\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 254, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/254\hfil Well-posedness and exponential stability]
{Well-posedness and exponential stability for a linear damped Timoshenko
 system with second sound and internal distributed delay}

\author[T. A. Apalara \hfil EJDE-2014/254\hfilneg]
{Tijani A. Apalara}  % in alphabetical order

\address{Tijani A. Apalara  \newline
Affiliated Colleges at Hafr Al-Batin,
General Science and Studies Unit-Mathematics,
King Fahd University of Petroleum and Minerals,
Hafr Al-Batin 31991, Saudi Arabia}
\email{tijani@kfupm.edu.sa}

\thanks{Submitted August 10, 2014. Published December 4, 2014.}
\subjclass[2000]{35B40, 74F05, 74F20, 93D15, 93D20}
\keywords{Timoshenko system; second sound well-posedness;
\hfill\break\indent exponential stability; distributed delay}

\begin{abstract}
 In this article we consider one-dimensional linear thermoelastic system
 of Timoshenko type with linear frictional damping and a distributed delay
 acting on the displacement equation. The heat flux of the system is
 governed by Cattaneo's law. Under suitable assumption on the weight
 of the delay and that of frictional damping, we establish the
 well-posedness result and prove that the system is exponentially
 stable regardless of the speeds of wave propagation.
\end{abstract}

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


\section{Introduction}

It is well-known that the model using classic Fourier's law of heat conduction
(which states that the heat flux is proportional to the gradient of temperature)
predicts the physical paradox of infinite speed of heat propagation.
In other words, any thermal disturbance at one point has an instantaneous
effect elsewhere in the body. To overcome this physical paradox but still
keeping the essentials of heat conduction process, many theories have merged.
One of which is the advent of the second sound effects observed experimentally
in materials at low temperature. This theory suggests replacing the classic
Fourier's law $\beta q+\theta_{x}=0$ by a modified law of heat conduction
called Cattaneo's law $\tau q_{t}+\beta q+\theta_{x}=0$.
Consequently, heat is transported by a wave propagation process instead
of the usual diffusion thereby eliminating the physical paradox of infinite
speed of heat propagation. We refer the reader to
 \cite{Cha, Cha1, Col, Jos, Mes02, Rac2, Tar} and the references therein,
for more discussion on Cattaneo's law and thermoelasticity with second sound.

 In this article we are concerned with the following thermoelastic
system of Timoshenko type with a linear frictional damping and an
internal distributed delay acting on the transverse displacement,
 where the heat flux is given by Cattaneo's law:
\begin{equation}\label{1.1}
\begin{gathered}
\rho_1\varphi_{tt}-\kappa(\varphi_{x}+\psi)_{x}
+\mu_1\varphi_{t}+\int_{\tau_1}^{\tau_2}\mu_2(s)\varphi_{t}(x, t-s)ds=0
\quad  \text{in }(0, 1)\times (0, \infty),
\\
\rho_2\psi_{tt}-b\psi_{xx}+\kappa(\varphi_{x}+\psi)
+\delta\theta_{x}=0 \quad \text{in }(0, 1)\times (0, \infty),
\\
\rho_3\theta_{t}+q_{x}+\delta\psi_{tx}=0 \quad \text{in }(0, 1)\times (0, \infty),
\\
\tau q_{t}+\beta q+\theta_{x}=0 \quad \text{in }(0, 1)\times (0, \infty),
\\
\varphi(x,0)=\varphi_0(x), \quad \varphi_{t}(x, 0)=\varphi_1(x),\quad
 \theta(x, 0)=\theta_0(x) \quad  \text{in } (0, 1),\\
\psi(x, 0)=\psi_0(x), \quad \psi_{t}(x, 0)=\psi_1(x) ,\quad
 q(x, 0)=q_0(x)\quad  \text{in } (0, 1),\\
\varphi(0, t)=\varphi(1, t)=\psi_{x}(0, t)=\psi_{x}(1, t)
=\theta(0, t)=\theta(1, t)=0 \quad\text{in }(0, \infty),\\
\varphi_{t}(x, -t)=f_0(x, t) \quad  \text{in } (0, 1)\times(0, \tau_2).
\end{gathered}
\end{equation}
Here $\varphi=\varphi(x, t)$ is the transverse displacement of the beam,
$\psi=\psi(x, t)$ is the rotation angle, $\theta=\theta(x, t)$ is
the difference temperature, $q=q(x, t)$ is the heat flux,
$\varphi_0, \varphi_1, \psi_0, \psi_1, \theta_0, q_0$ are initial data,
and $f_0$ is history function. The coefficients,
$\rho_1, \rho_2, b, \kappa,  \delta,  \beta, \mu_1$ are positive constants,
 and $\mu_2\colon [\tau_1, \tau_2]\rightarrow \mathbb{R}$ is a bounded function,
 where $\tau_1$ and  $\tau_2$ are two real numbers satisfying
$0\le \tau_1 < \tau_2$. The parameter $\tau >0$ is the relaxation time describing
the time lag in the response of the heat flux to a gradient in the temperature.
The purpose of this paper is to study the well-posedness and the asymptotic
behavior of the solution of \eqref{1.1} regardless of the speeds of wave propagation.

Delay effects arise in many applications and practical problems
(see for instance  \cite{Beu, Ric}) and it has attracted lots of attentions
from researchers in diverse fields of human endeavor such as mathematics,
engineering, science, and economics. It has been established that voluntary
introduction of delay can benefit the control (see \cite{Abd}). On the other hand,
it may not only destabilize a system which is asymptotically stable in
 the absence of delay but may also lead to ill-posedness (see \cite{Dat, Rac1}
 and the references therein). Therefore, the issue of well-posedness and the
stability result of systems with delay are of practical and theoretical importance.

Nicaise and Pignotti \cite{Nic1} considered wave equation with linear frictional
damping and internal distributed delay
\[
u_{tt}-\Delta u+\mu_1u_{t}+a(x)\int_{\tau_1}^{\tau_2}\mu_2(s)u_{t}(t-s)ds=0
\]
in $\Omega\times (0, \infty)$, with initial and mixed Dirichlet-Neumann
 boundary conditions and $a$ is a function chosen in an appropriate space.
They established exponential stability of the solution under the assumption that
\begin{equation}\label{1.2}
\|a\|_{\infty}\int_{\tau_1}^{\tau_2}\mu_2(s)ds<\mu_1.
\end{equation}
The authors also obtained the same result when the distributed delay acted
on the part of the boundary. Recently, Mustafa and Kafini \cite{Mus1}
considered a thermoelastic system with internal distributed delay
\begin{gather*}
au_{tt}-du_{xx}+\beta\theta_{x}=0\\
b\theta_{t}-\mu_1\theta_{xx}-\int_{\tau_1}^{\tau_2}\mu_2(s)\theta_{xx}(t-s)ds
+\beta u_{xt}=0
\end{gather*}
in $(0, L)\times (0, \infty)$, and proved that the damping effect via
heat conduction is strong enough to exponentially stabilize the system
provided \eqref{1.2} (with $a=1$) holds. See \cite{Mus2} for similar
result concerning boundary distributed delay. Interested reader is
referred to
 \cite{Apa, Kir, Kir2, Mes01, Mus, Mus3, Nic, Nic2, Nic3, Pig, SaiL, Sai4},
for more results concerning other types of delay (constant or time-varying delay).

In the absence of delay $(\mu_2\equiv 0)$, Messaoudi et  al \cite{Mes2}
considered \eqref{1.1} for both linear and nonlinear case and proved that the
system is exponentially stable without any restriction on the coefficients.
 Whereas, in the absence of both the frictional damping $(\mu_1=0)$ and delay,
Fern\'andez Sare and Racke \cite{Fer} proved that the system is no
longer exponentially stable even in the presence of viscoelastic damping
term of the form $\int_0^{\infty}g(s)\varphi_{xx}(t-s)ds$ in the second equation
of \eqref{1.1}. The results of \cite{Fer} were generalized by Guesmia et al
 \cite{Gue3} to the case where $g$ does not converge exponentially to zero.
On the other hand, if the infinite memory is considered in the first equation,
then it was proved in \cite{Gue3} that the uniform stability
(exponential, polynomial or others depending on the growth of $g$ at infinity)
 holds without any restriction on the parameters. Very recently,
Santos et al \cite{San} improved the result of \cite{Fer} (for $g=0$)
by introducing a new stability number
\[
\chi=\Big(\tau-\frac{\rho_1}{\kappa\rho_3}\Big)
\Big(\rho_2-\frac{b\rho_1}{\kappa}\Big)-\frac{\tau\rho_1\delta^2}{\kappa\rho_3}
\]
and proved that the corresponding semigroup associated to the system is
exponentially stable if and only if $\chi=0$, otherwise there is a lack
of exponential stability. In addition to the absence of frictional damping
and delay, if $\tau=0$ (classical thermoelasticity)
then Rivera and Racke \cite{Mu} proved that \eqref{1.1} is exponentially
stable if the propagation speeds are equal i.e.
$\frac{\kappa}{\rho_1}=\frac{b}{\rho_2}$, otherwise a weaker rate of decay
is obtained for strong solutions.

In this present work we consider \eqref{1.1} and prove the well-posedness
and establish exponential stability results regardless of the speeds of
wave propagation.
Our work extends the stability results in \cite{Mes2} to Timoshenko
systems with distributed delay acting on the displacement equation.

The rest of this article is organized as follows. In section $2$, we
introduce some transformations and state the assumption needed in our work.
In section $3$, we use the semigroup method to prove the well-posedness of
our problem. In the last section, we state and prove our stability result.
We use $c$ throughout this paper to denote a generic positive constant.

\section{Preliminaries}

As in \cite{Nic1}, we introduce the new variable
\begin{equation}\label{2.1}
z(x, \rho, s, t)=\varphi_{t}(x, t-\rho s) \quad  \text{in }  (0, 1)\times (0, 1)\times (\tau_1, \tau_2)\times (0, \infty).
\end{equation}
It is straight forward to check that $z$ satisfies
\[
sz_{t}(x, \rho, s, t)+z_{\rho}(x, \rho, s, t)=0 \quad  \text{in }
 (0, 1)\times (0, 1)\times (\tau_1, \tau_2)\times (0, \infty).
\]
Consequently, problem \eqref{1.1} is equivalent to
\begin{gather}
\rho_1\varphi_{tt}-\kappa (\varphi_{x}+\psi)_{x}+\mu_1\varphi_{t}
+\int_{\tau_1}^{\tau_2}\mu_2(s)z(x, 1,s, t)ds=0  \quad
\text{in }(0, 1)\times (0, \infty),
 \nonumber \\
\rho_2\psi_{tt}-b\psi_{xx}+\kappa(\varphi_{x}+\psi)
 +\delta\theta_{x}=0 \quad \text{in }(0, 1)\times (0, \infty),
\nonumber \\
\rho_3\theta_{t}+q_{x}+\delta\psi_{tx}=0 \quad \text{in }(0, 1)\times (0, \infty),
\nonumber \\
\tau q_{t}+\beta q+\theta_{x}=0 \quad \text{in }(0, 1)\times (0, \infty),
\nonumber \\
sz_{t}(x, \rho, s, t)+z_{\rho}(x, \rho, s, t)=0 \quad
 \text{in }(0, 1)\times (0, 1)\times (\tau_1, \tau_2)\times (0, \infty),
 \nonumber \\
z(x, 0, s, t)=\varphi_{t} \quad \text{in }  (0, 1)\times (\tau_1, \tau_2)
 \times (0, \infty)
\nonumber \\
\varphi(x,0)=\varphi_0(x), \quad \varphi_{t}(x, 0)=\varphi_1(x),\quad
 \theta(x, 0)=\theta_0(x) \quad  \text{in } (0, 1), \nonumber \\
\psi(x, 0)=\psi_0(x), \quad \psi_{t}(x, 0)=\psi_1(x) ,\quad
  q(x, 0)=q_0(x)\quad  \text{in } (0, 1),\nonumber \\
\varphi(0, t)=\varphi(1, t)=\psi_{x}(0, t)=\psi_{x}(1, t)=\theta(0, t)
=\theta(1, t)=0 \quad \text{in }(0, \infty), \nonumber \\
z(x, \rho, s, 0)=f_0(x, \rho s) \quad  \text{in }
(0, 1)\times (0, 1)\times(0, \tau_2). \label{2.2}
\end{gather}
Concerning the weight of the delay, we assume that
\begin{equation}\label{2.3}
\int_{\tau_1}^{\tau_2}|\mu_2(s)| ds  < \mu_1
\end{equation}
and establish the well-posedness as well as the exponential stability
results of the energy $E$, defined by
\begin{equation}\label{2.4}
\begin{aligned}
E(t) &= \frac{1}{2}\int_0^{1}\Big[\rho_1\varphi_{t}^2+\rho_2\psi_{t}^2
 + b\psi_{x}^2+\kappa(\varphi_{x}+\psi)^2
 +\rho_3\theta^2+\tau q^2\Big]\,dx\\
&\quad +\frac{1}{2}\int_0^{1}\int_0^{1}\int_{\tau_1}^{\tau_2}s
 |\mu_2(s)| z^2(x, \rho, s, t)\,ds\,d\rho\,dx.
\end{aligned}
\end{equation}
Meanwhile, using \eqref{2.2}$_2$, \eqref{2.2}$_4$, and the boundary conditions,
we conclude that
\begin{equation}\label{2.5}
\frac{d^2}{dt^2}\int_0^{1}\psi(x, t) +\frac{\kappa}{\rho_2}
\int_0^{1}\psi(x, t) = 0 \quad \text{and} \quad
\frac{d}{dt}\int_0^{1}q(x, t) +\frac{\beta}{\tau} \int_0^{1}q(x, t) = 0 .
\end{equation}
So, by solving \eqref{2.5} and using the initial data of $\psi$ and $q$, we obtain
\[
\int_0^{1}\psi(x, t)\,dx=\Big(\int_0^{1}\psi_0(x)\,dx\Big)
\cos\sqrt{\frac{\kappa}{\rho_2}}t+\sqrt{\frac{\rho_2}{\kappa}}
\Big(\int_0^{1}\psi_1(x)\,dx\Big)\sin\sqrt{\frac{\kappa}{\rho_2}}t
\]
and
\[
\int_0^{1}q(x, t)\,dx=\Big(\int_0^{1}q_0(x)\,dx\Big)\exp(-\frac{\beta}{\tau}t).
\]
Consequently, if we let
\begin{gather*}
\overline{\psi}(x, t)=\psi(x, t)-\Big(\int_0^{1}\psi_0(x)\,dx\Big)
\cos\sqrt{\frac{\kappa}{\rho_2}}t-\sqrt{\frac{\rho_2}{\kappa}}
\Big(\int_0^{1}\psi_1(x)\,dx\Big)\sin\sqrt{\frac{\kappa}{\rho_2}}t,
\\
\overline{q}(x, t)=q(x, t)-\Big(\int_0^{1}q_0(x)\,dx\Big)
\exp(-\frac{ \beta}{\tau}t).
\end{gather*}
Then it follows that
\[
\int_0^{1}\overline{\psi}(x, t)\,dx=0 \quad \text{and} \quad
\int_0^{1}\overline{q}(x, t)\,dx=0, \quad \forall t\ge 0.
\]
Therefore, the use of Poincar\' e's inequality for $\overline{\psi}$ is justified.
In addition, simple substitution shows that
$(\varphi, \overline{\psi}, \theta, \overline{q}, z)$
satisfies system \eqref{2.2} with initial data for $\overline{\psi}$ and
$\overline{q}$ given as
\begin{gather*}
\overline{\psi}_0(x)=\psi_0(x)-\int_0^{1}\psi_0(x)\,dx,
\quad \overline{\psi}_1(x)=\psi_1(x)-\int_0^{1}\psi_1(x)\,dx,
\\
\overline{q}_0(x)=q_0(x)-\int_0^{1}q_0(x)\,dx
\end{gather*}
instead of $\psi_0(x), \psi_1(x)$ for $\psi$, and $q_0$ for $q$,
respectively. Henceforth, we work with $\overline{\psi}$ and $\overline{q}$
instead of $\psi$ and $q$ but write $\psi$ and $q$ for simplicity of notation.

\section{Well-posedness of the problem}

In this section, we prove the existence and uniqueness of solutions for  \eqref{2.2} using semigroup theory.
Introducing the vector function $\Phi = (\varphi, u, \psi, v, \theta, q, z)^T$, where $u=\varphi_{t}$ and $v=\psi_{t}$, system \eqref{2.2} can be written as
\begin{equation}\label{3.1}
\begin{gathered}
\Phi^{\prime}(t) + \mathcal{A} \Phi (t) = 0, \quad  t > 0, \\
\Phi(0) = \Phi_0 = \big(\varphi_0, \varphi_1, \psi_0, \psi_1, \theta_0,
 q_0, f_0\big)^{T},
\end{gathered}
\end{equation}
where the operator $\mathcal{A}$ is defined by
\[
\mathcal{A}\Phi=
\begin{pmatrix}
-u\\
-\frac{\kappa}{\rho_1}(\varphi_{x}+\psi)_{x}+\frac{\mu_1}{\rho_1}u
+\frac{1}{\rho_1}\int_{\tau_1}^{\tau_2}\mu_2(s)z(x, 1, s)\,ds \\
-v\\
-\frac{b}{\rho_2}\psi_{xx} +\frac{\kappa}{\rho_2}(\varphi_{x}+\psi)
+ \frac{\delta}{\rho_2} \theta_{x}\\
\frac{1}{\rho_3}q_{x} +\frac{\delta}{\rho_3} v_{x}\\
\frac{\beta}{\tau}q +\frac{1}{\tau}\theta_{x}\\
\frac{1}{s}z_{\rho}(x, \rho, s)
\end{pmatrix}.
\]
We consider the following spaces
\begin{gather*}
 L^{2}_{\star}(0, 1)=\{w \in L^{2}(0, 1)  : \int_0^{1}w(s)\,ds = 0\},  \quad
H_{\star}^{1}(0, 1)= H^{1}(0, 1)\cap L^{2}_{\star}(0, 1),\\
 H^{2}_{\star}(0, 1)=\{w \in H^{2}(0, 1)  : w_{x}(0)=w_{x}(1)=0\}.
\end{gather*}
Let
\begin{align*}
\mathcal{H}&:=H_0^{1}(0, 1) \times L^{2}(0, 1)\times H_{\star}^{1}(0, 1)
\times L_{\star}^{2}(0, 1) \times L^{2}(0, 1) \\
&\quad \times L_{\star}^{2}(0, 1) \times L^{2}((0, 1)
 \times (0, 1) \times (\tau_1, \tau_2))
\end{align*}
be the Hilbert space equipped with the  inner product
\begin{align*}
(\Phi, \tilde{\Phi})_{\mathcal{H}}
&=\kappa\int_0^{1}(\varphi_{x}+\psi)(\tilde{\varphi}_{x}+\tilde{\psi})\,dx
+ \rho_1\int_0^{1} u \tilde{u}dx + b\int_0^{1} \psi_{x} \tilde{\psi}_{x}\,dx \\
&\quad + \rho_2 \int_0^{1} v \tilde{v}\,dx
+ \rho_3\int_0^{1} \theta \tilde{\theta}\,dx+\tau\int_0^{1}q\tilde{q}\,dx\\
&\quad +\int_0^{1}\int_0^{1} \int_{\tau_1}^{\tau_2}s| \mu_2(s)
| z(x, \rho, s)\tilde{z}(x, \rho, s)\,ds\,d\rho\,dx.
\end{align*}
The domain of $\mathcal{A}$ is
\begin{align*}
D(\mathcal{A})
=\Big\{&\Phi \in {\mathcal{H}} : \varphi \in {H^{2}(0, 1) \cap H_0^{1}(0, 1)}, \;
\psi \in {H_{\star}^{2}(0, 1) \cap H_{\star}^{1}(0, 1)},\\
& u, \theta \in H_0^{1}(0, 1), \;
 v, q \in H_{\star}^{1}(0, 1),\; z(x, 0, s)=u,\\
& z, z_{\rho} \in L^{2}((0, 1) \times (0, 1) \times (\tau_1, \tau_2))\Big\}
\end{align*}
Clearly, $D(\mathcal{A})$ is dense in ${\mathcal{H}}$.

We have the following existence and uniqueness result.

\begin{theorem}\label{thm3.1}
Let $\Phi_0 \in \mathcal{H}$, then there exists a unique solution
$\Phi \in C {(\mathbb{R}^+, \mathcal{H}})$ of problem \eqref{3.1}.
Moreover, if $\Phi_0 \in D(\mathcal{A})$, then
$\Phi \in C {(\mathbb{R}^+, D(\mathcal{A}))}\cap C^{1} {(\mathbb{R}^+, \mathcal{H})}$.
\end{theorem}

\begin{proof}
The result follows from Lumer-Phillips theorem provided we prove that $\mathcal{A}$
is a maximal monotone operator. In what follows, we prove that $\mathcal{A}$
is  monotone. For any  $\Phi \in D(\mathcal{A})$, and using the inner product,
we obtain
\begin{equation}\label{3.2}
\begin{aligned}
&(\mathcal{A}\Phi, \Phi)_{\mathcal{H}} \\
&= \beta\int_0^{1}q^{2}dx+\frac{1}{2}\int_0^{1} \int_{\tau_1}^{\tau_2}
|\mu_2(s)| z^{2}(x, 1, s) \,ds\,dx\\
&+(\mu_1-\frac{1}{2} \int_{\tau_1}^{\tau_2}| \mu_2(s)| ds)\int_0^{1}u^{2}dx
+\int_0^{1}u\int_{\tau_1}^{\tau_2} \mu_2(s) z(x, 1, s) \,ds\,dx.
\end{aligned}
\end{equation}
Using Young's inequality, the last term in \eqref{3.2}, we have
\begin{equation}\label{3.3}
\begin{aligned}
&-\int_0^{1}u\int_{\tau_1}^{\tau_2}\mu_2(s) z(x, 1, s) \,ds\,dx\\
&\le\frac{1}{2} \int_{\tau_1}^{\tau_2}| \mu_2(s)| \,ds\int_0^{1}u^{2}\,dx
+\frac{1}{2}\int_0^{1} \int_{\tau_1}^{\tau_2}|\mu_2(s)| z^{2}(x, 1, s) \,ds\,dx.
\end{aligned}
\end{equation}
Substituting \eqref{3.3} in \eqref{3.2} yields
\[
(\mathcal{A}\Phi, \Phi)_{\mathcal{H}}
\ge \beta\int_0^{1}q^{2}\,dx+\Big(\mu_1
-\int_{\tau_1}^{\tau_2}| \mu_2(s)| \,ds\Big)\int_0^{1}u^{2}\,dx.
\]
By \eqref{2.3}, it follows that $(\mathcal{A}\Phi, \Phi)_{\mathcal{H}} \ge 0$,
which implies that $\mathcal{A}$ is monotone.
Next, we prove that the operator $I + \mathcal{A}$ is surjective.
Given  $G = (g_1, g_2, g_3, g_4, g_5, g_6, g_7)^{T} \in \mathcal{H}$,
we prove that there exists $\Phi \in D(\mathcal{A})$ satisfying
\begin{equation}\label{3.4}
\Phi + \mathcal{A}\Phi = G;
\end{equation}
that is,
\begin{equation}\label{3.5}
\begin{gathered}
- u+\varphi = g_1  \in H_0^{1}(0, 1)\\
-\kappa(\varphi_{x}+\psi)_{x}+(\rho_1+\mu_1)u
+\int_{\tau_1}^{\tau_2}\mu_2(s) z(x, 1, s)ds=\rho_1 g_2  \in L^{2}(0, 1)\\
-  v+\psi = g_3  \in H_{\star}^{1}(0, 1)\\
- b\psi_{xx} +\kappa(\varphi_{x}+\psi)+ \delta \theta_{x}+\rho_2v
= \rho_2g_4  \in L_{\star}^{2}(0, 1) \\
 q_{x} +\delta v_{x}+\rho_3\theta=\rho_3g_5  \in L^{2}(0, 1) \\
 (\beta+\tau) q +\theta_{x}=\tau g_6  \in L_{\star}^{2}(0, 1) \\
 z_{\rho}(x, \rho, s) +s z(x, \rho, s)= sg_7(x, \rho, s)
 \in L^{2}((0, 1) \times (0, 1) \times (\tau_1, \tau_2)).
\end{gathered}
\end{equation}
We note that the last equation in \eqref{3.5} with $z(x, 0, s)=u$, has a unique
solution
\begin{equation}\label{3.6}
z(x, \rho, s)=e^{-s\rho}u+se^{-s\rho}\int_0^{\rho}e^{s\tau}g_7(x, \tau, s)\,d\tau.
\end{equation}
From the sixth equation in \eqref{3.5}, we define
\begin{equation}\label{3.7}
\theta=\tau\int_0^{x} g_6\,dx-(\beta+\tau)\int_0^{x}q\,dx,
\end{equation}
then $\theta(0)=\theta(1)=0$.
Inserting $u=\varphi- g_1$, $v=\psi- g_3$, and \eqref{3.7} in \eqref{3.5}$_2$,
\eqref{3.5}$_4$, and \eqref{3.5}$_5$, we obtain
\begin{equation}\label{3.8}
\begin{gathered}
-\kappa(\varphi_{x}+\psi)_{x}+\mu\varphi=h_1  \in L^{2}(0, 1) \\
-b\psi_{xx}+\kappa(\varphi_{x}+\psi)+\rho_2\psi-(\beta+\tau)\delta q =  h_2  \in L_{\star}^{2}(0, 1)\\
-q_{x}+(\beta+\tau)\rho_3\int_0^{x}q(y)dy-\delta\psi_{x}=h_3  \in L^{2}(0, 1),
\end{gathered}
\end{equation}
where
\begin{equation}\label{3.14}
\begin{gathered}
\mu=\mu_1+\rho_1+\int_{\tau_1}^{\tau_2}\mu_2(s)e^{-s}ds\\
h_1=\mu g_1+\rho_1g_2-\int_{\tau_1}^{\tau_2}s\mu_2(s) e^{-s}\int_0^{1}e^{s\tau}g_5(x, \tau, s)d\tau ds\\
h_2=\rho_2(g_3+g_4)-\tau  \delta  g_6 \\
h_3=-\delta g_{3x}-\rho_3\Big(g_5-\tau\int_0^{x}g_6(y)dy\Big).
\end{gathered}
\end{equation}
To solve \eqref{3.8} we consider
\begin{equation}\label{3.10}
B\big((\varphi, \psi, q), (\tilde{\varphi}, \tilde{\psi}, \tilde{q})\big)
=F\big(\tilde{\varphi}, \tilde{\psi}, \tilde{q} \big),
\end{equation}
where $B: {\big[H_0^{1}(0, 1) \times H^{1}_{\star}(0, 1)
\times L^{2}_{\star}(0, 1)\big]}^{2}\to \mathbb{R}$ is
the bilinear form
\begin{align*}
 B\big((\varphi, \psi, q), (\tilde{\varphi}, \tilde{\psi}, \tilde{q})\big)
&=\kappa\int_0^{1}(\varphi_{x}+\psi)(\tilde{\varphi}_{x}
+\tilde{\psi})\,dx+(\beta+\tau)\int_0^{1} q\tilde{q}\,dx\\
&\quad + b\int_0^{1}\psi_{x}\tilde{\psi}_{x}\,dx
 +\rho_2\int_0^{1}\psi\tilde{\psi}\,dx-\delta(\beta+\tau)\int_0^{1} q\tilde{\psi}\,dx\\
&\quad +\mu\int_0^{1}\varphi\tilde{\varphi}\,dx
 +\delta(\beta+\tau)\int_0^{1} \psi \tilde{q}\,dx\\
&\quad +\rho_3(\beta+\tau)^{2}\int_0^{1}
 \Big(\int_0^{x} q(y)dy\int_0^{x} \tilde{q}(y)dy\Big)\,dx\\
\end{align*}
and $F: \big[H_0^{1}(0, 1) \times H^{1}_{\star}(0, 1) \times L^{2}_{\star}(0, 1)
\big]\to \mathbb{R}$ is the linear form
\[
F(\tilde{\varphi}, \tilde{\psi}, \tilde{q})
= \int_0^{1}h_1\tilde{\varphi}\, dx+ \int_0^{1}h_2\tilde{\psi}\, dx
+ \int_0^{1}h_3\int_0^{x}\tilde{q}(y) \,dy\,dx.
\]
Now, for  $V= H_0^{1}(0, 1) \times H^{1}_{\star}(0, 1) \times L^{2}_{\star}(0, 1)$
equipped with the norm
\[
{\|(\varphi, \psi, q)\|}_V = {\|(\varphi_{x}+\psi)\|}_2^{2}
+{\|\varphi\|}_2^{2}+{\|\psi_{x}\|}_2^{2}+{\| q\|}_2^{2},
\]
one can easily see that $B$ and $F$ are bounded. Furthermore,
using integration by parts, we obtain
\begin{align*}
& B\big((\varphi, \psi, q), (\varphi, \psi, q)\big)\\
&=\kappa\int_0^{1}(\varphi_{x}+\psi)^{2}\,dx
+(\beta+\tau)\int_0^{1} q^{2}\,dx+ b\int_0^{1}\psi_{x}^{2}\,dx\\
&\quad +\rho_2\int_0^{1}\psi^{2}\,dx
+ \mu\int_0^{1}\varphi^{2}\,dx
+\rho_3(\beta+\tau)^{2}\int_0^{1}\Big(\int_0^{x} q(y)\,dy\Big)^{2}\,dx\\
&\ge c {\|(\varphi, \psi, q)\|}_V^{2}.
\end{align*}
Thus $B$ is coercive. Consequently, by Lax-Milgram Lemma, system \eqref{3.8}
has a unique solution
$$
\varphi\in H_0^{1}(0, 1), \quad \psi\in H_{\star}^{1}(0, 1),\quad
q \in L_{\star}^{2}(0, 1).
$$
Substituting $\varphi,\ \psi$, and $q$ in \eqref{3.5}$_1$, \eqref{3.5}$_3$,
and \eqref{3.5}$_6$, respectively, we obtain
 $$
u\in H_0^{1}(0, 1), \quad  v\in H_{\star}^{1}(0, 1), \quad \theta \in H_0^{1}(0, 1).
$$
Similarly, inserting $u$ in \eqref{3.6} and bearing in mind \eqref{3.6}$_7$, we obtain
$$
z, z_{\rho} \in L^{2}((0, 1) \times (0, 1) \times (\tau_1, \tau_2)).
$$
Now, if $(\tilde{\varphi},  \tilde{q}) \equiv (0,  0) \in H_0^{1}(0, 1)
\times L_{\star}^{2}(0, 1)$,
then \eqref{3.10} reduces to
\begin{equation}\label{3.11}
\begin{aligned}
&\kappa\int_0^{1}(\varphi_{x}+\psi)\tilde{\psi}\,dx
+  b\int_0^{1}\psi_{x}\tilde{\psi}_{x}\,dx
+\rho_2\int_0^{1}\psi\tilde{\psi}\,dx
-\delta(\beta+\tau)\int_0^{1} q\tilde{\psi}\,dx\\
&=\int_0^{1}h_2\tilde{\psi} \,dx, \quad \forall \tilde{\psi} \in H_{\star}^{1}(0, 1),
\end{aligned}
\end{equation}
which implies
\begin{equation}\label{3.12}
-b\psi_{xx}=-(\kappa+\rho_2)\psi-\kappa\varphi_{x}+(\beta+\tau)\delta q
+h_2 \in L^{2}(0,1).
\end{equation}
Consequently, by the regularity theory for the linear elliptic equations,
it follows that
$$
\psi \in H^{2}(0,1) \cap H_{\star}^{1}(0,1).
$$
Moreover, \eqref{3.11} is also true for any
$\phi\in C^{1}([0, 1]) \subset H_{\star}^{1}(0, 1)$. Hence, we have
\[
b\int_0^{1}\psi_{x}\phi_{x}dx+\int_0^{1}\Big(\kappa(\varphi_{x}+\psi)
+\rho_2\psi-\delta(\beta+\tau)q-h_2\Big)\phi dx =0
\]
for all $\phi \in C^{1}([0, 1])$.
Thus, using integration by parts and bearing in mind \eqref{3.12}, we obtain
$$
\psi_{x}(1)\phi(1)-\psi_{x}(0)\phi(0)=0, \quad \forall\phi\in C^{1}([0, 1]).
$$
Therefore,
$\psi_{x}(0)=\psi_{x}(1)=0$.
Consequently, we obtain
$$
\psi \in H_{\star}^{2}(0,1) \cap H_{\star}^{1}(0, 1).
$$
Similarly, we obtain
\begin{gather*}
- \kappa\varphi_{xx}= -\mu\varphi-\kappa\psi_{x}+h_1 \ \in L^{2}(0, 1)\\
-q_{x} =\delta\psi_{x}-(\beta+\tau)\rho_3\int_0^{x}q(y)\,dy+h_3 \ \in L^{2}(0, 1);
\end{gather*}
thus, we have
$$
\varphi\in H^{2}(0, 1)\cap H_0^{1}(0, 1), \quad q \in H_{\star}^{1}(0, 1).
$$
Finally, the application of the regularity theory for the linear elliptic
equations guarantees the existence of unique $\Phi \in D(\mathcal{A})$
such that \eqref{3.4} is satisfied. Consequently, $\mathcal{A}$ is a
maximal operator. Hence,  the result of Theorem \ref{thm3.1} follows
from Lumer-Phillips theorem (see \cite{Liu, Paz}).
\end{proof}


\section{Exponential Stability}

In this section, we state and prove our stability result for the energy of
the solution of system \eqref{2.2}, using the multiplier technique.
To achieve our goal, we need the following lemmas.

\begin{lemma}\label{lem4.1}
Let $ (\varphi, \psi, \theta, q, z) $ be the solution of \eqref{2.2} and
assume \eqref{2.3} holds. Then the energy functional, defined by \eqref{2.4}
satisfies
\begin{equation}\label{4.1}
E' (t) \le -m_0\int_0^{1} \varphi_{t}^2dx-\beta\int_0^{1} q^2dx\le 0, \quad
 \forall t \ge 0,
\end{equation}
for some positive constant $m_0$.
\end{lemma}

\begin{proof}
Multiplying \eqref{2.2}$_1$,   \eqref{2.2}$_2$, \eqref{2.2}$_3$, and
\eqref{2.2}$_4$ by $\varphi_{t}$, $\psi_{t}$, $\theta$, and $q$, respectively,
and integrating over $(0, 1)$, using integration by parts and the boundary
conditions, we obtain
\begin{equation}\label{4.2}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\int_0^{1}\Big[\rho_1\varphi_{t}^2
 +\rho_2\psi_{t}^2+b\psi_{x}^2+\kappa(\varphi_{x}+\psi)^2
+\rho_3\theta^2+\tau q^2\Big]\,dx\\
&=-\mu_1\int_0^{1}\varphi_{t}^2\,dx
-\beta\int_0^{1} q^2dx-\int_0^{1}\varphi_{t}\int_{\tau_1}^{\tau_2}
\mu_2(s) z(x, 1, s, t)\,ds\,dx.
\end{aligned}
\end{equation}
Multiplying \eqref{2.2}$_3$ by $|\mu_2(s)| z$, integrating the product over
 $(0, 1)\times (0, 1)\times (\tau_1, \tau_2)$, and recall that
$z(x, 0, s, t)=\varphi_{t}$, yield
\begin{equation}\label{4.3}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\int_0^{1}\int_0^{1}
 \int_{\tau_1}^{\tau_2}s|\mu_2(s)| z^2(x, \rho, s, t) \,ds\,d\rho\,dx \\
&= -\frac{1}{2}\int_0^{1}\int_{\tau_1}^{\tau_2}|\mu_2(s)| z^2(x, 1, s, t)\,ds\,dx
 +\frac{1}{2}\int_0^{1}\varphi_{t}^2\int_{\tau_1}^{\tau_2}|\mu_2(s)| \,ds\,dx.
\end{aligned}
\end{equation}
A combination of \eqref{4.2} and \eqref{4.3} gives
\begin{equation}\label{4.4}
\begin{aligned}
E' (t) &= -\Big(\mu_1-\frac{1}{2}\int_{\tau_1}^{\tau_2}|\mu_2(s)| ds\Big)
 \int_0^{1}\varphi_{t}^2dx-\beta\int_0^{1} q^2dx\\
&-\int_0^{1}\varphi_{t}\int_{\tau_1}^{\tau_2}\mu_2(s) z(x, 1, s, t)\,ds\,dx
-\frac{1}{2}\int_0^{1}\int_{\tau_1}^{\tau_2}|\mu_2(s)| z^2(x, 1, s, t)\,ds\,dx.
\end{aligned}
\end{equation}
Meanwhile, using Young's inequality, we have
\begin{equation}\label{4.5}
\begin{aligned}
&-\int_0^{1}\varphi_{t}\int_{\tau_1}^{\tau_2}\mu_2(s) z(x, 1, s, t)\,ds\,dx\\
&\le\frac{1}{2}\int_{\tau_1}^{\tau_2}|\mu_2(s)| ds \int_0^{1}\varphi_{t}^2dx
 +\frac{1}{2}\int_0^{1}\int_{\tau_1}^{\tau_2}|\mu_2(s)| z^2(x, 1, s, t)\,ds\,dx.
\end{aligned}
\end{equation}
Simple substitution of \eqref{4.5} into \eqref{4.4} and using \eqref{2.3}
give \eqref{4.1}, which concludes the proof.
\end{proof}

\begin{lemma}\label{lem 4.2}
Let $(\varphi, \psi, \theta, q, z)$ be the solution of \eqref{2.2}.
Then the functional
\[
F_1(t):=\rho_2\int_0^{1}\psi\psi_{t}\,dx
\]
satisfies, the estimate
\begin{equation}\label{4.6}
F_1' (t) \le-\frac{b}{2}\int_0^{1}\psi_{x}^2dx
+\rho_2\int_0^{1}\psi_{t}^2dx+c\int_0^{1}(\varphi_{x}+\psi)^2dx
+c\int_0^{1}\theta^2dx.
\end{equation}
\end{lemma}

\begin{proof}
A simple differentiation of $F_1$, using \eqref{2.2}$_2$, gives
\[
F_1' (t)=-b\int_0^{1}\psi_{x}^2dx+\rho_2\int_0^{1}\psi_{t}^2dx
+\delta\int_0^{1}\theta\psi_{x}dx-\kappa\int_0^{1}\psi(\varphi_{x}+\psi) dx.
\]
Using Young's and Poincar\'e inequalities,  estimate \eqref{4.6} is established.
\end{proof}

\begin{lemma}\label{lem 4.3}
Let $ (\varphi, \psi, \theta, q, z) $ be the solution of \eqref{2.2}.
Then the functional
\[
F_2(t):=-\frac{\rho_2\rho_3}{\delta}\int_0^{1}\theta\int_0^{x}\psi_{t}(y)\,dy\,dx
\]
satisfies, for any $\varepsilon_1>0$, the estimate
\begin{equation}\label{4.7}
\begin{aligned}
F_2' (t) &\le-\frac{\rho_2}{2}\int_0^{1}\psi_{t}^2dx
 +\varepsilon_1\int_0^{1}\psi_{x}^2dx+c\int_0^{1}q^2dx\\
&\quad +c\big(1+\frac{1}{\varepsilon_1}\big)\int_0^{1}\theta^2dx
 +c\int_0^{1}(\varphi_{x}+\psi)^2dx.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
By differentiating $F_2$, then exploiting the second and the third equations
 in \eqref{2.2}, and integrating by parts, we obtain
\[
\begin{aligned}
F_2' (t) &=-\rho_2\int_0^{1}\psi^2_{t}dx-\frac{\rho_2}{\delta}
 \int_0^{1}q\psi_{t}dx-\frac{b\rho_3}{\delta}\int_0^{1}\theta\psi_{x}dx\\
&\quad +\rho_3\int_0^{1}\theta^2dx+\frac{\rho_3\kappa}{\delta}
 \int_0^{1}\theta\Big(\varphi+\int_0^{x}\psi(y)dy\Big)\,dx.
\end{aligned}
\]
Using Poincar\'e and Young's inequalities with $\varepsilon_1>0$,
 we obtain estimate \eqref{4.7}.
\end{proof}

\begin{lemma}\label{lem4.4}
Let $ (\varphi, \psi, \theta, q, z) $ be the solution of \eqref{2.2}.
Then the functional
\[
F_3(t):=\rho_1\int_0^{1}\varphi_{t}\left(\varphi+\int_0^{x}\psi(y)dy\right)\,dx
\]
satisfies, for any $\varepsilon_2>0$, the estimate
\begin{equation}\label{4.8}
\begin{aligned}
F_3' (t)
&\le-\frac{\kappa}{2}\int_0^{1}(\varphi_{x}+\psi)^2dx
 +\varepsilon_2\int_0^{1}\psi_{t}^2dx\\
&+c\big(1+\frac{1}{\varepsilon_2}\big)
 \int_0^{1}\varphi_{t}^2dx+c\int_0^{1}\int_{\tau_1}^{\tau_2}
|\mu_2(s)| z^2(x, 1, s, t)\,ds\,dx.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
Taking the derivative of $F_3$, using \eqref{2.2}$_2$ and integration by parts,
 we obtain
\begin{equation}\label{4.9}
\begin{aligned}
F_3' (t)
&=\rho_1\int_0^{1}\varphi_{t}\int_0^{x}\psi_{t}(y)\,dy\,dx
-\int_0^{1}\Big(\varphi+\int_0^{x}\psi(y)dy\Big)\int_{\tau_1}^{\tau_2}
 \mu_2(s)z(x, 1, s, t)\,ds\,dx\\
&\quad -\kappa\int_0^{1}(\varphi_{x}+\psi)^2dx
 +\rho_1\int_0^{1}\varphi_{t}^2dx
 -\mu_1\int_0^{1}\varphi_{t}\Big(\varphi+\int_0^{x}\psi(y)dy\Big)\,dx.
\end{aligned}
\end{equation}
Now, we estimate the terms in the right hand side of \eqref{4.9} using Young's,
Poincar\'e, and Cauchy-Schwarz inequalities
\begin{equation}\label{4.10}
\rho_1\int_0^{1}\varphi_{t}\int_0^{x}\psi_{t}(y)dy)\,dy
\le\varepsilon_2\int_0^{1}\psi_{t}^2dx
+\frac{c}{\varepsilon_2}\int_0^{1}\varphi_{t}^2dx,
\end{equation}
where $\varepsilon_2>0$, and
\begin{equation}\label{4.11}
\begin{aligned}
&-\int_0^{1}\Big(\varphi+\int_0^{x}\psi(y)dy\Big)\int_{\tau_1}^{\tau_2}
 \mu_2(s)z(x, 1, s, t)\,ds\,dx\\
&\le\frac{\kappa}{4}\int_0^{1}\Big(\varphi+\int_0^{x}\psi(y)dy\Big)^2dx
 +\frac{1}{\kappa}\int_0^{1}\Big(\int_{\tau_1}^{\tau_2}\mu_2(s)z(x, 1, s, t)ds
 \Big)^2dx\\
&\le\frac{\kappa}{4}\int_0^{1}(\varphi_{x}+\psi)^2dx
 +\frac{1}{\kappa}\underbrace{\int_{\tau_1}^{\tau_2}|\mu_2(s)| ds}
 _{<\ \mu_1}\int_0^{1}\int_{\tau_1}^{\tau_2}|\mu_2(s)| z^2(x, 1, s, t)\,ds\,dx.
\end{aligned}
\end{equation}
\begin{equation}\label{4.12}
\begin{aligned}
-\mu_1\int_0^{1}\varphi_{t}\Big(\varphi+\int_0^{x}\psi(y)dy\Big)\,dx
&\le\frac{\kappa}{4}\int_0^{1}\Big(\varphi+\int_0^{x}\psi(y)dy\Big)^2dx
 +\frac{\mu_1^2}{\kappa}\int_0^{1}\varphi_{t}^2dx\\
&\le\frac{\kappa}{4}\int_0^{1}(\varphi_{x}+\psi)^2dx+c\int_0^{1}\varphi_{t}^2dx,
\end{aligned}
\end{equation}
Estimate \eqref{4.8} follows by substituting \eqref{4.10}--\eqref{4.12}
into \eqref{4.9}.
\end{proof}

\begin{lemma}\label{lem 4.5}
Let $ (\varphi, \psi, \theta, q, z) $ be the solution of \eqref{2.2}.
Then the functional
\[
F_4(t):=\tau\rho_3\int_0^{1}\theta\int_0^{x}q(y)\,dy\,dx
\]
satisfies, for any $\varepsilon_2>0$, the estimate
\begin{equation}\label{4.13}
F_4' (t) \le-\frac{\rho_3}{2}\int_0^{1}\theta^2dx
+\varepsilon_2\int_0^{1}\psi_{t}^2dx
+c\big(1+\frac{1}{\varepsilon_2}\big)\int_0^{1}q^2dx.
\end{equation}
\end{lemma}

\begin{proof}
Taking the derivative of $F_4$, using the third and the fourth equations
in \eqref{2.2} and integration by parts, we obtain
\begin{equation}\label{4.14}
\begin{aligned}
F_4' (t) =-\rho_3\int_0^{1}\theta^2dx
+\tau\beta\int_0^{1}q^2dx+\tau\delta\int_0^{1}q\psi_{t}dx
-\beta\rho_3\int_0^{1}\theta\int_0^{x}q(y)\,dy\,dx.
\end{aligned}
\end{equation}
We now use Cauchy-Schwarz and Young's inequalities with $\varepsilon_2>0$
on \eqref{4.14} to obtain \eqref{4.13}.
\end{proof}

\begin{lemma}\label{lem 4.6}
Let $ (\varphi, \psi, \theta, q, z)$ be the solution of \eqref{2.2}.
Then the functional
$$
F_5(t):=\int_0^{1} \int_0^{1}\int_{\tau_1}^{\tau_2}se^{-s\rho}|\mu_2(s)| z^2(x, \rho, s, t) \,ds\,d\rho\,dx
$$
satisfies, for some positive constant $m_1$, the following  estimate
\begin{equation}\label{4.15}
\begin{aligned}
F_5' (t)&\le - m_1\int_0^{1} \int_0^{1}\int_{\tau_1}^{\tau_2}s
 |\mu_2(s)| z^2(x, \rho, s, t) ds d\rho dx\\
&\quad -m_1\int_0^{1}\int_{\tau_1}^{\tau_2}|\mu_2(s)| z^2(x, 1, s, t) \,ds\,dx
+\mu_1\int_0^{1}\varphi_{t}^2dx.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
Differentiating $F_5$, and using the fifth equation in \eqref{2.2}, we obtain
\begin{align*}
F_5' (t)
&= -2\int_0^{1} \int_0^{1}\int_{\tau_1}^{\tau_2}e^{-s\rho}|\mu_2(s)
 | z(x, \rho, s, t) z_{\rho}(x, \rho, s, t)\,ds\,d\rho\,dx \\
&=-\frac{d}{d\rho} \int_0^{1} \int_0^{1}\int_{\tau_1}^{\tau_2}e^{-s\rho}
 |\mu_2(s)| z^2(x, \rho, s, t) \,ds\,d\rho\,dx\\
&\quad  -\int_0^{1} \int_0^{1}\int_{\tau_1}^{\tau_2}se^{-s\rho}|\mu_2(s)
 | z^2(x, \rho, s, t) \,ds\,d\rho\,dx\\
&=-\int_0^{1}\int_{\tau_1}^{\tau_2}|\mu_2(s)|[e^{-s} z^2(x, 1, s, t)
 -z^2(x, 0, s, t)] \,ds\,dx\\
&\quad  -\int_0^{1} \int_0^{1}\int_{\tau_1}^{\tau_2}se^{-s\rho}
 |\mu_2(s)| z^2(x, \rho, s, t) \,ds\,d\rho\,dx.
\end{align*}
Using the fact that $z(x, 0, s, t)=\varphi_{t}$ and
$e^{-s}\le e^{-s\rho}\le 1$, for all $\rho\in [0, 1]$, we obtain
\begin{align*}
F_5' (t)
&\le-\int_0^{1}\int_{\tau_1}^{\tau_2}e^{-s}|\mu_2(s)| z^2(x, 1, s, t)\,ds\,dx
 +\int_{\tau_1}^{\tau_2}|\mu_2(s)| ds\int_0^{1}\varphi_{t}^2dx\\
&\quad -\int_0^{1} \int_0^{1}\int_{\tau_1}^{\tau_2}se^{-s}|\mu_2(s)| z^2
 (x, \rho, s, t) \,ds\,d\rho\,dx.
\end{align*}
Because $-e^{-s}$ is an increasing function, we have $-e^{-s}\le-e^{-\tau_2}$,
for all $s\in [\tau_1, \tau_2]$. Finally, setting $m_1=e^{-\tau_2}$ and
recalling \eqref{2.3}, we obtain \eqref{4.15}.
\end{proof}

Next, we define a Lyapunov functional $\mathcal{L}$ and show that it is
equivalent to the energy functional $E$.

\begin{lemma}\label{lem4.7}
For $N$ sufficiently large, the functional defined by
\begin{equation}\label{4.16}
\mathcal{L}(t):=NE(t)+ F_1(t)+ 4F_2(t)+ N_1(F_3(t)+ F_4(t))+N_2 F_5(t),
\end{equation}
where $N_1$ and $N_2$ are positive real numbers to be chosen appropriately later,
satisfies
\begin{equation}\label{4.17}
c_1E(t) \le \mathcal{L}(t) \le c_2E(t), \quad  \forall t \ge 0,
\end{equation}
for two positive constants $c_1$ and $c_2$.
\end{lemma}

\begin{proof}
Let $\mathscr{L}(t)=F_1(t)+ 4F_2(t)+ N_1(F_3(t)+ F_4(t))+N_2 F_5(t)$
\begin{align*}
|\mathscr{L}(t)|
&\le  \rho_2\int_0^{1}|\psi\psi_{t}| dx
 +\rho_1N_1\int_0^{1}\big|\varphi_{t}\Big(\varphi
 +\int_0^{x}\psi(y)dy\Big)\big| dx\\
&\quad +\frac{4\rho_2\rho_3}{\delta}\int_0^{1}\big| \theta\int_0^{x}
  \psi_{t}(y)dy\big| dx
 +\tau\rho_3N_1\int_0^{1}\big| \theta\int_0^{x} q(y)dy\big| dx\\
&\quad +\int_0^{1} \int_0^{1}\int_{\tau_1}^{\tau_2}s|\mu_2(s)e^{-s\rho}
 | z^2(x, \rho, s, t) \,ds\,d\rho\,dx.
\end{align*}
Exploiting Young's, Poincar\'e, Cauchy-Schwarz inequalities, \eqref{2.4},
and the fact that $e^{-s\rho}\le 1$ for all $\rho \in [0, 1]$, we obtain
\begin{align*}
|\mathscr{L}(t)|
&\le c \int_0^{1}\Big(\varphi_{t} ^2+ \psi_{t}^2 + \psi_{x}^2
 + (\varphi_{x}+\psi)^2+\theta^2+q^2\Big)\,dx\\
&\quad +c\int_0^{1} \int_0^{1}\int_{\tau_1}^{\tau_2}s|\mu_2(s)|
 z^2(x, \rho, s, t) \,ds\,d\rho\,dx\\
&\le  cE(t).
\end{align*}
Consequently, $| \mathcal{L}(t)-NE(t)| \le cE(t)$,
which yields
$$
(N-c)E(t)\le \mathcal{L}(t)\le (N+c)E(t).
$$
Choosing $N$ large enough, we obtain estimate \eqref{4.17}.
\end{proof}

Now, we are ready to state and prove the main result of this section.

\begin{theorem}\label{thrm4.8}
Let $ (\varphi, \psi, \theta, q, z) $ be the solution of \eqref{2.2}.
Then the energy functional \eqref{2.4} satisfies,
\begin{equation}\label{4.18}
E(t)\le k_0e^{-k_1t}, \quad  \forall t \ge 0,
\end{equation}
where $ k_0$ and $k_1$ are positive constants.
\end{theorem}

\begin{proof}
By differentiating \eqref{4.16} and recalling \eqref{4.1}, \eqref{4.6}, \eqref{4.7},
\eqref{4.8}, \eqref{4.13}, and \eqref{4.15}, and letting
$\varepsilon_1=\frac{b}{16}$, we obtain
\begin{align*}
\mathcal{L}'(t)
&\le -\big[ m_0N- cN_1\big(1+\frac{1}{\varepsilon_2}\big)
 -\mu_1N_2 \big]\int_0^{1}\varphi_{t}^2dx
 -\big[\rho_2-2\varepsilon_2N_1\big]\int_0^{1} \psi_{t}^2dx\\
&\quad -[\frac{\rho_3}{2} N_1-c]\int_0^{1}\theta^2 dx
 - m_1N_2\int_0^{1} \int_0^{1}\int_{\tau_1}^{\tau_2}s|\mu_2(s)|
 z^2(x, \rho, s, t) \,ds\,d\rho\,dx\\
&\quad -[\frac{\kappa }{2}N_1-c]\int_0^{1}(\varphi_{x}+\psi)^2dx
-\big[\beta N-c-cN_1\big(1+\frac{1}{\varepsilon_2}\big)\big]\int_0^{1}q^2 dx\\
&\quad -\frac{b}{4}\int_0^{1}\psi_{x}^2dx-[m_1N_2-cN_1]
\int_0^{1}\int_{\tau_1}^{\tau_2}|\mu_2(s)| z^2(x, 1, s, t) \,ds\,dx.
\end{align*}
At this point, we set $\varepsilon_2=\rho_2/(4N_1)$,
then we choose $N_1$ large enough so that
$$
\gamma_0:=\frac{\rho_3}{2}N_1-c >0 \quad \text{and} \quad
\gamma_1:=\frac{\kappa }{2}N_1-c >0.
$$
Once $N_1$ is fixed, we then choose $N_2$ large enough so that
$m_1N_2-cN_1 >0$.

Finally, we choose $N$ large enough such that \eqref{4.17} remains valid and
$$
\gamma_2:= m_0N- cN_1\Big(1+\frac{1}{\varepsilon_2}\Big)
-\mu_1N_2>0, \quad
 \gamma_3:=\beta N-c-cN_1\Big(1+\frac{1}{\varepsilon_2}\Big)>0.
$$
Thus, by letting $\gamma_4:=m_1N_2$, we arrive at
\begin{align*}
\mathcal{L}'(t)
&\le -\int_0^{1}\Big(\gamma_2\varphi_{t}^2
 +\frac{\rho_2}{2}\psi_{t}^2+\frac{b}{4}\psi_{x}^2
 +\gamma_1(\varphi_{x}+\psi)^2+\gamma_0\theta^2 dx+\gamma_3q^2\Big) dx\\
&\quad - \gamma_4\int_0^{1} \int_0^{1}\int_{\tau_1}^{\tau_2}s|\mu_2(s)|
z^2(x, \rho, s, t) \,ds\,d\rho\,dx.
\end{align*}
By \eqref{2.4}, we obtain
\begin{equation}\label{4.19}
\mathcal{L}'(t)\le - \alpha_0E(t), \quad  \forall t \ge 0,
\end{equation}
for some $\alpha_0>0$.
A combination of \eqref{4.17} and \eqref{4.19} gives
\begin{equation}\label{4.20}
\mathcal{L}'(t)\le - k_1\mathcal{L}(t), \quad  \forall t \ge 0,
\end{equation}
where $k_1=\alpha_0/c_2$. A simple integration of \eqref{4.20}
over $(0, t)$ yields
\begin{equation}\label{4.21}
\mathcal{L}(t)\le\mathcal{L}(0)e^{-k_1t}, \quad  \forall t \ge 0.
\end{equation}
Finally, by combining \eqref{4.17} and \eqref{4.21} we obtain
 \eqref{4.18} with $k_0=\frac{c_2E(0)}{c_1}$,
which completes the proof.
\end{proof}

\subsection*{Acknowledgement}
The author wants to thank ACHB-KFUPM for its continuous support and to
the anonymous referees for their careful reading and valuable suggestions.

\begin{thebibliography}{00}

\bibitem{Abd} Abdallah, C.; Dorato, P.; Benitez-Read, J.; Byrne, R.;
\emph {Delayed positive feedback can stabilize oscillatory system},
 ACC. San Francisco, 3106--3107 (1993).

\bibitem{Apa} Apalara, T. A,; Messaoudi, S. A,;  Mustafa, M. I.;
\emph{Energy decay in thermoelastic type III with viscoelastic damping and delay},
Elect. J. Differ. Eqns. \textbf{2012} (128), 1--15 (2012).

\bibitem{Beu} Beuter, A.; B\'{e}lair, J.; Labrie, C.;
\emph{Feedback and delays in neurological diseases: a modeling study using
dynamical systems}, Bull. Math. Bio., \textbf{55} (3), 525--541 (1993).

\bibitem{Bre} Brezis, H.;
\emph{Functional analysis, Sobolev spaces and partial differential equations},
Springer (2010).

\bibitem{Cha} Chandrasekharaiah, D. S.;
\emph{Thermoelasticity with second sound: a review},  Appl. Mech. Rev.,
\textbf{39} (3), 355--376 (1986).

\bibitem{Cha1} Chandrasekharaiah, D. S.;
\emph{Hyperbolic thermoelasticity: a review of recent literature},
 Appl. Mech. Rev., \textbf{51} (12), 705--729 (1998).

\bibitem{Col} Coleman, B. D.; Hrusa, W. J.; Owen D. R.;
\emph{Stability of equilibrium for a nonlinear hyperbolic system describing
heat propagation by second sound in solids},
Arch. Rational Mech. Anal. \textbf{94}, 267--289 (1986).

\bibitem{Dat} Datko, R.; Lagnese, J.; Polis, M. P.;
 \emph{An example on the effect of time delays in boundary feedback stabilization
 of wave equations}, SIAM J. Control Optim. \textbf{24} (1), 152--156 (1986).

\bibitem{Fer} Fern\'andez Sare, H. D.;  Racke, R.;
\emph{On the stability of damped Timoshenko systems: Cattaneo versus Fourier law,}
 Arch. Rational Mech. Anal., \textbf{194} (1), 221--251 (2009).

\bibitem{Gue3} Guesmia, A.; Messaoudi, S. A.; Soufyane, A.;
\emph{Stabilization of a linear Timoshenko system with infinite history
and applications to the Timoshenko-Heat systems},
Elect. J. Diff. Equa., \textbf{2012} (193), 1--45 (2012).

\bibitem{Jos} Joseph, D. D.; Preziosi, L.;
 \emph{Heat waves}, Reviews of Modern Physics \textbf{61} (1), 41--73 (1989).

\bibitem{Kir} Kirane, M.; Said-Houari, B.;
 \emph {Existence and asymptotic stability of a viscoelastic wave equation
with a delay}, Z. Angew. Math. Phys. \textbf{62} (6), 1065--1082 (2011).

\bibitem{Kir2} Kirane, M.; Said-Houari, B.; Anwar, M.;
\emph{Stability result for the Timoshenko system with a time-varying delay
term in the internal feedbacks}, Commun. Pure Appl. Anal.,
\textbf{10} (2), 667--686 (2011).

\bibitem{Liu} Liu, Z; Zheng, S.;
\emph{Semigroups Associated with Dissipative Systems}, Chapman Hall/CRC:
 Boca, Raton, (1999).

\bibitem{Mes02} Messaoudi, S. A.; Said-Houari, B.;
\emph{Exponential stability in one-dimensional non-llinear thermoelasticity
with second sound}, Math. Meth. Appl. sci., \textbf{28} (2), 205-232 (2005).

\bibitem{Mes2} Messaoudi, S. A.; Pokojovy, M.; Said-Houari, B.;
\emph{Nonlinear damped Timoshenko systems with second sound-global
existence and exponential stability,}
 Math. Meth. Appl. Sci., \textbf{32} (5), 505--534 (2009).

\bibitem{Mes01} Messaoudi, S. A.; Apalara, T. A.;
\emph{Asymptotic stability of thermoelasticity type III with delay
term and infinite memory}, IMA J. Math. Contr. Info. dnt024 (2013).

\bibitem{Mu} Mu\~noz Rivera, J. E.; Racke, R.;
\emph{Mildly dissipative nonlinear Timoshenko systems-global existence
 and exponential stability}, J. Math. Analy. and Appli.,
\textbf{276} (1), 248--278 (2002).

\bibitem{Mus} Mustafa, M. I.;
\emph{Uniform stability for thermoelastic systems with boundary time-varying delay},
 J. Math. Anal. Appl. \textbf{383} (2), 490--498 (2011).

\bibitem{Mus3} Mustafa, M. I.;
\emph{Exponential decay in thermoelastic systems with boundary delay},
J. Abstr. Diff. Equa., \textbf{2} (1), 1--13 (2011).

\bibitem{Mus1} Mustafa, M. I.; Kafini, M.;
\emph{Exponential Decay in Thermoelastic Systems with Internal Distributed Delay},
 Palestine J. Math. \textbf{2} (2), 287--299 (2013).

\bibitem{Mus2} Mustafa, M. I.;
\emph{A uniform stability result for thermoelasticity of type III
with boundary distributed delay}, J. Abstr. Diff. Equa. Appl.,
\textbf{2} (1), 1--13 (2014).

\bibitem{Nic} Nicaise, S.; Pignotti, C.;
\emph{Stability and instability results of the wave equation with a delay
term in the boundary or internal feedbacks}. SIAM J. Control Optim.
\textbf{45} (5), 1561--1585 (2006).

\bibitem{Nic1} Nicaise, S.; Pignotti, C.;
\emph{Stabilization of the wave equation with boundary or internal
distributed delay}, Diff. Int. Equs. \textbf{21} (9--10), 935--958 (2008).

\bibitem{Nic2} Nicaise, S.; Valein, J.; Fridman, E.;
 \emph{Stability of the heat and of the wave equations with boundary
time-varying delays} Discrete Contin. Dyn. Syst. Ser.
S \textbf{2} (3), 559--581 (2009).

\bibitem{Nic3} Nicaise, S.; Pignotti, C.;
\emph{Interior feedback stabilization of wave equations with time dependent delay},
Elect. J. Differ. Eqns. \textbf{2011} (41), 1--20 (2011).

\bibitem{Paz} Pazy, A.;
\emph{Semigroups of LinearOperators and Applications to PartialDifferential
Equations}, Springer-Verlag: New York, (1983).

\bibitem{Pig} Pignotti, C.;
\emph{A note on stabilization of locally damped wave equations with time delay},
Systems and control letters \textbf{61} (1), 92--97 (2012).

\bibitem{Rac2} Racke, R.;
\emph{Thermoelasticity with second sound--exponential stability in linear
and non-linear $1-$d}, Math. Meth. Appl. Sci., \textbf{25} (5), 409--441 (2002).

\bibitem{Rac1} Racke, R.;
\emph{Instability of coupled systems with delay,} Comm. Pure, Appl. Anal.,
\textbf{11} (5), (2012).

\bibitem{Ric} Richard, J. P.;
\emph{Time-delay systems: an overview of some recent advances and open problems},
Automatica, \textbf{39} (10), 1667--1694 (2003).

\bibitem{SaiL} Said-Houari, B.; Laskri, Y.;
\emph{A stability result of a Timoshenko system with a delay term in the
 internal feedback}, Appl. Math. Comput., \textbf{217} (6), 2857--2869 (2010).

\bibitem{Sai4} Said-Houari, B.; Soufyane, A.;
\emph{Stability result of the Timoshenko system with delay and boundary feedback},
IMA J. Math. Contr. Info., \textbf{29} (3), 383--398 (2012).

\bibitem{San} Santos, M. L.; Almeida J\'unior, D. S.;  Mu\'noz Rivera, J. E.;
\emph{The stability number of the Timoshenko system with second sound},
J. Diff. Eqns., \textbf{253} (9), 2715--2733 (2012).

\bibitem{Tar} Tarabek, M. A.;
 \emph{On the existence of smooth solutions in one-dimensional nonlinear
thermoelasticity with second sound}, Quart. Appl. Math. \textbf{50} (4),
727--742 (1992).

\end{thebibliography}

\end{document}