volterra's inverse operator, the
boundary condition
\begin{equation}
u(x,t) =-\int_0^{t}g(t-s) \Big(\mu \frac{
\partial u}{\partial \nu }+(\mu +\lambda ) (\operatorname{div}u)
\nu \Big) (s) ds,\quad \text{on }\Gamma _1\times [0,+\infty ),  \label{2.1}
\end{equation}
can be transformed into
\[
\mu \frac{\partial u}{\partial \nu }+(\mu +\lambda ) (
\operatorname{div}u) \nu =-\frac{1}{g(0) }(u_{t}+k\ast
u_{t}) ,\quad \text{on }\Gamma _1\times [ 0,+\infty ),
\]
where * denotes the convolution product
\[
(\varphi \ast \psi )(t)=\int_0^{t}\varphi (t-s)\psi (s)\,ds,
\]
and $k$ is the resolvent kernel of $(-g'/g(0))$ which satisfies
\[
k+\frac{1}{g(0) }(g'\ast k) =-\frac{1}{g(0) }g'.
\]

Taking $\ \eta =1/g(0)$, we arrive at
\begin{equation}
\mu \frac{\partial u}{\partial \nu }+(\mu +\lambda ) (
\operatorname{div}u) \nu
=-\eta (u_{t}+k(0)u-k(t)u_0+k'\ast u),\quad\text{on }
\Gamma _1\times [ 0,+\infty ).  \label{2.2}
\end{equation}
Then, we will use the boundary relation \eqref{2.2}  instead of
the fourth equation in \eqref{1.3}.
Let us define
\begin{gather*}
(\varphi \circ \psi ) (t) =\int_0^{t}\varphi
(t-s)| \psi (t) -\psi (s) | ^2ds,
\\
(\varphi \Diamond \psi ) (t) =\int_0^{t}\varphi
(t-s)(\psi (t) -\psi (s) ) ds.
\end{gather*}
By using H\^{o}lder's inequality, we have
\begin{equation}
| (\varphi \Diamond \psi ) (t) |^2
\leq \Big(\int_0^{t}| \varphi (s)| ds\Big)(| \varphi | \circ \psi ) (t).
\label{1.5}
\end{equation}

\begin{lemma}[\cite{Mag}]
If $\varphi, \psi \in C^{1}(\mathbb{R}^{+})$, then
\begin{equation}
(\varphi \ast \psi ) \psi _{t}=-\frac{1}{2}\varphi (t)|
\psi (t)| ^2+\frac{1}{2}\varphi '\circ \psi
 -\frac{1}{2} \frac{d}{dt}\Big(\varphi \circ \psi -\Big(\int_0^{t}\varphi
(s)ds\Big) | \psi (t)| ^2\Big) .  \label{1.6}
\end{equation}
\end{lemma}

Let us define
\[
V=\{ v\in (H^{1}(\Omega ) ) : v=0\quad \text{on } \Gamma _0\} .
\]
The well-posedness of system \eqref{1.3}  is presented in the
following theorem, which can be proved, using the Galerkin method as in
 \cite{AND,CAVD} and the reference therein.

\begin{theorem}
Let $k\in W^{2,1}(\mathbb{R}^{+}) \cap W^{1,\infty }(\mathbb{R}^{+})$,
$u_0\in ((H^2(\Omega ) )\cap V) ^n$,
$\theta _0\in H_0^{1}(\Omega )$,
$q_0\in (H^{1}(\Omega ) ) ^n$, and $u_1\in V^n$, with
\begin{equation}
\frac{\partial u_0}{\partial \nu }+\eta u_0=0\quad \text{on  }\Gamma_1.  \label{1.7}
\end{equation}
Then there exists a unique strong solution $u$ of system \eqref{1.3},
such that
\begin{gather*}
u \in C(\mathbb{R}^{+};(H^2(\Omega ) \cap
V) ^n) \cap C^{1}(\mathbb{R}^{+};V^n) , \\
\theta \in C(\mathbb{R}^{+};H_0^{1}(\Omega ) )
\cap C^{1}(\mathbb{R}^{+};L^2(\Omega ) ) , \\
q \in C(\mathbb{R}^{+};(H^{1}(\Omega ) )
^n) \cap C^{1}(\mathbb{R}^{+};(L^2(\Omega) ) ^n) .
\end{gather*}
\end{theorem}

\section{Decay of solutions}

In this section we discuss the asymptotic behavior of the solutions of system
\eqref{1.3} when the resolvent kernel $k$ satisfies the assumption
\begin{itemize}
\item[(B1)] $k:\mathbb{R}^{+}\to \mathbb{R}^{+}$ is a $C^2$
function such that
\[
k(0)>0,\quad \lim_{t\to \infty }k(t)=0,\quad k'(t)\leq 0,
\]
and there exists a positive function $H\in C^{1}(\mathbb{R}^{+})$, with
$H(0)=0$, and $H$ is linear or strictly increasing and strictly convex $C^2$
function on $(0,r]$, $r<1$, such that
\[
k''(t)\geq H(-k'(t)),\quad \forall t>0.
\]
\end{itemize}

It is a routine procedure to define the first-order energy of system
\eqref{1.3}  by (see Lemma \ref{lem3.2} below).
\begin{equation}
\begin{aligned}
E_1(t) &= \frac{1}{2}\int_{\Omega }\big[ |
u_{t}| ^2+\mu | \nabla u| ^2+(\mu +\lambda ) (\operatorname{div}u) ^2
+c\theta ^2+\tau _0q^2\big] dx   \\
&\quad -\frac{\eta }{2}\int_{\Gamma _1}(k'\circ u)(t)d\Gamma _1+
\frac{\eta }{2}\int_{\Gamma _1}k(t) | u|^2d\Gamma _1.
\end{aligned} \label{3.3}
\end{equation}
Now, we differentiate \eqref{1.3}, with respect to $t$, to obtain
\begin{equation}
\begin{gathered}
u_{ttt}-\mu \Delta u_{t}-(\mu +\lambda ) \nabla (
\operatorname{div}u_{t}) +\beta \nabla \theta _{t}=0 \quad \text{in }\Omega \times
(0,+\infty ) \\
c\theta _{tt}+\kappa \operatorname{div}q_{t}+\beta \operatorname{div}u_{tt}=0
\quad \text{in }\Omega \times (0,+\infty ) \\
\tau _0q_{tt}+q_{t}+\kappa \nabla \theta _{t}=0 \quad \text{in }\Omega
\times (0,+\infty )
\end{gathered}  \label{3.4}
\end{equation}
and the boundary condition \eqref{2.2}  to obtain
\begin{equation}
\mu \frac{\partial u_{t}}{\partial \nu }+(\mu +\lambda ) (
\operatorname{div}u_{t}) \nu =-\eta (u_{tt}+k(0)u_{t}+k'\ast
u_{t}) ,\quad\text{on }\Gamma _1\times \mathbb{R}^{+}.
\label{3.5}
\end{equation}
Consequently, similar computations yield the second-order energy of system
\eqref{1.3}:
\begin{align*}
E_2(t) &= \frac{1}{2}\int_{\Omega }\big[ |
u_{tt}| ^2+\mu | \nabla u_{t}| ^2+(\mu +\lambda ) (\operatorname{div}u_{t}) ^2
 +c\theta _{t}^2+\tau_0q_{t}^2\big] dx \\
&\quad -\frac{\eta }{2}\int_{\Gamma _1}(k'\circ u_{t})(t)d\Gamma _1+
\frac{\eta }{2}\int_{\Gamma _1}k(t) |u_{t}| ^2d\Gamma _1.
\end{align*}

\begin{theorem}\label{thm3.1}
Given $(u_0,u_1,\theta _0,q_0) \in (H^2(\Omega ) \cap V) ^n\times V^n\times
H_0^{1}(\Omega ) \times (H^{1}(\Omega )) ^n$, 
we assume that {\rm(A1)} and {\rm (B1)} hold. Then there exist positive
constants $c_1$, $c_2$, $k_1$, $k_2$, $k_3$, $\varepsilon _0$ and
 $t_1$ such that:

(I) In the special case $H(t) =ct^{p}$, where
$1\leq p<3/2$, the solution of \eqref{1.3} satisfies
\begin{equation}
E_1(t) \leq \Big(\frac{c_1+c_2\int_{t_1}^{t}[
k(s) \int_{\Gamma _1}| u_0| ^2d\Gamma
_1] ^{2p-1}ds}{t}\Big) ^{\frac{1}{2p-1}}-\frac{\eta }{2}\Big(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big)
\int_{t}^{\infty }k^2(s) ds  \label{solution1}
\end{equation}
for all $t\geq t_1$.

(II) In the general case, the solution of \eqref{1.3}
satisfies
\begin{equation} \label{solution1b}
\begin{aligned}
E_1(t) &\leq k_1H_1^{-1}\Big(\frac{k_2+k_3(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_{t_1}^{t}H_0(k(s) ) ds}{t}\Big) \\
&\quad -\frac{\eta }{2}\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma
_1\Big) \int_{t}^{\infty }k^2(s) ds\quad \text{for all } t\geq t_1,
\end{aligned}
\end{equation}
where
\[
H_1(t) =tH_0'(\varepsilon _0t) ,\quad  H_0(t) =H(D(t) ),
\]
provided that $D$ is a positive $C^{1}$ function, with
$D(0) =0$, for which $H_0$ is strictly increasing and strictly convex $C^2$
function on $(0,r]$ and
\begin{equation}
\int_0^{+\infty }\frac{-k'(s) }{H_0^{-1}(k''(s) ) }ds<+\infty .  \label{2.4}
\end{equation}
\end{theorem}

\subsection*{Remarks}
\textbf{1.} If $u_0=0$ on $\Gamma _1$, we then obtain the results in
Mustafa \cite{Mus}.

\textbf{2.} If $\int_0^{\infty }H_0(k(s) )ds<+\infty $, then \eqref{solution1b}
 reduces to
\[
E_1(t) \leq k_1H_1^{-1}(\frac{c}{t}) -\frac{
\eta }{2}\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma
_1\Big) \int_{t}^{\infty }k^2(s) ds,
\]
which clearly shows that $\lim_{t\to \infty}E_1(t) =0$, in this case.

\textbf{3.} The usual decay rate estimate, already proved for $k$ satisfying
$k''\geq d(-k')^{p}$, $1\leq p<3/2$, is a special
case of our result. We will provide a ``simpler'' proof for this special case.

\textbf{4.} The condition $k''\geq d(-k')^{p}$, $1\leq p<3/2$ assumes
$-k'(t)\leq \omega e^{-dt}$ when $p=1$ and $-k'(t)\leq \omega /t^{\frac{1}{p-1}}$
when $1<p<3/2$. Our result
allows resolvent kernels whose derivatives are not necessarily of
exponential or polynomial decay. For instance, if
\[
k'(t)=-\exp (-\sqrt{t}),
\]
then $k''(t)=H(-k'(t))$ where, for $t\in (0,r]$, $r<1 $,
\[
H(t)=\frac{t}{2\ln (1/t)}\,.
\]
Since
\[
H'(t)=\frac{1+\ln (1/t)}{2[\ln (1/t)]^2}\quad \text{and}\quad
H''(t)=\frac{\ln (1/t)+2}{2t[\ln (1/t)]^{3}},
\]
the function $H$ satisfies hypothesis (B1) on the interval $(0,r]$ for
any $0<r<1$. Also, by taking $D(t)=t^{\alpha }$, \eqref{2.4} is satisfied
for any $\alpha >1$. Therefore, an explicit rate of decay can be obtained by
Theorem \ref{thm3.1}. The function $H_0(t)=H(t^{\alpha })$ has
derivative
\[
H_0'(t)=\frac{\alpha t^{\alpha -1}[1+\ln (1/t^{\alpha })]}{2[\ln
(1/t^{\alpha })]^2}.
\]
Then, we do some direct calculations and use \eqref{solution1b} to deduce that
\begin{align*}
E_1(t)
&\leq k_1\Big(\frac{k_2+k_3\big(\int_{\Gamma _1}| u_0| ^2d\Gamma _1\big)
\int_{t_1}^{t}H_0(k(s) ) ds}{t}\Big) ^{1/(2\alpha )}\\
&\quad -\frac{\eta }{2}\Big(\int_{\Gamma _1}|
u_0| ^2d\Gamma _1\Big) \int_{t}^{\infty }k^2(
s) ds, \quad \forall t \geq t_1,
\end{align*}
for any $\alpha >1$, where
$H_0(k(s) ) =\frac{(k(s) ) ^{\alpha }}{2\ln (1/(k(s) ) ^{\alpha }) }$.
 Therefore, taking $\alpha \to 1$, the energy decays at the following rate
\begin{equation}
\begin{aligned}
E_1(t) &\leq k_1\Big(\frac{k_2+k_3(\int_{\Gamma_1}| u_0| ^2d\Gamma _1)
\int_{t_1}^{t}H(k(s) ) ds}{t}\Big) ^{1/2}\\
&\quad -\frac{\eta }{2}\Big(\int_{\Gamma _1}| u_0|
^2d\Gamma _1\Big) \int_{t}^{\infty }k^2(s) ds,
\quad \forall t \geq t_1.
\end{aligned}  \label{2.5}
\end{equation}
If $\int_0^{\infty }H(k(s) ) ds<+\infty $, then
equation \eqref{2.5} reduces to
\[
E_1(t) \leq \frac{C}{t^{\frac{1}{2}}}-\frac{\eta }{2}\Big(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big)
\int_{t}^{\infty }k^2(s) ds,\quad \forall t\geq t_1.
\]

\textbf{5.} The well-known Jensen's inequality will be of essential use in
establishing our main result. If $F$ is a convex function on
$[ a,b]$, $f:\Omega \to [ a,b] $ and $h$ are
integrable functions on $\Omega , h(x) \geq 0$, and
$\int_{\Omega }h(x)dx=k>0$, then Jensen's inequality states that
\[
F\Big[ \frac{1}{k}\int_{\Omega }f(x) h(x) dx\Big]
\leq \frac{1}{k}\int_{\Omega }F[ f(x)] h(x)dx.
\]

\textbf{6.} As in \cite{Mus} and since $\lim_{t\to \infty }k(t)=0$,
 $\lim_{t\to +\infty }(-k'(t) )$ cannot be equal to a positive number,
and so it is natural to assume that
$\lim_{t\to +\infty }(-k'(t) ) =0$, in the same way, we deduce that
$\lim_{t\to +\infty }k''(t) =0.$ Hence there is $t_1>0$ large enough such
that $k'(t_1)<0$ and
\begin{equation}
\max \{ k(t) ,-k'(t),k''(t)\}
<\min \{ r,H(r) ,H_0(r) \} ,\quad \forall t\geq t_1.  \label{2.6}
\end{equation}

As $k'$ is nondecreasing, $k'(0) <0$ and $k'(t_1) <0$, then $k'(t) <0$
for any $t\in [ 0,t_1] $ and
\[
0<-k'(t_1)\leq -k'(t)\leq -k'(0),\quad \forall t\in [ 0,t_1] .
\]
Therefore, since $H$ is a positive continuous function, we have
\[
a\leq H(-k'(t)) \leq b,\quad \forall t\in [ 0,t_1] ,
\]
for some positive constants $a$ and $b$. Consequently, for all
$t\in [0,t_1] $,
\[
k''(t) \geq H(-k'(t)) \geq a=\frac{a}{k'(0) }k'(0) \geq \frac{a}{k'(0) }k'(t)
\]
which gives
\begin{equation}
k''(t) \geq d(-k'(t)) ,\quad \forall t\in [ 0,t_1] ,  \label{2.7}
\end{equation}
for some positive constant $d$.

\begin{lemma} \label{lem3.2}
Under the assumptions of Theorem \ref{thm3.1}, the energies of the
solution of \eqref{1.3} satisfy
\begin{gather}
\begin{aligned}
E_1'(t)
&\leq -\int_{\Omega }| q|^2dx-\frac{\eta }{2}\int_{\Gamma _1}| u_{t}|
^2d\Gamma _1+\frac{\eta }{2}k'(t) \int_{\Gamma_1}| u| ^2d\Gamma _1   \\
&\quad -\frac{\eta }{2}\int_{\Gamma _1}(k''\circ u)
(t) d\Gamma _1+\frac{\eta }{2}k^2(t)
\int_{\Gamma _1}| u_0| ^2d\Gamma _1.
\end{aligned} \label{3.7}
\\
E_2'(t) \leq -\int_{\Omega }| q_{t}| ^2dx\leq 0.  \label{3.8}
\end{gather}
\end{lemma}

\begin{proof}
Multiplying \eqref{1.3}$_1$ by $u_{t}$,
\eqref{1.3}$_2$ by $\theta $, and \eqref{1.3}$_3$ by
$q$ and integrating over $\Omega $, using integration by parts, the
boundary conditions \eqref{2.2}  and \eqref{1.6}, one can easily find that
\begin{align*}
E_1'(t)
&= -\int_{\Omega }| q|^2dx-\eta \int_{\Gamma _1}| u_{t}| ^2d\Gamma _1+
\frac{\eta }{2}k'(t) \int_{\Gamma _1}|u| ^2d\Gamma _1 \\
&\quad -\frac{\eta }{2}\int_{\Gamma _1}(k''\circ u)
(t) d\Gamma _1+\eta \int_{\Gamma _1}k(t) (u_0.u_{t}) d\Gamma _1
\end{align*}
Young's inequality then yields
\[
\int_{\Gamma _1}k(t) (u_0.u_{t}) d\Gamma
_1\leq \frac{1}{2}\int_{\Gamma _1}| u_{t}|
^2d\Gamma _1+\frac{1}{2}k^2(t) \int_{\Gamma
_1}| u_0| ^2d\Gamma _1,
\]
and consequently, we obtain \eqref{3.7} for strong solutions.
Estimate \eqref{3.8} is established in a similar way using \eqref{3.4} and
\eqref{3.5}.
\end{proof}

\begin{remark} \label{rmk3.3} \rm
If $u_0=0$ on $\Gamma _1$, then $E_1'(t) \leq 0$, hence $E_1(t) \leq E_1(0)$.
If $u_0\neq 0$ on $\Gamma _1$, then
\begin{equation}
E_1(t) \leq E_1(0) +\frac{\eta }{2}\Big(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big)
\int_0^{t}k^2(s) ds\leq A,  \label{4.9}
\end{equation}
for some $A>0$.
\end{remark}

\begin{lemma} \label{lem3.4}
Under the assumptions {\rm (A1)} and {\rm (B1)}, the solution of \eqref{1.3}
satisfies: for any $\epsilon >0$,
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\int_{\Omega }u_{t}.[ M+(n-1) u] dx
 \\
&\leq -\int_{\Omega }| u_{t}| ^2dx-\mu \int_{\Omega
}| \nabla u| ^2dx-\frac{\mu +\lambda }{2}\int_{\Omega
}(\operatorname{div}u) ^2dx+C\int_{\Omega }| \nabla \theta
| ^2dx   \\
&\quad -\frac{\mu \delta }{2}\int_{\Gamma _1}| \nabla u|
^2d\Gamma _1-(\mu +\lambda ) \delta \int_{\Gamma _1}(
\operatorname{div}u) ^2d\Gamma _1+C(1+\frac{1}{\epsilon })
\int_{\Gamma _1}| u_{t}| ^2d\Gamma _1   \\
&\quad +Ck^2(t) \int_{\Gamma _1}| u|
^2d\Gamma _1-\frac{C}{\epsilon }\int_{\Gamma _1}(k'\circ u) (t) d\Gamma _1   \\
&\quad +\epsilon \int_{\Gamma _1}| u| ^2d\Gamma
_1+C(1+\frac{1}{\epsilon }) k^2(t) \int_{\Gamma
_1}| u_0| ^2d\Gamma _1,
\end{aligned} \label{4.10}
\end{equation}
where
\[
M=(M_1,M_2,...,M_{n})^{T},\quad \text{such that } M_{i}
=2m.\nabla u^{i},
\]
and $C$ is a ``generic'' positive
constant independent of $\epsilon$.
\end{lemma}

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

\section{Proof of main results}

In this section we prove our main result.

\begin{proof}[Proof of Theorem \ref{thm3.1}]
Taking $E(t) =E_1(t) +E_2(t) $, we define
\begin{equation}
L(t) =NE(t) +\int_{\Omega }u_{t}.[ M+(n-1) u] dx.  \label{4.21}
\end{equation}
From \eqref{3.7}, \eqref{3.8}, and \eqref{4.10}, we obtain
\begin{align*}
L'(t) &\leq -N\int_{\Omega }| q|^2dx-N\int_{\Omega }| q_{t}| ^2dx-\frac{N\eta }{2}
\int_{\Gamma _1}| u_{t}| ^2d\Gamma _1-\frac{N}{2}
\eta \int_{\Gamma _1}(k''\circ u) (t) d\Gamma _1 \\
&\quad -\int_{\Omega }| u_{t}| ^2dx-\mu \int_{\Omega
}| \nabla u| ^2dx-\frac{\mu +\lambda }{2}\int_{\Omega
}(\operatorname{div}u) ^2dx \\
&\quad -\frac{\mu \delta }{2}\int_{\Gamma _1}| \nabla u|
^2d\Gamma _1-(\mu +\lambda ) \delta \int_{\Gamma _1}(
\operatorname{div}u) ^2d\Gamma _1+C\int_{\Gamma _1}| u_{t}|
^2d\Gamma _1 \\
&\quad +\frac{C}{\epsilon }k^2(t) \int_{\Gamma _1}|
u| ^2d\Gamma _1-\frac{C}{\epsilon }\int_{\Gamma _1}(
k'\circ u) (t) d\Gamma _1+\epsilon \int_{\Gamma
_1}| u| ^2d\Gamma _1+C\int_{\Omega }|
\nabla \theta | ^2dx \\
&\quad +C(1+\frac{1}{\epsilon }) k^2(t) \int_{\Gamma
_1}| u_0| ^2d\Gamma _1+N\frac{\eta }{2}
k^2(t) \int_{\Gamma _1}| u_0|
^2d\Gamma _1.
\end{align*}
By using \eqref{1.3}$_3$ and
\begin{equation}
\int_{\Gamma _1}| u| ^2d\Gamma _1
\leq c_0\int_{\Omega }| \nabla u| ^2dx  \label{4}
\end{equation}
and $\int_{\Omega }| \theta | ^2dx\leq
c_{p}\int_{\Omega }| \nabla \theta | ^2dx$, for some
positive constants $c_0$ and $c_{p}$, we arrive at
\begin{equation}
\begin{aligned}
L'(t) &\leq -(N-C_1) \int_{\Omega}| q| ^2dx-(N-C_1) \int_{\Omega
}| q_{t}| ^2dx-\int_{\Omega }|u_{t}| ^2dx\\
&\quad -\int_{\Gamma _1}k(t) |u| ^2d\Gamma _1
-\Big(\mu -\epsilon c_0-\frac{C}{\epsilon }k^2(t)
-c_0k(t) \Big) \int_{\Omega }| \nabla u|^2dx   \\
&\quad -\frac{\mu +\lambda }{2}\int_{\Omega }(\operatorname{div}u) ^2dx
-\big(\frac{N}{2}\eta -C\big)
 \int_{\Gamma _1}| u_{t}|^2d\Gamma _1   \\
&\quad -C\int_{\Gamma _1}(k'\circ u) (t)
d\Gamma _1-\int_{\Omega }| \theta | ^2dx   \\
&\quad +\big[ \frac{N}{2}\eta +C(1+\frac{1}{\epsilon }) \big]
k^2(t) \int_{\Gamma _1}| u_0|^2d\Gamma _1.
\end{aligned} \label{3.22}
\end{equation}
At this point, we choose our constants carefully.
We first, fix $\epsilon $ so small that $\epsilon c_0=\frac{1}{2}\mu $
and pick $N$ large enough so that $L\sim E$,
\[
a_1=\frac{N}{2}\eta -C\geq 0\quad \text{and}\quad a_2=N-C_1>0.
\]
Thus, \eqref{3.22} simplifies to
\begin{align*}
L'(t) &\leq -\int_{\Omega }| u_{t}|
^2dx-\Big(\frac{\mu }{2}-\frac{C}{\epsilon }k^2(t)
-c_0k(t) \Big) \int_{\Omega }| \nabla u|^2dx \\
&\quad -\frac{\mu +\lambda }{2}\int_{\Omega }(\operatorname{div}u) ^2dx
 -\int_{\Gamma _1}k(t) | u| ^2d\Gamma
_1-a_2\int_{\Omega }| q| ^2dx-\int_{\Omega
}| \theta | ^2dx \\
&\quad -C\int_{\Gamma _1}(k'\circ u) (t)
d\Gamma _1+Ck^2(t) \int_{\Gamma _1}|
u_0| ^2d\Gamma _1.
\end{align*}
Using the fact that $\lim_{t\to +\infty }k(t) =0$,
we obtain
\begin{equation}
L'(t) \leq -mE_1(t) +Ck^2(
t) \int_{\Gamma _1}| u_0| ^2d\Gamma
_1-c\int_{\Gamma _1}(k'\circ u) (t)
d\Gamma _1,\quad \forall t\geq t_1,  \label{3.26}
\end{equation}
for some $t_1$ large enough and some positive constants $m$ and $c$.

Now we use \eqref{2.7}, and \eqref{3.7} to
conclude that, for any $t\geq t_1$,
\begin{equation}
\begin{aligned}
&-\int_0^{t_1}k'(s)\int_{\Gamma _1}| u(
t) -u(t-s) | ^2d\Gamma _1ds   \\
&\leq \frac{1}{d}\int_0^{t_1}k''(s)
\int_{\Gamma _1}| u(t) -u(t-s)
| ^2d\Gamma _1ds   \\
&\leq -c[ E_1'(t) -\frac{\eta }{2}k^2(
t) \int_{\Gamma _1}| u_0| ^2d\Gamma _1] .
\end{aligned} \label{3.9}
\end{equation}
Next we take $F(t) =L(t) +cE_1(t) $,
which is clearly equivalent to $E(t) $, and use \eqref{3.26} and \eqref{3.9},
 to obtain: for all $t\geq t_1$ with some new positive constant $C>0$,
\begin{equation}
F'(t) \leq -mE_1(t) +Ck^2(t) \int_{\Gamma _1}| u_0| ^2d\Gamma
_1-c\int_{t_1}^{t}k'(s)\int_{\Gamma _1}| u(
t) -u(t-s) | ^2d\Gamma _1ds.  \label{3.10}
\end{equation}
Similarly to \cite{Mus} we consider two cases:

\noindent \textbf{(I)} $H(t)=ct^{p}$ and $1\leq p<3/2$:
If $ 1<p<3/2$, one can easily show that
$\int_0^{+\infty }[-k'(s)]^{1-\delta _0}ds<+\infty $ for any $\delta _0<2-p$.
Using this fact, \eqref{3.7}, \eqref{4.9}  and \eqref{4} and
choosing $t_1$ even larger if needed, we deduce that, for all $t\geq t_1$
\begin{equation}
\begin{aligned}
\eta (t)
&= \int_{t_1}^{t}[-k'(s)]^{1-\delta _0}\int_{\Gamma_1}|u(t)-u(t-s)|^2\,d\Gamma _1ds   \\
&\leq 2\int_{t_1}^{t}[-k'(s)]^{1-\delta _0}\int_{\Gamma
_1}(|u(t)|^2+|u(t-s)|^2)\,d\Gamma _1ds   \\
&\leq cA\int_0^{+\infty }[-k'(s)]^{1-\delta _0}ds<1.
\end{aligned} \label{e3.20}
\end{equation}
Then, Jensen's inequality, \eqref{3.7}, hypothesis (B1), and \eqref{e3.20}
lead to
\begin{align*}
& -\int_{t_1}^{t}k'(s)\int_{\Gamma
_1}|u(t)-u(t-s)|^2\,d\Gamma _1ds \\
& =\int_{t_1}^{t}[ -k'(s) ] ^{\delta _0}
[ -k'(s) ] ^{1-\delta _0}\int_{\Gamma
_1}|u(t)-u(t-s)|^2\,d\Gamma _1ds \\
& =\int_{t_1}^{t}[ -k'(s) ] ^{p-1+\delta
_0(\frac{\delta _0}{p-1+\delta _0}) }[ -k'(s) ] ^{1-\delta _0}\int_{\Gamma
_1}|u(t)-u(t-s)|^2\,d\Gamma _1ds \\
& \leq \eta (t)\Big[\frac{1}{\eta (t)}\int_{t_1}^{t}[ -k'(s) ] ^{(p-1+\delta _0) }[ -k'(s) ] ^{1-\delta _0}\int_{\Gamma
_1}|u(t)-u(t-s)|^2\,d\Gamma _1ds\Big]^{\frac{\delta _0}{p-1+\delta
_0}} \\
& \leq \Big[\int_{t_1}^{t}[ -k'(s) ]
^{p}\int_{\Gamma _1}|u(t)-u(t-s)|^2\,d\Gamma _1ds\Big]^{\frac{\delta
_0}{p-1+\delta _0}} \\
& \leq c\Big[\int_{t_1}^{t}k''(s)\int_{\Gamma
_1}|u(t)-u(t-s)|^2\,d\Gamma _1ds\Big]^{\frac{\delta _0}{p-1+\delta
_0}} \\
& \leq c\Big[ -E_1'(t)+\frac{\eta }{2}k^2(t)
\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big)
\Big] ^{\frac{\delta _0}{p-1+\delta _0}}.
\end{align*}
Then, particularly for $\delta _0=1/2$, we find that \eqref{3.10} becomes
\[
F'(t) \leq -mE_1(t) +c\Big[
-E_1'(t)+\frac{\eta }{2}k^2(t) (\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \Big] ^{\frac{1}{2p-1}}
+Ck^2(t) \int_{\Gamma _1}| u_0|^2d\Gamma _1,
\]
for all $t\geq t_1$.
However,
\begin{align*}
&\Big(F(t) +\frac{\eta }{2}\Big(\int_{\Gamma _1}|
u_0| ^2d\Gamma _1\Big) \int_{t}^{\infty }k^2(s) ds\Big) '\\
&\leq F'(t) \\
&\leq -mE_1(t) +c\Big[ -E_1'(t)+\frac{\eta }{2}
k^2(t) \Big(\int_{\Gamma _1}| u_0|^2d\Gamma _1\Big) \Big] ^{\frac{1}{2p-1}}
+Ck^2(t) \int_{\Gamma _1}| u_0|^2d\Gamma _1,
\end{align*}
hence,  for all $t\geq t_1$, we have
\begin{align}
&\Big(F(t) +\frac{\eta }{2}\Big(\int_{\Gamma _1}|
u_0| ^2d\Gamma _1\Big) \int_{t}^{\infty }k^2(s) ds\Big) ' \nonumber \\
&\leq -m\Big[ E_1(t) +\frac{\eta }{2}\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma
_1\Big) \int_{t}^{\infty }k^2(s) ds\Big]  \nonumber  \\
&\quad +c\Big[ -E_1'(t)+\frac{\eta }{2}k^2(t)
\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big) \Big]^{\frac{1}{2p-1}}
+Ck^2(t) \int_{\Gamma _1}| u_0|^2d\Gamma _1  \nonumber  \\
&\quad +m\frac{\eta }{2}\Big(\int_{\Gamma _1}| u_0|
^2d\Gamma _1\Big) \int_{t}^{\infty }k^2(s) ds. \label{G2}
\end{align}  
Using hypothesis (B1),  for all $t\geq t_1$, we have
\begin{equation}
\int_{t}^{\infty }k^2(s) ds\leq k(t)
\int_{t}^{\infty }k(s) ds\leq k(t) \int_0^{\infty
}k(s) ds,\ \text{and }k^2(t) \leq c'k(t) .  \label{use}
\end{equation}
Then \eqref{G2} becomes, for some positive constant $C$
\begin{align*}
&\Big(F(t) +\frac{\eta }{2}\Big(\int_{\Gamma _1}|
u_0| ^2d\Gamma _1\Big) \int_{t}^{\infty }k^2(
s) ds\Big) ' \\
&\leq -m\Big[ E_1(t) +\frac{
\eta }{2}\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma
_1\Big) \int_{t}^{\infty }k^2(s) ds\Big] \\
&\quad+c\Big[ -E_1'(t)+\frac{\eta }{2}k^2(t)
\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big) \Big]
^{\frac{1}{2p-1}}
+Ck(t) \int_{\Gamma _1}| u_0| ^2d\Gamma _1.
\end{align*}
Now we multiply by $\big[ E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds\big] ^{2p-2}$,  using that
$E_1'(t) \leq \frac{\eta }{2}k^2(t)
(\int_{\Gamma _1}| u_0| ^2d\Gamma _1)$ we obtain
\begin{align*}
&\Big(\Big[ F(t) +\frac{\eta }{2}\Big(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1\Big) \int_{t}^{\infty }k^2(s) ds\Big]
\Big[ E_1(t)\\
&+\frac{\eta }{2}\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big)
\int_{t}^{\infty }k^2(s) ds\Big] ^{2p-2}\Big) '
\\
&\leq \Big(F(t)+\frac{\eta }{2}(\int_{\Gamma _1}|
u_0| ^2d\Gamma _1) \int_{t}^{\infty }k^2(
s) ds\Big) '\\
&\quad\times \Big[ E_1(t)+\frac{\eta }{2}\Big(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big)
\int_{t}^{\infty }k^2(s) ds\Big] ^{2p-2}
\\
&\leq -m\Big[ E_1(t) +\frac{\eta }{2}\Big(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1\Big) \int_{t}^{\infty}k^2(s) ds\Big] ^{2p-1} \\
&\quad +c\Big[ E_1(t)+\frac{\eta }{2}\Big(\int_{\Gamma _1}|
u_0| ^2d\Gamma _1\Big) \int_{t}^{\infty }k^2(s) ds\Big] ^{2p-2}\\
&\quad\times \Big[ -E_1'(t)+\frac{\eta }{2}
k^2(t) (\int_{\Gamma _1}| u_0|
^2d\Gamma _1) \Big] ^{\frac{1}{2p-1}} \\
&\quad +C\Big[ k(t) \int_{\Gamma _1}| u_0|
^2d\Gamma _1\Big] \Big[ E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty}k^2(s) ds\Big] ^{2p-2}.
\end{align*}
Then, applying Young's inequality, with $\sigma =2p-1$, and $\sigma
'=\frac{2p-1}{2p-2}$, for
\[
\Big[ -E_1'(t)+\frac{\eta }{2}k^2(t) \Big(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big) \Big]
^{\frac{1}{2p-1}}
\Big[ E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty}k^2(s) ds\Big] ^{2p-2}
\]
and
\[
\Big[ k(t) \int_{\Gamma _1}| u_0|
^2d\Gamma _1\Big] \Big[ E_1(t)+\frac{\eta }{2}\Big(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1\Big) \int_{t}^{\infty
}k^2(s) ds\Big] ^{2p-2}
\]
we obtain, for $C>0$,
\begin{align*}
&\Big(\Big[ F(t) +\frac{\eta }{2}\Big(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1\Big) \int_{t}^{\infty }k^2(s) ds\Big] \\
&\times \Big[ E_1(t)+\frac{\eta }{2}(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_{t}^{\infty }k^2(s) ds\Big] ^{2p-2}\Big) ' \\
&\leq -m\Big[ E_1(t) +\frac{\eta }{2}\Big(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1\Big) \int_{t}^{\infty
}k^2(s) ds\Big] ^{2p-1}\\
&\quad +2\varepsilon  \Big[ E_1\Big(
t\Big) +\frac{\eta }{2}\Big(\int_{\Gamma _1}|
u_0| ^2d\Gamma _1\Big) \int_{t}^{\infty }k^2(s) ds\Big] ^{2p-1} \\
&\quad +C_{\varepsilon }\Big[ -E_1'(t)+\frac{\eta }{2}k^2(
t) \Big(\int_{\Gamma _1}| u_0| ^2d\Gamma_1\Big) \Big] +C[ k(t)
 \int_{\Gamma _1}|u_0| ^2d\Gamma _1] ^{2p-1}.
\end{align*}
Consequently, for $2\varepsilon <m$, we obtain
\[
F_0'(t)\leq -m'\Big[ E_1(t) +\frac{\eta }{
2}\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma_1\Big)
\int_{t}^{\infty }k^2(s) ds\Big] ^{2p-1}
+C\Big[k(t) \int_{\Gamma _1}| u_0| ^2d\Gamma_1\Big] ^{2p-1},
\]
where $m'$ is some positive constant and
\begin{align*}
F_0(t)
&= \Big[ F(t) +\frac{\eta }{2}\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big)
\int_{t}^{\infty }k^2(s) ds\Big] \\
&\quad\times \Big[ E_1(t)+\frac{\eta
}{2}(\int_{\Gamma _1}| u_0| ^2d\Gamma
_1) \int_{t}^{\infty }k^2(s) ds\Big] ^{2p-2} \\
&\quad +C_{\varepsilon }\Big[ E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds\Big] .
\end{align*}
Also, it is easy to show that this inequality is true for $p=1$.
 Once again, we use the fact that $E_1'(t) \leq \frac{\eta
}{2}k^2(t) (\int_{\Gamma _1}|
u_0| ^2d\Gamma _1) $ to deduce that
\begin{align*}
&\Big(t\Big[ E_1(t) +\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds\Big] ^{2p-1}\Big) '\\
 &\leq \Big[
E_1(t) +\frac{\eta }{2}\Big(\int_{\Gamma _1}|u_0| ^2d\Gamma _1\Big)
 \int_{t}^{\infty }k^2(s) ds\Big] ^{2p-1} \\
&\leq -\frac{1}{m'}F_0'(t) +\frac{C}{ m'}\Big[ k(t) \int_{\Gamma _1}|
u_0| ^2d\Gamma _1\Big] ^{2p-1},
\end{align*}
for all$\ t\geq t_1.$ A simple integration over $(t_1,t) $
yields
\begin{align*}
&t\Big[ E_1(t) +\frac{\eta }{2}\Big(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1\Big) \int_{t}^{\infty
}k^2(s) ds\Big] ^{2p-1} \\
&\leq \frac{1}{m'} F_0(t_1) +\frac{C}{m'}\int_{t_1}^{t}
\Big[k(s) \int_{\Gamma _1}| u_0| ^2d\Gamma_1\Big] ^{2p-1}ds \\
&\quad +t_1\Big[ E_1(t_1) +\frac{\eta }{2}\Big(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1\Big)\int_{t_1}^{\infty }k^2(s) ds\Big] ^{2p-1} \\
&\leq c_1+\frac{C}{m'}\int_{t_1}^{t}\Big[ k(s)
\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big] ^{2p-1},
\end{align*}
hence
\begin{align*}
&\Big[ E_1(t) +\frac{\eta }{2}\Big(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1\Big) \int_{t}^{\infty}k^2(s) ds\Big] ^{2p-1}\\
&\leq \frac{c_1}{t}+\frac{C}{
m't}\int_{t_1}^{t}\Big[ k(s) \int_{\Gamma_1}| u_0| ^2d\Gamma _1\Big] ^{2p-1}ds.
\end{align*}
Therefore,
\begin{align*}
E_1(t) &\leq \Big(\frac{c_1+c_2\int_{t_1}^{t}[
k(s) \int_{\Gamma _1}| u_0| ^2d\Gamma
_1] ^{2p-1}ds}{t}\Big) ^{\frac{1}{2p-1}}\\
&\quad -\frac{\eta }{2}\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big)
\int_{t}^{\infty }k^2(s) ds.
\end{align*}

\noindent\textbf{(II) The general case:}
As in \cite{Mus}, we define
\[
I(t)=\int_{t_1}^{t}\frac{-k'(s) }{H_0^{-1}(k''(s) ) }\int_{\Gamma _1}|
u(t) -u(t-s) | ^2d\Gamma _1ds
\]
where $H_0$ is such that \eqref{2.4} is satisfied. As in
\eqref{e3.20}, we find that for all $t\geq t_1$, $I(t)$ satisfies
\begin{equation}
I(t)<1.  \label{3.13}
\end{equation}
We also assume, without loss of generality that $I(t)\geq \beta _0>0$,
for all $t\geq t_1$; otherwise \eqref{3.10} yields an
explicit decay. In addition, we define$\ \lambda (t) $ by
\[
\lambda (t) =\int_{t_1}^{t}k''(s)
\frac{-k'(s)}{H_0^{-1}(k''(s)) }\int_{\Gamma _1}| u(t) -u(t-s)
| ^2d\Gamma _1ds,
\]
and infer from (B1) and the properties of $H_0$ and $D$ that
\[
\frac{-k'(s)}{H_0^{-1}(k''(s)
) }\leq \frac{-k'(s)}{H_0^{-1}(H(-k'(s) ) ) }=\frac{-k'(s)}{D^{-1}(
-k'(s)) }\leq k_0,
\]
for some positive constant$ k_0.$ Then, using \eqref{2.6}, \eqref{3.7}, and
 \eqref{4.9}, one can
easily see that for all $t\geq t_1$, $\lambda (t)$ satisfies
\begin{equation}
\begin{aligned}
\lambda (t)
&\leq k_0\int_{t_1}^{t}k''(s) \int_{\Gamma _1}| u(t) -u(t-s)
| ^2d\Gamma _1ds   \\
&\leq cA\int_{t_1}^{t}k''(s)ds\leq cA(-k'(t_1))   \\
&< \min \{ r,H(r),H_0(r)\},
\end{aligned} \label{3.14}
\end{equation}
for $t_1$ even larger (if needed). In addition, we can easily see that
\begin{equation}
\lambda (t) \leq -c\Big[ E_1'(t) -\frac{
\eta }{2}k^2(t) \int_{\Gamma _1}| u_0|
^2d\Gamma _1\Big] ,\quad \forall t\geq t_1.  \label{b}
\end{equation}
Since $H_0$ is strictly convex on $(0,r] $, and $H_0(0)=0$, we have
\[
H_0(\theta x) \leq \theta H_0(x) ,
\]
provided $0\leq \theta \leq 1$, and $x\in (0,r]$. Then using
this fact, \eqref{3.13} $, \eqref{3.14} $, and
Jensen's inequality leads to
\begin{align*}
\lambda (t)
&= \frac{1}{I(t) }\int_{t_1}^{t}I(t)H_0[ H_0^{-1}(k''(
s) ) ] \frac{-k'(s)}{H_0^{-1}(k''(s) ) }\int_{\Gamma _1}| u(
t) -u(t-s) | ^2d\Gamma _1ds \\
&\geq \frac{1}{I(t) }\int_{t_1}^{t}H_0[ I(t) H_0^{-1}(k''(s) ) ]
\frac{-k'(s)}{H_0^{-1}(k''(s)) }\int_{\Gamma _1}| u(t) -u(t-s)
| ^2d\Gamma _1ds \\
&\geq H_0\Big(\frac{1}{I(t)}\int_{t_1}^{t}I(t)H_0^{-1}(
k''(s) ) \frac{-k'(s)}{
H_0^{-1}(k''(s) ) }\int_{\Gamma_1}| u(t) -u(t-s) | ^2d\Gamma_1ds\Big) \\
&= H_0\Big(-\int_{t_1}^{t}k'(s) \int_{\Gamma_1}| u(t) -u(t-s) | ^2
d\Gamma_1ds\Big) ,\quad \forall t\geq t_1.
\end{align*}
This implies
\[
-\int_{t_1}^{t}k'(s) \int_{\Gamma _1}|
u(t) -u(t-s) | ^2d\Gamma _1ds\leq
H_0^{-1}(\lambda (t) ) ,\quad \forall t\geq t_1,
\]
and \eqref{3.10}  becomes
\begin{equation}
F'(t)\leq -mE_1(t)+Ck^2(t) \int_{\Gamma
_1}| u_0| ^2d\Gamma _1+cH_0^{-1}(\lambda
(t) ) ,\quad \forall t\geq t_1.  \label{3.15}
\end{equation}
Now, using  that $H_0'>0$, $H_0''>0$,
$\varepsilon _0<r$,  $c_0>0$, and
\[
E_1'(t) \leq \frac{\eta }{2}k^2(t)(\int_{\Gamma _1}| u_0| ^2d\Gamma _1),
\]
we find that the functional
\begin{align*}
F_1(t) &= H_0'\Big(\varepsilon _0\frac{E_1(t)+
\frac{\eta }{2}(\int_{\Gamma _1}| u_0|
^2d\Gamma _1) \int_{t}^{\infty }k^2(s) ds}{E_1(0)+
\frac{\eta }{2}(\int_{\Gamma _1}| u_0|
^2d\Gamma _1) \int_0^{\infty }k^2(s) ds}\Big)
F(t) \\
&\quad +c_0\Big(E_1(t)+\frac{\eta }{2}(\int_{\Gamma _1}|
u_0| ^2d\Gamma _1) \int_{t}^{\infty }k^2(s) ds\Big)
\end{align*}
satisfies
\begin{align*}
F_1'(t)
&= \Big(\varepsilon _0\frac{E_1'(t)-\frac{\eta }{2}(\int_{\Gamma _1}| u_0|
^2d\Gamma _1) k^2(t)}{E_1(0)+\frac{\eta }{2}
(\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_0^{\infty }k^2(s) ds}\Big) \\
&\quad \times H_0''\Big(\varepsilon _0\frac{E_1(t)+\frac{
\eta }{2}(\int_{\Gamma _1}| u_0| ^2d\Gamma
_1) \int_{t}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{
2}(\int_{\Gamma _1}| u_0| ^2d\Gamma
_1) \int_0^{\infty }k^2(s) ds}\Big) F(t)
 \\
&\quad +H_0'\Big(\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}
(\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_{t}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{2}(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_0^{\infty }k^2(s) ds}\Big) F'(t)
\\
&\quad +c_0\Big[ E_1'(t)-\frac{\eta }{2}k^2(t) (
\int_{\Gamma _1}| u_0| ^2d\Gamma _1) \Big]
\\
&\leq H_0'\Big(\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}
(\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_{t}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{2}(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_0^{\infty }k^2(s) ds}\Big) F'(t) \\
&\quad +c_0\Big[ E_1'(t)-\frac{\eta }{2}k^2(t) (
\int_{\Gamma _1}| u_0| ^2d\Gamma _1) \Big]
,\quad \forall t\geq t_1.
\end{align*}
Hence, using \eqref{3.15},  for all $\ t\geq t_1$, we obtain
\begin{equation}
\begin{aligned}
&F_1'(t) \\
&\leq -mE_1(t)H_0'\Big(
\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds}{E_1(0)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_0^{\infty}k^2(s) ds}\Big)
 \\
&\quad +C\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma_1\Big) k^2(t)
 H_0'\Big(\varepsilon _0
\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma _1}|
u_0| ^2d\Gamma _1) \int_{t}^{\infty }k^2(
s) ds}{E_1(0)+\frac{\eta }{2}(\int_{\Gamma _1}|
u_0| ^2d\Gamma _1) \int_0^{\infty }k^2(s) ds}\Big)
 \\
&\quad +cH_0^{-1}(\lambda (t) ) H_0'\Big(
\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds}{E_1(0)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_0^{\infty}k^2(s) ds}\Big)
  \\
&\quad +c_0\Big[ E_1'(t)-\frac{\eta }{2}k^2(t) (
\int_{\Gamma _1}| u_0| ^2d\Gamma _1) \Big].
\end{aligned}  \label{3.17}
\end{equation}
Let $H_0^{\ast }$ be the convex conjugate of $H_0$ in the sense of
Young\ (see  \cite[p. 61-64]{ARN}), then
\begin{equation}
H_0^{\ast }(s) =s(H_0') ^{-1}(s) -H_0[ (H_0') ^{-1}(s)] ,\quad
\text{if }s\in (0,H_0'(r) ],
\label{3.18}
\end{equation}
and $H_0^{\ast }$ satisfies the  Young inequality
\begin{equation}
AB\leq H_0^{\ast }(A) +H_0(B) ,\quad \text{if }
A\in (0,H_0'(r) ] ,\; B\in (0,r] ,  \label{3.19}
\end{equation}
with
\[
A=H_0'\Big(\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_{t}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{2}(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_0^{\infty }k^2(s) ds}\Big), \quad B=H_0^{-1}(\lambda (t) )\,.
\]
Using \eqref{3.14}, \eqref{3.17}, and \eqref{3.19}, we arrive at
\begin{align*}
F_1'(t)
&\leq -mE_1(t)H_0'\Big(\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds}{E_1(0)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_0^{\infty}k^2(s) ds}\Big) \\
&\quad +C(\int_{\Gamma _1}| u_0| ^2d\Gamma_1) k^2(t)
H_0'\Big(\varepsilon _0
\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma _1}|
u_0| ^2d\Gamma _1) \int_{t}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{2}
\Big(\int_{\Gamma _1}|u_0| ^2d\Gamma _1\Big) \int_0^{\infty }k^2(s) ds}\Big) \\
&\quad +c\lambda (t) +cH_0^{\ast }\Big(H_0'\Big(
\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds}{E_1(0)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_0^{\infty}k^2(s) ds}\Big) \Big) \\
&\quad +c_0\Big[ E_1'(t)-\frac{\eta }{2}k^2(t)
\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big) \Big].
\end{align*}
Then using \eqref{b} and \eqref{3.18}, we obtain that
for all $t\geq t_1$,
\begin{align*}
F_1'(t)
&\leq -mE_1(t)H_0'\Big(
\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds}{E_1(0)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_0^{\infty}k^2(s) ds}\Big) \\
&\quad +Ck^2(t) \Big(\int_{\Gamma _1}|u_0| ^2d\Gamma _1\Big)
 H_0'\Big(\varepsilon_0\frac{E_1(t)+\frac{\eta }{2}
 \Big(\int_{\Gamma _1}|u_0| ^2d\Gamma _1\Big)
 \int_{t}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{2}
 \Big(\int_{\Gamma _1}|u_0| ^2d\Gamma _1\Big)
 \int_0^{\infty }k^2(s) ds}\Big) \\
&\quad -c\Big[ E_1'(t)-\frac{\eta }{2}k^2(t)
\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big] \\
&\quad +c(\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}
\big(\int_{\Gamma_1}| u_0| ^2d\Gamma _1\Big)
\int_{t}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{2}
\Big(\int_{\Gamma_1}| u_0| ^2d\Gamma _1\Big)
\int_0^{\infty }k^2(s) ds}) \\
&\quad \times H_0'\Big(\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}
(\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_{t}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{2}(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_0^{\infty }k^2(s) ds}\Big) \\
&\quad +c_0\Big[ E_1'(t)-\frac{\eta }{2}k^2(t) (
\int_{\Gamma _1}| u_0| ^2d\Gamma _1) \Big].
\end{align*}
Consequently, with a suitable choice of $c_0$, we obtain, for all
$t\geq t_1$,
\begin{align}
F_1'(t)
&\leq -m\Big(\frac{E_1(t)+\frac{\eta }{2}
(\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_{t}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{2}(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_0^{\infty }k^2(s) ds}\Big) \nonumber \\
&\quad\times H_0'\Big(
\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds}{E_1(0)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_0^{\infty}k^2(s) ds}\Big)  \nonumber \\
&\quad +m\Big(\frac{\eta }{2}\Big(\int_{\Gamma _1}|
u_0| ^2d\Gamma _1\Big) \int_{t}^{\infty }k^2(s) ds\Big) \nonumber \\
&\quad \times H_0'\Big(\varepsilon _0\frac{E_1(t)+
\frac{\eta }{2}(\int_{\Gamma _1}| u_0|
^2d\Gamma _1) \int_{t}^{\infty }k^2(s) ds}{E_1(0)+
\frac{\eta }{2}(\int_{\Gamma _1}| u_0|
^2d\Gamma _1) \int_0^{\infty }k^2(s) ds}\Big)
\nonumber \\
&\quad +C(k^2(t) \int_{\Gamma _1}|u_0| ^2d\Gamma _1)
 H_0'(\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}
\Big(\int_{\Gamma _1}|u_0| ^2d\Gamma _1\Big)
\int_{t}^{\infty }k^2( s) ds}{E_1(0)+\frac{\eta }{2}
(\int_{\Gamma _1}|u_0| ^2d\Gamma _1)
\int_0^{\infty }k^2(s) ds}) \nonumber \\
&\quad +c\Big(\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds}{E_1(0)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_0^{\infty}k^2(s) ds}\Big) \nonumber  \\
&\quad \times H_0'\Big(\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}
(\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_{t}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{2}(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_0^{\infty }k^2(s) ds}\Big) .  \label{G}
\end{align} 
Using \eqref{use}, for some positive constant $C$, \eqref{G} becomes
\begin{equation}
\begin{aligned}
F_1'(t)
&\leq -m\Big(\frac{E_1(t)+\frac{\eta }{2}
(\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_{t}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{2}(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_0^{\infty }k^2(s) ds}\Big) \\
&\quad\times H_0'\Big(\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds}{E_1(0)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_0^{\infty}k^2(s) ds}\Big)
\\
&\quad +C\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma
_1\Big) k(t)  H_0'\Big(\varepsilon _0\frac{
E_1(t)+\frac{\eta }{2}(\int_{\Gamma _1}| u_0|
^2d\Gamma _1) \int_{t}^{\infty }k^2(s) ds}{E_1(0)+
\frac{\eta }{2}(\int_{\Gamma _1}| u_0|
^2d\Gamma _1) \int_0^{\infty }k^2(s) ds}\Big)
 \\
&\quad +c\Big(\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds}{E_1(0)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_0^{\infty
}k^2(s) ds}\Big)   \\
&\quad \times H_0'\Big(\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}
(\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_{t}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{2}(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_0^{\infty }k^2(s) ds}\Big) .  
\end{aligned}  \label{R}
\end{equation}
In a similar way, using \eqref{3.18} and \eqref{3.19},
we find that, for $t_1$ even larger (if needed),
\begin{align*}
&k(t) \Big[ H_0'(\varepsilon _0\frac{
E_1(t)+\frac{\eta }{2}(\int_{\Gamma _1}| u_0|
^2d\Gamma _1) \int_{t}^{\infty }k^2(s) ds}{E_1(0)+
\frac{\eta }{2}(\int_{\Gamma _1}| u_0|
^2d\Gamma _1) \int_0^{\infty }k^2(s) ds})\Big] \\
&\leq \Big(\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_{t}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{2}(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_0^{\infty }k^2(s) ds}\Big) \\
&\quad\times H_0'\Big(\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds}{E_1(0)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_0^{\infty}k^2(s) ds}\Big) \\
&\quad +H_0(k(t) ),\quad \forall t\geq t_1.
\end{align*}
So,  with a suitable choice of $\varepsilon _0$, \eqref{R} becomes
\begin{align*}
F_1'(t)
&\leq -\ell \Big(\frac{E_1(t)+\frac{\eta
}{2}(\int_{\Gamma _1}| u_0| ^2d\Gamma
_1) \int_{t}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{
2}(\int_{\Gamma _1}| u_0| ^2d\Gamma
_1) \int_0^{\infty }k^2(s) ds}\Big) \\
&\quad\times  H_0'\Big(\varepsilon _0\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds}{E_1(0)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_0^{\infty}k^2(s) ds}\Big) \\
&\quad +C(\int_{\Gamma _1}| u_0| ^2d\Gamma
_1) H_0(k(t) ) ,
\end{align*}
for all $t\geq t_1$.

Hence,
\begin{equation}
\begin{aligned}
F_1'(t) 
&\leq -\ell H_1\Big(\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma _1}| u_0| ^2d\Gamma
_1) \int_{t}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{
2}(\int_{\Gamma _1}| u_0| ^2d\Gamma
_1) \int_0^{\infty }k^2(s) ds}\Big) \\
&\quad +C\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big)
H_0(k(t) ) ,\quad \forall t\geq t_1,
\end{aligned} \label{3.20}
\end{equation}
where $H_1(t) =tH_0'(\varepsilon _0t)$.

Since $H_1'(t) =H_0'(\varepsilon_0t) +\varepsilon _0tH_0''(\varepsilon_0t) $, 
then using the strict convexity of $H_0$ on $(0,r] $, we find that 
$H_1'(t) $, $H_1(t) >0, $ on $(0,1]$. Thus, taking in account that 
$E_1'(t) \leq \frac{\eta }{2}k^2(t)\big(\int_{\Gamma _1}| u_0| ^2d\Gamma _1\big)$ 
and \eqref{3.20},  for all $t\geq t_1$, we have
\begin{align*}
&\Big[ tH_1\Big(\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds}{E_1(0)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_0^{\infty
}k^2(s) ds}\Big) \Big] ' \\
&\leq H_1\Big(\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma _1}| u_0|
^2d\Gamma _1) \int_{t}^{\infty }k^2(s) ds}{E_1(0)+
\frac{\eta }{2}(\int_{\Gamma _1}| u_0|
^2d\Gamma _1) \int_0^{\infty }k^2(s) ds}\Big) \\
&\leq -\frac{1}{\ell }F_1'(t) +\frac{C}{\ell }\Big(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big)H_0(k(t) ) .
\end{align*}
A simple integration over $(t_1,t) $ yields
\begin{align*}
&tH_1\Big(\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds}{E_1(0)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_0^{\infty
}k^2(s) ds}\Big) \\
&\leq t_1H_1(\frac{E_1(t_1)+\frac{\eta }{2}\Big(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big)
\int_{t_1}^{\infty }k^2(s) ds}{E_1(0)+\frac{\eta }{2}
(\int_{\Gamma _1}| u_0| ^2d\Gamma _1)\int_0^{\infty }k^2(s) ds}) \\
&\quad +\frac{1}{\ell }F_1(t_1) +\frac{C}{\ell }\Big(\int_{\Gamma _1}| u_0|
^2d\Gamma _1\Big) \int_{t_1}^{t}H_0(k(s) )ds.
\end{align*}
This gives, for some positive constant $k_2$ and for all $t\geq t_1$,
\[
H_1\Big(\frac{E_1(t)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_{t}^{\infty
}k^2(s) ds}{E_1(0)+\frac{\eta }{2}(\int_{\Gamma
_1}| u_0| ^2d\Gamma _1) \int_0^{\infty}k^2(s) ds}\Big) 
\leq \frac{k_2}{t}+\frac{C}{\ell t}\Big(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1\Big)
\int_{t_1}^{t}H_0(k(s) ) ds.
\]
Therefore,  for some positive constants $k_1$ and $k_3$, we obtain
\begin{align*}
E_1(t) &\leq k_1H_1^{-1}\Big(\frac{k_2+k_3(
\int_{\Gamma _1}| u_0| ^2d\Gamma _1)
\int_{t_1}^{t}H_0(k(s) ) ds}{t}\Big) \\
&\quad -\frac{ \eta }{2}\Big(\int_{\Gamma _1}| u_0| ^2d\Gamma_1\Big) 
\int_{t}^{\infty }k^2(s) ds.
\end{align*}
This completes the proof.
\end{proof}

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