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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 193, pp. 1--45.\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-2012/193\hfil Stabilization of a linear Timoshenko system]
{Stabilization of a linear Timoshenko system with infinite history 
 and applications to the Timoshenko-heat systems}

\author[A. Guesmia, S. A. Messaoudi, A. Soufyane\hfil EJDE-2012/193\hfilneg]
{Aissa Guesmia, Salim A. Messaoudi, Abdelaziz Soufyane}  % in alphabetical order

\address{Aissa Guesmia \newline
Laboratory of Mathematics and Applications of Metz\\
Bat. A, Lorraine - Metz University\\
Ile de Sauley, 57045 Metz Cedex 01, France}
\email{guesmia@univ-metz.fr}

\address{Salim A. Messaoudi \newline
Mathematical Sciences Department\\
KFUPM, Dhahran 31261, Saudi Arabia}
\email{messaoud@kfupm.edu.sa}

\address{Abdelaziz Soufyane \newline
College of Engineering and Applied Sciences \\
Alhosn University, P.O. Box 38772, Abu Dhabi, UAE}
\email{a.soufyane@alhosnu.ae}

\thanks{Submitted April 9, 2012. Published November 6, 2012.}
\subjclass[2000]{35B37, 35L55, 74D05, 93D15, 93D20}
\keywords{General decay; infinite history; relaxation function; 
\hfill\break\indent Timoshenko;  thermoelasticity}

\begin{abstract}
 In this article, we, first, consider a vibrating system of Timoshenko
 type in a one-dimensional bounded domain with an
 infinite history acting in the equation of the rotation angle. We establish
 a general decay of the solution for the case of equal-speed wave propagation
 as well as for the nonequal-speed case. We, also, discuss the well-posedness
 and smoothness of solutions using the semigroup theory. Then, we give
 applications to the coupled Timoshenko-heat systems (under Fourier's,
 Cattaneo's and Green and Naghdi's theories). To establish our results, we
 adopt the method introduced,  in \cite{g5} with some necessary
 modifications imposed by the nature of our problems since they do not fall
 directly in the abstract frame of the problem treated in \cite{g5}. Our results
 allow a larger class of kernels than those considered in \cite{m9,m10,m11}, and
 in some particular cases, our decay estimates improve the results of
 \cite{m9,m10}. Our approach can be applied to many other systems with an infinite
 history.
\end{abstract}

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

\section{Introduction}

In the present work, we are concerned with the well-posedness, smoothness
and asymptotic behavior of the solution of the  Timoshenko system
\begin{equation}
\begin{gathered}
\rho _1\varphi _{tt}-k_1(\varphi _x+\psi )_x=0, \\
\rho _2\psi _{tt}-k_2\psi _{xx}+k_1(\varphi _x+\psi
)+\int_0^{\infty }g(s)\psi _{xx}(t-s)ds=0, \\
\varphi (0,t)=\psi (0,t)=\varphi (L,t)=\psi (L,t)=0, \\
\varphi (x,0)=\varphi _0(x),\quad \varphi _t(x,0)=\varphi _1(x), \\
\psi (x,-t)=\psi _0(x,t),\quad \psi _t(x,0)=\psi _1(x),
\end{gathered} \label{eP}
\end{equation}
where $(x,t)\in ]0,L[\times \mathbb{R}_{+}$,
$\mathbb{R}_{+}=[0,+\infty [ $, $g:\mathbb{R}_{+}\to \mathbb{R}_{+}$
is a given function (which will
be specified later on), $L,\,\rho _i,\,k_i$ ($i=1,2$) are positive
constants, $\varphi _0$, $\varphi _1$, $\psi _0$ and $\psi _1$ are
given initial data, and $(\varphi ,\psi )$ is the state of \eqref{eP}. The
infinite integral in \eqref{eP} represents the infinite history. The derivative
of a generic function $f$ with respect to a variable $y$ is denoted by
$f_{y} $ or $\partial _{y}f$. When $f$ has only one variable $y$, the
derivative of $f$ is noted by $f'$. To simplify the notation, we
omit, in general, the space and time variables (or we note only the time
variable when it is necessary).

In 1921, Timoshenko \cite{t1} introduced the system \eqref{eP} with $g=0$ to describe
the transverse vibration of a thick beam, where $t$ denotes the time
variable, $x$ is the space variable along the beam of length $L$, in its
equilibrium configuration, $\varphi $ is the transverse displacement of the
beam, and $-\psi $ is the rotation angle of the filament of the beam. The
positive constants $\rho _1$, $\rho _2$, $k_1$ and $k_2$ denote,
respectively, the density (the mass per unit length), the polar moment of
inertia of a cross section, the shear modulus and Young's modulus of
elasticity times the moment of inertia of a cross section.

During the last few years, an important amount of research has been devoted
to the issue of the stabilization of the Timoshenko system and search for
the minimum dissipation by which the solutions decay uniformly to the stable
state as time goes to infinity. To achieve this goal, diverse types of
dissipative mechanisms have been introduced and several stability results
have been obtained. Let us mention some of these results (for further
results, we refer the reader to the list of references of this paper, which
is not exhaustive, and the references therein).

In the  presence of controls on both the rotation angle and the
transverse displacement, studies show that the Timoshenko system is stable
for any weak solution and without any restriction on the constants
 $\rho_1 $, $\rho _2$, $k_1$ and $k_2$. Many decay estimates were obtained
in this case; see for example \cite{k1,m5,r1,x1,y1,y2}.

In the case of only one control on the rotation angle, the rate of decay
depends heavily on the constants $\rho _1$, $\rho _2$, $k_1$ and
$k_2 $. Precisely, if
\begin{equation}
{\frac{{k_1}}{{\rho _1}}}={\frac{{k_2}}{{\rho _2}}},\label{e1.1}
\end{equation}
holds (that is, the speeds of wave propagation are equal), the results show
that we obtain similar decay rates as in the presence of two controls. We
quote in this regard \cite{a1,a3,g6,g7,m6,m7,m10,m14,m15,m16,s2}.
However, if \eqref{e1.1} does not hold, a situation which is more interesting from
the physics point of view, then it has been shown that the Timoshenko system
is not exponentially stable even for exponentially decaying relaxation
functions. Whereas, some polynomial decay estimates can be obtained for the
strong solution in the presence of dissipation. This has been demonstrated
in \cite{a1} for the case of an internal feedback, and in \cite{m10,m11}
 for the case of an infinite history.

For Timoshenko system coupled with the heat equation, we mention the pioneer
work of Mu\~noz and Racke \cite{m13}, where they considered the system
\begin{gather*}
\rho _1\varphi _{tt}-\sigma (\varphi _x,\psi )_x=0,\quad\text{in } ]0,L[\times \mathbb{R}_{+}, \\
\rho _2\psi _{tt}-b\psi _{xx}+k(\varphi _x+\psi )+\gamma \theta _x=0,\quad
\text{in }]0,L[\times \mathbb{R}_{+}, \\
\rho _3\theta _t-k\theta _{xx}+\gamma \psi _{tx}=0,\quad\text{in }
]0,L[\times \mathbb{R}_{+}.
\end{gather*}
Under appropriate conditions on $\sigma $, $\rho _i$, $b$, $k$ and
$\gamma$ they established well posedness and exponential decay results for the
linearized system with several boundary conditions. They also proved a non
exponential stability result for the case of different wave speeds. In
addition, the nonlinear case was discussed and an exponential decay was
established. These results were later pushed by Messaoudi et al. \cite{m4} to the
situation, where the heat propagation is given by Cattaneo's law, and by
Messaoudi and Said-Houari \cite{m9} to the situation, where the heat propagation
is given by Green and Naghdi's theory \cite{g1,g2,g3}.

The main problem concerning the stability in the presence of infinite
history is determining the largest class of kernels $g$ which guarantee the
stability and the best relation between the decay rates of $g$ and the
solutions of the considered system.

When $g$ satisfies
\begin{equation}
\exists \delta _1,\,\delta _2>0: -\delta _1g(s)\leq g'(s)\leq -\delta _2g(s),\quad \forall s\in \mathbb{R}_{+},\label{e1.2}
\end{equation}
Mu\~noz and Fern\'{a}ndez Sare \cite{m11} proved that \eqref{eP} is exponentially stable
if and only if \eqref{e1.1} holds, and it is polynomially stable in general. In
addition, the decay rate depends on the smoothness of the initial data. When
$g$ satisfies
\begin{equation}
\exists \delta >0,\,\exists p\in [ 1,{\frac{3}{2}}[\,:g'(s)\leq
-\delta \,g^{p}(s),\quad \forall s\in \mathbb{R}_{+},\label{e1.3}
\end{equation}
it was proved in \cite{m1} that \eqref{eP} is exponentially stable when $p=1$ and
\eqref{e1.1} holds, and it is polynomially stable otherwise, where the decay rate
is better in the case \eqref{e1.1} than in that of opposite case. No
relationship between the decay rate and the smoothness of the initial data
was given in \cite{m1}. Similar results were proved for \eqref{eP1} (see Section 6)
and \eqref{eP5} (see Section 7), respectively, in \cite{f1} under \eqref{e1.2}
 and \cite{m9} under \eqref{e1.3}.
Recently Ma et al. \cite{m1} proved the exponential stability of \eqref{eP5}
under \eqref{e1.1} and \eqref{e1.2} using the semigroup method. On the other hand,
Fern\'andez Sare and Racke \cite{f1} proved that \eqref{eP4} (see Section 6) is not
exponentially stable even if \eqref{e1.1} holds and $g$ satisfies \eqref{e1.2}.

The infinite history was also used to stabilize the semigroup associated to
a general abstract linear equation in \cite{c1,g5,m12,p1} (see also the
references therein for more details on the existing results in this
direction). In \cite{m12}, some decay estimates were proved depending on the
considered operators provided that $g$ satisfies \eqref{e1.2},
while in \cite{p1}, it
was proved that the exponential stability still holds even if $g$ has
horizontal inflection points or even flat zones provided that $g$ is equal
to a negative exponential except on a sufficiently small set where $g$ is
flat. In \cite{g5}, the weak stability was proved for the (much) larger class of
$g$ satisfying (H2) below. The author of \cite{c1} proved that the exponential
stability does not hold if the following condition is not satisfied:
\begin{equation}
\exists \delta _1\geq 1,\,\exists \delta _2>0:g(t+s)\leq \delta
_1e^{-\delta _2t}g(s),\quad \forall t\in \mathbb{R}_{+},\,
\text{for a.e. }s\in \mathbb{R}_{+}.\label{e1.4}
\end{equation}

The stability of Timoshenko systems with a finite history (that is the
infinite integral $\int_0^{+\infty }$ in \eqref{eP} is replaced
with the finite one $\int_0^t$) have attracted a
considerable attention in the recent years and many authors have proved
different decay estimates depending on the relation \eqref{e1.1} and the growth of
the kernel $g$ at infinity (see for example \cite{g3} and the references therein
for more details). Using an approach introduced in \cite{m2} for a viscoelastic
equation, a general estimate of stability of \eqref{eP} with finite history and
under \eqref{e1.1} was obtained in \cite{g7} for kernels satisfying
\begin{equation}
g'(s)\leq -\xi (s)g(s),\quad \forall s\in \mathbb{R}_{+},
\label{e1.5}
\end{equation}
where $\xi $ is a positive and non-increasing function. The decay result in
\cite{g7} improves earlier ones in the literature in which only the exponential
and polynomial decay rates are obtained (see \cite{g7}). The case where \eqref{e1.1}
does not hold was studied in \cite{g8} for kernels satisfying
\begin{equation}
g'(s)\leq -\xi (s)g^{p}(s),\quad \forall s\in \mathbb{R}
_{+},\label{e1.6}
\end{equation}
where $\xi $ is a positive and non-increasing function and $p\geq 1$.

Concerning the stability of abstract equations with a finite history, we
mention the results in \cite{a2} (see the references therein for more results),
where a general and sufficient condition under which the energy converges to
zero at least as fast as the kernel at infinity was given by assuming the
following condition:
\begin{equation}
g'(s)\leq -H(g(s)),\quad \forall s\in \mathbb{R}_{+},
\label{e1.7}
\end{equation}
where $H$ is a non-negative function satisfying some hypotheses. Recently,
the asymptotic stability of Timoshenko system with a finite history was
considered in \cite{m8} under \eqref{e1.7} with weaker conditions on $H$ than those
imposed in \cite{a2}. The general relation between the decay rate for the energy
and that of $g$ obtained in \cite{m8} holds without imposing restrictive
assumptions on the behavior of $g$ at infinity.

Condition \eqref{e1.3} implies that $g$ converges to zero at infinity faster than
$t^{-2}$. For Timoshenko system with an infinite history, \eqref{e1.3} is, to our
best knowledge, the weakest condition considered in the literature \cite{m9,m10}
on the growth of $g$ at infinity.

Our aim in this work is to establish a general decay estimate for the
solutions of systems \eqref{eP} in the case \eqref{e1.1} as well as in the opposite one,
and give applications to coupled Timoshenko-heat systems \eqref{eP1}-\eqref{eP4}
(see Section 6) and Timoshenko-thermoelasticity systems of type III
\eqref{eP5}-\eqref{eP6}
(see Section 7). We prove that the stability of these systems holds for
kernels $g$ having more general decay (which can be arbitrary close to
$t^{-1}$), and we obtain general decay results from which the exponential and
polynomial decay results of \cite{f1,m9,m10,m11} are only special cases.
In addition, we improve the results of \cite{m9,m10} by getting, in some particular
cases, a better decay rate of solutions. The proof is based on the
multipliers method and a new approach introduced by the first author in \cite{g5}
for a class of abstract hyperbolic systems with an infinite history.

The paper is organized as follows. In Section 2, we state some hypotheses
and present our stability results for \eqref{eP}. The proofs of these stability
results for \eqref{eP} will be given in Sections 3 when \eqref{e1.1} holds, and in
Section 4 when \eqref{e1.1} does not hold. In Section 5, we discuss the
well-posedness and smoothness of the solution of \eqref{eP}. Our stability
results of $\eqref{eP1}-\eqref{eP4}$ and $\eqref{eP5}-\eqref{eP6}$ will be given and proved in Sections
6 and 7, respectively. Finally, we conclude our paper by giving some general
comments in Section 8.

\section{Preliminaries}

In this section, we state our stability results for problem \eqref{eP}. For this
purpose, we start with the following hypotheses:
\begin{itemize}
\item[(H1)] $g: \mathbb{R}_{+}\to \mathbb{R}_{+}$ is
a non-increasing differentiable function such that $g (0)>0$ and
\begin{equation}
l =k_2 -\int_0^{+\infty }g (s)ds >0.\label{e2.1}
\end{equation}

\item[(H2)] There exists an increasing strictly convex function
 $G:\mathbb{R}_{+}\to \mathbb{R}_{+}$ of class
$C^1 (\mathbb{R}_{+})\cap C^2(]0,+\infty [)$ satisfying
\[
G(0)=G'(0)=0\quad \text{and}\quad \lim_{t\to +\infty}G'(t)=+\infty
\]
such that
\begin{equation}
\int_0^{+\infty }{\frac{{g(s)}}{{G^{-1}(-g'(s))}}}ds
+\sup_{s\in \mathbb{R}_{+}}{\frac{{g(s)}}{{G^{-1}(-g'(s))}}}<+\infty.\label{e2.2}
\end{equation}
\end{itemize}

\begin{remark} \label{rmk2.1} \rm
Hypothesis (H2), which was
introduced by the first author in \cite{g5}, is weaker than the classical one
\eqref{e1.3} considered in \cite{m9,m10}. Indeed, \eqref{e1.3} implies
 (H2) with $G(t)=t^{2p}$
(because \eqref{e1.3} implies that $\int_0^{+\infty }\sqrt{g(t)}
dt<+\infty $; see \cite{m10}). On the other hand, for example, for
$g(t)=q_0(1+t)^{-q}$ with $q_0>0$ and $q\in ]1,2]$, (H2) is satisfied
with $G(t)=t^{r}$ for all $r>{\frac{{q+1}}{{q-1}}}$, but \eqref{e1.3} is not
satisfied.

In general, any positive function $g$ of class
$C^{1}(\mathbb{R}_{+})$ with $g'<0$ satisfies (H2) if it is integrable on
$\mathbb{R}_{+}$. However, it does not satisfy \eqref{e1.3} if it does
not converge to zero at infinity faster than $t^{-2}$. Because the
integrability of $g$ on $\mathbb{R}_{+}$ is necessary for the
well-posedness of \eqref{eP}, then (H2) seems to be very realistic. In addition,
(H2) allows us to improve the results of \cite{m9,m10} by getting, in some
particular cases, stronger decay rates (see Examples \ref{examp2.1}--\ref{examp8.1} below).
\end{remark}

 We consider, as in \cite{m9,m10}, the classical energy functional
associated with \eqref{eP} as follows:
\begin{equation}
E(t)  =  \frac{1}{2}\int_0^{L}\Bigl(\rho_1\varphi _t^2+\rho_2\psi_t^2
+k_1 (\varphi_x+\psi)^2+\Bigl(k_2-\int_0^{+\infty}g (s)ds\Bigr)\psi _x^2\Bigr) dx
+\frac{1}{2}g\circ \psi _x, \label{e2.3}
\end{equation}
where, for $v:\mathbb{R}\to L^2(]0,L[)$ and
$\phi :\mathbb{R}_{+}\to \mathbb{R}_{+}$,
\begin{equation}
\phi \circ v=\int_0^{L}\int_0^{+\infty }\phi (s)(v(t)-v(t-s))^2\,ds\,dx.
\label{e2.4}
\end{equation}
Thanks to \eqref{e2.1}, the expression
 $\int_0^{L}\Bigl(k_1(\varphi _x+\psi )^2+\Bigl(k_2-\int_0^{+\infty
}g(s)ds\Bigr)\psi _x^2\Bigr)dx$ defines a norm on
$\bigl(H_0^{1}(]0,L[) \bigr)^2$, for $(\varphi ,\psi )$, equivalent
to the one induced by $\bigl(H^{1}(]0,L[)\bigr)^2$, where
$H_0^{1}(]0,L[)=\{v\in H^{1}(]0,L[),\,v(0)=v(L)=0\}$.

 Here, we define the energy space $\mathcal{H}$ (for more
details see Section 5) by:
\[
\mathcal{H}:=\bigl(H_0^{1}(]0,L[)\bigr)^2\times \bigl(L^2(]0,L[)\bigr)
^2\times L_{g}
\]
with
\[
L_{g}=\{v:\mathbb{R}_{+}\to
H_0^{1}(]0,L[),\,\int_0^{L}\int_0^{+\infty
}g(s)v_x^2(s)\,ds\,dx<+\infty \}.
\]

Now, we give our first main stability result, which concerns the case \eqref{e1.1}.

\begin{theorem} \label{thm2.1}
Assume that \eqref{e1.1}, {\rm (H1)} and
{\rm (H2)} are satisfied, and let $U_0 \in \mathcal{H}$ (see Section 5) such that
\begin{equation}
\exists M_0 \geq 0 : \| \eta_{0x} (s)\|_{L^2 (]0,L[)}\leq M_0
,\quad\forall s>0.\label{e2.5}
\end{equation}
Then there exist positive constants $c',\,c''$ and
$\epsilon_0$ (depending continuously on $E(0)$) for which $E$ satisfies
\begin{equation}
E(t)\leq c''G_1^{-1} (c't),\quad \forall t\in \mathbb{R}_{+}, \label{e2.6}
\end{equation}
where
\begin{equation}
G_1 (s)=\int_{s}^{1}{\frac{1}{{\tau G'(\epsilon_0 \tau)}}}d\tau\,\,
(s\in ]0,1]).\label{e2.7}
\end{equation}
\end{theorem}

  \begin{remark} \label{rmk2.2} \rm
1. Because $\lim_{t\to 0^+}G_1 (t)=+\infty$, we have the strong stability
 of \eqref{eP}; that is,
\begin{equation}
\lim_{t\to +\infty }E(t)=0.\label{e2.8}
\end{equation}

2. The decay rate given by \eqref{e2.6} is weaker than the exponential
decay
\begin{equation}
E(t)\leq c''e^{-{c'}t},\quad \forall t\in \mathbb{R}_{+}.\label{e2.9}
\end{equation}
The estimate \eqref{e2.6} coincides with \eqref{e2.9} when $G=Id$.
\end{remark}

 Now, we treat the case when \eqref{e1.1} does not hold.

 \begin{theorem} \label{thm2.2}
Assume that {\rm (H1)} and {\rm (H2)} are
satisfied, and let $U_0\in D(A)$ (see Section 5) such that
\begin{equation}
\exists M_0\geq 0:\max \{\| \eta _{0x}(s)\|
_{L^2(]0,L[)},\,\| \partial _{s}\eta _{0x}(s)\|
_{L^2(]0,L[)}\}\leq M_0,\quad \forall s>0.\label{e2.10}
\end{equation}
Then there exist positive constants $C$ and $\epsilon _0$ (depending
continuously on $\| U_0\| _{D(A)}$) such that
\begin{equation}
E(t)\leq G_0^{-1}({\frac{{C}}{{t}}}),\quad \forall t>0,\label{e2.11}
\end{equation}
where
\begin{equation}
G_0(s)=s{G}'(\epsilon _0s)\,\,(s\in \mathbb{R}_{+}).
\label{e2.12}
\end{equation}
\end{theorem}

  \begin{remark} \label{rmk2.3} \rm
Estimate \eqref{e2.11} implies \eqref{e2.8}
but it is weaker than
\begin{equation}
E(t)\leq {\frac{{C}}{{t}}},\quad \forall t>0,\label{e2.13}
\end{equation}
which, in turn, coincides with \eqref{e2.11} when $G=Id$. When $g$ satisfies the
classical condition \eqref{e1.3} with $p=1$ (that is $g$ converges exponentially to
zero at infinity), it is well known (see \cite{m11}) that \eqref{e2.9} and \eqref{e2.13} are
satisfied (without the restrictions \eqref{e2.5} and \eqref{e2.10}).
\end{remark}

 \begin{example} \label{examp2.1} \rm
 Let us give two examples to illustrate our
general decay estimates and show how they generalize and improve the ones
known in the literature. For other examples, see \cite{g5}.
\end{example}

Let $g(t)={\frac{{d}}{{(2+t)(\ln (2+t))^{q}}}}$ for $q>1$, and $d>0$ small
enough so that \eqref{e2.1} is satisfied. The classical condition \eqref{e1.3} is not
satisfied, while (H2) holds with
\[
G(t)=\int_0^ts^{1/p}e^{-s^{-1/p}}ds\quad \text{for any } p\in ]0,q-1[.
\]
Indeed, here \eqref{e2.2} depends only on the growth of $G$ near zero. Using the
fact that $G(t)\leq t^{{\frac{1}{p}}+1}e^{-t^{-1/p}}$, we can
see that $G(t(\ln t)^{r}g(t))\leq -g'(t)$, for $t$ near infinity
and for any $r\in ]1,q-p[$, which implies \eqref{e2.2}.
Then \eqref{e2.6} takes the form
\begin{equation}
E(t)\leq {\frac{{C}}{{(\ln (t+2))^{p}}}},\quad \forall t\in
\mathbb{R}_{+},\; \forall p\in ]0,q-1[.\label{e2.14}
\end{equation}
Because $G_0(s)\geq e^{-cs^{-1/p}}$, for some positive
constant $c$ and for $s$ near zero, then also \eqref{e2.11} implies \eqref{e2.14}. \vskip
0,1truecm 2. Let $g(t)=de^{-(\ln (2+t))^{q}}$, for $q>1$, and $d>0$ small
enough so that \eqref{e2.1} is satisfied. Hypothesis (H2) holds with
\[
G(t)=\int_0^t(-\ln s)^{1-{\frac{1}{p}}}e^{-(-\ln s)^{1/p} }ds
\quad \text{for $t$ near zero and for any $p\in ]1,q[$},
\]
since condition \eqref{e2.2} depends only on the growth of $G$ at zero, and when $t$
goes to infinity and $p\in ]1,q[$, $G(t^{r}g(t))\leq -g'(t)$, for
any $r>1$. Then \eqref{e2.6} becomes
\begin{equation}
E(t)\leq ce^{-C(\ln (1+t))^{p}},\quad \forall t\in \mathbb{R}
_{+},\;\forall p\in ]1,q[.\label{e2.15}
\end{equation}
Condition \eqref{e2.3} holds also with $G(s)=s^{p}$, for any $p>1$. Then \eqref{e2.11}
gives
\begin{equation}
E(t)\leq {\frac{{C}}{{(t+1)^{\frac{{1}}{{p}}}}}},\quad \forall t\in
\mathbb{R}_{+},\;\forall p>1.\label{e2.16}
\end{equation}
Here, the decay rates of $E$ in \eqref{e2.15} and \eqref{e2.16} are arbitrary
 close to the one of $g$ and $t^{-1}$, respectively. This improves the
results of \cite{m9,m10}
in case \eqref{e1.1}, where only the polynomial decay was obtained.

\section{Proof of Theorem \ref{thm2.1}}

We will use $c$ (sometimes $c_{\tau}$, which depends on some parameter $\tau
$), throughout this paper, to denote a generic positive constant, which
depends continuously on the initial data and can be different from step to
step.

Following the classical energy method (see \cite{a1,a3,f1,g7,m9,m10,m11,m12}
 for example), we will construct a Lyapunov function $F$, equivalent to $E$ and
satisfying \eqref{e3.24} below. For this purpose we establish several lemmas, for
all $U_0\in D(A)$ satisfying \eqref{e2.5}, so all the calculations are justified.
By a simple density argument ($D(A)$ is dense in $\mathcal{H}$; see Section
5), \eqref{e2.6} remains valid, for any $U_0\in \mathcal{H}$ satisfying \eqref{e2.5}. On
the other hand, if $E(t_0)=0$, for some $t_0\in \mathbb{R}
_{+} $, then $E(t)=0$, for all $t\geq t_0$ ($E$ is non-increasing thanks
to \eqref{e3.1} below) and thus the stability estimates \eqref{e2.6} and \eqref{e2.11} are
satisfied. Therefore, without loss of generality, we assume that $E(t)>0$,
for all $t\in \mathbb{R}_{+}$.

To obtain estimate \eqref{e3.18} below, we prove Lemmas \ref{lem3.1}-\ref{lem3.10},
where the
proofs are inspired from the classical multipliers method used in
\cite{a1,a3,e1,f1,g7,k2,l1,m1,m2,m9,m10,m11,m12}.
 Our main contribution in this
section is the use of the new approach of \cite{g5} to prove \eqref{e3.19}
 below under assumption (H2).

\begin{lemma} \label{lem3.1}
The energy functional $E$ defined by \eqref{e2.3} satisfies
\begin{equation}
E'(t)={\frac{1}{2}}g'\circ \psi _x\leq 0.\label{e3.1}
\end{equation}
\end{lemma}

 \begin{proof}
 By multiplying the first two equations in
\eqref{eP}, respectively, by $\varphi _t$ and $\psi _t$, integrating over
$]0,L[$, and using the boundary conditions, we obtain \eqref{e3.1} (note that $g$ is
non-increasing). The estimate \eqref{e3.1} shows that \eqref{eP} is dissipative, where
the entire dissipation is given by the infinite history.
\end{proof}

 \begin{lemma}[\cite{g6}] \label{lem3.2}
The following inequalities
hold, where $g_0=\int_0^{+\infty }g(s)ds$:
\begin{gather}
\Bigl(\int_0^{+\infty }g(s)(\psi _x(t)-\psi _x(t-s))ds\Bigr)^2\leq
g_0\int_0^{+\infty }g(s)(\psi _x(t)-\psi _x(t-s))^2ds,\label{e3.2}
\\
\Bigl(\int_0^{+\infty }g'(s)(\psi _x(t)-\psi _x(t-s))ds\Bigr)
^2\leq -g(0)\int_0^{+\infty }g'(s)(\psi _x(t)-\psi
_x(t-s))^2ds.\label{e3.3}
\end{gather}
\end{lemma}

 As in \cite{g7,m10}, we consider the following case.

\begin{lemma}[\cite{m7,m10}] \label{lem3.3}
The functional
\begin{equation}
I_1(t)=-\rho _2\int_0^{L}\psi _t\int_0^{+\infty }g(s)(\psi
(t)-\psi (t-s))\,ds\,dx\label{e3.4}
\end{equation}
satisfies, for any $\delta >0$,
\begin{equation}
\begin{split}
I_1'(t) & \leq  -\rho_2 \Bigl(\int_0^{+\infty}g (s)ds
 -\delta \Bigr)\int_0^{L}\psi_t^2dx\\
& \quad +\delta\int_0^L (\psi_x^2+(\varphi_x +\psi)^2 )dx
+ c_{\delta} g \circ\psi_x-c_{\delta} g'\circ \psi_x .
\end{split}\label{e3.5}
\end{equation}
\end{lemma}

  As in \cite{m10,m11}, we consider the following result.

\begin{lemma}[\cite{a3,m7,m10}] \label{lem3.4.}
The functional
\[
I_2(t)=-\int_0^{L}(\rho _1\varphi \varphi _t+\rho _2\psi \psi _t)dx
\]
satisfies
\begin{equation}
I_2'(t)\leq -\int_0^{L}(\rho _1\varphi _t^2+\rho _2\psi
_t^2)dx+\int_0^{L}(k_1(\varphi _x+\psi )^2+c\psi
_x^2)dx+cg\circ \psi _x.\label{e3.6}
\end{equation}
\end{lemma}

  Similarly to \cite{a3}, we consider the following result.

 \begin{lemma} \label{lem3.5}
The functional
\[
I_3(t)=\rho _2\int_0^{L}\psi _t(\varphi _x+\psi )dx+{\frac{{
k_2\rho _1}}{{k_1}}}\int_0^{L}\psi _x\varphi _tdx-{\frac{{\rho
_1}}{{k_1}}}\int_0^{L}\varphi _t\int_0^{+\infty }g(s)\psi
_x(t-s)\,ds\,dx
\]
satisfies, for any $\epsilon >0$,
\begin{equation}
\begin{split}
I_3'(t) &\leq  \frac{1}{2\epsilon} \Bigl(k_2 \psi_x (L,t)
-\int_0^{+\infty}g (s)\psi_x(L,t-s)ds\Bigr)^2
\\
&\quad +\frac{1}{2\epsilon} \Bigl(k_2 \psi_x (0,t)
 -\int_0^{+\infty}g (s)\psi_x(0,t-s)ds\Bigr)^2
\\
&\quad  +\frac{\epsilon}{2} (\varphi_x^2 (L,t)+\varphi_x^2 (0,t))
 -k_1 \int_0^{L}(\varphi_x+\psi)^2dx+\rho_2\int_0^{L}\psi_t^2dx
\\
&\quad +\epsilon \int_0^{L}\varphi_t^2dx-c_{\epsilon} g'\circ \psi_x
 +({{k_2\rho_1}\over{k_1}}-\rho_2)\int_0^L \varphi_t\psi_{xt} dx.
\end{split}\label{e3.7}
\end{equation}
\end{lemma}

\begin{proof} Using the equations in \eqref{eP} and arguing as
before, we have
\begin{align*}
I_3'(t) &=  \rho_2\int_0^{L}(\varphi_{xt}+\psi_t)\psi_tdx
+{{k_2\rho_1}\over{k_1}}\int_0^{L}\psi_{xt}\varphi_tdx
\\
&\quad  +\int_0^{L}(\varphi_x+\psi)\Bigl(k_2\psi_{xx}
 -\int_0^{+\infty}g (s)\psi_{xx} (t-s) ds-k_1 (\varphi_x+\psi)\Bigr)dx
\\
&\quad  +k_2 \int_0^{L}\psi _x(\varphi_x+\psi)_x dx
-\int_0^{L} (\varphi_x+\psi)_x\Bigl(\int_0^{+\infty}g(s)\psi_x(t-s)ds\Bigr)dx
\\
&\quad -{{\rho_1}\over{k_1}}\int_0^{L}\varphi_t\Bigl(g(0)\psi_x+
\int_0^{+\infty}g'(s)\psi_x(t-s)ds\Bigr)dx
\\
&=  -k_1\int_0^{L}(\varphi_x+\psi)^2dx +\rho_2\int_0^{L}\psi_t^2dx
+({{k_2\rho_1}\over{k_1}}-\rho_2)\int_0^L \varphi_t\psi_{xt} dx
\\
&\quad  +\Bigl[\Bigl(k_2\psi_x-\int_0^{+\infty} g (s)\psi_x(t-s)ds\Bigr)
(\varphi_x +\psi)\Bigr]_{x=0}^{x=L}
\\
&\quad  +{{\rho_1}\over {k_1}}\int_0^{L}\varphi_t\int_0^{+\infty}g'(s)(\psi_x(t)
 -\psi_x(t-s))\,ds\,dx.
\end{align*}
By using \eqref{e3.3} and Young's inequality (for the last three terms of this
equality), \eqref{e3.7} is established.
\end{proof}

To estimate the boundary terms in \eqref{e3.7}, we proceed as in \cite{a3}.

\begin{lemma}[\cite{a3}] \label{lem3.6}
Let $m(x)=2-{\frac{{4}}{{L}}}x$. Then, for any
$\epsilon >0$, the functionals
\begin{gather*}
I_4=\rho _2\int_0^{L}m(x)\psi _t\Bigl(k_2\psi
_x-\int_0^{+\infty }g(s)\psi _x(t-s)ds\Bigr)dx,
\\
I_5=\rho _1\int_0^{L}m(x)\varphi _t\varphi _xdx
\end{gather*}
satisfy
\begin{equation}
\begin{aligned}
I_4'(t)
 &\leq  -\Bigl(k_2\psi_x(L,t)-\int_0^{+\infty}g (s)\psi_x(L,t-s)ds\Bigr)^2\\
&\quad -\Bigl(k_2 \psi_x(0,t)-\int_0^{+\infty}g (s)\psi_x(0,t-s)ds\Bigr)^2
 +\epsilon k_1\int_0^{L}(\varphi_x+\psi)^2dx\\
&\quad +c(1+{\frac{1}{{\varepsilon}}})\int_0^{L}\psi_x^2dx  +c_{\epsilon} g\circ \psi_x
+c\int_0^{L} \psi_t^2 dx -cg'\circ \psi_x
\end{aligned}\label{e3.8}
\end{equation}
and
\begin{equation}
I_5'(t)\leq -k_1(\varphi _x^2(L,t)+\varphi
_x^2(0,t))+c\int_0^{L}(\varphi _t^2+\varphi _x^2+\psi
_x^2)dx.\label{e3.9}
\end{equation}
\end{lemma}

\begin{lemma} \label{lem3.7}
For any $\epsilon \in ]0,1[$, the functional
\[
I_6(t)=I_3(t)+\frac{1}{{2\epsilon }}I_4(t)+{\frac{{\epsilon }}{{2k_1}
}}I_5(t)
\]
satisfies
\begin{equation}
\begin{aligned}
I_6'(t)
&\leq -({\frac{k_1}{2}}-c\epsilon)\int_0^{L}(\varphi_x+\psi)^2dx
+c\epsilon \int_0^{L}\varphi_t^2dx+{\frac{c}{\epsilon}}\int_0^{L}\psi_t^2dx
\\
&\quad  +{\frac{c}{\epsilon^2}}\int_0^{L}\psi_x^2dx
+c_{\epsilon}(g\circ \psi_x-g'\circ \psi_x)
+({\frac{\rho_1 k_2}{k_1}}-\rho_2)\int_0^{L}\varphi_t\psi_{xt}dx.
\end{aligned}\label{e3.10}
\end{equation}
\end{lemma}

\begin{proof}  By using Poincar\'e's inequality for
$\psi $, we have
\[
\int_0^{L}\varphi _x^2dx\leq 2\int_0^{L}(\varphi _x+\psi)^2dx+2\int_0^{L}\psi ^2dx.
\]
Then \eqref{e3.7}-\eqref{e3.9} imply \eqref{e3.10}.
\end{proof}

\begin{lemma} \label{lem3.8}
The functional $I_7(t)=I_6(t)+{\frac{1}{{8}}}I_2(t)$ satisfies
\begin{equation}
\begin{aligned}
I_7'(t)
&\leq  -{\frac{k_1}{4}}\int_0^{L}(\varphi_x+\psi)^2dx
-{{\rho_1}\over{16}} \int_0^{L}\varphi_t^2dx+c\int_0^{L}(\psi_t^2 +\psi_x^2)dx
\\
&\quad  +c(g\circ\psi_x-g'\circ \psi_x)
+({{\rho_1 k_2}\over{k_1}}-\rho_2)\int_0^{L}\varphi_t\psi_{xt}dx.
\end{aligned}\label{e3.11}
\end{equation}
\end{lemma}

\begin{proof}
 Inequalities \eqref{e3.10} (with $\epsilon \in ]0,1[$ small enough) and
\eqref{e3.6} imply \eqref{e3.11}.
\end{proof}
Now, as in \cite{a3}, we use a function $w$ to get a crucial estimate.

\begin{lemma} \label{lem3.9}
The function
\begin{equation}
w(x,t)=-\int_0^{x}\psi (y,t)dy+{\frac{1}{L} }\Bigl(\int_0^{L}\psi
(y,t)dy\Bigr)x\label{e3.12}
\end{equation}
satisfies the estimates
\begin{gather}
\int_0^{L}w_x^2dx\leq c\int_0^{L}\psi ^2dx,\quad \forall t\geq 0,
\label{e3.13} \\
\int_0^{L}w_t^2dx\leq c\int_0^{L}\psi _t^2dx,\quad \forall t\geq
0.\label{e3.14}
\end{gather}
\end{lemma}

\begin{proof}
 We just have to calculate $w_x$ and use H\"older's inequality
to get \eqref{e3.13}. Applying \eqref{e3.13} to $w_t$, we obtain
\[
\int_0^{L}w_{xt}^2dx\leq c\int_0^{L}\psi _t^2dx,\quad \forall t\geq 0.
\]
Then, using Poincar\'e's inequality for $w_t$ (note that
$w_t(0,t)=w_t(L,t)=0$), we arrive at \eqref{e3.14}.
\end{proof}

\begin{lemma}[\cite{a3,m10}] \label{lem3.10}
For any $\epsilon \in ]0,1[$, the functional
\[
I_8(t)=\int_0^{L}(\rho _2\psi \psi _t+\rho _1w\varphi _t)dx
\]
satisfies
\begin{equation}
I_8'(t)\leq -{\frac{{l}}{2}}\int_0^{L}\psi _x^2dx+{\frac{c}{
{\epsilon }}}\int_0^{L}\psi _t^2dx+\epsilon \int_0^{L}\varphi
_t^2dx+cg\circ \psi _x,\label{e3.15}
\end{equation}
where $l$ is defined by \eqref{e2.1}.
\end{lemma}

 Now, for $N_1,N_2,N_3>0 $, let
\begin{equation}
I_{9}(t)=N_1E(t)+N_2I_1(t)+N_3I_8(t)+I_7(t).\label{e3.16}
\end{equation}
By combining \eqref{e3.1}, \eqref{e3.5}, \eqref{e3.11} and \eqref{e3.15},
taking $\delta ={\frac{{k_1}}{{8N_2}}}$ in \eqref{e3.5} and noting that
$g_0=\int_0^{+\infty}g(s)ds<+\infty $ (thanks to (H1)), we obtain
\begin{equation}
\begin{aligned}
I_9'(t)
&\leq  -(\frac{l{N_3}}{{2}}-c)\int_0^{L}\psi_x^2dx
-(\frac{\rho_1}{16}-\epsilon N_3)\int_0^{L}\varphi _t^2dx
\\
&\quad  -\int_0^{L}\Bigl(N_2\rho_2 g_0-\frac{{cN_3}}{{\epsilon }}-c\Bigr)\psi_t^2dx
 -{\frac{{k_1}}{{8}}}\int_0^{L}(\varphi_x +\psi)^2dx
\\
&\quad  +c_{N_2, N_3 }g\circ \psi_x+({\frac{{N_1}}{{2}}}-c_{N_2} )g'\circ \psi_x
+({{\rho_1 k_2}\over {k_1}}-\rho_2)\int_0^L\varphi_t\psi_{xt}dx.
\end{aligned}\label{e3.17}
\end{equation}

At this point, we choose $N_3$ large enough so that
 $\frac{l{N_3}}{{2}} -c>0$, then $\epsilon\in ]0,1[$ small enough so that
 $\frac{\rho_1}{16}-\epsilon N_3>0$. Next, we pick $N_2$ large enough so
that $N_2\rho_2 g_0-\frac{{cN_3}}{{\epsilon }}-c>0$.

On the other hand, by definition of the functionals $I_1 -I_8$ and $E$,
there exists a positive constant $\beta$ satisfying
$|N_2I_1 +N_3I_8+I_7 |\leq \beta E$, which implies that
\[
(N_1 -\beta) E \leq I_9\leq (N_1 +\beta)E,
\]
then we choose $N_1$ large enough so that
${\frac{{N_1}}{{2}}}-c_{N_2}\geq0$ and $N_1 >\beta$ (that is $I_9 \sim E$).
Consequently, using the definition \eqref{e2.3} of $E$, from \eqref{e3.17}
we obtain
\begin{equation}
I_9'(t)\leq -cE(t)+cg\circ \psi_x+({\frac{{\rho_1 k_2}}{{k_1}}}
-\rho_2)\int_0^L\varphi_t\psi_{xt}dx.\label{e3.18}
\end{equation}

Now, we estimate the term $g\circ \psi _x$ in \eqref{e3.18} in function of
$E'$ by exploiting (H2). This is the main difficulty in treating the
infinite history term.

\begin{lemma} \label{lem3.11}
For any $\epsilon _0>0$, the following inequality holds:
\begin{equation}
G'(\epsilon _0E(t))g\circ \psi _x\leq -cE'(t)+c\epsilon _0E(t)G'(\epsilon _0E(t)).
\label{e3.19}
\end{equation}
\end{lemma}

\begin{proof}
This lemma was proved by the first author (see \cite[Lemma 3.4]{g5}) for an abstract
system with infinite history. The proof is based on some classical properties
of convex functions (see \cite{a4,e1} for example), in particular, the general
Young inequality. Let us give a short proof of \eqref{e3.19} in the particular
case \eqref{eP} (see \cite{g5} for details).

Because $E$ is non-increasing, then ($\eta_0$ is defined in Section 5)
\begin{align*}
\int_0^L (\psi_x (t)-\psi_x (t-s))^2 dx
&\leq  4\sup_{\tau\in \mathbb{R}} \int_0^L \psi_x^2 (\tau) dx
\\
&\leq  4\sup_{\tau >0} \int_0^L \psi_{0x}^2 (\tau ) dx+cE(0)
\\
&\leq  c\sup_{\tau>0} \int_0^L \eta_{0x}^2 (\tau) dx+cE(0).
\end{align*}
Thus, thanks to \eqref{e2.5}, there exists a positive constant $m_1 =c(M_0^2 +E(0))$
(where $M_0$ is defined in \eqref{e2.5}) such that
\[
\int_0^L (\psi_x (t)-\psi_x (t-s))^2 dx\leq m_1 , \quad\forall t,\,s\in
\mathbb{R}_+ .
\]
Let $\epsilon_0 ,\tau_1 , \tau_2 >0$ and $K(s)=s/G^{-1}(s)$
which is non-decreasing. Then,
\[
K\Bigl(-\tau_2 g'(s)\int_0^L (\psi_x (t)-\psi_x (t-s))^2 dx\Bigr)
\leq K(-m_1 \tau_2 g'(s)).
\]
Using this inequality, we arrive at
\begin{align*}
g\circ \psi_x
&=  {{1}\over{\tau_1 G' (\epsilon_0 E(t))}}\int_0^{+\infty}G^{-1}
 \Bigl(-\tau_2 g'(s)\int_0^L (\psi_x (t)-\psi_x (t-s))^2 dx\Bigr)
\\
&\quad  \times {{\tau_1 G'(\epsilon_0 E(t))g(s)}\over{-\tau_2 g'(s)}}K
 \Bigl(-\tau_2 g'(s)\int_0^L (\psi_x (t)-\psi_x (t-s))^2 dx\Bigr)ds
\\
&\leq  {{1}\over{\tau_1 G' (\epsilon_0 E(t))}}\int_0^{+\infty}G^{-1}
 \Bigl(-\tau_2 g'(s)\int_0^L (\psi_x (t)-\psi_x (t-s))^2 dx\Bigr)
\\
&\quad  \times {{\tau_1 G'(\epsilon_0 E(t))g(s)}\over{-\tau_2 g'(s)}}
 K(-m_1 \tau_2 g'(s))ds
\\
&\leq  {{1}\over{\tau_1 G' (\epsilon_0 E(t))}}\int_0^{+\infty}G^{-1}
 \Bigl(-\tau_2 g'(s)\int_0^L (\psi_x (t)-\psi_x (t-s))^2 dx\Bigr)
\\
&\quad  \times {{m_1\tau_1 G'(\epsilon_0 E(t))g(s)}\over{G^{-1}(-m_1\tau_2 g'(s))}}ds.
\end{align*}
We denote by $G^*$ the dual function of $G$ defined by
\[
G^* (t)=\sup_{s\in \mathbb{R}_{+}} \{ts-G(s)\} = t{G'}^{-1}
(t)-G({G'}^{-1} (t)),\quad\forall t\in \mathbb{R}_{+} .
\]
Using Young's inequality: $t_1 t_2\leq G(t_1)+G^* (t_2)$, for
\[
t_1 =G^{-1}\Bigl(-\tau_2 g'(s)\int_0^L (\psi_x (t)-\psi_x (t-s))^2
dx\Bigr),\quad
t_2 ={\frac{{m_1\tau_1 G'(\epsilon_0 E(t))g(s)}}{{G^{-1}(-m_1\tau_2 g'(s))}}},
\]
we obtain
\[
g\circ \psi_x 
\leq {\frac{{-\tau_2}}{{\tau_1 G'(\epsilon_0 E(t))}}
}g'\circ \psi_x +{\frac{1}{{\tau_1 G'(\epsilon_0 E(t))}}}
\int_0^{+\infty}G^{*}\Bigl({\frac{{m_1\tau_1 G'(\epsilon_0 E(t))g(s)
}}{{G^{-1}(-m_1\tau_2 g'(s))}}}\Bigr)ds.
\]
Using \eqref{e3.1} and the fact that $G^* (t)\leq t{G'}^{-1} (t)$, we obtain
\begin{align*}
g\circ \psi_x
&\leq {\frac{{-2\tau_2}}{{\tau_1 G'(\epsilon_0 E(t))}}}E'(t) \\
&\quad +m_1\int_0^{+\infty}{\frac{{g(s)}}{{G^{-1}(-m_1\tau_2
g'(s))}}}{G'}^{-1} 
\Bigl({\frac{{m_1\tau_1 G'(\epsilon_0 E(t))g(s)}}{{G^{-1}
(-m_1\tau_2 g'(s))}}}\Bigr)ds.
\end{align*}
Thanks to \eqref{e2.2}, $\sup_{s\in \mathbb{R}_+}{\frac{{g (s)}}{{
G^{-1}(-g'(s))}}}=m_2 <+\infty$. Then, using the fact that
${G'}^{-1}$ is non-decreasing (thanks to (H2)) and choosing
$\tau_2 = 1/m_1$, we obtain
\[
g\circ \psi_x\leq {\frac{{-2}}{{m_1\tau_1 G'(\epsilon_0 E(t))}}}
E'(t) +m_1 {G'}^{-1}\Bigl(m_1 m_2 \tau_1 G'(\epsilon_0 E(t))\Bigr) 
\int_0^{\infty} {\frac{{g(s)}}{{G^{-1}(-g'(s))}}}ds.
\]
Now, choosing $\tau_1 = {\frac{{1}}{{m_1 m_2}}}$ and using the fact that
$\int_0^{+\infty}{\frac{{g(s)}}{{G^{-1}(-g'(s))}}}ds=m_3 <+\infty$
(thanks to \eqref{e2.2}), we obtain
\[
g\circ \psi_x\leq {\frac{{-c}}{{G'(\epsilon_0 E(t))}}}E'(t)+c\epsilon_0 E(t),
\]
which implies \eqref{e3.19} with $c=\max \{2m_2 ,m_1 m_3\}$.
\end{proof}

 Now, going back to the proof of Theorem \ref{thm2.1}, multiplying \eqref{e3.18} by
$G'(\epsilon_0 E(t))$ and using \eqref{e3.19}, we obtain
\begin{align*}
G' (\epsilon_0 E(t))I_{9}' (t)
&\leq  -(c-c\epsilon_0 )E(t)G' (\epsilon_0 E(t))-cE'(t)
\\
&\quad  +({{\rho_1 k_2}\over {k_1}}-\rho_2)G' (\epsilon_0 E(t))
\int_0^L\varphi_t\psi_{xt}dx.
\end{align*}
Choosing $\epsilon_0$ small enough, we obtain
\begin{equation}
\begin{aligned}
G' (\epsilon_0 E(t))I_{9}' (t)+cE'(t) &\leq  -cE(t)G' (\epsilon_0 E(t))
\\
&\quad  +({{\rho_1 k_2}\over {k_1}}-\rho_2)G' (\epsilon_0 E(t))
\int_0^L\varphi_t\psi_{xt}dx.
\end{aligned}\label{e3.20}
\end{equation}
Let
$$
F=\tau \Bigl(G'(\epsilon_0 E)I_{9}+cE\Bigr),\label{e3.21}
$$
where $\tau>0$. We have $F\sim E$ (because $I_{9}\sim E$ and $G'(\epsilon_0 E)$
is non-increasing) and, using \eqref{e3.20},
\begin{equation}
F'(t)\leq -c\tau E(t)G'(\epsilon_0 E(t))+\tau ({\frac{{
\rho_1 k_2}}{{k_1}}}-\rho_2)G'(\epsilon_0
E(t))\int_0^L\varphi_t\psi_{xt}dx. \label{e3.22}
\end{equation}
Now, thanks to \eqref{e1.1}, the last term of \eqref{e3.22} vanishes.
Then, for $\tau>0$ small enough such that
\begin{equation}
F\leq E\quad\text{and}\quad F(0)\leq 1,\label{e3.23}
\end{equation}
we obtain, for $c'=c\tau >0$,
\begin{equation}
F'\leq -c'FG'(\epsilon_0 F).\label{e3.24}
\end{equation}
This implies that $(G_1 (F))'\geq c'$, where $G_1$
is defined by \eqref{e2.7}. Then, by integrating over $[0,t]$, we obtain
\[
G_1 (F (t))\geq c't+G_1 (F (0)).
\]
Because $F (0)\leq 1$, $G_1 (1)=0$ and $G_1$ is decreasing, we obtain $G_1
(F (t))\geq c't$, which implies that $F (t)\leq G_1^{-1} (c't)$.
The fact that $F\sim E$ gives \eqref{e2.6}. This completes the proof of
Theorem \ref{thm2.1}.

\section{Proof of Theorem \ref{thm2.2}}

In this section, we treat the case when \eqref{e1.1} does not hold, which is more
realistic from the physics point of view. We will estimate the last term of
\eqref{e3.22} using the following system  resulting from differentiating \eqref{eP}
with respect to time,
\begin{equation}
\begin{gathered}
\rho _1\varphi _{ttt}-k_1(\varphi _{xt}+\psi _t)_x=0, \\
\rho _2\psi _{ttt}-k_2\psi _{xxt}+k_1(\varphi _{xt}+\psi
_t)+\int_0^{+\infty }g(s)\psi _{xxt}(t-s)ds=0, \\
\varphi _t(0,t)=\psi _t(0,t)=\varphi _t(L,t)=\psi _t(L,t)=0.
\end{gathered} \label{etP}
\end{equation}
System \eqref{etP} is well posed for initial data $U_0\in D(A)$ (see
Section 5). Let ${\tilde{E}}$ be the second-order energy (the energy of 
\eqref{etP}) defined by ${\tilde{E}}(t)=E(U_t(t))$, where $E(U(t))=E(t)$
and $E$ is defined by \eqref{e2.3}. A simple calculation (as in \eqref{e3.1}) implies that
\begin{equation}
{\tilde{E}}'(t)={\frac{1}{2}}g'\circ \psi _{xt}\leq 0.\label{e4.1}
\end{equation}

The energy of high order is widely used in the literature to estimate some
terms (see \cite{a1,g4,m11,m12} for example). Our main contribution in this
section is obtaining estimate \eqref{e4.5} below under (H2).
Now, we proceed as in \cite{m11} to establish the following lemma.

\begin{lemma} \label{lem4.1}
For any $\epsilon >0$, we have
\begin{equation}
({\frac{{\rho _1k_2}}{{k_1}}}-\rho _2)\int_0^{L}\varphi _t\psi
_{xt}dx\leq \epsilon E(t)+c_{\epsilon }(g\circ \psi _{xt}-g'\circ
\psi _x).\label{e4.2}
\end{equation}
\end{lemma}

\begin{proof} By recalling that $g_0=\int_0^{+\infty}g(s)ds$, we have
\begin{equation}
\begin{aligned}
({{\rho_1 k_2}\over {k_1}}-\rho_2)\int_0^L\varphi_t\psi_{xt}dx
 &=  {{{{\rho_1 k_2}\over {k_1}}-\rho_2}\over{g_0}}\int_0^L \varphi_t\int_0^{+\infty}
 g(s)(\psi_{xt} (t)-\psi_{xt} (t-s))\,ds\,dx
\\
&\quad  +{{{{\rho_1 k_2}\over {k_1}}-\rho_2}\over{g_0}}\int_0^L
\varphi_t\int_0^{+\infty} g(s)\psi_{xt} (t-s)\,ds\,dx.
\end{aligned}\label{e4.3}
\end{equation}
Using Young's inequality and \eqref{e3.2} (for $\psi _{xt}$ instead of
$\psi _x $), we obtain, for all $\epsilon >0$,
\begin{align*}
&{\frac{{{\frac{{\rho _1k_2}}{{k_1}}}-\rho _2}}{{g_0}}}
\int_0^{L}\varphi _t\int_0^{+\infty }g(s)(\psi _{xt}(t)-\psi
_{xt}(t-s))\,ds\,dx
\\
&\leq c\int_0^{L}|\varphi _t|\int_0^{+\infty }g(s)|\psi _{xt}(t)-\psi
_{xt}(t-s)|\,ds\,dx\\
&\leq {\frac{{\epsilon }}{2}}E(t)+c_{\epsilon }g\circ \psi_{xt}.
\end{align*}
On the other hand, by integrating by parts and using \eqref{e3.3}, we obtain
\begin{align*}
&{\frac{{{\frac{{\rho _1k_2}}{{k_1}}}-\rho _2}}{{g_0}}}
\int_0^{L}\varphi _t\int_0^{+\infty }g(s)\psi _{xt}(t-s)\,ds\,dx\\
&=  {{{{\rho_1 k_2}\over {k_1}}-\rho_2}\over{g_0}}
 \int_0^L \varphi_t\Bigl(g(0)\psi_x+\int_0^{+\infty} g' (s)\psi_x (t-s)ds\Bigr)dx
\\
&=  {{{{\rho_1 k_2}\over {k_1}}-\rho_2}\over{g_0}}\int_0^L \varphi_t\int_0^{+\infty}
 (-g' (s))(\psi_x (t)-\psi_x (t-s))\,ds\,dx
\\
&\leq  \frac{\epsilon}{2} E(t)-c_{\epsilon}g'\circ\psi_x.
\end{align*}
Inserting these last two inequalities into \eqref{e4.3},
we obtain \eqref{e4.2}.
\end{proof}

 Now, going back to the proof of Theorem \ref{thm2.2}, choosing
$\tau =1$ in \eqref{e3.21}, using \eqref{e3.22} and \eqref{e4.2} and
choosing $\epsilon $ small enough,
we obtain
\[
F'(t)\leq -cE(t)G'(\epsilon _0E(t))+cG'(\epsilon _0E(t))(g\circ \psi _{xt}
-g'\circ \psi _x),
\]
which implies, using \eqref{e3.1} and the fact that $G'(\epsilon _0E)$
is non-increasing,
\begin{equation}
E(t)G'(\epsilon _0E(t))\leq -cG'(\epsilon
_0E(0))E'(t)-cF'(t)+cG'(\epsilon_0E(t))g\circ \psi _{xt}.\label{e4.4}
\end{equation}
Now, we estimate the last term in \eqref{e4.4}. Similarly to the case of
$g\circ \psi _x$ in Lemma \ref{lem3.11} (for $g\circ \psi _{xt}$ instead of
$g\circ \psi _x$), we obtain, using \eqref{e2.10} and \eqref{e4.1},
\begin{equation}
G'(\epsilon _0E(t))g\circ \psi _{xt}\leq -c{\tilde{E}}'(t)
+c\epsilon _0E(t)G'(\epsilon _0E(t)),\quad \forall \epsilon_0>0. \label{e4.5}
\end{equation}
Then \eqref{e4.4} and \eqref{e4.5} with $\epsilon _0$ chosen small enough imply that
\begin{equation}
E(t)G'(\epsilon _0E(t))\leq -cG'(\epsilon
_0E(0))E'(t)-cF'(t)-c{\tilde{E}}'(t).\label{e4.6}
\end{equation}
Using the fact that $F\sim E$ and $EG'(\epsilon _0E)$ is
non-increasing, we deduce that, for all $T\in \mathbb{R}_{+}$
 ($G_0$ is defined by \eqref{e2.12}),
\begin{equation}
G_0(E(T))T\leq \int_0^{T}G_0(E(t))dt\leq c(G'(\epsilon
_0E(0))+1)E(0)+c{\tilde{E}}(0),\label{e4.7}
\end{equation}
which gives \eqref{e2.11} with
$C=c(G'(\epsilon _0E(0))+1)E(0)+c{\tilde{E }}(0)$. This completes
the proof of Theorem \ref{thm2.2}.


\section{Well-Posedness and Smoothness}

In this section, we discuss the existence, uniqueness and smoothness of
solution of \eqref{eP} under hypothesis (H1). We use the semigroup theory and
some arguments of \cite{d1} (see also \cite{m10,m11}).
 Following the idea of \cite{d1}, let
\begin{equation}
\eta (x,t,s)=\psi (x,t)-\psi (x,t-s)\quad \text{for }(x,t,s)\in
]0,L[\times \mathbb{R}_{+}\times \mathbb{R}_{+}\label{e5.1}
\end{equation}
($\eta $ is the relative history of $\psi $, and it was introduced first in
\cite{d1}). This function satisfies the initial conditions
\begin{equation}
\eta (0,t,s)=\eta (L,t,s)=0,\quad \text{in}\,\,\mathbb{R}
_{+}\times \mathbb{R}_{+},
\eta (x,t,0)=0,\quad \text{in}
\,\,]0,L[\times \mathbb{R}_{+}\label{e5.2}
\end{equation}
and the equation
\begin{equation}
\eta _t+\eta _{s}-\psi _t=0,\quad \text{in }]0,L[\times \mathbb{R}_{+}
\times \mathbb{R}_{+}.\label{e5.3}
\end{equation}
Then the second equation of \eqref{eP} can be formulated as
\[
\rho _2\psi _{tt}-k_2\psi _{xx}+\Bigl(\int_0^{+\infty }g(s)ds\Bigr)
\psi _{xx}-\int_0^{+\infty }g(s)\eta _{xx}ds+k_1(\varphi _x+\psi )=0.
\]
Let $\eta _0(x,s)=\eta (x,0,s)=\psi _0(x,0)-\psi _0(x,s)$ for
$(x,s)\in ]0,L[\times \mathbb{R}_{+}$. This means that the history
is considered as an initial data for $\eta $. Let
\begin{equation}
\mathcal{H}=\bigl(H_0^{1}(]0,L[)\bigr)^2\times \bigl(L^2(]0,L[)\bigr)
^2\times L_{g}\label{e5.4}
\end{equation}
with
\begin{equation}
L_{g}=\{v:\mathbb{R}_{+}\to
H_0^{1}(]0,L[),\,\int_0^{L}\int_0^{+\infty
}g(s)v_x^2(s)\,ds\,dx<+\infty \}.\label{e5.5}
\end{equation}
The set $L_{g}$ is a Hilbert space endowed with the inner product
\begin{equation}
\langle v,w\rangle _{L_{g}}=\int_0^{L}\int_0^{+\infty
}g(s)v_x(s)w_x(s)\,ds\,dx.\label{e5.6}
\end{equation}
Then $\mathcal{H}$ is also a Hilbert space endowed with the inner product
defined, for $V=(v_1,v_2,v_3,v_4,v_5)^{T},
\,W=(w_1,w_2,w_3,w_4,w_5)^{T}\in \mathcal{H}$, by
\begin{equation}
\begin{aligned}
\langle V,W\rangle_{\mathcal{H}}
&=  \int_0^L \Bigl(\Bigl(k_2 -\int_0^{+\infty} g(s)ds\Bigr)
\partial_x v_2 \partial_x w_2 +\rho_1 v_3 w_3 +\rho_2 v_4 w_4\Bigr)dx
\\
&\quad  +\langle v_5,w_5\rangle_{L_g}
+k_1 \int_0^L (\partial_x v_1 +v_2)(\partial_x w_1 +w_2)dx.
\end{aligned}\label{e5.7}
\end{equation}
Now, for $U=(\varphi ,\psi ,\varphi _t,\psi _t,\eta )^{T}$ and
$U_0=(\varphi _0,\psi _0(\cdot ,0),\varphi _1,\psi _1,\eta _0)^{T}$,
\eqref{eP} is equivalent to the  abstract linear first order Cauchy
problem
\begin{equation}
\begin{gathered}
U_t (t)+AU(t)=0\quad\text{on}\,\, \mathbb{R}_{+} ,\cr
U(0)=U_0 , \cr
\end{gathered}\label{e5.8}
\end{equation}
where $A$ is the linear operator defined by
$AV=(f_1,f_2,f_3,f_4,f_5)$, for any
$V=(v_1,v_2,v_3,v_4,v_5)^{T}\in D(A)$, where
\begin{gather*}
f_1=-v_3,\quad f_2=-v_4,\quad f_3=-{\frac{{k_1}}{{\rho _1}}}\partial
_x(\partial _xv_1+v_2), \\
f_4=-{\frac{1}{{\rho _2}}}\Bigl(k_2-\int_0^{+\infty }g(s)ds\Bigr)
\partial _{xx}v_2-{\frac{{1}}{{\rho _2}}}\int_0^{+\infty }g(s)\partial
_{xx}v_5(s)ds+{\frac{{k_1}}{{\rho _2}}}(\partial _xv_1+v_2), \\
f_5=-v_4+\partial _{s}v_5.
\end{gather*}
The domain $D(A)$ of $A$ given by
$D(A)=\{V\in \mathcal{H},\,AV\in \mathcal{H}\text{ and }v_5(0)=0\}$
and endowed with the graph norm
\begin{equation}
\| V\| _{D(A)}=\| V\| _{\mathcal{H}}+\| AV\| _{\mathcal{H}} \label{e5.9}
\end{equation}
can be characterized by
\begin{align*}
D(A)=\Bigl\{&V=(v_1,v_2,v_3,v_4,v_5)^{T}\in \bigl(
H^2(]0,L[)\cap H_0^{1}(]0,L[)\bigr)\times \bigl(H_0^{1}(]0,L[)\bigr)
^3\times \mathcal{L}_{g},\\
&\Bigl(k_2-\int_0^{+\infty }g(s)ds\Bigr)\partial_{xx}v_2
+\int_0^{+\infty }g(s)\partial _{xx}v_5(s)ds\in L^2(]0,L[)
\Bigr\}
\end{align*}
and it is dense in $\mathcal{H}$ (see also \cite{m11,m12} and the reference
therein), where
\begin{equation}
\mathcal{L}_{g}=\{v\in L_{g},\,\partial _{s}v\in L_{g},\,v(x,0)=0\}.\label{e5.10}
\end{equation}
Now, we prove that $A:D(A)\to \mathcal{H}$ is a maximal monotone
operator; that is $-A$ is dissipative and $Id+A$ is surjective. Indeed, a
simple calculation implies that, for any
$V=(v_1,v_2,v_3,v_4,v_5)^{T}\in D(A)$,
\begin{equation}
\langle AV,V\rangle _{\mathcal{H}}=-{\frac{1}{2}}\int_0^{L}\int_0^{+
\infty }g'(s)(\partial _xv_5(s))^2\,ds\,dx\geq 0,\label{e5.11}
\end{equation}
since $g$ is non-increasing. This implies that $-A$ is dissipative.

On the other hand, we prove that $Id+A$ is surjective; that is, for any
$F=(f_1 ,f_2 ,f_3 ,f_4 ,f_5)^T\in \mathcal{H}$, there exists
$V=(v_1 ,v_2,v_3 ,v_4 ,v_5)^T\in D(A)$ satisfying
\begin{equation}
(Id+A)V=F.\label{e5.12}
\end{equation}
The first two equations of system \eqref{e5.12} are equivalent to
\begin{equation}
v_1 =v_3 +f_1 \quad\text{and}\quad v_2 =v_4 +f_2 .\label{e5.13}
\end{equation}
The last equation of system \eqref{e5.12} is equivalent to
\[
v_5 +\partial_s v_5 =v_4 +f_5 ,
\]
then, by integrating with respect to $s$ and noting that $v_5 (0)=0$, we obtain
\begin{equation}
v_5 (s)=\Bigl(\int_0^s (v_4 +f_5 (\tau))e^{\tau} d\tau\Bigr)e^{-s} .\label{e5.14}
\end{equation}
Now, we look for $(v_3 ,v_4 )\in \bigl( H_0^1 (]0,L[)\bigr)^2 $. To simplify
the formulations, we put $\mathcal{H}_1 =\bigl( H_0^1 (]0,L[)\bigr)^2 $ and
$\mathcal{H}_2 =\bigl(L^2 (]0,L[)\bigr)^2$ endowed with the inner products
\begin{equation}
\begin{aligned}
\langle (z_1 ,z_2)^T ,(w_1 ,w_2 )^T\rangle_{\mathcal{H}_1}
&=  \int_0^L \Bigl(k_2 -\int_0^{+\infty} e^{-s} g(s)ds\Bigr)
\partial_x z_2 \partial_x w_2 dx
\\
&\quad  + k_1 \int_0^L (\partial_x z_1 +z_2)(\partial_x w_1 +w_2)dx
\end{aligned} \label{e5.15}
\end{equation}
and
\begin{equation}
\langle (z_1 ,z_2)^T ,(w_1 ,w_2 )^T\rangle_{\mathcal{H}_2}=\int_0^L (\rho_1
z_1 w_1 +\rho_2 z_2 w_2)dx.\label{e5.16}
\end{equation}
Thanks to \eqref{e2.1} and Poincar\'e's inequality,
$\langle ,\rangle_{\mathcal{H} _1}$ defines a norm on $\mathcal{H}_1$
equivalent to the norm induced by $\Bigl( H^1 (]0,L[)\Bigr)^2 $.
On the other hand, the inclusion $\mathcal{H}_1 \subset \mathcal{H}_2$
is dense and compact.

Now, inserting \eqref{e5.13} and \eqref{e5.14} into the third and the fourth
equations of system \eqref{e5.12}, multiplying them, respectively,
by $\rho _1w_3$ and $\rho_2w_4$, where $(w_3,w_4)^{T}\in \mathcal{H}_1$,
and then integrating their sum over $]0,L[$, we obtain, for all
 $(w_3,w_4)^{T}\in\mathcal{H}_1$,
\begin{equation}
\begin{aligned}
&\langle (v_3,v_4)^{T},(w_3,w_4)^{T}\rangle _{\mathcal{H}
_2}+\langle (v_3,v_4)^{T},(w_3,w_4)^{T}\rangle _{\mathcal{H}_1}
\\
&=  \langle (f_3 ,f_4)^T ,(w_3 ,w_4 )^T\rangle_{\mathcal{H}_2}
-\langle (f_1 ,f_2)^T ,(w_3 ,w_4 )^T\rangle_{\mathcal{H}_1}
\\
&\quad  +\Bigl(\int_0^{+\infty}(1-e^{-s})g(s)ds\Bigr)
 \int_0^L \partial_x f_2 \partial_x w_4 dx
\\
&\quad  -\int_0^L \partial_x \Bigl(\int_0^{+\infty}e^{-s} g(s)
\Bigl(\int_0^s f_5 (\tau)e^{\tau} d\tau\Bigr)ds\Bigr)\partial_x w_4 dx.
\end{aligned}\label{e5.17}
\end{equation}
We have just to prove that \eqref{e5.17} has a solution
$(v_3,v_4)^{T}\in \mathcal{H}_1$, and then, using \eqref{e5.13}, \eqref{e5.14}
and regularity arguments, we find \eqref{e5.12}.
 Following the method in \cite[page 95]{k2}, let $\mathcal{H}_1'$ be the dual space
of $\mathcal{H}_1$ and $A_1:\mathcal{H}_1\to \mathcal{H}_1'$ be the duality mapping.
We consider the map $B_1:\mathcal{H}_1\to \mathcal{H}_1'$ defined by
\[
\langle B_1(z_1,z_2)^{T},(w_1,w_2)^{T}\rangle _{\mathcal{H}
_1',\mathcal{H}_1}=\int_0^{L}\partial _xz_2\partial_xw_2dx.
\]
We identify $\mathcal{H}_2$ with its dual space $\mathcal{H}_2'$
and we set
\[
{\tilde{f}_5}=\Bigl(\int_0^{+\infty }(1-e^{-s})g(s)ds\Bigr)
f_2-\int_0^{+\infty }e^{-s}g(s)\Bigl(\int_0^{s}f_5(\tau )e^{\tau
}d\tau \Bigr)ds.
\]
We have $B_1(0,{\tilde{f}_5})^{T}\in \mathcal{H}_1'$ and
\eqref{e5.17} becomes
\begin{equation} \label{e5.18}
\begin{aligned}
&\langle (Id+A_1)(v_3,v_4)^{T},(w_3,w_4)^{T}\rangle _{\mathcal{H}
_1',\mathcal{H}_1}
\\
&=\langle (f_3,f_4)^{T}-A_1(f_1,f_2)^{T}+B_1(0,{\tilde{f}_5}
)^{T},(w_3,w_4)^{T}\rangle _{\mathcal{H}_1',\mathcal{H}
_1},
\end{aligned}
\end{equation}
for all $(w_3,w_4)^{T}\in \mathcal{H}_1$.
Let
\[
({\tilde{f}}_1,{\tilde{f}}_2)^{T}=(f_3,f_4)^{T}-A_1(f_1,f_2)^{T}
+B_1(0,{\tilde{f}_5})^{T}.
\]
Because $\mathcal{H}_2=\mathcal{H}_2'\subset \mathcal{H}_1'$,
then $({\tilde{f}}_1,{\tilde{f}}_2)^{T}\in \mathcal{H}_1'$.
Therefore, \eqref{e5.18} is well-defined and equivalent to
\begin{equation}
(Id+A_1)(v_3,v_4)^{T}=({\tilde{f}}_1,{\tilde{f}}_2)^{T}.\label{e5.19}
\end{equation}
It is sufficient to show that \eqref{e5.19} has a solution
$(v_3,v_4)^{T}\in \mathcal{H}_1$.
Let $\Gamma :\mathcal{H}_1\to \mathbb{R}$ defined by
\begin{equation}
\Gamma ((z_1 ,z_2)^T )
=  {1\over 2}\| (z_1 ,z_2)^T\|_{\mathcal{H}_2}^2 +{1\over 2}
 \| (z_1 ,z_2)^T\|_{\mathcal{H}_1}^2
 -\langle ({\tilde f}_1 ,{\tilde f}_2)^T ,(z_1 ,z_2)^T
\rangle_{\mathcal{H}'_1 ,\mathcal{H}_1}.
\label{e5.20}
\end{equation}
The map $\Gamma $ is well-defined and differentiable such that
\begin{equation} \label{e5.21}
\begin{aligned}
&\Gamma '((z_1,z_2)^{T})(w_1,w_2)^{T}\\
&=\langle (Id+A_1)(z_1,z_2)^{T}-({\tilde{f}}_1,{\tilde{f}}_2)^{T},
(w_1,w_2)^{T}\rangle _{\mathcal{H}_1',\mathcal{H}_1},
\end{aligned}
\end{equation}
for all $(z_1,z_2)^{T},(w_1,w_2)^{T}\in \mathcal{H}_1$.

On the other hand, using Cauchy-Schwarz inequality to minimize the last term
in \eqref{e5.20}, we have
\begin{equation} \label{e5.22}
\Gamma ((z_1,z_2)^{T})\geq \Bigl({\frac{1}{2}}\|
(z_1,z_2)^{T}\| _{\mathcal{H}_1}-\| ({\tilde{f}}_1,{\tilde{f}}
_2)^{T}\| _{\mathcal{H}_1'}\Bigr)\|
(z_1,z_2)^{T}\| _{\mathcal{H}_1}.
\end{equation}
This implies that $\Gamma $ goes to infinity when
$\| (z_1,z_2)^{T}\| _{\mathcal{H}_1}$ goes to infinity, and therefore,
$\Gamma $ reaches its minimum at some point
$(v_3,v_4)^{T}\in \mathcal{H}_1$. This point satisfies
$\Gamma '((v_3,v_4)^{T})=0$, which
solves \eqref{e5.19} thanks to \eqref{e5.21} with the choice
$(z_1,z_2)^{T}=(v_3,v_4)^{T}$.

Finally, using Lummer-Phillips theorem (see \cite{p2}), we deduce that $A$ is an
infinitesimal generator of a contraction semigroup in $\mathcal{H}$, which
implies the following results of existence, uniqueness and smoothness of the
solution of \eqref{eP} (see \cite{k2,p2}).

 \begin{theorem} \label{thm5.1}
Assume that {\rm (H1)} is satisfied.

1. For any $U_0\in \mathcal{H}$, \eqref{eP} has a unique weak solution
\begin{equation}
U\in C(\mathbb{R}_{+};\mathcal{H}).\label{e5.23}
\end{equation}

2. If $U_0\in D(A^{n})$ for $n\in \mathbb{N}^{\ast }$, then the
solution $U$ has the regularity
\begin{equation}
U\in \cap _{j=0}^{n}C^{n-j}(\mathbb{R}_{+};D(A^{j})),\label{e5.24}
\end{equation}
where $D(A^{j})$ is endowed with the graph norm
$\| V\| _{D(A^{j})}=\sum_{m=0}^{j}\| A^{m}V\| _{\mathcal{H}}$.
\end{theorem}


\section{Timoshenko-heat: Fourier's and Cattaneo's laws}

In this section, we give applications of our results of Section 2 to the
case of coupled Timoshenko-heat systems on $]0,L[$ under Fourier's law of
heat conduction and with an infinite history acting on the second equation:
\begin{equation}
\begin{gathered}
\rho_1 \varphi_{tt}-k_1 (\varphi_x+\psi)_x=0, \\
\rho_2 \psi _{tt}-k_2 \psi_{xx}+k_1 (\varphi_x+\psi)+k_4 \theta_x +
\int_0^{+\infty}g (s)\psi_{xx} (x,t-s)ds =0, \\
\rho_3 \theta_t-k_3 \theta_{xx} +k_4 \psi_{xt}=0, \\
\varphi (0,t)=\psi (0,t)=\theta (0,t)=\varphi (L,t)=\psi (L,t)=\theta
(L,t)=0, \\
\varphi (x ,0)=\varphi _0 (x),\quad \varphi _t(x ,0)=\varphi _1 (x), \\
\psi (x ,-t)=\psi _0 (x,t),\quad \psi _t(x ,0)=\psi _1 (x),\quad
 \theta (x,0)=\theta_0 (x)
\end{gathered}
\label{eP1}
\end{equation}
and with an infinite history acting on the first equation:
\begin{equation}
\begin{gathered}
\rho_1 \varphi_{tt}-k_1 (\varphi_x+\psi)_x +
\int_0^{+\infty}g (s)\varphi_{xx} (x,t-s)ds=0, \\
\rho_2 \psi _{tt}-k_2 \psi_{xx}+k_1 (\varphi_x+\psi)+k_4 \theta_x =0, \\
\rho_3 \theta_t-k_3 \theta_{xx} +k_4 \psi_{xt}=0, \\
\varphi (0,t)=\psi (0,t)=\theta_x (0,t)=\varphi (L,t)=\psi (L,t)=\theta_x
(L,t)=0, \\
\varphi (x ,-t)=\varphi _0 (x,t),\quad \varphi _t(x ,0)=\varphi _1 (x), \\
\psi (x ,0)=\psi _0 (x),\quad \psi _t(x ,0)=\psi _1 (x),\quad
 \theta (x,0)=\theta_0 (x).
\end{gathered} \label{eP2}
\end{equation}
We also consider systems under Cattaneo's law and with an infinite history
acting on the first equation:
\begin{equation}
\begin{gathered}
\rho_1 \varphi_{tt}-k_1 (\varphi_x+\psi)_x +
\int_0^{+\infty}g (s)\varphi_{xx} (x,t-s)ds =0, \\
\rho_2 \psi _{tt}-k_2 \psi_{xx}+k_1 (\varphi_x+\psi)+k_4 \theta_x =0, \\
\rho_3 \theta_t +k_3 q_x +k_4 \psi_{xt}=0, \\
\rho_4 q_t +k_5 q +k_3 \theta_x=0, \\
\varphi (0,t)=\psi (0,t)=q (0,t)=\varphi (L,t)=\psi (L,t)=q (L,t)=0, \\
\varphi (x ,-t)=\varphi _0 (x,t),\quad \varphi _t(x ,0)=\varphi _1 (x), \\
\psi (x ,0)=\psi _0 (x),\quad \psi _t(x ,0)=\psi _1 (x),\quad \theta
(x,0)=\theta_0 (x),\quad  q (x,0)=q_0 (x),
\end{gathered} \label{eP3}
\end{equation}
and with an infinite history acting on the second equation:
\begin{equation}
\begin{gathered}
\rho_1 \varphi_{tt}-k_1 (\varphi_x+\psi)_x =0, \\
\rho_2 \psi _{tt}-k_2 \psi_{xx}+k_1 (\varphi_x+\psi)+k_4 \theta_x +
\int_0^{+\infty}g (s)\psi_{xx} (x,t-s)ds=0, \\
\rho_3 \theta_t +k_3 q_x +k_4 \psi_{xt}=0, \\
\rho_4 q_t +k_5 q +k_3 \theta_x=0, \\
\varphi (0,t)=\psi (0,t)=q (0,t)=\varphi (L,t)=\psi (L,t)=q (L,t)=0, \\
\varphi (x,0)=\varphi _0 (x),\quad \varphi _t(x ,0)=\varphi _1 (x), \\
\psi (x ,-t)=\psi_0 (x,t),\quad \psi_t(x ,0)=\psi_1 (x),\quad  \theta
(x,0)=\theta_0 (x),\quad  q (x,0)=q_0 (x),
\end{gathered} \label{eP4}
\end{equation}
where $\rho_i$ and $k_i$ are also positive constants, and $\theta$ and $q$
denote, respectively, the temperature difference and the heat flux vector.
Systems \eqref{eP3} and \eqref{eP4} (Cattaneo law), with $\rho_4 =0$, implies,
respectively, \eqref{eP2} and \eqref{eP1} (Fourier's law).

\subsection{Well-posedness}

We start our analysis by showing without details,
using the semigroup theory (as for \eqref{eP} in Section 5), how to prove that
\eqref{eP1}--\eqref{eP4} are well-posed under the following hypothesis:
\begin{itemize}
\item[(H3)] $g:\mathbb{R}_{+}\to \mathbb{R}_{+}$ is a non-increasing
differentiable function satisfying $g(0)>0$ such
that \eqref{e2.1} holds in case \eqref{eP1} and \eqref{eP4}, and
\begin{equation}
\int_0^{+\infty }g(s)ds<{\frac{{k_1k_2}}{{k_2+k_0k_1}}}\label{e6.1}
\end{equation}
in case \eqref{eP2} and \eqref{eP3}, where $k_0$ is the smallest positive
constant satisfying, for all $v\in H_0^{1}(]0,L[)$ (Poincar\'e's
inequality),
\begin{equation}
\int_0^{L}v^2dx\leq k_0\int_0^{L}v_x^2dx.\label{e6.2}
\end{equation}
\end{itemize}

\begin{remark} \label{rmk6.1} \rm
Thanks to Poincar\'e's inequality (applied for $\psi $), we have
\[
k_1\int_0^{L}(\varphi _x+\psi )^2dx\geq k_1(1-\epsilon
)\int_0^{L}\varphi _x^2dx+k_0k_1(1-{\frac{1}{{\epsilon }}}
)\int_0^{L}\psi _x^2dx,
\]
for any $0<\epsilon <1$. Then, thanks to \eqref{e6.1}, we can choose
\[
{\frac{{k_0k_1}}{{k_2+k_0k_1}}}<\epsilon <{\frac{{1}}{{k_1}}}
\Bigl(k_1-\int_0^{+\infty }g(s)ds\Bigr)
\]
and we obtain
\[
{\hat{c}}\int_0^{L}(\varphi _x^2+\psi _x^2)dx\leq \int_0^{L}
\Bigl(-\Bigl(\int_0^{+\infty }g(s)ds\Bigr)\varphi _x^2+k_2\psi
_x^2+k_1(\varphi _x+\psi )^2\Bigr)dx,
\]
where ${\hat{c}}=\min \{k_1(1-\epsilon )-\int_0^{+\infty
}g(s)ds,k_2+k_0k_1(1-{\frac{1}{{\epsilon }}})\}>0$. Thus,
\[
\int_0^{L}\Bigl(-\Bigl(\int_0^{+\infty }g(s)ds\Bigr)\varphi
_x^2+k_2\psi _x^2+k_1(\varphi _x+\psi )^2\Bigr)dx
\]
defines a norm on $\bigl(H_0^{1}(]0,L[)\bigr)^2$, for $(\varphi ,\psi )
$, equivalent to the one induced by $\bigl(H^{1}(]0,L[)\bigr)^2$.
\end{remark}

 Now, following the idea of \cite{d1}, let, as in Section 5, $\eta $
be the relative history of $\psi $ in cases of \eqref{eP1} and \eqref{eP4}, and of
$\varphi $ in cases of \eqref{eP2} and \eqref{eP3}, defined by
\[
\eta (x,t,s)=\begin{cases}
\psi (x,t)-\psi (x,t-s),& \text{in  cases  \eqref{eP1} and \eqref{eP4}} \\
\varphi (x,t)-\varphi (x,t-s),& \text{in cases  \eqref{eP2} and \eqref{eP3}},
\end{cases}
\]
for $(x,t,s)\in ]0,L[\times \mathbb{R}_{+}\times \mathbb{R}_{+}$.
 This function satisfies \eqref{e5.2} and \eqref{e5.3} in case  \eqref{eP1} and
\eqref{eP4}, and satisfies \eqref{e5.2} and
\[
\eta _t+\eta _{s}-\varphi _t=0,\quad \text{in }]0,L[\times
\mathbb{R}_{+}\times \mathbb{R}_{+}
\]
in case  \eqref{eP2} and \eqref{eP3}. Then the second equation of \eqref{eP1}
and \eqref{eP4}, and the first one of \eqref{eP2} and \eqref{eP3} can be formulated,
 respectively, as
\[
\rho _2\psi _{tt}-\Bigl(k_2-\int_0^{+\infty }g(s)ds\Bigr)\psi
_{xx}-\int_0^{+\infty }g(s)\eta _{xx}ds+k_1(\varphi _x+\psi
)+k_4\theta _x=0
\]
and
\[
\rho _1\varphi _{tt}-k_1(\varphi _x+\psi )_x+\Bigl(\int_0^{+\infty
}g(s)ds\Bigr)\varphi _{xx}-\int_0^{+\infty }g(s)\eta _{xx}ds=0.
\]
Let
\[
\mathcal{H}=\begin{cases}
\bigl(H_0^{1}(]0,L[)\bigr)^2\times \bigl(L^2(]0,L[)\bigr)^3\times
L_{g},& \text{for \eqref{eP1}}
\\
\bigl(H_0^{1}(]0,L[)\bigr)^2\times \bigl(L^2(]0,L[)\bigr)^2\times
L_{\ast }^2(]0,L[)\times L_{g}, & \text{for \eqref{eP2}}
\\
\bigl(H_0^{1}(]0,L[)\bigr)^2\times \bigl(L^2(]0,L[)\bigr)^2\times
L_{\ast }^2(]0,L[)\times L^2(]0,L[)\times L_{g},
&\text{for  \eqref{eP3}-\eqref{eP4}},
\end{cases}
\]
where $L_{\ast }^2(]0,L[)=\{v\in L^2(]0,L[)$,
$\int_0^{L}v(x)dx=0\}$ and $L_{g}$ is defined by \eqref{e5.5} and endowed with the
inner product \eqref{e5.6}. Then, thanks to \eqref{e2.1} and \eqref{e6.1}
(see Remark \ref{rmk6.1}), $\mathcal{H}$ is also a Hilbert space endowed with the
inner product defined, for $V,W\in \mathcal{H}$, by
\begin{align*}
\langle V,W\rangle_{\mathcal{H}}
&=  \langle v_6,w_6\rangle_{L_g} +k_1 \int_0^L (\partial_x v_1 +v_2)
(\partial_x w_1 +w_2)dx
\\
&\quad  +\int_0^L \Bigl(\Bigl(k_2 -\int_0^{+\infty} g(s)ds\Bigr)
\partial_x v_2 \partial_x w_2 +\rho_1 v_3 w_3 +\rho_2 v_4 w_4 +\rho_3 v_5 w_5\Bigr)dx
\end{align*}
in case \eqref{eP1},
\begin{align*}
\langle V,W\rangle_{\mathcal{H}}
&=  \langle v_6,w_6\rangle_{L_g} +\int_0^L \Bigl(k_1 (\partial_x v_1 +v_2)
 (\partial_x w_1 +w_2)+k_2 \partial_x v_2 \partial_x w_2 \Bigr)dx
\\
&\quad  +\int_0^L \Bigl(-\Bigl(\int_0^{+\infty} g(s)ds\Bigr)
\partial_x v_1 \partial_x w_1 +\rho_1 v_3 w_3 +\rho_2 v_4 w_4 +\rho_3 v_5 w_5\Bigr)dx
\end{align*}
in case \eqref{eP2},
\begin{align*}
&\langle V,W\rangle_{\mathcal{H}}\\
 &=  \langle v_7,w_7\rangle_{L_g} +\int_0^L
\Bigl( k_1 (\partial_x v_1 +v_2)(\partial_x w_1 +w_2)+k_2 \partial_x v_2
\partial_x w_2 \Bigr)dx
\\
&\quad  +\int_0^L \Bigl(-\Bigl(\int_0^{+\infty} g(s)ds\Bigr)
\partial_x v_1 \partial_x w_1 +\rho_1 v_3 w_3 +\rho_2 v_4 w_4
+\rho_3 v_5 w_5 +\rho_4 v_6 w_6\Bigr)dx
\end{align*}
in case \eqref{eP3}, and
\begin{align*}
\langle V,W\rangle_{\mathcal{H}}
 &=  \langle v_7,w_7\rangle_{L_g} +k_1 \int_0^L (\partial_x v_1 +v_2)
(\partial_x w_1 +w_2)dx
\\
&\quad  +\int_0^L \Bigl(\Bigl(k_2 -\int_0^{+\infty} g(s)ds\Bigr)
\partial_x v_2 \partial_x w_2 +\rho_1 v_3 w_3 \\
&\quad +\rho_2 v_4 w_4 +\rho_3 v_5 w_5+\rho_4 v_6 w_6\Bigr)dx
\end{align*}
in case \eqref{eP4}. Now, let $\eta _0(x,s)=\eta (x,0,s)$,
\[
U_0=\begin{cases}
(\varphi _0,\psi _0(\cdot ,0),\varphi _1,\psi _1,\theta _0,\eta
_0)^{T}, &\text{in case \eqref{eP1} }
\\
(\varphi _0(\cdot ,0),\psi _0,\varphi _1,\psi _1,\theta _0,\eta
_0)^{T}, &\text{in case \eqref{eP2}}
\\
(\varphi _0(\cdot ,0),\psi _0,\varphi _1,\psi _1,\theta
_0,q_0,\eta _0)^{T}, & \text{in case \eqref{eP3}}
 \\
(\varphi _0,\psi _0(\cdot ,0),\varphi _1,\psi _1,\theta
_0,q_0,\eta _0)^{T}, &\text{in case \eqref{eP4}}
\end{cases}
\]
and
\[
U=\begin{cases}
(\varphi ,\psi ,\varphi _t,\psi _t,\theta ,\eta )^{T},
&\text{in cases \eqref{eP1} and \eqref{eP2}}
\\
(\varphi ,\psi ,\varphi _t,\psi _t,\theta ,q,\eta )^{T} , &
\text{in cases \eqref{eP3} and \eqref{eP4}}
\end{cases}
\]
Systems \eqref{eP1}--\eqref{eP4} can be rewritten as the abstract
 problem \eqref{e5.8}, where $A$
is the linear operator defined, for any $V\in D(A)$, by $AV=F$ and
\begin{gather*}
f_1=-v_3,\quad f_2=-v_4,\quad f_3=-{\frac{{k_1}}{{\rho _1}}}\partial
_x(\partial _xv_1+v_2), \\
\begin{aligned}
f_4&=-{\frac{1}{{\rho _2}}}\Bigl(k_2-\int_0^{+\infty }g(s)ds\Bigr)
\partial _{xx}v_2+{\frac{{k_1}}{{\rho _2}}}(\partial _xv_1+v_2)\\
&\quad +{\frac{{k_4}}{{\rho _2}}}\partial _xv_5-{\frac{{1}}{{\rho _2}}}
\int_0^{+\infty }g(s)\partial _{xx}v_6(s)ds, 
\end{aligned} \\
f_5=-{\frac{{k_3}}{{\rho _3}}}\partial _{xx}v_5+{\frac{{k_4}}{{
\rho _3}}}\partial _xv_4,\quad f_6=-v_4+\partial _{s}v_6,
\end{gather*}
in case \eqref{eP1},
\begin{gather*}
f_1=-v_3,\quad f_2=-v_4, \\
f_3=-{\frac{{k_1}}{{\rho _1}}}\partial _x(\partial _xv_1+v_2)+{
\frac{1}{{\rho _1}}}\Bigl(\int_0^{+\infty }g(s)ds\Bigr)\partial
_{xx}v_1-{\frac{{1}}{{\rho _1}}}\int_0^{+\infty }g(s)\partial
_{xx}v_6(s)ds, \\
f_4=-{\frac{{k_2}}{{\rho _2}}}\partial _{xx}v_2+{\frac{{k_1}}{{
\rho _2}}}(\partial _xv_1+v_2)+{\frac{{k_4}}{{\rho _2}}}\partial
_xv_5, \\
f_5=-{\frac{{k_3}}{{\rho _3}}}\partial _{xx}v_5+{\frac{{k_4}}{{
\rho _3}}}\partial _xv_4,\quad f_6=-v_3+\partial _{s}v_6,
\end{gather*}
in case \eqref{eP2},
\begin{gather*}
f_1=-v_3,\quad f_2=-v_4, \\
f_3=-{\frac{{k_1}}{{\rho _1}}}\partial _x(\partial _xv_1+v_2)+{
\frac{1}{{\rho _1}}}\Bigl(\int_0^{+\infty }g(s)ds\Bigr)\partial
_{xx}v_1-{\frac{{1}}{{\rho _1}}}\int_0^{+\infty }g(s)\partial
_{xx}v_7(s)ds, \\
f_4=-{\frac{{k_2}}{{\rho _2}}}\partial _{xx}v_2+{\frac{{k_1}}{{
\rho _2}}}(\partial _xv_1+v_2)+{\frac{{k_5}}{{\rho _2}}}\partial _xv_5,\quad
f_5={\frac{{k_3}}{{\rho _3}}}\partial _xv_6+{\frac{{
k_5}}{{\rho _3}}}\partial _xv_4, \\
f_6={\frac{{k_4}}{{\rho _4}}}v_6+{\frac{{k_3}}{{\rho _4}}}
\partial _xv_5,\ f_7=-v_3+\partial _{s}v_7,
\end{gather*}
in case \eqref{eP3}, and
\begin{gather*}
f_1=-v_3,\quad f_2=-v_4,\quad f_3=-{\frac{{k_1}}{{\rho _1}}}
\partial _x(\partial _xv_1+v_2) \\
\begin{aligned}
f_4&=-{\frac{{1}}{{\rho _2}}}\Bigl(k_2-\int_0^{+\infty }g(s)ds\Bigr)
\partial _{xx}v_2+{\frac{{k_1}}{{\rho _2}}}(\partial _xv_1+v_2)\\
&\quad +{\frac{{k_4}}{{\rho _2}}}\partial _xv_5-{\frac{{1}}{{\rho _2}}}
\int_0^{+\infty }g(s)\partial _{xx}v_7(s)ds, 
\end{aligned} \\
f_5={\frac{{k_3}}{{\rho _3}}}\partial _xv_6+{\frac{{k_4}}{{\rho
_3}}}\partial _xv_4,\ \ f_6={\frac{{k_5}}{{\rho _4}}}v_6+{
\frac{{k_3}}{{\rho _4}}}\partial _xv_5,\quad f_7=-v_4+\partial
_{s}v_7,
\end{gather*}
in case \eqref{eP4}.

The domain $D(A)$ of $A$ is endowed with the norm \eqref{e5.9} and it is given by
$D(A)=\{V\in \mathcal{H},\,AV\in \mathcal{H}\text{ and }v_6 (0)=0\}$ in
case \eqref{eP1} and \eqref{eP2}, and
$D(A)=\{V\in \mathcal{H},\,AV\in \mathcal{H}\text{ and }v_7 (0)=0\}$
in case \eqref{eP3} and \eqref{eP4}. The operator $A$ is
maximal monotone (the proof is similar to the one of Section 5), and then $A$
is an infinitesimal generator of a contraction semigroup in $\mathcal{H}$,
which implies the well-posedness results of
Theorem \ref{thm5.1} for \eqref{eP1}-\eqref{eP4}.

\subsection{Stability}
Similarly to \eqref{eP} and under the
hypotheses (H2) and (H3), we prove that \eqref{e2.6} (when \eqref{e1.1} holds)
and \eqref{e2.11} (when \eqref{e1.1} does not hold) remain valid for \eqref{eP1}.
For \eqref{eP2} and \eqref{eP3},
we prove that \eqref{e2.6} holds independently of \eqref{e1.1}.
Finally, for \eqref{eP4}, we
prove only \eqref{e2.11} even if \eqref{e1.1} holds.

We start by the system \eqref{eP1} and we consider its energy functional defined
by (we recall \eqref{e2.4})
\begin{equation}
\begin{split}
E_1 (t) &=  \frac{1}{2}\int_0^{L}\Bigl(\rho_1\varphi _t^2+\rho_2\psi_t^2
 +\rho_3\theta^2 +k_1 (\varphi_x+\psi)^2\\
&\quad +\Bigl(k_2-\int_0^{+\infty}g (s)ds\Bigr)\psi _x^2\Bigr) dx
 + \frac{1}{2}g\circ \psi _x .
\end{split}\label{e6.3}
\end{equation}

\begin{theorem}[\eqref{e1.1} holds]  \label{thm6.1}
Assume that \eqref{e1.1}, {\rm (H2),  (H3)} are satisfied,
and let $U_0\in \mathcal{H}$ satisfying \eqref{e2.5}.
Then there exist positive constants $c',c'',\epsilon _0$ (depending continuously on
$E_1(0)$) for which $E_1$ satisfies \eqref{e2.6}.
\end{theorem}

\begin{theorem}[\eqref{e1.1} does not hold] \label{thm6.2}
Assume that {\rm (H2)} and {\rm (H3)} are
satisfied, and let $U_0\in D(A)$ satisfying \eqref{e2.10}. Then there exist
positive constants $C$ and $\epsilon _0$ (depending continuously on
 $\|U_0\| _{D(A)}$) such that $E_1$ satisfies \eqref{e2.11}.
\end{theorem}

 The energy functionals of \eqref{eP2}, \eqref{eP3} and \eqref{eP4} are,
respectively, defined by
\begin{gather}
\begin{aligned}
E_2 (t) &=  \frac{1}{2}\int_0^{L}\Bigl(\rho_1\varphi _t^2
+\rho_2\psi_t^2 +\rho_3 {\tilde {\theta}}^2 +k_1 (\varphi_x+\psi)^2 +k_2 \psi_x^2 \\
&\quad - \Bigl(\int_0^{+\infty}g (s)ds\Bigr)\varphi_x^2\Bigr) dx
 + \frac{1}{2}g\circ \varphi _x ,
\end{aligned}\label{e6.4}
\\
\begin{aligned}
E_3 (t) &=  \frac{1}{2}\int_0^{L}\Bigl(\rho_1\varphi_t^2+\rho_2\psi_t^2
 +\rho_3 {\tilde {\theta}}^2 +\rho_4 q^2 +k_1 (\varphi_x+\psi)^2 +k_2 \psi_x^2\\
&\quad  -\Bigl(\int_0^{+\infty}g (s)ds\Bigr)\varphi_x^2\Bigr) dx
   +\frac{1}{2}g\circ \varphi _x,
\end{aligned}\label{e6.5}\\
\begin{aligned}
E_4 (t) &=  \frac{1}{2}\int_0^{L}\Bigl(\rho_1\varphi_t^2+\rho_2\psi_t^2
  +\rho_3 {\tilde {\theta}}^2 +\rho_4 q^2 +k_1 (\varphi_x+\psi)^2 \\
&\quad +\Bigl( k_2 -\int_0^{+\infty}g (s)ds\Bigr)\psi_x^2\Bigr) dx
 +\frac{1}{2}g\circ \psi _x ,
\end{aligned}\label{e6.6}
\end{gather}
where
\begin{equation}
{\tilde{\theta}}(x,t)=\theta (x,t)-{\frac{1}{L}}\int_0^{L}\theta _0(y)dy.
\label{e6.7}
\end{equation}

\begin{theorem} \label{thm6.3}
 Assume that {\rm (H2)} and {\rm (H3)} are
satisfied, and let $U_0\in \mathcal{H}$ satisfying \eqref{e2.5}. Then there exist
positive constants $c',\,c'',\epsilon _0$
(depending continuously on $E_2(0)$ in case \eqref{eP2}, and on $E_3(0)$
in case \eqref{eP3}) for which $E_2$ and $E_3$ satisfy \eqref{e2.6}.
\end{theorem}

 \begin{theorem} \label{thm6.4}
Assume that {\rm (H2)} and {\rm (H3)} are
satisfied, and let $U_0\in D(A)$ satisfying \eqref{e2.10}. Then there exist
positive constants $C$ and $\epsilon _0$ (depending continuously on
$\|U_0\| _{D(A)}$) such that $E_4$ satisfies \eqref{e2.11}.
\end{theorem}

\subsection{Proof of Theorems \ref{thm6.1} and \ref{thm6.2}}
The proofs are very similar to the ones of Theorems \ref{thm2.1}
 and \ref{thm2.2}, respectively.

\begin{lemma} \label{lem6.1}
The energy functional $E_1$ defined by \eqref{e6.3} satisfies
\begin{equation}
E_1'(t)={\frac{1}{2}}g'\circ \psi
_x-k_3\int_0^{L}\theta _x^2dx\leq 0.\label{e6.8}
\end{equation}
\end{lemma}

\begin{proof} By multiplying the first three equations
of \eqref{eP1} by $\varphi _t$, $\psi _t$ and $\theta $, respectively,
integrating over $]0,L[$, and using the boundary conditions, we obtain \eqref{e6.8}
(which implies that \eqref{eP1} is dissipative).
\end{proof}

Now, we consider the functionals $I_1-I_8$ defined in Section 3 and
\begin{equation}
I_{10}(t)=N_1E_1(t)+N_2I_1(t)+N_3I_8(t)+I_7(t),\label{e6.9}
\end{equation}
where $N_1,N_2,N_3>0$. Using \eqref{e6.8} and the same computations as in
Section 3 (where we keep the terms depending on $\theta $), we obtain (instead
of \eqref{e3.17} and with $g_0=\int_0^{+\infty }g(s)ds$ and
$\delta ={\frac{{k_1}}{{8N_2}}}$ in \eqref{e3.5})
\begin{equation}
\begin{split}
I_{10}'(t)
&\leq  -(\frac{l {N_3}}{{2}}-c)\int_0^{L}\psi_x^2dx
-(\frac{\rho_1}{16}-\epsilon N_3)\int_0^{L}\varphi _t^2dx
\\
&\quad  -\Bigl(N_2\rho_2 g_0 -\frac{{cN_3}}{{\epsilon }}-c\Bigr)
\int_0^{L}\psi_t^2dx-{\frac{{k_1}}{{8}}}\int_0^{L}(\varphi_x
+\psi)^2dx+c_{N_2,N_3}g\circ \psi_x
\\
&\quad  +({\frac{{N_1}}{{2}}}-c_{N_2})g'\circ \psi_x-N_1 k_3
\int_0^L \theta_x^2 dx+({{\rho_1 k_2}\over{k_1}}-\rho_2)\int_0^L\varphi_t\psi_{xt}dx
\\
&\quad  +k_4 \int_0^{L}\theta_x \Bigl(({1\over 8}-N_3)\psi
-(\varphi_x +\psi)+N_2 \int_0^{+\infty}g(s)(\psi (t)-\psi (t-s))ds
\\
&\quad  -\frac{1}{2\epsilon} m(x)\Bigl(k_2 \psi_x -\int_0^{+\infty}
g(s)\psi_x (t-s)ds\Bigr)\Bigr)dx.
\end{split}\label{e6.10}
\end{equation}
Using Young's inequality, \eqref{e3.2}, \eqref{e6.2} (for $\psi $)
and \eqref{e6.3},  for
any $\epsilon _1>0$, we have
\begin{align*}
&k_4\int_0^{L}\theta _x\Bigl(({\frac{1}{8}}-N_3)\psi -(\varphi
_x+\psi )+N_2\int_0^{+\infty }g(s)(\psi (t)-\psi (t-s))ds
\\
&-{\frac{1}{{2\epsilon }}}m(x)\Bigl(k_2\psi _x-\int_0^{+\infty
}g(s)\psi _x(t-s)ds\Bigr)\Bigr)dx
\\
&\leq \epsilon _1E_1(t)+c_{N_2,N_3,\epsilon ,\epsilon
_1}\int_0^{L}\theta _x^2dx.
\end{align*}
Then, with the same choice of $N_3$, $\epsilon $ and $N_2$ as for
\eqref{e3.17},  from \eqref{e6.10}, we obtain
\begin{align*}
I_{10}'(t)
&\leq  -c \Bigl(g\circ \psi_x+\int_0^{L}\Bigr(\varphi _t^2 +\psi_t^2
+\theta^2 +(\varphi_x +\psi)^2 +\psi_x^2\Bigl)dx\Bigr)
\\
&\quad  +\epsilon_1 E_1 (t)+cg\circ \psi_x +({{\rho_1 k_2}\over {k_1}}-\rho_2)
\int_0^L\varphi_t\psi_{xt}dx
\\
&\quad  +c\int_0^{L}\theta^2 dx -(N_1 k_3-c_{\epsilon_1})\int_0^L
\theta_x^2 dx+({\frac{{N_1}}{{2}}}-c)g'\circ \psi_x .
\end{align*}
Therefore, using the definition of $E_1$ and Poincar\'e's inequality
\eqref{e6.2} for $\theta $,
\begin{equation}
\begin{split}
I_{10}'(t) &\leq  -(c-\epsilon_1 )E_1 (t)+cg\circ \psi_x
+({{\rho_1 k_2}\over {k_1}}-\rho_2)\int_0^L\varphi_t\psi_{xt}dx
\\
&\quad  -(N_1 k_3 -c_{\epsilon_1})\int_0^L \theta_x^2 dx+({\frac{{N_1}}{{2}}}-c)g'\circ \psi_x .
\end{split}\label{e6.11}
\end{equation}
On the other hand, there exists a positive constant $\alpha $ (which does
not depend on $N_1$) satisfying $|N_2I_1+N_3I_8+I_7|\leq \alpha
E_1$, which implies
\[
(N_1-\alpha )E_1\leq I_{10}\leq (N_1+\alpha )E_1.
\]
Then, by choosing $\epsilon _1$ small enough so that $c-\epsilon _1>0$,
and $N_1$ large enough so that ${\frac{{N_1}}{{2}}}-c\geq 0$,
$N_1k_3-c_{\epsilon _1}\geq 0$ and $N_1>\alpha $, we deduce that
$I_{10}\sim E_1$ and
\begin{equation}
I_{10}'(t)\leq -cE_1(t)+cg\circ \psi _x+({\frac{{\rho _1k_2}
}{{k_1}}}-\rho _2)\int_0^{L}\varphi _t\psi _{xt}dx,\label{e6.12}
\end{equation}
which is similar to \eqref{e3.18}. Then the proof of Theorems \ref{thm6.1}
 and \ref{thm6.2} can be completed as in Sections 3 and 4, respectively.

\subsection{Proof of Theorem \ref{thm6.3}}
 First, we consider the case \eqref{eP2}.

\begin{lemma} \label{lem6.2}
The energy functional $E_2$ defined by \eqref{e6.4} satisfies
\begin{equation}
E_2'(t)={\frac{1}{2}}g'\circ \varphi
_x-k_3\int_0^{L}\theta _x^2dx\leq 0.\label{e6.13}
\end{equation}
\end{lemma}

 \begin{proof} Note that, thanks to the fact that
${\tilde{\theta}}_x=\theta _x$ and ${\tilde{\theta}}_t=\theta _t$,
\eqref{eP2} is also satisfied with ${\tilde{\theta}}$ (defined by \eqref{e6.7}) and
$\theta _0-{\frac{1}{L}}\int_0^{L}\theta _0(y)dy$ instead
of $\theta $ and $\theta _0$, respectively. Then, as in case \eqref{e6.8}, by
multiplying the first three equations of \eqref{eP2}, respectively, by $\varphi
_t$, $\psi _t$ and ${\tilde{\theta}}$, integrating over $]0,L[$, and
using the boundary conditions, we obtain \eqref{e6.13}.
\end{proof}

\begin{lemma} \label{lem6.3}
The functional
\[
J_1(t)=-\rho _1\int_0^{L}\varphi _t\int_0^{+\infty }g(s)(\varphi
(t)-\varphi (t-s))\,ds\,dx
\]
satisfies, for any $\delta >0$,
\begin{equation}
\begin{split}
J_1'(t) &\leq  -\rho_1 \Bigl(\int_0^{+\infty}g (s)ds-\delta \Bigr)
\int_0^{L} \varphi_t^2\,dx
\\
&\quad  +\delta\int_0^L (\psi_x^2 +(\varphi_x +\psi)^2 )dx+ c_{\delta}
g \circ\varphi_x-c_{\delta} g'\circ \varphi_x.
\end{split}\label{e6.14}
\end{equation}
\end{lemma}

 The proof of the above lemma is similar to the proof of Lemma \ref{lem3.3},
and is omitted.

 \begin{lemma} \label{lem6.4}
The functional
\[
J_2(t)=\int_0^{L}(\rho _1\varphi \varphi _t+\rho _2\psi \psi_t)dx
\]
satisfies, for some positive constants $c$ and ${\tilde{c}}$,
\begin{equation}
\begin{split}
J_2'(t) &\leq  \int_0^{L} (\rho_1\varphi_t^2 +\rho_2\psi_t^2)dx
  -c \int_0^{L} ((\varphi_x+\psi )^2 +\psi_x^2 )dx\\
&\quad +{\tilde c}\Bigl(\int_0^L {\tilde{\theta}}^2 dx +g \circ \varphi_x \Bigr).
\end{split}\label{e6.15}
\end{equation}
\end{lemma}

\begin{proof} Because  system \eqref{eP2} is still
satisfied with ${\tilde{\theta}}$ and
$\theta _0-{\frac{1}{L}}\int_0^{L}\theta _0(y)dy$ instead of $\theta $
and $\theta _0$,
respectively, then, by exploiting the first two equations of \eqref{eP2} and
integrating over $]0,L[$, we obtain
\begin{align*}
J_2'(t) &=  \int_0^{L} (\rho_1\varphi_t^2 +\rho_2\psi_t^2)dx
-k_1\int_0^{L}(\varphi_x+\psi)^2 dx
\\
&\quad -k_2\int_0^{L}\psi_x^2dx+\int_0^L \varphi_x \int_0^{+\infty} g(s)
\varphi_x (t-s)\,ds\,dx+k_4 \int_0^{L} \psi_x {\tilde  {\theta}} dx.
\end{align*}
Let $l_1=k_1-\int_0^{+\infty }g(s)ds$
($l_1>0$ thanks to \eqref{e6.1}). Using \eqref{e3.2} for $\varphi $
and Young's inequality, for any $\epsilon >0$,  we obtain
\begin{align*}
&\int_0^L \varphi_x \int_0^{+\infty} g(s)\varphi_x (t-s)\,ds\,dx\\
&=  \int_0^L \varphi_x \int_0^{+\infty} g(s)(\varphi_x (t-s)-\varphi_x (t)
+\varphi_x (t))\,ds\,dx
\\
&=  \int_0^{+\infty }g (s)ds\int_0^L \varphi_x^2 dx
-\int_0^L \varphi_x (t)\int_0^{+\infty} g(s)(\varphi_x (t)-\varphi_x (t-s))ds dx
\\
&\leq  (k_1 -l_1 +\epsilon)\int_0^L \varphi_x^2 dx + c_{\epsilon}g \circ \varphi_x.
\end{align*}
Similarly, for any $\epsilon '>0$,  we have
\[
k_4\int_0^{L}\psi _x{\tilde{\theta}}dx\leq \epsilon '\int_0^{L}\psi _x^2dx
+c_{\epsilon '}\int_0^{L}{\tilde{\theta}}^2dx.
\]
On the other hand, using \eqref{e6.2} for $\psi $,  for any $\epsilon
''>0$, we have
\[
\int_0^{L}\varphi _x^2dx=\int_0^{L}(\varphi _x+\psi -\psi
)^2dx\leq (1+\epsilon '')\int_0^{L}(\varphi _x+\psi
)^2dx+(1+{\frac{{1}}{{\epsilon ''}}})k_0\int_0^{L}\psi
_x^2dx.
\]
Inserting these three estimates into the previous equality, we obtain
\begin{align*}
J_2'(t) &\leq  \int_0^{L} (\rho_1\varphi_t^2+\rho_2\psi_t^2)dx
-\Bigl(l_1-\epsilon-\epsilon''(k_1-l_1 +\epsilon)\Bigr)
\int_0^L (\varphi_x +\psi)^2 dx
\\
&\quad  -\Bigl(k_2-\epsilon'-k_0 (1+{1\over{\epsilon''}})
(k_1-l_1+\epsilon)\Bigr)\int_0^L \psi_x^2 dx
+c_{\epsilon'}\int_0^{L} {\tilde  {\theta}}^2 dx+c_{\epsilon} g \circ\varphi_x.
\end{align*}
Thanks to \eqref{e6.1}, we can choose
${\frac{{k_0(k_1-l_1)}}{{k_2-k_0(k_1-l_1)}}}<\epsilon ''<{\frac{{l_1}}{{
k_1-l_1}}}$ and $\epsilon ,\epsilon '>0$ small enough such
that
\[
\min \{l_1-\epsilon -\epsilon ''(k_1-l_1+\epsilon
),k_2-\epsilon '-k_0(1+{\frac{1}{{\epsilon ''}}}
)(k_1-l_1+\epsilon )\}>0,
\]
therefore, we obtain \eqref{e6.15}.
\end{proof}

 The Neumann boundary conditions considered on $\theta $ in \eqref{eP2}
 do not allow the use of Poincar\'e's inequality \eqref{e6.2} for $\theta $.
To overcome this difficulty we use the following classical argument
 (see \cite{m13}):
 by integrating the third equation of \eqref{eP2} and using the boundary conditions,
 we obtain
\begin{align*}
\partial_t \Bigl(\int_0^L \theta (x,t)dx\Bigr)
&=  \int_0^L \theta_t (x,t)dx={1\over {\rho_3}}\int_0^L (k_3 \theta_{xx} (x,t)
-k_4 \psi_{xt} (x,t))dx
\\
&=  {1\over {\rho_3}}\Bigl[k_3 \theta_x (x,t)-k_4 \psi_t (x,t)\Bigr]_{x=0}^{x=L} =0,
 \quad\forall t\in \mathbb{R}_{+}.
\end{align*}
Then
\[
\int_0^{L}\theta (x,t)dx=\int_0^{L}\theta (x,0)dx=\int_0^{L}\theta
_0(x)dx,\quad \forall t\in \mathbb{R}_{+}.
\]
Therefore, the functional ${\tilde{\theta}}$ defined by \eqref{e6.7} satisfies
$\int_0^{L}{\tilde{\theta}}(x,t)dx=0$, and then \eqref{e6.2} is also
applicable for ${\tilde{\theta}}$. Now, as in \cite{m13}, we have the following
lemma.

\begin{lemma} \label{lem6.5}
The functional
\[
J_3(t)=\rho _2\rho _3\int_0^{L}\psi _t\Bigl(\int_0^{x}{\tilde{
\theta}}(y,t)dy\Bigr)dx
\]
satisfies, for any $\epsilon >0$,
\begin{equation}
J_3'(t)\leq -(\rho _2k_4-\epsilon )\int_0^{L}\psi
_t^2+\epsilon \int_0^{L}(\psi _x^2+(\varphi _x+\psi
)^2)dx+c_{\epsilon }\int_0^{L}\theta _x^2dx.\label{e6.16}
\end{equation}
\end{lemma}

Following the same arguments as in \cite{m13}
one can prove easily our lemma \ref{lem6.5}.

Now, we go back to the proof of Theorem \ref{thm6.3} in case \eqref{eP2}. Let
$N_1,N_2,N_3,N_4>0$ and, as before,
$g_0= \int_0^{+\infty }g(s)ds$. We put
\begin{equation}
J_4(t)=N_1E_2(t)+N_2J_1(t)+N_3J_2(t)+N_4J_3(t).\label{e6.17}
\end{equation}
Then, using Poincar\'e's inequality \eqref{e6.2} for ${\tilde{\theta}}$,
combining \eqref{e6.13}-\eqref{e6.16} and choosing $\delta ={\frac{{1}}{{N_2}}}$ and
$\epsilon ={\frac{{1}}{{N_4}}}$ in \eqref{e6.14} and \eqref{e6.16},
respectively, we find
\begin{equation}
\begin{split}
J_4'(t) &\leq  -(cN_3 -2)\int_0^{L}(\psi_x^2+(\varphi_x +\psi)^2)dx
-(N_4 \rho_2 k_4 -\rho_2 N_3 -1)\int_0^{L}\psi _t^2dx
\\
&\quad  -(N_2\rho_1 g_0 -\rho_1 N_3-\rho_1 )\int_0^{L}\varphi_t^2dx
-(N_1 k_3 -c_{N_3,N_4})\int_0^L \theta_x^2 dx
\\
&\quad  +({\frac{{N_1}}{{2}}}-c_{N_2})g'\circ \varphi_x+c_{N_2,N_3}g\circ \varphi_x.
\end{split}\label{e6.18}
\end{equation}
So, we choose $N_3$ large enough so that $cN_3-2>0$, then $N_4$ large
enough so that $N_4k_4\rho _2-\rho _2N_3-1>0$. Next, we choose
$N_2$ large enough so that
\[
N_2\rho _1g_0-\rho _1N_3-\rho _1>0.
\]
Consequently, using again Poincar\'e's inequality \eqref{e6.2} for
 ${\tilde{\theta}}$,  from \eqref{e6.18}, we obtain
\begin{align*}
J_4'(t) &\leq  -c \Bigl(g\circ \varphi_x
+\int_0^{L}\Bigr(\varphi _t^2 +\psi_t^2 +{\tilde{\theta}}^2
+(\varphi_x +\psi)^2 +\psi_x^2\Bigl)dx\Bigr)
\\
&\quad  +cg\circ \varphi_x -(N_1 k_3-c)\int_0^L \theta_x^2 dx
+({\frac{{N_1}}{{2}}}-c)g'\circ \varphi_x .
\end{align*}
Therefore, using the definition of $E_2$,
\[
J_4'(t)\leq -cE_2(t)+cg\circ \varphi
_x-(N_1k_3-c)\int_0^{L}\theta _x^2dx+({\frac{{N_1}}{{2}}}
-c)g'\circ \varphi _x.
\]
Now, as in Section 3, choosing $N_1$ large enough so that ${\frac{{N_1}}{
{2}}}-c\geq 0$, $N_1k_3-c\geq 0$ and $J_4\sim E_2$, we deduce that
\begin{equation}
J_4'(t)\leq -cE_2(t)+cg\circ \varphi _x.\label{e6.19}
\end{equation}
To estimate the term $g\circ \varphi _x$ in \eqref{e6.19},
we apply Lemma \ref{lem3.11}
for $\varphi $ and $E_2$ instead of $\psi $ and $E$, respectively,
and we obtain (similarly to \eqref{e3.19})
\begin{equation}
G'(\epsilon _0E_2(t))g\circ \varphi _x\leq -cE_2'(t)
+c\epsilon _0E_2(t)G'(\epsilon _0E_2(t)),\quad \forall
t\in \mathbb{R}_{+},\,\,\forall \epsilon _0>0.\label{e6.20}
\end{equation}
By multiplying \eqref{e6.19} by $G'(\epsilon _0E_2(t))$, inserting
\eqref{e6.20} and choosing $\epsilon _0$ small enough, we obtain
\begin{equation}
G'(\epsilon _0E_2(t))J_4'(t)+cE_2'(t)\leq
-cE_2(t)G'(\epsilon _0E_2(t)),\label{e6.21}
\end{equation}
and then the proof can be finalized exactly as for \eqref{e3.20} in Section 3,
which shows \eqref{e2.6} for $E_2$.

Now, we consider the case \eqref{eP3} and we prove \eqref{e2.6} for $E_3$.

\begin{lemma} \label{lem6.6}
The energy functional $E_3$ defined by \eqref{e6.5} satisfies
\begin{equation}
E_3'(t)={\frac{1}{2}}g'\circ \varphi
_x-k_5\int_0^{L}q^2dx\leq 0.\label{e6.22}
\end{equation}
\end{lemma}

\begin{proof} Because \eqref{eP3} holds with ${\tilde{\theta}}$ and
$\theta _0-{\frac{1}{L}}\int_0^{L}\theta _0(y)dy$
instead of $\theta $ and $\theta _0$, respectively, then by multiplying
the first--fourth equations of \eqref{eP3} by $\varphi _t$, $\psi _t$, ${
\tilde{\theta}}$ and $q$, respectively, and integrating over $]0,L[$, we obtain
\eqref{e6.22}.
\end{proof}

As in \cite{m13} and similarly to \eqref{e6.16}, we
prove the following estimate.

\begin{lemma} \label{lem6.7}
The functional
\[
Q_1(t)=\rho _2\rho _3\int_0^{L}\psi _t\Bigl(\int_0^{x}{\tilde{
\theta}}(y,t)dy\Bigr)dx
\]
satisfies, for any $\epsilon _1>0$,
\begin{equation}
\begin{split}
Q_1'(t) &\leq  -(\rho_2 k_4 -\epsilon_1)\int_0^{L} \psi_t^2 dx
+\epsilon_1 \int_0^{L}(\psi_x^2 +(\varphi_x+\psi )^2 )dx
\\
&\quad  +({{c}\over{\epsilon_1}} +\rho_3 k_4)\int_0^L {\tilde {\theta}}^2 dx
+c_{\epsilon_1}\int_0^L q^2 dx.
\end{split} \label{e6.23}
\end{equation}
\end{lemma}

\begin{proof}
By exploiting the second and the third
equations of \eqref{eP3} and integrating over $]0,L[$  (note also that
$\int_0^{L}{\tilde{\theta}}(x,t)dx=0$, ${\tilde{\theta}}
_x=\theta _x$ and ${\tilde{\theta}}_t=\theta _t$), we obtain
\begin{align*}
Q_1'(t) &=  \rho_3\int_0^{L} (k_2 \psi_{xx} -k_1 (\varphi_x +\psi)
-k_4 {\tilde{\theta}}_x)\Bigl(\int_0^x {\tilde {\theta}} (y,t)dy\Bigr)dx
\\
&\quad  +\rho_2\int_0^{L} \psi_t\Bigl(\int_0^x (-k_3 q_x (y,t)-k_4 \psi_{xt} (y,t))dy
\Bigr)dx
\\
&=  -\rho_2 k_4\int_0^{L} \psi_t^2 dx+\rho_3 k_4 \int_0^{L} {\tilde {\theta}}^2 dx-\rho_3 k_2\int_0^{L} \psi_x{\tilde {\theta}} dx\\
\\
&\quad  -\rho_3 k_1\int_0^{L} (\varphi_x +\psi)
\Bigl(\int_0^x {\tilde {\theta}} (y,t)dy\Bigr)dx-\rho_2 k_3\int_0^{L} \psi_t q dx.
\end{align*}
Using Young's and H\"older's inequalities to estimate the last three
integrals, we obtain \eqref{e6.23}.
\end{proof}

 As in \cite{m4}, we consider the following functional.

\begin{lemma} \label{lem6.8}
The functional
\[
Q_2(t)=-\rho _3\rho _4\int_0^{L}q\Bigl(\int_0^{x}{\tilde{\theta}}
(y,t)dy\Bigr)dx
\]
satisfies, for any $\epsilon _2>0$,
\begin{equation}
Q_2'(t)\leq -(\rho _3k_3-\epsilon _2)\int_0^{L}{\tilde{
\theta}}^2dx+\epsilon _2\int_0^{L}\psi _t^2dx+c_{\epsilon
_2}\int_0^{L}q^2dx.\label{e6.24}
\end{equation}
\end{lemma}

\begin{proof} By using the third and the fourth
equations of \eqref{eP3} and integrating over $]0,L[$ (note also that
$\int_0^{L}{\tilde{\theta}}(x,t)dx=0$, ${\tilde{\theta}}
_x=\theta _x$ and ${\tilde{\theta}}_t=\theta _t$), we obtain
\begin{align*}
Q_2'(t)
&=  \rho_3\int_0^{L} (k_5 q+k_3\theta_x)
 \Bigl(\int_0^x {\tilde {\theta}} (y,t)dy\Bigr)dx\\
&\quad +\rho_4\int_0^{L}
q\Bigl(\int_0^x (k_3 q_x (y,t)+k_4 \psi_{xt} (y,t))dy\Bigr)dx
\\
&=  -\rho_3 k_3 \int_0^{L} {\tilde {\theta}}^2 dx+\rho_3 k_5
\int_0^{L} q\Bigl(\int_0^x {\tilde {\theta}} (y,t)dy\Bigr)dx
+\rho_4\int_0^{L} q(k_3 q+k_4 \psi_t)dx.
\end{align*}
Using Young's and H\"older's inequalities to estimate the last two
integrals, we obtain \eqref{e6.24}.
\end{proof}

Now, we complete the proof of Theorem \ref{thm6.3} in case \eqref{eP3}.
 Let $N_1,N_2,N_3,N_4,N_5$ be positive, $g_0=\int_0^{+\infty }g(s)ds$ and
\begin{equation}
Q_3(t)=N_1E_3(t)+N_2J_1(t)+N_3J_2(t)+N_4Q_1(t)+N_5Q_2,
\label{e6.25}
\end{equation}
where $J_1$ and $J_2$ are defined in Lemmas \ref{lem6.3} and \ref{lem6.4},
respectively.
The estimates \eqref{e6.14} and \eqref{e6.15} hold also for \eqref{eP3}
because we used only the first two equations of \eqref{eP2} and the boundary
conditions on $\varphi $ and $\psi $, which are the same as in \eqref{eP3}.
Then, by combining \eqref{e6.22}-\eqref{e6.24}, \eqref{e6.14} and \eqref{e6.15},
choosing $\delta =1/N_2$,
$\epsilon _1=1/N_4$ and $\epsilon _2=1/N_5$ in \eqref{e6.14}, \eqref{e6.23} 
and \eqref{e6.24}, respectively, we obtain
\begin{equation}
\begin{split}
Q_3'(t) &\leq  -(cN_3 -2)\int_0^{L}(\psi_x^2
 +(\varphi_x +\psi)^2)dx-(N_4 k_4\rho_2-\rho_2 N_3 -2)\int_0^{L}\psi _t^2dx
\\
&\quad  -(N_2\rho_1 g_0 -\rho_1 N_3-\rho_1 )\int_0^{L}\varphi_t^2dx\\
&\quad -(N_5 \rho_3 k_3 -{\tilde c}N_3-cN_4^2 -\rho_3 k_4 N_4 -1)
 \int_0^L {\tilde{\theta}}^2 dx
\\
&\quad -(N_1 k_5 -c_{N_4,N_5})\int_0^L q^2 dx+({\frac{{N_1}}{{2}}}
-c_{N_2})g'\circ \varphi_x+c_{N_2,N_3}g\circ \varphi_x.
\end{split}\label{e6.26}
\end{equation}
We choose $N_3$ large enough so that $cN_3-2>0$, then $N_4$ large
enough so that $N_4k_4\rho _2-\rho _2N_3-2>0$. After, we choose
$N_2$ large enough so that
\[
N_2\rho _1g_0-\rho _1N_3-\rho _1>0.
\]
Next, we pick $N_5$ large enough so that
 $N_5k_3\rho _3-{\tilde{c}} N_3-cN_4^2-\rho _3k_4N_4-1>0$.
Then \eqref{e6.26} implies
\begin{align*}
Q_3'(t)
&\leq  -c \Bigl(g\circ \varphi_x+\int_0^{L}\Bigr(\varphi _t^2 +\psi_t^2
+{\tilde{\theta}}^2 +q^2 +(\varphi_x +\psi)^2 +\psi_x^2\Bigl)dx\Bigr)
\\
&\quad  +cg\circ \varphi_x -(N_1 k_5 -c)\int_0^L q^2 dx+({\frac{{N_1}}{{2}}}-c)
g'\circ \varphi_x .
\end{align*}
Now, as before, choosing $N_1$ large enough so that
 ${\frac{{N_1}}{{2}}}-c\geq 0$, $N_1k_5-c\geq 0$ and $Q_3\sim E_3$, and using the
definition of $E_3$, we conclude that
\begin{equation}
Q_3'(t)\leq -cE_3(t)+cg\circ \varphi _x.\label{e6.27}
\end{equation}
The estimate \eqref{e6.27} is similar to \eqref{e6.19}, and then we complete the proof
exactly as for \eqref{eP2} to get \eqref{e2.6} for $E_3$.
This completes  the proof of Theorem \ref{thm6.3}.


\subsection*{Proof of Theorem \ref{thm6.4}}
As in Section 4, we will show that $E_4$ satisfies the
inequality \eqref{e6.41} below, which implies \eqref{e2.11} for $E_4$.
As for \eqref{e6.22}, we can see that the energy functional $E_4$ defined
by \eqref{e6.6} satisfies
\begin{equation}
E_4'(t)={\frac{1}{2}}g'\circ \psi
_x-k_5\int_0^{L}q^2dx\leq 0.\label{e6.28}
\end{equation}
As in Section 4, we consider the second-order energy ${\tilde{E}}_4$ of
the system resulting from differentiating \eqref{eP4} with respect to time (which
is well posed for initial data $U_0\in D(A)$); that is,
 ${\tilde{E}}_4(t)=E_4(U_t(t))$, where $E_4(U(t))=E_4(t)$ and $E_4$ is
defined by \eqref{e6.6}. A simple calculation (as for \eqref{e6.28}) implies that
\begin{equation}
{\tilde{E}}_4'(t)={\frac{1}{2}}g'\circ \psi
_{xt}-k_5\int_0^{L}q_t^2dx\leq 0.\label{e6.29}
\end{equation}
Now, let $N_1,N_2,N_3>0$ and
\begin{equation}
I_{11}(t)=N_1(E_4(t)+{\tilde{E}}
_4(t))+N_2I_1(t)+N_3I_8(t)+I_7(t),\label{e6.30}
\end{equation}
where $I_1$, $I_7$ and $I_8$ are defined in Lemmas \ref{lem3.3},
\ref{lem3.8} and \ref{lem3.10}, respectively.
Then (as for $I_{10}$ in \eqref{e6.10}), by combining \eqref{e6.28} and
\eqref{e6.29}, and using the same computations as in Section 3 (we keep the terms
depending on $\theta _x$), we obtain (instead of \eqref{e3.17} and with $g_0=
\int_0^{+\infty }g(s)ds$ and $\delta ={\frac{{k_1}}{{8N_2}}}$ in \eqref{e3.5})
\begin{equation}
\begin{split}
I_{11}'(t)n
&\leq  -(\frac{l {N_3}}{{2}}-c)\int_0^{L}\psi_x^2dx
 -(\frac{\rho_1}{16}-\epsilon N_3)\int_0^{L}\varphi _t^2dx
\\
&\quad  -(N_2\rho_2 g_0 -\frac{{cN_3}}{{\epsilon }}-c)\int_0^{L}\psi_t^2dx
-{\frac{{k_1}}{{8}}}\int_0^{L}(\varphi_x +\psi)^2dx\\
&\quad +({\frac{{N_1}}{{2}}}-c_{N_2})g'\circ \psi_x
  +{\frac{{N_1}}{{2}}} g'\circ \psi_{xt}+c_{N_2,N_3}g\circ \psi_x\\
&\quad +({{\rho_1 k_2}\over {k_1}}-\rho_2)\int_0^L\varphi_t\psi_{xt}dx-k_5 N_1
\int_0^L (q^2 +q_t^2 )dx
\\
&\quad  +k_4 \int_0^{L}\theta_x \Bigl(({1\over 8}-N_3)\psi -(\varphi_x +\psi)\\
&\quad +N_2 \int_0^{+\infty}g(s)(\psi (t)-\psi (t-s))ds
\\
&\quad  -\frac{1}{2\epsilon} m(x)\Bigl(k_2 \psi_x -\int_0^{+\infty}
g(s)\psi_x (t-s)ds\Bigr)\Bigr)dx.
\end{split}\label{e6.31}
\end{equation}
Using Young's inequality, \eqref{e3.2}, \eqref{e6.2} (for $\psi $) and \eqref{e6.6},
 for any $\epsilon _1>0$, we have
\begin{align*}
&k_4\int_0^{L}\theta _x\Bigl(({\frac{1}{8}}-N_3)\psi -(\varphi
_x+\psi )+N_2\int_0^{+\infty }g(s)(\psi (t)-\psi (t-s))ds
\\
&-{\frac{1}{{2\epsilon }}}m(x)\Bigl(k_2\psi _x-\int_0^{+\infty
}g(s)\psi _x(t-s)ds\Bigr)\Bigr)dx
\\
&\leq \epsilon _1E_4(t)+c_{N_2,N_3,\epsilon ,\epsilon
_1}\int_0^{L}\theta _x^2dx.
\end{align*}
We choose $N_3$, $\epsilon $ and $N_2$ as for \eqref{e3.17} (to get negative
coefficients of the first three integrals of \eqref{e6.31}) and using \eqref{e6.2}
for ${
\tilde{\theta}}$ (note also that ${\tilde{\theta}}_x=\theta _x$), we obtain
\begin{equation}
\begin{split}
I_{11}'(t)
 &\leq  -c \Bigl(g\circ \psi_x+\int_0^{L}\Bigr(\varphi_t^2 +\psi_t^2
 +{\tilde \theta}^2 +q^2 +(\varphi_x +\psi)^2 +\psi_x^2\Bigl)dx\Bigr)
\\
&\quad  +\epsilon_1 E_4 (t)+cg\circ \psi_x+({{\rho_1 k_2}\over {k_1}}
 -\rho_2)\int_0^L\varphi_t\psi_{xt}dx
\\
&\quad  +c_{\epsilon_1}\int_0^L \theta_x^2 dx+({\frac{{N_1}}{{2}}}-c)
g'\circ \psi_x+{\frac{{N_1}}{{2}}}g'\circ \psi_{xt}
\\
&\quad  -(k_5 N_1 -c)\int_0^L q^2 dx-k_5 N_1 \int_0^L q_t^2 dx.
\end{split}\label{e6.32}
\end{equation}
On the other hand, the fourth equation of \eqref{eP4} implies that
\begin{equation}
\int_0^{L}\theta _x^2dx\leq c\int_0^{L}(q^2+q_t^2)dx.\label{e6.33}
\end{equation}
Therefore, by combining \eqref{e6.32} and \eqref{e6.33}, using \eqref{e6.6}
and choosing $\epsilon _1$ small enough,
\begin{equation}
\begin{split}
I_{11}'(t) &\leq  -cE_4 (t)+cg\circ \psi_x+({{\rho_1 k_2}\over {k_1}}
-\rho_2)\int_0^L\varphi_t\psi_{xt}dx
\\
&\quad  -(N_1 k_5 -c)\int_0^L (q^2 +q_t^2 )dx
+({\frac{{N_1}}{{2}}}-c)g'\circ \psi_x+{\frac{{N_1}}{{2}}}g'\circ \psi_{xt}.
\end{split}\label{e6.34}
\end{equation}
Now, exactly as in Lemma \ref{lem4.1}, we have, for any $\epsilon >0$,
\begin{equation}
({\frac{{\rho _1k_2}}{{k_1}}}-\rho _2)\int_0^{L}\varphi _t\psi
_{xt}dx\leq \epsilon E_4(t)+c_{\epsilon }(g\circ \psi _{xt}-g'\circ \psi _x).\label{e6.35}
\end{equation}
Then, by combining \eqref{e6.34} and \eqref{e6.35}, and choosing $\epsilon $
small enough,
\begin{equation}
\begin{aligned}
I_{11}'(t)
&\leq  -cE_4 (t)+c(g\circ \psi_x +g\circ \psi_{xt})+{\frac{{N_1}}{{2}}}
 g'\circ \psi_{xt}
\\
&\quad  -(N_1 k_5 -c)\int_0^L (q^2 +q_t^2 )dx+({\frac{{N_1}}{{2}}}-c)g'\circ \psi_x.
\end{aligned}\label{e6.36}
\end{equation}
Then, choosing $N_1$ large enough so that $N_1k_5-c\geq 0$ and
 ${\frac{ {N_1}}{{2}}}-c\geq 0$, we find
\begin{equation}
I_{11}'(t)\leq -cE_4(t)+c(g\circ \psi _x+g\circ \psi _{xt}).
\label{e6.37}
\end{equation}
Now, similarly to the case $g\circ \psi _x$ and $g\circ \psi _{xt}$ in
Sections 3 and 4 (estimates \eqref{e3.19} and \eqref{e4.5})
and using \eqref{e2.10}, \eqref{e6.28} and
\eqref{e6.29}, we obtain the following two estimates:
\begin{gather}
G'(\epsilon _0E_4(t))g\circ \psi _x\leq -cE_4'(t)+c\epsilon _0E_4(t)G'(\epsilon _0E_4(t)),\quad \forall
\epsilon _0>0, \label{e6.38}
\\
G'(\epsilon _0E_4(t))g\circ \psi _{xt}\leq -c{\tilde{E}}
_4'(t)+c\epsilon _0E_4(t)G'(\epsilon
_0E_4(t)),\quad \forall \epsilon _0>0.\label{e6.39}
\end{gather}
Multiplying \eqref{e6.37} by $G'(\epsilon _0E_4(t))$, inserting \eqref{e6.38}
and \eqref{e6.39}, and choosing $\epsilon _0$ small enough, we deduce that
\begin{equation}
\begin{split}
E_4(t)G'(\epsilon _0E_4(t))
&\leq -c\Bigl(G'(\epsilon_0E_4(t))I_{11}(t)+c\epsilon _0E_4'(t)
G''(\epsilon _0E_4(t))I_{11}(t)\\
&\quad -c(E_4'(t)+{\tilde{E}}_4'(t))\Bigr)'.
\end{split}\label{e6.40}
\end{equation}
Then, by integrating \eqref{e6.40} over $[0,T]$ and using the fact that
$I_{11}\leq c(E_4+{\tilde{E}}_4)$ (thanks to \eqref{e6.30}) and
$E_4'(t)G''(\epsilon _0E_4(t))\leq 0$ (thanks to \eqref{e6.28} and (H3)),
\begin{equation}
\int_0^{T}G_0(E_4(t))dt\leq c(G'(\epsilon
_0E_4(0))+1)(E_4(0)+{\tilde{E}}_4(0)),\quad \forall T\in
\mathbb{R}_{+},\label{e6.41}
\end{equation}
where $G_0$ is defined by \eqref{e2.12}. The fact that $G_0(E_4)$ is
non-increasing and \eqref{e6.41} imply \eqref{e2.11} for $E_4$ with
 $C=c(G'(\epsilon _0E_4(0))+1)(E_4(0)+{\tilde{E}}_4(0))$. This completes the
proof of Theorem \ref{thm6.4}.

\section{Timoshenko-heat: Green and Naghdi's theory}

In this section, we consider coupled Timoshenko-thermoelasticity systems of
type III on $]0,L[$ with an infinite history acting on the second equation:
\begin{equation}
\begin{gathered}
\rho _1\varphi _{tt}-k_1(\varphi _x+\psi )_x=0, \\
\rho _2\psi _{tt}-k_2\psi _{xx}+k_1(\varphi _x+\psi )+k_4\theta
_{xt}+\int_0^{+\infty }g(s)\psi _{xx}(x,t-s)ds=0, \\
\rho _3\theta _{tt}-k_3\theta _{xx}+k_4\psi _{xt}-k_5\theta _{xxt}=0,
\\
\varphi (0,t)=\psi (0,t)=\theta _x(0,t)=\varphi (L,t)=\psi (L,t)=\theta
_x(L,t)=0, \\
\varphi (x,0)=\varphi _0(x),\quad \varphi _t(x,0)=\varphi _1(x), \\
\psi (x,-t)=\psi _0(x,t),\quad \psi _t(x,0)=\psi _1(x),\quad
 \theta (x,0)=\theta _0(x),\,\theta _t(x,0)=\theta _1(x),
\end{gathered}\label{eP5}
\end{equation}
and with an infinite history acting on the first equation:
\begin{equation}
\begin{gathered}
\rho _1\varphi _{tt}-k_1(\varphi _x+\psi )_x+\int_0^{+\infty
}g(s)\varphi _{xx}(x,t-s)ds=0, \\
\rho _2\psi _{tt}-k_2\psi _{xx}+k_1(\varphi _x+\psi )+k_4\theta
_{xt}=0, \\
\rho _3\theta _{tt}-k_3\theta _{xx}+k_4\psi _{xt}-k_5\theta _{xxt}=0,
\\
\varphi (0,t)=\psi (0,t)=\theta _x(0,t)=\varphi (L,t)=\psi (L,t)=\theta
_x(L,t)=0, \\
\varphi (x,-t)=\varphi _0(x,t),\quad \varphi _t(x,0)=\varphi _1(x), \\
\psi (x,0)=\psi _0(x),\quad \psi _t(x,0)=\psi _1(x),\quad \theta (x,0)=\theta
_0(x),\quad \theta _t(x,0)=\theta _1(x).
\end{gathered} \label{eP6}
\end{equation}
Systems \eqref{eP5} and \eqref{eP6} model the transverse vibration of a thick beam,
taking in account the heat conduction given by Green and Naghdi's theory
\cite{g1,g1,g3}.

We prove the stability of \eqref{eP5} and \eqref{eP6} under (H2) and the following
hypothesis:
\begin{itemize}

\item[(H4)] $g:\mathbb{R}_{+}\to \mathbb{R}_{+}$ is a non-increasing
differentiable function satisfying $g(0)>0$ such that \eqref{e2.1}
and \eqref{e6.1} hold in cases of \eqref{eP5} and \eqref{eP6}, respectively.
\end{itemize}

\subsection{Well-posedness}

 We discuss briefly here the well-posedness of \eqref{eP5} and \eqref{eP6}
under (H4). As in Section 5 and following the idea of \cite{d1}, we consider
\[
\eta (x,t,s)=\begin{cases}
\psi (x,t)-\psi (x,t-s) &\text{in case  \eqref{eP5}} \\
\varphi (x,t)-\varphi (x,t-s) &\text{in case  \eqref{eP6}}
\end{cases}
\]
$\eta _0(x,s)=\eta (x,0,s)$,
$U=(\varphi ,\psi ,\theta ,\varphi _t,\psi_t,\theta _t,\eta )^{T}$, and
\[
U_0=\begin{cases}
(\varphi _0,\psi _0(\cdot ,0),\theta _0,\varphi _1,\psi _1,\theta
_1,\eta _0)^{T} &\text{in case \eqref{eP5}} \\
(\varphi _0(\cdot ,0),\psi _0,\theta _0,\varphi _1,\psi _1,\theta
_1,\eta _0)^{T} &\text{in case \eqref{eP6}}
\end{cases}
\]
Then \eqref{eP5} and \eqref{eP6} can be formulated in the form \eqref{e5.8},
where $A$ is the linear operator given by
$A(v_1,v_2,v_3,v_4,v_5,v_6,v_7)=(f_1,f_2,f_3,f_4,f_5,f_6,f_7)^{T}$,
 where
\begin{gather*}
f_1=-v_4,\quad f_2=-v_5,\ f_3=-v_6,\quad
f_4=-{\frac{{k_1}}{{ \rho _1}}}\partial _x(\partial _xv_1+v_2), 
\\
\begin{aligned}
f_5&=-{\frac{1}{{\rho _2}}}\Bigl(k_2-\int_0^{+\infty }g(s)ds\Bigr)
\partial _{xx}v_2+{\frac{{k_1}}{{\rho _2}}}(\partial _xv_1+v_2)\\
&\quad +{\frac{{k_4}}{{\rho _2}}}\partial _xv_6-{\frac{{1}}{{\rho _2}}}
\int_0^{+\infty }g(s)\partial _{xx}v_7(s),
\end{aligned} \\
f_6=-{\frac{{k_3}}{{\rho _3}}}\partial _{xx}v_3+{\frac{{k_4}}{{
\rho _3}}}\partial _xv_5-{\frac{{k_5}}{{\rho _3}}}\partial
_{xx}v_6,\quad  f_7=-v_5+\partial _{s}v_7,
\end{gather*}
in case \eqref{eP5}, and
\begin{gather*}
f_1=-v_4,\quad  f_2=-v_5,\quad  f_3=-v_6,  \\
f_4=-{\frac{{k_1}}{{\rho _1}}}\partial _x(\partial _xv_1+v_2)+{
\frac{1}{{\rho _1}}}\Bigl(\int_0^{+\infty }g(s)ds\Bigr)\partial
_{xx}v_1-{\frac{{1}}{{\rho _1}}}\int_0^{+\infty }g(s)\partial
_{xx}v_7(s), \\
f_5=-{\frac{{k_2}}{{\rho _2}}}\partial _{xx}v_2+{\frac{{k_1}}{{
\rho _2}}}(\partial _xv_1+v_2)+{\frac{{k_4}}{{\rho _2}}}\partial
_xv_6, \\
f_6=-{\frac{{k_3}}{{\rho _3}}}\partial _{xx}v_3+{\frac{{k_4}}{{
\rho _3}}}\partial _xv_5-{\frac{{k_5}}{{\rho _3}}}\partial
_{xx}v_6,\quad f_7=-v_5+\partial _{s}v_7,
\end{gather*}
in case \eqref{eP6}. Let
\[
\mathcal{H}=\bigl(H_0^{1}(]0,L[)\bigr)^2\times H_{\ast
}^{1}(]0,L[)\times \bigl(L^2(]0,L[)\bigr)^2\times L_{\ast
}^2(]0,L[)\times L_{g},
\]
where $L_{g}$ is defined by \eqref{e5.5} and endowed with the inner
 product \eqref{e5.6},
\begin{gather*}
H_{\ast }^{1}(]0,L[)=\{v\in H^{1}(]0,L[),\,\int_0^{L}vdx=0\},\\
L_{\ast }^2(]0,L[)=\{v\in L^2(]0,L[),\,\int_0^{L}vdx=0\}.
\end{gather*}
The set $\mathcal{H}$, together with the inner product defined, for
$V,W\in \mathcal{H}$, by
\begin{align*}
\langle V,W\rangle_{\mathcal{H}}
 &=  \langle v_7,w_7\rangle_{L_g} +\int_0^L \Bigl( k_1 (\partial_x v_1 +v_2)
(\partial_x w_1 +w_2)+k_3 \partial_x v_3 \partial_x w_3 \Bigr)dx
\\
&\quad  +\int_0^L \Bigl(\Bigl(k_2 -\int_0^{+\infty} g(s)ds\Bigr)\partial_x v_2
\partial_x w_2 +\rho_1 v_4 w_4 +\rho_2 v_5 w_5 +\rho_3 v_6 w_6\Bigr)dx
\end{align*}
in case \eqref{eP5}, and
\begin{align*}
\langle V,W\rangle_{\mathcal{H}}
&=  \langle v_7,w_7\rangle_{L_g} +\int_0^L \Bigl( k_1 (\partial_x v_1 +v_2)
(\partial_x w_1 +w_2)+k_3 \partial_x v_3 \partial_x w_3 \Bigr)dx
\\
&\quad  +\int_0^L \Bigl(-\Bigl(\int_0^{+\infty} g(s)ds\Bigr)\partial_x v_1
\partial_x w_1 +k_2 \partial_x v_2 \partial_x w_2 +\rho_1 v_4 w_4\\
&\quad +\rho_2 v_5 w_5 +\rho_3 v_6 w_6\Bigr)dx
\end{align*}
in case \eqref{eP6}, is a Hilbert space. The domain $D(A)$ of $A$ endowed with
the norm \eqref{e5.9} is given by
 $D(A)=\{V\in \mathcal{H},\,AV\in \mathcal{H}\text{ and }v_7(0)=0\}$
and $A$ is a maximal monotone operator (the proof
is similar to the one of Section 5), and then $A$ is an infinitesimal
generator of a contraction semigroup in $\mathcal{H}$, which implies the
well-posedness results of Theorem \ref{thm5.1} for \eqref{eP5} and \eqref{eP6}.

\subsection*{Stability}

We introduce the energy functionals in \eqref{eP5} and
\eqref{eP6}, respectively, as
\begin{equation}
\begin{split}
E_5 (t) &=  \frac{1}{2}\int_0^{L}\Bigl(\rho_1\varphi_t^2+\rho_2\psi_t^2
+k_1 (\varphi_x+\psi)^2+\Bigl(k_2 -\int_0^{+\infty} g(s)ds\Bigr)\psi_x^2\Bigr)dx
\\
&\quad  +\frac{1}{2}\int_0^{L}(\rho_3 {\tilde \theta }_t^2
+k_3 {\tilde \theta }_x^2)dx+\frac{1}{2}g\circ \psi_x
\end{split}\label{e7.1}
\end{equation}
and
\begin{equation}
\begin{split}
E_6 (t) &=  \frac{1}{2}\int_0^{L}\Bigl(\rho_1\varphi_t^2+\rho_2\psi_t^2
+k_1 (\varphi_x+\psi)^2-\Bigl(\int_0^{+\infty} g(s)ds\Bigr)\varphi_x^2\Bigr)dx
\\
&\quad  +\frac{1}{2}\int_0^{L}(k_2 \psi_x^2 +\rho_3 {\tilde \theta }_t^2
+k_3 {\tilde \theta }_x^2)dx+\frac{1}{2}g\circ \varphi_x ,
\end{split}\label{e7.2}
\end{equation}
where
\begin{equation}
{\tilde{\theta}}(x,t)=\theta (x,t)-{\frac{{t}}{{L}}}\int_0^{L}\theta
_1(y)dy-{\frac{1}{L}}\int_0^{L}\theta _0(y)dy.\label{e7.3}
\end{equation}

\begin{remark} \label{rmk7.1} \rm
Using the third equation in
\eqref{eP5} and \eqref{eP6} and the boundary conditions, we easily verify that
\[
\partial _{tt}\Bigl(\int_0^{L}\theta (x,t)dx\Bigr)=\int_0^{L}\theta
_{tt}(x,t)dx={\frac{1}{{\rho _3}}}\Bigl[k_3\theta _x-k_4\psi
_t+k_5\theta _{xt}\Bigr]_{x=0}^{x=L}=0,\quad \forall t\in \mathbb{R}_{+},
\]
which implies that, using the initial data of $\theta $,
\begin{equation}
\int_0^{L}\theta (x,t)dx=t\int_0^{L}\theta _1(x)dx+\int_0^{L}\theta
_0(x)dx,\quad \forall t\in \mathbb{R}_{+}.\label{e7.4}
\end{equation}
Therefore, \eqref{e7.3} and \eqref{e7.4} imply that
$\int_0^{L}{\tilde{\theta}}(x,t)dx=0$, and then Poincar\'e's
inequality \eqref{e6.2} is applicable
also for ${\tilde{\theta}}$ and, in addition, \eqref{eP5} and \eqref{eP6} are
satisfied with ${\tilde{\theta}}$,
$\theta _0-{\frac{1}{L} } \int_0^{L}\theta _0(y)dy$ and
 $\theta _1-{\frac{1}{L} } \int_0^{L}\theta _1(y)dy$ instead of $\theta $,
 $\theta_0 $ and $\theta _1$, respectively. In the sequel, we work with
${\tilde{\theta}}$ instead of $\theta $.
\end{remark}

For the stability of \eqref{eP5}, we distinguish two cases depending
on \eqref{e1.1}.

\begin{theorem}[\eqref{e1.1} holds]  \label{thm7.1}.
Assume that {\rm (H2), (H4)} and \eqref{e1.1}
hold, and let $U_0\in \mathcal{H}$ satisfying \eqref{e2.5}. Then there exist
positive constants $c',\,c'',\epsilon _0$
(depending continuously on $E_5(0)$) for which $E_5$ satisfies \eqref{e2.6}.
\end{theorem}

\begin{theorem}[\eqref{e1.1} does not hold] \label{thm7.2}
Assume that (H2) and (H4) hold, and let $U_0\in D(A)$ satisfying \eqref{e2.10}. Then
there exist positive constants $C$ and $\epsilon _0$ (depending
continuously on $\| U_0\| _{D(A)}$) such that $E_5$ satisfies
\eqref{e2.11}.
\end{theorem}

Concerning \eqref{eP6}, the estimate \eqref{e2.6} holds for $E_6$ independently
of \eqref{e1.1}.

\begin{theorem} \label{thm7.3}
Assume that {\rm (H2)} and {\rm (H4)} hold, and let $U_0\in \mathcal{H}$
satisfying \eqref{e2.5}. Then there exist positive constants $c',\,c'',\epsilon _0$
(depending continuously on $E_6(0)$) for which $E_6$ satisfies \eqref{e2.6}.
\end{theorem}

\subsection{Proof of Theorems \ref{thm7.1} and \ref{thm7.2}}

 As for $E_1-E_4$ and by multiplying the
first equation in \eqref{eP5} by $\varphi _t$, the second one by $\psi _t$
and the third one (with ${\tilde{\theta}}$ instead of $\theta $) by
${\tilde{\theta}}_t$, integrating over $]0,L[$ and using the boundary conditions,
we obtain
\begin{equation}
E_5'(t)={\frac{1}{2}}g'\circ \psi _x-k_5\int_0^{L}{
\tilde{\theta}}_{xt}^2dx\leq 0.\label{e7.5}
\end{equation}
Now, we consider the functionals $I_1$--$I_8$ defined in Section 3 and (as
in \eqref{e6.9})
\begin{equation}
I_{12}(t)=N_1E_5(t)+N_2I_1(t)+N_3I_8(t)+I_7(t).\label{e7.6}
\end{equation}
Using \eqref{e7.5} and the same computations as in Section 3 (for \eqref{eP5} with
${\tilde{\theta}}$ instead of $\theta $), we obtain (instead of \eqref{e3.17}
and with $g_0=\int_0^{+\infty }g(s)ds$ and
$\delta ={\frac{{k_1}}{{8N_2}}}$ in \eqref{e3.5}),
\begin{equation}
\begin{split}
&I_{12}'(t)\\
&\leq  -(\frac{l {N_3}}{{2}}-c)\int_0^{L}\psi_x^2dx
-(\frac{\rho_1}{16}-\epsilon N_3)\int_0^{L}\varphi _t^2dx
-{\frac{{k_1}}{{8}}}\int_0^{L}(\varphi_x +\psi)^2dx
\\
&\quad  -\Bigl(N_2\rho_2 g_0 -\frac{{cN_3}}{{\epsilon }}-c\Bigr)
\int_0^{L}\psi_t^2dx 
\\
&\quad  -\frac{1}{2\epsilon} m(x)\Bigl(k_2 \psi_x
-\int_0^{+\infty} g(s)\psi_x (t-s)ds\Bigr)\Bigr)dx
 -N_1 k_5 \int_0^L {\tilde\theta}_{xt}^2 dx
\\
&\quad  +({\frac{{N_1}}{{2}}}-c_{N_2})g'\circ \psi_x
+c_{N_2,N_3}g\circ \psi_x+({{\rho_1 k_2}\over {k_1}}-\rho_2)
 \int_0^L\varphi_t\psi_{xt}dx
\\
&\quad  +k_4 \int_0^{L}{\tilde\theta}_{xt} \Bigl(({1\over 8}-N_3)\psi
-(\varphi_x +\psi)+N_2 \int_0^{+\infty}g(s)(\psi (t)-\psi (t-s))ds\,.
\end{split}\label{e7.7}
\end{equation}
Using Young's inequality, \eqref{e3.2}, \eqref{e6.2} (for $\psi $) and \eqref{e7.1},
 for any $\epsilon _1>0$, we have
\begin{align*}
&k_4\int_0^{L}{\tilde{\theta}}_{xt}\Bigl(({\frac{1}{8}}-N_3)\psi
-(\varphi _x+\psi )+N_2\int_0^{+\infty }g(s)(\psi (t)-\psi (t-s))ds
\\
&-{\frac{1}{{2\epsilon }}}m(x)\Bigl(k_2\psi _x-\int_0^{+\infty
}g(s)\psi _x(t-s)ds\Bigr)\Bigr)dx
\\
&\leq \epsilon _1E_5(t)+c_{N_2,N_3,\epsilon ,\epsilon
_1}\int_0^{L}{\tilde{\theta}}_{xt}^2dx.
\end{align*}
Then, with the same choice of $N_3$, $\epsilon $ and $N_2$ as in Section
3, we obtain, from \eqref{e7.7},
\begin{equation}
\begin{split}
I_{12}'(t)
&\leq  -c \Bigl(g\circ \psi_x+\int_0^{L}(\varphi_t^2 +\psi_t^2 +{\tilde\theta}_t^2
+(\varphi_x +\psi)^2 +\psi_x^2 )dx\Bigr)
\\
&\quad  +\epsilon_1 E_5 (t)+cg\circ \psi_x+({{\rho_1 k_2}\over {k_1}}
-\rho_2)\int_0^L\varphi_t\psi_{xt}dx+c\int_0^{L}{\tilde\theta}_t^2 dx
\\
&\quad  -(N_1 k_5-c_{\epsilon_1})\int_0^L {\tilde\theta}_{xt}^2 dx
+({\frac{{N_1}}{{2}}}-c)g'\circ \psi_x .
\end{split}\label{e7.8}
\end{equation}
On the other hand, let us set
\begin{equation}
R_1(t)=\int_0^{L}(\rho _3{\tilde{\theta}}_t{\tilde{\theta}}
+k_4\psi _x{\tilde{\theta}}+{\frac{{k_5}}{2}}{\tilde{\theta}}
_x^2)dx.\label{e7.9}
\end{equation}
By differentiating $R_1$ and using the third equation in \eqref{eP5} (with
 ${\tilde{\theta}}$ instead of $\theta $), we obtain
\[
R_1'(t)=-k_3\int_0^{L}{\tilde{\theta}}_x^2dx+\rho
_3\int_0^{L}{\tilde{\theta}}_t^2dx+k_4\int_0^{L}\psi _x{\tilde{
\theta}}_tdx.
\]
Young's inequality and the definition of $E_5$ then yield, for any
$\epsilon _1>0$,
\begin{equation}
R_1'(t)\leq -k_3\int_0^{L}{\tilde{\theta}}_x^2dx+c_{
\epsilon _1}\int_0^{L}{\tilde{\theta}}_t^2dx+\epsilon _1E_5(t).
\label{e7.10}
\end{equation}
Let $R_2=I_{12}+R_1$. Then, using the definition of $E_5$ and applying
Poincar\'e's inequality \eqref{e6.2} for ${\tilde{\theta}}_t$, we deduce from
\eqref{e7.8} and \eqref{e7.10} that
\begin{equation}
\begin{split}
R_2' (t) &\leq  -(c-2\epsilon_1 )E_5 (t)+cg\circ \psi_x
+({{\rho_1 k_2}\over {k_1}}-\rho_2)\int_0^L\varphi_t\psi_{xt}dx
\\
&\quad  -(N_1 k_5-c_{\epsilon_1})\int_0^L {\tilde\theta}_{xt}^2 dx
+({\frac{{N_1}}{{2}}}-c)g'\circ \psi_x .
\end{split}\label{e7.11}
\end{equation}
Then, by choosing $\epsilon _1$ small enough so that $c-2\epsilon _1>0$,
and $N_1$ large enough so that ${\frac{{N_1}}{{2}}}-c\geq 0$,
$N_1k_5-c_{\epsilon _1}\geq 0$ and $R_2\sim E_5$ (which is possible
thanks to the definition of $E_5$, $I_{12}$, $R_1$ and $R_2$), we
deduce that
\begin{equation}
R_2'(t)\leq -cE_5(t)+cg\circ \psi _x+({\frac{{\rho _1k_2}
}{{k_1}}}-\rho _2)\int_0^{L}\varphi _t\psi _{xt}dx\label{e7.12}
\end{equation}
which is similar to \eqref{e3.18}. Then the proof of Theorems \ref{thm7.1}
 and \ref{thm7.2} can be
completed as in Sections 3 and 4, respectively.


\subsection{Proof of Theorem \ref{thm7.3}} First, as for $E_5$, we have
\begin{equation}
E_6'(t)={\frac{1}{2}}g'\circ \varphi
_x-k_5\int_0^{L}{\tilde{\theta}}_{xt}^2dx\leq 0.\label{e7.13}
\end{equation}
Now, we consider the functionals $T_1=J_1$ and $T_2=J_2$, where
$J_1$ and $J_2$ are defined in Lemmas \ref{lem6.3} and \ref{lem6.4}, respectively.
 Then,
exactly as for \eqref{e6.14} and \eqref{e6.15} (where we used the first
two equations in \eqref{eP2}, which are the same as in \eqref{eP6}
with $\theta _{xt}$ instead of $\theta _x$), we have, for any $\delta >0$,
\begin{equation}
\begin{split}
T_1'(t) &\leq  -\rho_1 \Bigl(\int_0^{+\infty}g (s)ds-\delta \Bigr)
\int_0^{L} \varphi_t^2\,dx
\\
&\quad  +\delta\int_0^L (\psi_x^2 +(\varphi_x +\psi)^2 )dx
+ c_{\delta} g \circ\varphi_x-c_{\delta} g'\circ \varphi_x ,
\end{split}\label{e7.14}
\end{equation}
and for some positive constants $c$ and ${\tilde{c}}$,
\begin{equation}
\begin{split}
T_2'(t) &\leq  \int_0^{L} (\rho_1\varphi_t^2 +\rho_2\psi_t^2)dx
\\
&\quad  -c \int_0^{L} ((\varphi_x+\psi )^2 +\psi_x^2 )dx
+{\tilde c}\Bigl(\int_0^L {\tilde{\theta}}_t^2 dx +g \circ \varphi_x \Bigr).
\end{split}\label{e7.15}
\end{equation}
On the other hand, we we have the following  lemma.

\begin{lemma} \label{lem7.1}
The functional
\[
T_3(t)=\rho _2\rho _3\int_0^{L}\psi _t\Bigl(\int_0^{x}{\tilde{
\theta}}_t(y,t)dy\Bigr)dx-\rho _2k_3\int_0^{L}{\tilde{\theta}}
_x\psi dx
\]
satisfies, for any $\epsilon >0$,
\begin{equation}
T_3'(t)\leq -(\rho _2k_4-\epsilon )\int_0^{L}\psi
_t^2+\epsilon \int_0^{L}(\psi _x^2+(\varphi _x+\psi
)^2)dx+c_{\epsilon }\int_0^{L}{\tilde{\theta}}_{xt}^2dx.\label{e7.16}
\end{equation}
\end{lemma}

\begin{proof} By using the second and the third
equations in \eqref{eP6} and integrating over $]0,L[$, we obtain
(note that
$\int_0^{L}{\tilde{\theta}}(x,t)dx=0$, ${\tilde{\theta}}
_{tt}=\theta _{tt}$ and ${\tilde{\theta}}_x=\theta _x$)
\begin{align*}
T_3'(t) &= \rho_2\int_0^{L} \psi_t\Bigl(\int_0^x (k_3 {\tilde  {\theta}}_{xx} (y,t)
-k_4 \psi_{xt} (y,t)+k_5 {\tilde  {\theta}}_{xxt} (y,t))dy\Bigr)dx
\\
&\quad  +\rho_3\int_0^{L} (k_2 \psi_{xx} -k_1 (\varphi_x +\psi)
 -k_4 {\tilde{\theta}}_{xt})\Bigl(\int_0^x {\tilde  {\theta}}_t (y,t)dy\Bigr)dx\\
&\quad -\rho_2 k_3\int_0^{L} ({\tilde  {\theta}}_x \psi_t
 +{\tilde  {\theta}}_{xt} \psi)dx
\\
&=  -\rho_2 k_4 \int_0^{L} \psi_t^2 dx+\rho_2 k_5 \int_0^{L}
 \psi_t {\tilde  {\theta}}_{xt} dx+\rho_3\int_0^{L} (-k_2 \psi_x
 +k_4 {\tilde  {\theta}}_t){\tilde  {\theta}}_t dx
\\
&\quad  -\rho_3 k_1\int_0^{L} (\varphi_x +\psi)
 \Bigl(\int_0^x {\tilde  {\theta}}_t (y,t)dy\Bigr)dx
 -\rho_2 k_3\int_0^{L} {\tilde  {\theta}}_{xt} \psi dx.
\end{align*}
Using Young's and H\"older's inequalities for
$\int_0^{x}{ \tilde{\theta}}_t(y,t)dy$ and Poincar\'e's inequality for
${\tilde{\theta }}_t$ and $\psi $ to estimate the last four integrals,
we obtain \eqref{e7.16}.
\end{proof}

 Now, we come back to the proof of Theorem \ref{thm7.3}. Let
$N_1,N_2,N_3,N_4>0$ and
\begin{equation}
T_5(t)=N_1E_6(t)+N_2T_1(t)+N_3T_2(t)+N_4T_3(t)+T_4(t),
\label{e7.17}
\end{equation}
where $T_4=R_1$ and $R_1$ is defined by \eqref{e7.9}. Note that, exactly as
for \eqref{e7.10} (where we used the third equation and the boundary conditions in
\eqref{eP5}, which are the same as in \eqref{eP6}), $T_4$ satisfies, for any
$\epsilon _1>0$,
\begin{equation}
T_4'(t)\leq -k_3\int_0^{L}{\tilde{\theta}}_x^2dx+c_{
\epsilon _1}\int_0^{L}{\tilde{\theta}}_t^2dx+\epsilon _1E_6(t).
\label{e7.18}
\end{equation}
Then, using Poincar\'e's inequality \eqref{e6.2} for ${\tilde{\theta}}_t$,
combining \eqref{e7.13}-\eqref{e7.16} and \eqref{e7.18}, and choosing
$\delta ={\frac{{1}}{{N_2}}}$ and $\epsilon ={\frac{{1}}{{N_4}}}$
in \eqref{e7.14} and \eqref{e7.16}, respectively, we find
\begin{equation}
\begin{split}
T_5'(t) &\leq  -(cN_3 -2)\int_0^{L}(\psi_x^2+(\varphi_x +\psi)^2)dx
+\epsilon_1 E_6 (t)
\\
&\quad  -(N_4 \rho_2 k_4 -\rho_2 N_3 -1)\int_0^{L}\psi _t^2dx
-k_3 \int_0^{L} {\tilde  {\theta}}_x^2 dx
\\
&\quad  -(N_2\rho_1 g_0 -\rho_1 N_3-\rho_1 )\int_0^{L}\varphi_t^2dx
+c_{N_2,N_3}g\circ \varphi_x
\\
&\quad -(N_1 k_5 -c_{N_3,N_4 ,\epsilon_1})\int_0^L {\tilde  {\theta}}_{xt}^2 dx
+({\frac{{N_1}}{{2}}}-c_{N_2})g'\circ \varphi_x.
\end{split}\label{e7.19}
\end{equation}
So, we choose $N_3$ large enough so that $cN_3-2>0$, then $N_2$ and
$N_4$ large enough so that $N_2\rho _1g_0-\rho _1N_3-\rho _1>0$
and $N_4\rho _2k_4-\rho _2N_3-1>0$. Consequently, using again
Poincar\'e's inequality \eqref{e6.2} for ${\tilde{\theta}}_t$, from
\eqref{e7.19}, we obtain
\begin{align*}
T_5'(t)
&\leq  -c \Bigl(g\circ \varphi_x+\int_0^{L}
\Bigr(\varphi _t^2 +\psi_t^2 +{\tilde{\theta}}_t^2
+(\varphi_x +\psi)^2 +\psi_x^2 +{\tilde{\theta}}_x^2\Bigl)dx\Bigr)
\\
&\quad  +\epsilon_1 E_6 (t)+cg\circ \varphi_x
-(N_1 k_5-c_{\epsilon_1})\int_0^L {\tilde  {\theta}}_{xt}^2 dx
+({\frac{{N_1}}{{2}}}-c)g'\circ \varphi_x .
\end{align*}
Therefore, using the definition of $E_6$,
\[
T_5'(t)\leq -(c-\epsilon _1)E_6(t)+cg\circ \varphi
_x-(N_1k_5-c_{\epsilon _1})\int_0^{L}{\tilde{\theta}}_{xt}^2dx+({
\frac{{N_1}}{{2}}}-c)g'\circ \varphi _x.
\]
Now, as in previous sections, choosing $\epsilon _1<c$ and $N_1$ large
enough so that ${\frac{{N_1}}{{2}}}-c\geq 0$, $N_1k_5-c_{\epsilon
_1}\geq 0$ and $T_5\sim E_6$, we deduce that
\begin{equation}
T_5'(t)\leq -cE_6(t)+cg\circ \varphi _x,\label{e7.20}
\end{equation}
which is similar to \eqref{e6.19}. Then the proof can be ended exactly as for
\eqref{e6.19} in Section 6, which shows \eqref{e2.6} for $E_6$.

\section{General comments}

\textbf{Comment 1.} Our results hold if we consider the following Neumann
boundary conditions (instead of the corresponding Dirichlet ones):
\begin{equation}
\varphi _x(0,t)=\varphi _x(L,t)=0\quad
\text{for \eqref{eP}, \eqref{eP1}, \eqref{eP4}, \eqref{eP5}}\label{e8.1}
\end{equation}
and
\begin{equation}
\theta _x(0,t)=\theta _x(L,t)=0\quad \text{for}\,\eqref{eP1}.\label{e8.2}
\end{equation}
The energy is defined with ${\tilde{\varphi}}$ instead of $\varphi $ in case
\eqref{e8.1}, and with ${\tilde{\theta}}$ instead of $\theta $ in case \eqref{e8.2},
 where
${\tilde{\theta}}$ is defined by \eqref{e6.7} and
\begin{equation}
{\tilde{\varphi}}(x,t)=\varphi (x,t)-{\frac{{t}}{{L}}}\int_0^{L}\varphi
_1(y)dy-{\frac{1}{L}}\int_0^{L}\varphi _0(y)dy.\label{e8.3}
\end{equation}
Thanks to \eqref{e6.7} and \eqref{e8.3}, we have
$\int_0^{L}{\tilde{\varphi}} (x,t)dx=\int_0^{L}{\tilde{\theta}}(x,t)dx=0$,
and then Poincar\'e's inequality \eqref{e6.2} is still applicable for
${\tilde{\varphi}}$ and ${\tilde{\theta}}$.

In case \eqref{eP2} and \eqref{eP6}, the following boundary conditions:
\begin{equation}
\varphi (0,t)=\varphi (L,t)=\psi _x(0,t)=\psi _x(L,t)=\theta
(0,t)=\theta (L,t)=0\label{e8.4}
\end{equation}
can be considered. The energy functionals $E_2$ and $E_6$ are now
defined with ${\tilde{\psi}}$ and $\theta $ instead of $\psi $ and
${\tilde{\theta}}$, respectively, where
\[
{\tilde{\psi}}(x,t)=\psi (x,t)-{\frac{1}{L} }\sqrt{{\frac{{\rho _2}}{{
k_1}}}}\Bigl(\int_0^{L}\psi _1(y)dy\Bigr)\sin (\sqrt{{\frac{{k_1}}{{
\rho _2}}}}t)-{\frac{1}{L}}\Bigl(\int_0^{L}\psi _0(y)dy\Bigr)\cos (
\sqrt{{\frac{{k_1}}{{\rho _2}}}}t).
\]
Note that systems \eqref{eP2} and \eqref{eP6} are satisfied with ${\tilde{\psi}}$, 
$\psi _0-{\frac{1}{L} }\int_0^{L}\psi _0(y)dy$ and
 $\psi _1-{\frac{1}{L}} \int_0^{L}\psi _1(y)dy$ instead of $\psi $, $\psi _0$ and $\psi _1$,
respectively. On the other hand, by integrating the second equation of \eqref{eP2}
and \eqref{eP6} over $]0,L[$ and using \eqref{e8.4}, we obtain that $\int_0^{L}{\tilde{
\psi}}(x,t)dx=0$, and then \eqref{e6.2} is applicable for ${\tilde{\psi}}$. In this
case \eqref{e8.4}, and as in \cite{m13}, the functionals $J_3$ (Lemma \ref{lem6.5}) and $T_3$
(Lemma \ref{lem7.1}) are now defined by
\[
J_3(t)=-\rho _2\rho _3\int_0^{L}\theta \Bigl(\int_0^{x}{\tilde{\psi
}}_t(y,t)dy\Bigr)dx
\]
and
\[
T_3(t)=-\rho _2\rho _3\int_0^{L}\theta _t\Bigl(\int_0^{x}{\tilde{
\psi}}_t(y,t)dy\Bigr)dx-\rho _2k_3\int_0^{L}\theta _x{\tilde{\psi}}
dx.
\]

 \textbf{Comment 2.} Our results remain true if we consider
variable coefficients $\rho _i(x)$ ($i=1,\dots ,4$) and $k_i(x)$ (
$i=1,\dots ,5$) satisfying some smallness and smoothness conditions. On the
other hand, our approach can be adapted to different kind of systems with an
infinite history to get their stability with kernels satisfying the weaker
hypothesis (H2) (see \cite{g5}).

\textbf{Comment 3.} Our
stability results concerning \eqref{eP1} (Theorems \ref{thm6.1} and \ref{thm6.2}) hold for this
porous thermoelastic system with an infinite history
\begin{equation}
\begin{gathered}
\rho _1\varphi _{tt}-k_1(\varphi _x+\psi )_x+k_4\theta _x=0, \\
\rho _2\psi _{tt}-k_2\psi _{xx}+k_1(\varphi _x+\psi )-k_5\theta
+\int_0^{+\infty }g(s)\psi _{xx}(t-s)ds=0, \\
\rho _3\theta _t-k_3\theta _{xx}+k_4\varphi _{xt}+k_5\psi _t=0,
\\
\varphi (0,t)=\psi (0,t)=\theta (0,t)=\varphi (L,t)=\psi (L,t)=\theta
(L,t)=0, \\
\varphi (x,0)=\varphi _0(x),\quad \varphi _t(x,0)=\varphi _1(x),\quad
\theta (0,t)=\theta _0(x), \\
\psi (x,-t)=\psi _0(x,t),\quad \psi _t(x,0)=\psi _1(x).
\end{gathered} \label{eP7}
\end{equation}
This system was considered in \cite{m3,s1} under condition \eqref{e1.1} and with a
finite history (that is the infinite integral in \eqref{eP7} is replaced by the
finite one $\int_0^t$), and exponential and polynomial
decay estimates were proved in \cite{s1} under condition \eqref{e1.3}, and a general
decay estimate was proved in \cite{m3} under condition \eqref{e1.5}.

\textbf{Comment 4.} Our stability results hold if we consider a finite
history of type $\int_0^t$ (instead of
$\int_0^{+\infty }$) with $\psi _0=0$ in case \eqref{eP}, \eqref{eP1}, \eqref{eP4}
and \eqref{eP5}, and $\varphi _0=0$ in case \eqref{eP2}, \eqref{eP3} and \eqref{eP6}. In
this case, the restrictions \eqref{e2.5} and \eqref{e2.10} are automatically satisfied.
 In \cite{g8}, the stability of \eqref{eP} with a finite history was proved under
condition \eqref{e1.6} but independently of $\psi _0=0$ and \eqref{e1.1}. The arguments
of \cite{g8} can be applied to \eqref{eP1}, \eqref{eP4} and \eqref{eP5} with a finite history to
get their stability under \eqref{e1.6} even if $\psi _0\neq 0$ and \eqref{e1.1} does not
hold.

\textbf{Comment 5.} To the best of our knowledge,
getting a decay estimate for the solution of \eqref{eP} with (finite or infinite)
history (or even with a frictional damping) acting only on the first
equation is an open problem. But if we consider an infinite history on both
first and second equations of \eqref{eP}, then the energy functional of \eqref{eP}
satisfies \eqref{e2.6} without the restriction \eqref{e1.1}. Let us consider this
situation
\begin{equation}
\begin{gathered}
\rho _1\varphi _{tt}-k_1(\varphi _x+\psi )_x+\int_0^{+\infty
}g_1(s)\varphi _{xx}(t-s)ds=0, \\
\rho _2\psi _{tt}-k_2\psi _{xx}+k_1(\varphi _x+\psi
)+\int_0^{+\infty }g_2(s)\psi _{xx}(t-s)ds=0, \\
\varphi (0,t)=\psi (0,t)=\varphi (L,t)=\psi (L,t)=0, \\
\varphi (x,-t)=\varphi _0(x,t),\quad \varphi _t(x,0)=\varphi _1(x), \\
\psi (x,-t)=\psi _0(x,t),\quad \psi _t(x,0)=\psi _1(x)
\end{gathered} \label{eP8}
\end{equation}
under the following hypothesis:
\begin{itemize}
\item[(H5)] $g_i :\mathbb{R}_{+}\to \mathbb{R}_{+}$ ($i=1,2$)
is differentiable non-increasing function such that $g_i (0)>0$,
\begin{gather}
l_2 =k_2 -\int_0^{+\infty }g_2 (s)ds >0,\label{e8.5}
\\
l_1 =k_1 -\int_0^{+\infty }g_1 (s)ds > {\frac{{k_0 k_1}}{{l_2}}}
\int_0^{+\infty }g_1 (s)ds,\label{e8.6}
\end{gather}
where $k_0$ is defined in \eqref{e6.2}.
\end{itemize}

As for \eqref{eP} in Section 5 and following the idea of \cite{d1}, we can prove that,
under (H5), \eqref{eP8} is well-posed according to Theorem \ref{thm5.1} by considering
\begin{gather*}
\eta ^{1}(x,t,s)=\varphi (x,t)-\varphi (x,t-s),\quad \text{in }
]0,L[\times \mathbb{R}_{+}\times \mathbb{R}_{+},
\\
\eta ^2(x,t,s)=\psi (x,t)-\psi (x,t-s),\quad \text{in }]0,L[\times
\mathbb{R}_{+}\times \mathbb{R}_{+},
\\
\eta _0^{i}(x,s)=\eta ^{i}(x,0,s),\quad
U=(\varphi ,\psi ,\varphi _t,\psi _t,\eta ^{1},\eta ^2)^{T},\\
U_0=(\varphi _0(\cdot ,0),\psi _0(\cdot ,0),\varphi _1,\psi _1,
\eta _0^{1},\eta _0^2)^{T}
\end{gather*}
 and
\[
\mathcal{H}=\bigl(H_0^{1}(]0,L[)\bigr)^2\times \bigl(L^2(]0,L[)\bigr)
^2\times L_{g_1}\times L_{g_2},
\]
where $L_{g_i}$ is defined by \eqref{e5.5} and endowed with the inner product
\eqref{e5.6} (with $g_i$ instead of $g$).
Then \eqref{eP8} is equivalent to \eqref{e5.8}, where
$A(v_1,v_2,v_3,v_4,v_5,v_6)=(f_1,f_2,f_3,f_4,f_5,f_6)^{T}
$ and
\begin{gather*}
f_1=-v_3,\quad f_2=-v_4, \\
f_3=-{\frac{{k_1}}{{\rho _1}}}\partial _x(\partial _xv_1+v_2)+{
\frac{1}{{\rho _1}}}\Bigl(\int_0^{+\infty }g_1(s)ds\Bigr)\partial
_{xx}v_1\,-{\frac{{1}}{{\rho _1}}}\int_0^{+\infty }g_1(s)\partial
_{xx}v_5(s)ds, \\
f_4=-{\frac{1}{{\rho _2}}}\Bigl(k_2-\int_0^{+\infty }g_2(s)ds\Bigr)
\partial _{xx}v_2-{\frac{{1}}{{\rho _2}}}\int_0^{+\infty
}g_2(s)\partial _{xx}v_6(s)ds+{\frac{{k_1}}{{\rho _2}}}(\partial
_xv_1+v_2), \\
f_5=-v_3+\partial _{s}v_5,\quad f_6=-v_4+\partial _{s}v_6.
\end{gather*}
The domain $D(A)$ of $A$ (endowed with the graph norm \eqref{e5.9}) is given by
$D(A)=\{V\in \mathcal{H},\,AV\in \mathcal{H}\text{ and }v_5(0)=v_6(0)=0
\}$. The proof of Theorem \ref{thm5.1} for \eqref{eP8} can be completed as in Section 5 by
proving that $A$ is a maximal monotone operator.

Now, the energy functional associated with \eqref{eP8} is defined by
\begin{equation}
\begin{split}
E_8 (t) &=  \frac{1}{2}\int_0^{L}\Bigl(\rho_1\varphi _t^2+\rho_2\psi_t^2
 -\Bigl(\int_0^{+\infty}g_1 (s)ds\Bigr)\varphi _x^2 \\
&\quad +\Bigl(k_2-\int_0^{+\infty}g_2 (s)ds\Bigr)\psi _x^2
 +k_1 (\varphi_x+\psi)^2\Bigr) dx
 +\frac{1}{2}(g_1\circ \varphi_x+g_2\circ \psi _x).
\end{split} \label{e8.7}
\end{equation}

 \begin{remark} \label{rmk8.1} \rm
Similarly to Remark \ref{rmk6.1}, we mention here that \eqref{e6.2} implies
\[
k_1 \int_0^L (\varphi_x+\psi)^2 dx\geq k_1 (1-\epsilon)\int_0^L
\varphi_x^2 dx+k_0 k_1 (1-{\frac{1}{{\epsilon}}})\int_0^L\psi_x^2 dx,
\]
for any $0<\epsilon <1$. Then, thanks to \eqref{e8.5} and \eqref{e8.6},
we can choose ${\frac{{k_0 k_1}}{{l_2+k_0 k_1}}}<\epsilon<{\frac{{l_1}}{{k_1}}}$
and obtain
\begin{equation} \label{e8.8}
\begin{split}
{\hat c}\int_0^L (\varphi_x^2 +\psi_x^2)dx
& \leq \int_0^{L}\Bigl(-\Bigl(\int_0^{+\infty}g_1 (s)ds\Bigr)\varphi
_x^2\\
&\quad +\Bigl(k_2-\int_0^{+\infty}g_2 (s)ds\Bigr)\psi _x^2+k_1
(\varphi_x+\psi)^2\Bigr) dx,
\end{split}
\end{equation}
where ${\hat c}=\min\{l_1-\epsilon k_1, l_2+(1-{\frac{1}{{\epsilon}}})k_0
k_1\}>0$. This implies that the expression
\[
\int_0^{L}\Bigl(-\Bigl(\int_0^{+\infty}g_1 (s)ds\Bigr)\varphi _x^2+
\Bigl(k_2-\int_0^{+\infty}g_2 (s)ds\Bigr)\psi _x^2+k_1
(\varphi_x+\psi)^2\Bigr) dx
\]
defines a norm on $\bigl(H_0^{1}(]0,L[)\bigr)^2$, for $(\varphi,\psi )$,
equivalent to the one induced by $\bigl(H^{1}(]0,L[)\bigr)^2$. Consequently,
the energy $E_8$ defines a norm on $\mathcal{H}$ for $U$, and therefore,
$\mathcal{H}$ equipped with the inner product that induces this energy norm
is a Hilbert space.
\end{remark}

 \begin{theorem} \label{thm8.1}
Assume that {\rm (H5)} holds and $g_i$ ($i=1,2$) satisfies {\rm (H2)}
(instead of $g$). Let $U_0\in \mathcal{H}$ such that
\begin{equation}
\exists M_i \geq 0 :\; \|\eta^i_{0x} (s)\|_{L^2 (]0,L[)}<M_i,\quad
\forall s>0 \quad (i=1,2).\label{e8.9}
\end{equation}
Then there exist positive constants $c',\,c'',\epsilon_0$
(depending continuously on $E_8 (0)$) for which $E_8$
satisfies \eqref{e2.6}.
\end{theorem}

\begin{proof}
First, as for \eqref{eP}, we have that $E_8$ satisfies
$$
E'_8 (t)=\frac{1}{2} g_1' \circ \varphi_x+\frac{1}{2}
g_2' \circ \psi_x \leq 0 . \label{e8.10}
$$
Second, we consider the functionals
\begin{gather*}
D_1 (t)=-\rho_1\int_0^{L}\varphi_t\int_0^{+\infty}g_1 (s)(\varphi
(t)-\varphi (t-s))\,ds\,dx,
\\
D_2 (t)=-\rho_2\int_0^{L}\psi _t\int_0^{+\infty} g_2 (s)(\psi (t)-\psi
(t-s))\,ds\,dx,
\\
D_3 (t)=\int_0^{L}(\rho_1\varphi \varphi_t +\rho_2 \psi \psi_t)dx.
\end{gather*}
As in the previous sections, we can prove that, for any positive constant
$\delta$, $D_1 -D_3$ satisfy
\begin{equation}
\begin{split}
D_1'(t) &\leq  \delta \int_0^{L} (\psi_x^2 +(\varphi_x+\psi)^2 )dx
 -\rho_1 \Bigl(\int_0^{+\infty}g_1 (s)ds-\delta \Bigr)\int_0^{L} \varphi_t^2dx
\\
&\quad  +c_{\delta} (g_1 \circ\varphi_x - g_1'\circ \varphi_x ),
\end{split} \label{e8.11}
\end{equation}
\begin{equation}
\begin{split}
D_2'(t) &\leq  \delta\int_0^L (\psi_x^2 +(\varphi_x+\psi)^2 )dx
-\rho_2 \Bigl(\int_0^{+\infty}g_2 (s)ds-\delta \Bigr)\int_0^{L} \psi_t^2dx
\\
&\quad  +c_{\delta} (g_2 \circ\psi_x -g_2'\circ \psi_x)
\end{split} \label{e8.12}
\end{equation}
and, for some positive constants $\delta_1$ and $\delta_2$,
\begin{equation}
\begin{split}
D_3'(t) &\leq  \int_0^{L} (\rho_1\varphi_t^2 +\rho_2\psi_t^2)dx
-\delta_1 \int_0^{L} (\psi_x^2 +(\varphi_x+\psi)^2 ) dx
\\
&\quad  +\delta_2 (g_1 \circ\varphi_x +g_2 \circ\psi_x).
\end{split} \label{e8.13}
\end{equation}
Now, let $g_0=\min\big\{\int_0^{+\infty}g_1 (s)ds,
\int_0^{+\infty}g_2 (s)ds\big\}$, $N_1,N_2 >0$ and
\[
D_4 =N_1E_8 +N_2 (D_1 +D_2 )+D_3.
\]
By combining \eqref{e8.10}--\eqref{e8.13} and taking $\delta ={\frac{1}{{N_2^2}}}$ in
\eqref{e8.11} and \eqref{e8.12}, we obtain
\begin{align*}
D_4'(t)
 &\leq  -(\delta_1 -{{2}\over{N_2}})\int_0^{L}\psi_x^2dx
 -(\delta_1 -{{2}\over{N_2}})\int_0^{L}(\varphi_x +\psi)^2dx
\\
&\quad  -\rho_1 \Bigl(N_2g_0 -{1\over{N_2}}-1\Bigr)\int_0^{L}\varphi_t^2dx
-\rho_2 \Bigl(N_2g_0 -{1\over{N_2}}-1\Bigr)\int_0^{L}\psi_t^2dx
\\
&\quad  +({\frac{{N_1}}{{2}}}-c_{N_2})(g_1'\circ \varphi_x+g_2'\circ \psi_x)
+c_{N_2}(g_1\circ \varphi_x+g_2\circ \psi_x ).
\end{align*}
At this point, we choose $N_2$ large enough so that
\[
\min\{\delta_1 -{\frac{{2}}{{N_2}}},N_2g_0 -{\frac{1}{{N_2}}}-1 \}>0.
\]
Using \eqref{e6.2} (for $\varphi$ and $\psi$) and \eqref{e8.8},
we can find that there
exists a positive constant $M_{N_2}$ (depending on $N_2$) such that
\begin{equation}
(N_1 -M_{N_2}) E_8 \leq D_4 \leq (N_1 +M_{N_2}) E_8 .\label{e8.14}
\end{equation}
Thus, choosing $N_1$ large enough so that
${\frac{{N_1}}{{2}}} -c_{N_2}\geq 0$ and $N_1 >M_{N_2}$,
\begin{equation}
D_4'(t)\leq -c\int_0^{L}\Bigl(\varphi_t^2 +\psi_t^2
+\psi_x^2 +(\varphi_x +\psi)^2 \Bigr)dx+c(g_1\circ
\varphi_x+g_2\circ \psi_x ). \label{e8.15}
\end{equation}
Then, by using \eqref{e8.7}, Equation \eqref{e8.15} implies
\begin{equation}
D_4'(t)\leq -cE_8 (t)+c(g_1\circ \varphi_x+g_2\circ \psi_x) .
\label{e8.16}
\end{equation}
Using (H2) and \eqref{e8.9}, we have (as for \eqref{e3.19} and \eqref{e6.20})
\begin{gather*}
G'(\epsilon_0 E_8 (t))g_1 \circ \varphi_x\leq -c E_8'(t)
+c \epsilon_0 E_8 (t)G'(\epsilon_0 E_8 (t)),\quad\forall t\in
\mathbb{R}_{+} ,\;\forall \epsilon_0 >0,
\\
G'(\epsilon_0 E_8 (t))g_2 \circ \psi_x\leq -c E_8'(t)+c
\epsilon_0 E_8 (t)G'(\epsilon_0 E_8 (t)),\quad\forall t\in
 \mathbb{R}_{+} ,\;\forall \epsilon_0 >0.
\end{gather*}
The proof of Theorem \ref{thm8.1} can be finalized as in Section 3.
\end{proof}

 \textbf{Comment 6.}
For \eqref{eP}, \eqref{eP1} and \eqref{eP5} when \eqref{e1.1} does
not hold, and for \eqref{eP4}, the estimate \eqref{e2.11} proved in
Theorems \ref{thm2.2}, \ref{thm6.2}, \ref{thm6.4} and \ref{thm7.2}
 can be generalized by giving a
relationship between the smoothness of the initial data and the decay
rate of the energy. Indeed, let
us consider the case \eqref{eP}. We have the following result.

\begin{theorem} \label{thm8.2}
Assume that {\rm (H1)} and {\rm (H2)} hold and let
 $n\in \mathbb{N}^*$ and $U_0 \in D(A^n)$ satisfying
\begin{equation}
\exists M_0\geq 0: \max_{m\in\{0,\dots,n\}}\{\| \partial_s^{(m)}
\eta_{0x} (s)\|_{L^2 (]0,L[)}\}\leq M_0 ,\quad\forall s>0.
\label{e8.17}
\end{equation}
Then there exist positive constants $c_n$ and $\epsilon_0$ (depending
continuously on $\| U_0\|_{D(A^n)}$) such that
\begin{equation}
E(t)\leq \phi_n ({\frac{{c_n}}{{t}}}),\quad \forall t> 0, \label{e8.18}
\end{equation}
where $\phi_1 =G_0^{-1}$, $G_0$ is defined by \eqref{e2.12} and
$\phi_m (s)=G_0^{-1} (s\phi_{m-1}(s))$, for $m=2,3,\dots ,n$ and
$s\in \mathbb{R}_+$.
\end{theorem}


\begin{remark} \label{rmk8.2} \rm
1. Under the hypotheses of Theorem \ref{thm8.2}, $E_1$, $E_4$ and $E_5$ satisfy
\eqref{e8.18}. The proof
is exactly the same one given below.

2. The estimate \eqref{e8.18} is weaker than
\begin{equation}
E(t)\leq {\frac{{c_{n}}}{{t^{n}}}},\quad \forall t>0.\label{e8.19}
\end{equation}
The estimate \eqref{e8.18} coincides with \eqref{e8.19} when $G=Id$,
and \eqref{e8.18}  generalizes \eqref{e8.19} proved in \cite{m11}
(under \eqref{e1.2}) and the one proved in \cite{m9,m10}
(under \eqref{e1.3} and $n=1$).
\end{remark}

\begin{example} \label{examp8.1} \rm
Let us consider here a simple example to illustrate how the smoothness of
$U_0$ improves the decay rate in \eqref{e8.18}.
 Let $g(t)=d/(1+t)^q$, for $q>1$, and $d>0$ small
enough so that \eqref{e2.1} is satisfied. The
classical condition \eqref{e1.3} is not satisfied if $1<q\leq 2$, while (H2) holds
with $G(t)=t^{p}$, for any $q>1$ and $p>{\frac{{q+1}}{{q-1}}}$.
 Then $\phi_{n}(s)=cs^{r_{n}}$, where $c$ is some positive constant and
$r_{n}=\sum_{m=1}^{n}{\frac{1}{{p^{m}}}}$. Therefore, \eqref{e8.18} takes the form
\begin{equation}
E(t)\leq {\frac{{c_{n}}}{{t^{r_{n}}}}},\quad \forall t>0,\;
\forall p>{\frac{{q+1}}{{q-1}}}.\label{e8.20}
\end{equation}
The decay rate $r_{n}$ increases when $n$ increases or $p$ decreases, and it
converges to $n$ (which is the decay rate in \eqref{e8.19}) when $p$ converges to 1
(that is, when $q$ converges to $+\infty $).
\end{example}


\begin{proof}[Proof of Theorem \ref{thm8.2}]
 We prove \eqref{e8.18} by induction on $n$. For $n=1$, condition \eqref{e8.17}
coincides with \eqref{e2.10}, and \eqref{e8.18} is exactly \eqref{e2.11} given
in Theorem \ref{thm2.2} and proved in Section 4.

Now, suppose that \eqref{e8.18} holds and let $U_0\in D(A^{n+1})$ satisfying
\eqref{e8.17}, for $n+1$ instead of $n$. We have $U_t(0)\in D(A^{n})$ (thanks to
Theorem \ref{thm5.1}), $U_t(0)$ satisfies \eqref{e8.17} (because $U_0$ satisfies \eqref{e8.17},
for $n+1$) and $U_t$ satisfies the first two equations and the boundary
conditions of \eqref{eP}, and then the energy ${\tilde{E}}$ of \eqref{etP}
(defined in Section 4) also satisfies
\begin{equation}
{\tilde{E}}(t)\leq \phi _{n}({\frac{{{\tilde{c}}_{n}}}{{t}}}),\quad \forall
t>0,\label{e8.21}
\end{equation}
where ${\tilde{c}}_{n}$ is a positive constant depending continuously on
$\| U_0\| _{D(A^{n+1})}$. Now, integrating \eqref{e4.6} over $[T,2T]$, for
$T\in \mathbb{R}_{+}$, and using the fact that $F\sim E$ and
$G_0(E)$ is non-increasing, we deduce that
\begin{equation}
G_0(E(2T))T\leq \int_{T}^{2T}E(t)G'(\epsilon _0E(t))dt\leq
c(E(T)+{\tilde{E}}(T)).\label{e8.22}
\end{equation}
By combining \eqref{e8.18}, \eqref{e8.21} and \eqref{e8.22}, we obtain that
for all $T>0$,
\[
E(2T)\leq G_0^{-1}\Bigl({\frac{{2c}}{{2T}}}\Bigl(\phi _{n}({\frac{{2c_{n}}
}{{2T}}})+\phi _{n}({\frac{{2{\tilde{c}}_{n}}}{{2T}}})\Bigr)\Bigr),
\]
which implies, for $t=2T$ and $c_{n+1}=\max \{2c_{n},2{\tilde{c}}_{n},4c\}$
(note that $G_0^{-1}$ and $\phi _{n}$ are non-decreasing),
\[
E(t)\leq G_0^{-1}\Bigl({\frac{{c_{n+1}}}{{t}}}\phi _{n}({\frac{{c_{n+1}}}{{
t}}})\Bigr)=\phi _{n+1}({\frac{{c_{n+1}}}{{t}}}),\quad \forall t>0.
\]
This proves \eqref{e8.18}, for $n+1$. The proof of Theorem \ref{thm8.2} is complete
\end{proof}


\subsection*{Acknowledgements}
 This work was initiated during the
visit of the first author to the Sultan Qaboos (Sultanate of Oman), Alhosn
(UAE), KFUPM (Saudi Arabia) and Konstanz (Germany) Universities during
spring and summer 2010. The first author wishes to thank these Universities
for their kind support and hospitality. The first author would like to
express his gratitude to R. Racke and B. Chentouf for useful discussions.
This work is partially supported by KFUPM under project SB100003.
The authors want to thank the anonymous referees for their valuable suggestions.

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\end{document}