2021

[1]Abdelfeteh Fareh These authors contributed equally to this work.

1]Laboratory of operators theory and PDEs: foundations and applications,
University of El Oued, P.O.B. 789, El Oued, 39000, Algeria

# On the time behavior of a porous thermoelastic system with only thermal
dissipation given by Gurtin-Pipkin law

Afaf Ahmima<EMAIL_ADDRESS><EMAIL_ADDRESS>[

###### Abstract

In the present paper we consider a porous thermoelastic system with only one
dissipative mechanism generated by the heat conductivity modelled by the
Gurtin-Pipkin thermal law. By the use of a semigroup approach and the Lumer-
Phillips theorem we prove the existence of a unique solution. We introduce a
stability number $\chi_{g}$ depends on the coefficients of the system, and
establish an exponential stability result provided that $\chi_{g}=0$.
Otherwise, if $\chi_{g}\neq 0$, we prove the lack of exponential decay. Our
result improves and generalizes the previous results in the literature
obtained for Fourier’s and Cattaneo’s laws of thermal conductivity.

###### keywords:

Porous material, Gurtin-Pipkin law, well-posedness, stability number,
exponential stability.

###### pacs:

[

MSC Classification]35B40, 47D03, 74D05, 74F15

##  1 Introduction

In this paper we are concerned by the following porous thermelastic system

$\left\\{\begin{array}[]{ll}\rho
u_{tt}=au_{xx}+b\phi_{x}&{in}~{}~{}(0,\pi)\times(0,\infty),\\\
J\phi_{tt}=\alpha\phi_{xx}-bu_{x}-\xi\phi-\beta\theta_{x}&{in}~{}~{}(0,\pi)\times(0,\infty),\\\
c\theta_{t}=-q_{x}-\beta\phi_{xt}&{in}~{}~{}(0,\pi)\times(0,\infty),\\\
q=-\displaystyle\int_{-\infty}^{0}g(t-s)\theta(x,s)ds&{in}~{}~{}(0,\pi)\times(0,\infty).\end{array}\right.$
(1)

This system models a one-dimensional porous elastic material of length $\pi$,
subjected to thermal effects modelled by the Gurtin-Pipkin law of thermal
conductivity. Here, $u$ is the transversal displacement, $\phi$ is the volume
fraction, $\theta$ is the difference of temperature from an equilibrium
reference value and $q$ is the heat flux. The coefficients
$\rho,J,c,\mu,b,\alpha,\xi$ are positive constitutive constants. Moreover, to
ensure that the energy associated to the solution of (1) is a positive
definite form, we assume that $b,\mu,\xi$ satisfy $\mu\xi>b^{2}$. The
coefficient $\beta$ is a coupling constant that is different from zero but its
sign does not matter in the analysis.

The basic evolution equations in the theory of thermoelastic materials with
voids are developed by Ieşan Iesan86 ; Iesan . In the one dimensional case
these equations are written as follows

$\rho u_{tt}=T_{x},~{}~{}J\varphi_{tt}=H_{x}-G,~{}~{}\rho\eta_{t}=q_{x},$ (2)

where, $T$ is the stress tensor, $H$ is the equilibrated stress vector, $G$ is
the equilibrated body force and $\eta$ is the entropy.

In the linear theory, and assuming that the body is free from stresses and has
zero intrinsic equilibrated body force, the constitutive equations are Iesan86
; Iesan

$\begin{array}[]{ll}T=\mu u_{x}+b\varphi,&H=\alpha\varphi_{x}-\beta\theta,\\\
G=bu_{x}+\xi\varphi,&\rho\eta=-c\theta-\beta\varphi_{x}.\end{array}$ (3)

The substitution of the constitutive equation (3) in the evolution equations
(2) leads to the first three equation of the system (1). The forth equation of
(1) represents the heat conduction modelled by Gurtin-Pipkin thermal law (16)
which will be introduced later.

Much has been written on the study of the longtime behavior of porous
thermoelastic systems. Quintanilla Quin1  proved the slow decay of the
solution of the system

$\left\\{\begin{array}[]{l}\rho u_{tt}=\mu u_{xx}+\beta\varphi_{x},\\\
\rho\kappa\varphi_{tt}=\alpha\varphi_{xx}-\beta
u_{x}-\xi\varphi-\tau\varphi_{t},\end{array}\right.$ (4)

subjected to the Dirichlet-Neumann or the Neumann-Dirichlet boundary
conditions.
Later in 2017, Apalara Apalara  and, independently, Santos et _al._ Santos1
established the exponential stability of (4) provided that the wave speeds of
the equations of (21) are equal

$\frac{\mu}{\rho}=\frac{\alpha}{\rho\kappa}.$ (5)

The same result was obtained by Santos _et al._ Santos1  when the damping
$-\tau\varphi_{t}$ is replaced by the elastic damping $\gamma u_{t}$ in the
first equation.

During the last three decades, several dissipative mechanisms have been
examined to exponentially stabilize the solution of (4). First, Magaña and
Quintanilla replace the visco-porous damping $-\tau\varphi_{t}$ by a visco-
elastic one of the form $\lambda u_{txx}$ in the first equation of (4) and
proved the non-exponential decay of the solution. Casas and Quintanilla Casas1
, coupled the system (4) with the heat equation. They examined the system

$\left\\{\begin{array}[]{l}\rho u_{tt}=\mu
u_{xx}+b\varphi_{x}-\beta\theta_{x},\\\
J\varphi_{tt}=\alpha\varphi_{xx}-bu_{x}-\xi\varphi+m\theta-\tau\varphi_{t},\\\
c\theta_{t}=k^{\star}\theta_{xx}-\beta u_{xt}-m\varphi_{t},\end{array}\right.$
(6)

with the boundary conditions

$u(0,t)=u(\pi,t)=\varphi_{x}(0,t)=\varphi_{x}(\pi,t)=\theta_{x}(0,t)=\theta_{x}(\pi,t)=0,$

and proved that the combination of the visco-porous and thermal dissipations
lead to an exponential rate of decay. The same rate of decay was obtained when
micro-temperature and viscoelastic dissipations are combined Magana . However,
removing the visco-porous damping $-\tau\varphi_{t}$ from the system (6), or
replacing it by the viscoelastic damping $\lambda u_{txx}$ in the first
equation of (6), lead to a lose of the exponential decay Casas2 ; Magana .

Apalara Apalara1  and Feng Feng1  examined, respectively, the effect of the
dissipation introduced by a memory term or a past history. They obtained a
general rate of decay depending on the rate of the relaxation function.

Note that Magaña and Quintanilla Magana  are summarized the effects of the
above dissipative mechanisms as follows:
if a thermal or a viscoelastic damping is combined with a micro-thermal or a
visco-porous damping, the solution decays exponentially, otherwise, the
solution decays in a slow way.

The rate of decay of Timoshenko type systems is quite similar to that of
porous thermoelastic systems. We cite here some of them. In a pioneer work,
Soufyane Soufyane  considered the Timoshenko system

$\left\\{\begin{array}[]{l}\rho_{1}\varphi_{tt}=\kappa(\varphi_{x}+\psi)_{x},\\\
\rho_{2}\psi_{tt}=\alpha\varphi_{xx}-\kappa(\varphi_{x}+\psi)-b(x)\psi_{t},\end{array}\right.$
(7)

where $b\in L^{\infty}(0,L)$ is a strictly positive function, and showed that
the system is exponentially stable if and only if

$\frac{\kappa}{\rho_{1}}-\frac{\alpha}{\rho_{2}}=0.$ (8)

The strong damping $b(x)\psi_{t}$ was replaced by the memory term
$\int_{0}^{t}g(t-s)\psi_{xx}(x,s)ds$ in Ammar ; Mustafa ; Tatar  and an
exponential stability result was obtained for an exponentially decaying
relaxation function $g$.

Rivera and Racke Racke  examined the heat dissipation produced by the coupling
with the heat equation. They considered the system

$\left\\{\begin{array}[]{l}\rho_{1}\varphi_{tt}=\kappa(\varphi_{x}+\psi)_{x},\\\
\rho_{2}\psi_{tt}=b\varphi_{xx}-\kappa(\varphi_{x}+\psi)-\gamma\theta_{x},\\\
\rho_{3}\theta_{t}=k\theta_{xx}-\gamma\psi_{tx},\end{array}\right.$ (9)

with different boundary conditions, and proved that the solution decays
exponentially, if and only if (8) holds. The same result was obtained by
Almeida Junior _et al._ Almeida  for the following Timoshenko type system with
a thermoelastic coupling in shear force

$\left\\{\begin{array}[]{ll}\rho_{1}\varphi_{tt}-\kappa\left(\varphi_{x}+\psi\right)_{x}+\sigma\theta_{x}=0,&in\;\left]0,L\right[\times\mathbb{R}_{+},\\\
\rho_{2}\psi_{tt}-b\psi_{xx}+\kappa\left(\varphi_{x}+\psi\right)-\sigma\theta=0,&in\;\left]0,L\right[\times\mathbb{R}_{+},\\\
\rho_{3}\theta_{t}-\gamma\theta_{xx}+\sigma\left(\varphi_{x}+\psi\right)_{t}=0,&in\;\left]0,L\right[\times\mathbb{R}_{+}.\end{array}\right.$
(10)

We can summarize the rate of decay of Timosenko type systems is as follows:

  * $\cdot$

Two dissipative mechanisms lead, always, to an exponential stability result
regardless any assumption on the coefficients of the system Kim ; Raposo ; Shi
.

  * $\cdot$

A single dissipation mechanism produce an exponential rate of decay, if and
only if the wave speeds of the hyperbolic part of the system are equal.

Note that the heat conduction in (6) and (10) is given through Fourier’s law

$q=-\kappa\theta_{x},$ (11)

which leads to a parabolic equation and consequently, the heat propagates with
an infinite speed, which is unrealistic. To overcome this physical paradox,
many alternative theories were developed. Lord and Shulman Lord  replaced the
Fourier law (11) by the Cattaneo one

$\tau_{0}q_{t}+q+\kappa\theta_{x}=0,$ (12)

where the positive constant $\tau_{0}$ represents the time lag in the response
of the heat flux to the temperature gradient. According to this theory,
(called second sound thermoelasticity), the system becomes fully hyperbolic
and the heat propagates as a wave with a finite speed. Moreover, the speed of
the heat equation is involved in the exponential stability condition.

In the context of this theory, Messaoudi and Fareh Fareh5  considered the
porous thermoelastic system of second sound

$\left\\{\begin{array}[]{l}\rho u_{tt}=\mu u_{xx}+b\phi_{x},\\\
J\phi_{tt}=\alpha\phi_{xx}-bu_{x}-\xi\phi-\beta\theta_{x},\\\
c\theta_{t}=-q_{x}-\beta\phi_{tx}-\delta\theta,\\\
\tau_{0}q_{t}+q+\kappa\theta_{x}=0,\end{array}\right.$ (13)

where $\rho,\mu,J,\alpha,\xi,\beta,c,\kappa$ and $\tau_{0}$ are positive
constants and $\mu\xi=b^{2}$. They introduced the stability number

$\chi=\beta^{2}-\left(\frac{c\alpha\mu}{\rho}-\frac{\alpha\kappa}{\tau_{0}}\right)\left(\frac{J}{\alpha}-\frac{\rho}{\mu}\right)$

and proved that the solution of (13) is exponentially stable if and only if
$\chi=0$.

To the best of our knowledge Fareh5  is the only contribution that introduces
the stability number for porous thermoelastic problems. However, for
Timoshenko type systems, the stability number was introduced first by Santos
_et al._ Santos  for Timoshenko systems. They considered the following system
with second sound thermoelasticity

$\left\\{\begin{array}[]{ll}\rho_{1}\varphi_{tt}-\kappa(\varphi_{x}+\psi)_{x}=0,&~{}\mbox{in}~{}(0,l)\times\mathbb{R}_{+},\\\
\rho_{2}\psi_{tt}-b\psi_{xx}+\kappa(\varphi_{x}+\psi)+\delta\theta_{x}=0,&~{}\mbox{in}~{}(0,l)\times\mathbb{R}_{+},\\\
\rho_{3}\theta_{t}+q_{x}+\delta\psi_{tx}=0,&~{}\mbox{in}~{}(0,l)\times\mathbb{R}_{+},\\\
\tau q_{t}+\beta
q+\theta_{x}=0,&~{}\mbox{in}~{}(0,l)\times\mathbb{R}_{+},\end{array}\right.$
(14)

and showed the exponential stability of the solution of (14), if and only if

$\chi_{0}=\left(\tau-\frac{\rho_{1}}{\rho_{3}\kappa}\right)\left(\rho_{2}-\frac{b\rho_{1}}{\kappa}\right)-\frac{\tau\rho_{1}\delta^{2}}{\rho_{3}\kappa}=0.$
(15)

In fact, the stability number $\chi$ defined in (8) is recovered from
$\chi_{0}$ at the limit $\tau\rightarrow 0,$ when (14) collapses to (9).

It is worth noting that the theory of thermoelasticity with second sound is
incapable to depict the memory effect which prevails in some materials,
particularly at low temperature. This fact leads to the search for a more
general constitutive hypothesis that links the heat flux to the thermal
memory. Gurtin and Pipkin Gurtin  supposed that the heat flux depends on the
accumulated history of the temperature gradient weighted by a relaxation
function called the heat flux kernel. They formulated a general nonlinear
theory for which thermal perturbations disseminate with finite speed.
According to this theory, the linearized constitutive equation for $q$ writes

$q=-\int_{-\infty}^{t}g\left(t-s\right)\theta_{x}\left(x,s\right)ds,$ (16)

where $g(s)$ is the heat conductivity relaxation kernel. The presence of
convolution term (16) in the heat equation renders the system into a fully
hyperbolic system, which allows the heat to propagate with finite speed and
accepts to describe the memory effect of heat conduction. We notice that
several constitutive models arise from different choices of $g\left(s\right).$
In particular, if we take $g\left(s\right)=\kappa\delta\left(s\right),$ where
$\delta$ is the Dirac mass weighted at $0,$ then (16) becomes Fourier’s law
(11), and if we let

$g\left(s\right)=\frac{\kappa}{\tau_{0}}e^{-\frac{s}{\tau_{0}}},\;\tau>0,$

we recover Cattaneo’s law (12). Therefore, (16) is a generalized form of
Fourier’s and Cattaneo’s laws.

In the context of Gurtin-Pipkin theory, Pata and Vuk Pata  considered the
linear thermoelastic system

$\left\\{\begin{array}[]{l}u_{tt}(x,t)=u_{xx}(x,t)-\theta_{x}(x,t),\\\
\theta_{t}\left(x,t\right)=-u_{tx}\left(x,t\right)-q_{x}\left(x,t\right),\end{array}\right.$

where the heat flux $q$ is modeled by (16). They used the semigroup theory and
established an exponential stability result under some assumptions on
$\mu\left(s\right)=-g^{\prime}\left(s\right)$. Fatori and Muñoz Rivera Fatouri
studied the system

$\left\\{\begin{array}[]{l}u_{tt}-au_{xx}+\alpha\theta_{x}=0\;in\;(0\mathchar
24635\relax\;L)\times\mathbb{R}_{+}\\\ \theta_{t}-k\ast\theta_{xx}+\alpha
u_{xt}=0\;in\;(0\mathchar
24635\relax\;L)\times\mathbb{R}_{+},\end{array}\right.$

where

$\left(k\ast\theta_{xx}\right)\left(t\right)=\int_{0}^{t}k\left(t-\tau\right)\theta_{xx}\left(\tau\right)d\tau.$

They obtained an exponential decay result provided that the kernel $k$ is
positive definite and decays exponentially.

Concerning Timoshenko type systems with heat flux given by Gurtin-Pipkin law,
Dell’Oro an Pata Oro  analyzed the following system

$\left\\{\begin{array}[]{l}\rho_{1}\varphi_{tt}-\kappa\left(\varphi_{x}+\psi\right)_{x}=0,\\\
\rho_{2}\psi_{tt}-b\psi_{xx}+\kappa\left(\varphi_{x}+\psi\right)+\delta\theta_{x}=0,\\\
\rho_{3}\theta_{t}-\displaystyle\frac{1}{\beta}\int_{0}^{+\infty}g\left(s\right)\theta_{xx}\left(t-s\right)ds+\delta\psi_{tx}=0.\end{array}\right.$
(17)

They introduced the stability constant

$\chi_{g}=\left[\frac{\rho_{1}}{\rho_{3}\kappa}-\frac{\beta}{g\left(0\right)}\right]\left[\frac{\rho_{1}}{\kappa}-\frac{\rho_{2}}{b}\right]-\frac{\beta}{g\left(0\right)}\frac{\rho_{1}\delta^{2}}{\rho_{3}\kappa
b}$

and proved that the solution of the system is exponentially stable if and only
if $\chi_{g}=0.$

Recently, Fareh FarehRV  considered the porous thermoelastic system

$\left\\{\begin{array}[]{l}\rho u_{tt}=\mu
u_{xx}+b\varphi_{x}-\beta\theta_{x},\\\
J\varphi_{tt}=\alpha\varphi_{xx}-bu_{x}-\xi\varphi+m\theta-\tau\varphi_{t},\\\
c\theta_{t}=-\displaystyle\int_{-\infty}^{0}k\left(t-s\right)\theta_{xx}\left(s\right)ds-\beta
u_{tx}-m\varphi_{t},\end{array}\right.$ (18)

where the heat conduction is modeled by Gurtin-Pipkin thermal law (16). He
proved the exponential stability of the solution of (18) without any
restriction on the coefficients of the system.

The system (1) considered in the present paper is a (weak) version of (18),
where the coupling porosity-temperature is direct, however, the coupling
elasticity-temperature is through the porosity equation. Moreover, system (1)
contain only one dissipative mechanism produced by the heat equation of
hyperbolic type.

Here, we introduce a stability number $\chi_{g}$, for which the solution of
(1) is exponentially stable if and only if $\chi_{g}=0$.

The system (1) is supplemented by the following initial and boundary
conditions

$\left\\{\begin{array}[]{c}u(x,0)=u_{0}(x),~{}\phi(x,0)=\phi_{0}(x),~{}\theta(x,0)=\theta_{0}(x),\\\
u_{t}(x,0)=u_{1}(x),~{}\phi_{t}(x,0)=\phi_{1}(x),\end{array}\right.$ (19)
$u(0,t)=u(\pi,t)=\phi_{x}(0,t)=\phi_{x}(\pi,t)=\theta(0,t)=\theta(\pi,t)=0.$
(20)

To write the problem in the semigroup setting, we introduce the new variables

$\theta^{t}(x,s):=\theta(x,t-s),\;s\geq 0,$

and

$\eta(x,s)=\eta^{t}(x,s):=\int_{0}^{s}\theta^{t}(x,\tau)d\tau,\;s\geq 0,$

which denote the past history and the integrated past history of $\theta$ up
to $t$, respectively.

Clearly $\eta^{t}\left(x,s\right)$ satisfies the boundary conditions

$\eta(0,s)=\eta(\pi,s)=0.$

Moreover, we assume that $g\left(\infty\right)=0$ and
$\eta(x,0)=\underset{s\longrightarrow 0}{\lim}\eta^{t}(x,s)=0,$ then

$q=-\int_{-\infty}^{t}g\left(t-s\right)\theta_{x}\left(x,s\right)ds=\int_{0}^{+\infty}g^{\prime}\left(s\right)\eta_{x}^{t}\left(x,s\right)ds.$

Further, we have

$\eta_{t}(x,s)=\theta-\eta_{s}(x,s).$ (21)

Setting $\kappa\left(s\right)=-g^{\prime}\left(s\right),$ system (1) becomes

$\left\\{\begin{array}[]{ll}\rho u_{tt}=\mu
u_{xx}+b\phi_{x}&{in}~{}~{}(0,\pi)\times(0,\infty),\\\
J\phi_{tt}=\alpha\phi_{xx}-bu_{x}-\xi\phi-\beta\theta_{x}&{in}~{}~{}(0,\pi)\times(0,\infty),\\\
c\theta_{t}=-\beta\phi_{xt}+\displaystyle\int_{0}^{+\infty}\kappa(s)\eta_{xx}^{t}(x,s)ds&{in}~{}~{}(0,\pi)\times(0,\infty),\\\
\eta_{t}^{t}=\theta-\eta_{s}^{t}&{in}~{}~{}(0,\pi)\times(0,\infty),\end{array}\right.$
(22)

From (22)2 and the boundary conditions (20), we have

$J\frac{d^{2}}{dt^{2}}\int_{0}^{\pi}\phi\left(x,t\right)dx+\xi\int_{0}^{\pi}\phi\left(x,t\right)dx=0,$

which gives

$\int_{0}^{\pi}\phi\left(x,t\right)dx=\left(\int_{0}^{\pi}\phi_{0}\left(x\right)dx\right)\cos\left(\sqrt{\frac{\xi}{J}}t\right)+\sqrt{\frac{\xi}{J}}\left(\int_{0}^{\pi}\phi_{1}\left(x\right)dx\right)\sin\left(\sqrt{\frac{\xi}{J}}t\right).$

Consequently, if we set

$\displaystyle\overline{\phi}\left(x,t\right)=$
$\displaystyle\phi\left(x,t\right)-\left(\int_{0}^{\pi}\phi_{0}\left(x\right)dx\right)\cos\left(\sqrt{\frac{\xi}{J}}t\right)$
$\displaystyle-\sqrt{\frac{\xi}{J}}\left(\int_{0}^{\pi}\phi_{1}\left(x\right)dx\right)\sin\left(\sqrt{\frac{\xi}{J}}t\right),$

we find

$\int_{0}^{\pi}\overline{\phi}\left(x,t\right)dx=0.$

Moreover, we check easily that $\left(u,\overline{\phi},\theta,\eta\right)$
solves (22) subjected to the boundary conditions (20), initial data

$\left\\{\begin{array}[]{l}u(x,0)=u_{0}(x),~{}\overline{\phi}(x,0)=\phi_{0}(x)-\displaystyle\int_{0}^{\pi}\phi(x)dx,~{}\theta(x,0)=\theta_{0}(x),\\\
u_{t}(x,0)=u_{1}(x),~{}\overline{\phi}_{t}(x,0)=\phi_{1}(x)-\displaystyle\sqrt{\frac{\xi}{J}}\int_{0}^{\pi}\phi_{1}\left(x\right)dx,~{}\eta^{0}(x,s)=\eta_{0}(x,s),\end{array}\right.$
(23)

and more importantly, Poincaré’s inequality can be applied for
$\overline{\phi}$. In the sequel we work with
$\left(u,\overline{\phi},\theta,\eta\right)$ but we write
$\left(u,\phi,\theta,\eta\right)$ for convenience.

Regarding the memory kernel, we assume the following set of hypotheses:

  * (h1)

$\kappa\in C\left(R^{+}\right)\cap L^{1}\left(R^{+}\right).$

  * (h2)

$\kappa\left(s\right)>0,\kappa^{{}^{\prime}}\left(s\right)\leq 0,\forall s\geq
0.$

  * (h3)

$\int_{0}^{+\infty}\kappa(s)ds=g\left(0\right),$

  * (h4)

there exists $\delta>0$ such that
$\kappa^{\prime}\left(s\right)\leq-\delta\kappa\left(s\right),\forall s\geq
0.$

To the best of our knowledge, the only contribution that examined the behavior
of porous thermoelastic systems with Gurtin-Pipkin law was done by Fareh
FarehRV . Here, we improves the results of Fareh FarehRV  in the sense that we
consider the direct coupling only between porosity and temperature and omit
the strong dissipation given by $-\tau\phi_{t}$. Moreover, our result improves
that of Oro  obtained for Timoshenko systems and generalizes the results of
Fareh5  and MunozQuintanilla  since (16) subsumes (11 ) and (12).

The rest of the paper is organized as follow: in Section 2, we introduce some
functional preliminaries. Section 3 is devoted to the prove of the existence
of a unique solution to (22), (20), (23). In Section 4, we state and prove our
stability result provided that $\chi_{g}=0$. Finally, in Section 5, we prove
the lack of the exponential decay whenever $\gamma_{g}=0$ or $\chi_{g}\neq 0.$

##  2 Preliminaries

First, let $A:D(A)\subset L^{2}\left(0,\pi\right)\rightarrow
L^{2}\left(0,\pi\right)$ be the operator defined by $Au=-D^{2}u$. For
Dirichlet boundary conditions $D\left(A\right)=H^{2}\cap H_{0}^{1}$ and the
operator $A$ is a self-adjoint and positive operator. Therefore, it is
possible to define the powers $A^{r}$ of $A$ for $r\in\mathbb{R}$ and the
Hilbert space $V_{r}=D\left(A^{r/2}\right)$ equipped with the inner product

$\left\langle u,v\right\rangle_{r}=\left\langle
A^{r/2}u,A^{r/2}v\right\rangle$

and let $\left\|u\right\|_{r}$ be the associated norm. In particular,
$V_{0}=L^{2},$ $V_{-1}=H^{-1},$ $V_{1}=H_{0}^{1}$ and

$\left\langle A^{1/2}u,A^{1/2}v\right\rangle=\left\langle
Du,Dv\right\rangle,\;\forall u,v\in H_{0}^{1}.$

For $r_{1}>r_{2}$ the injection $V_{r_{1}}\hookrightarrow V_{r_{2}}$ is
continuous.

Next, we introduce the weighted Hilbert space

$\mathcal{V}=L_{\kappa}^{2}((0,+\infty)\mathchar
24635\relax\;H_{0}^{1}(0,\pi)),$

with inner product

$\langle\eta,\zeta\rangle_{\mathcal{V}}=\int_{0}^{+\infty}\kappa(s)\langle\eta_{x}(s),\zeta_{x}(s)\rangle
ds,$

and the norm

$\|\eta\|_{\mathcal{V}}^{2}=\int_{0}^{+\infty}\kappa(s)\|\eta_{x}(s)\|^{2}ds.$

At this point, we introduce the energy functional associated to the solution
$\left(u,\phi,\theta,\eta\right)$ of (22), (20), (23), by

$\displaystyle E\left(t\right)$ $\displaystyle:$
$\displaystyle=\frac{1}{2}\int_{0}^{\pi}\left[\rho u_{t}^{2}+J\phi_{t}^{2}+\mu
u_{x}^{2}+\xi\phi^{2}+2bu_{x}\phi+\alpha\phi_{x}^{2}+c\theta^{2}\right]dx$
$\displaystyle+\int_{0}^{+\infty}\kappa(s)\int_{0}^{\pi}\eta_{x}^{2}(s)dxds.$

The energy space is then

$\mathcal{H}:=H_{0}^{1}\left(0,\pi\right)\times L^{2}\left(0,\pi\right)\times
H_{\ast}^{1}\left(0,\pi\right)\times L^{2}\left(0,\pi\right)\times
L^{2}\left(0,\pi\right)\times\mathcal{V}.$

It is a Hilbert space with respect to the inner product

$\begin{array}[]{cc}\langle
U,U^{\ast}\rangle=&\rho\displaystyle\int_{0}^{\pi}vv^{\ast}dx+\mu\int_{0}^{\pi}u_{x}u_{x}^{\ast}dx+\xi\int_{0}^{\pi}\phi\phi^{\ast}dx+\alpha\int_{0}^{\pi}\phi_{x}\phi_{x}^{\ast}dx\\\
&+J\displaystyle\int_{0}^{\pi}\psi\psi^{\ast}dx+b\int_{0}^{\pi}\phi
u_{x}^{\ast}dx+b\int_{0}^{\pi}u_{x}\phi^{\ast}dx\\\
&+c\displaystyle\int_{0}^{\pi}\theta\theta^{\ast}dx+\int_{0}^{+\infty}\kappa(s)\int_{0}^{\pi}\eta_{x}(s)\eta_{x}^{\ast}(s)dxds,\end{array}$

where, $U=(u,v,\phi,\psi,\eta)^{T}$ and
$U^{\ast}=(u^{\ast},v^{\ast},\phi^{\ast},\psi^{\ast},\eta^{\ast})^{T}$. The
associated norm is

$\|U\|_{\mathcal{H}}^{2}=\mu\|u_{x}\|^{2}+\xi\|\phi\|^{2}+2b\langle\phi,u_{x}\rangle+\rho\|v\|^{2}+\alpha\|\phi_{x}\|^{2}+J\|\psi\|^{2}+c\|\theta\|^{2}+\|\eta\|_{\mathcal{V}}^{2}.$
(25)

######  Remark 1.

Under the hypothesis $\mu\xi>b^{2}$ we have

$\int_{0}^{\pi}\left(\mu
u_{x}^{2}+\xi\phi^{2}+2bu_{x}\phi\right)dx=\left\|\sqrt{\mu}u_{x}+\dfrac{b}{\sqrt{\mu}}\phi\right\|^{2}+\left(\xi-\dfrac{b^{2}}{\mu}\right)\|\phi\|^{2}.$
(26)

######  Lemma 1.

The energy $E(t)$ defined by (2), satisfies along the solution
$(u,\phi,\theta,\eta)$ the estimate

$\frac{d}{dt}E(t)=\langle T\eta,\eta\rangle.$ (27)

Proof: Taking the $L^{2}-$inner product of the first three equations of (22)
by $u_{t},\phi_{t},\theta$ respectively and the $\mathcal{V-}$inner product of
the forth equation by $\eta_{t},$ then adding the obtained equations and using
integration by parts, (27) follows immediately.

##  3 Well posedness

In this section we prove that the problem given by (22),(20) and (23) has a
unique solution. The prove is based on the theory of semigroups and Hille-
Yoside Theorem.
To rewrite the problem in the semigroup setting, we introduce two new
independent variables $u_{t}=v$ and $\phi_{t}=\psi$, then the system (22) and
the initial conditions (20) can written as follows

$\left\\{\begin{array}[]{c}U_{t}+\mathcal{A}U=0,\\\
U\left(0\right)=U_{0},\end{array}\right.$ (28)

where $\mathcal{A}$ is the operator defined on $\mathcal{H}$ by

$\mathcal{A}U=\left(\begin{array}[]{c}-v\\\
-\dfrac{\mu}{\rho}u_{xx}-\dfrac{b}{\rho}\phi_{x}\\\ -\psi\\\
-\dfrac{\alpha}{J}\phi_{xx}+\dfrac{\xi}{J}\phi+\dfrac{b}{J}u_{x}+\dfrac{\beta}{J}\theta\\\
\dfrac{\beta}{c}\psi-\dfrac{1}{c}\displaystyle\int_{0}^{+\infty}\kappa(s)\eta_{xx}\left(s\right)ds\\\
-\theta+\eta_{s}(s)\end{array}\right),$ (29)

with domain

$D\left(\mathcal{A}\right)=\left\\{\begin{array}[]{c}U\in\mathcal{H}:u,\phi\in
H^{2}(0,\pi),v,\theta\in H_{0}^{1}(0,\pi),\psi\in H_{\ast}^{1}(0,\pi),\\\
\eta\in H_{\kappa}^{1}((0,+\infty)\mathchar 24635\relax\;H_{0}^{1}),\\\
\displaystyle\int_{0}^{+\infty}\kappa(s)\eta_{xx}(s)ds\in
L^{2}(0,\pi),\eta(0)=0.\end{array}\right\\}$

The well-posedness result reads as follow:

######  Theorem 1.

Suppose that $\kappa$ satisfies the hypothesis (h1)-(h4), then for any
$U_{0}=\left(u_{0},u_{1},\phi_{0},\phi_{1}\right.$,$\left.\theta_{0},\eta_{0}\right)^{T}\in\mathcal{H}$
the problem (28) has a unique solution $U\in
C\left(\left(0,+\infty\right)\mathchar 24635\relax\;\mathcal{H}\right)$.
Moreover, if $U_{0}\in D\left(\mathcal{A}\right)$, then the solution $U$
satisfies

$U\in C\left(\left(0,+\infty\right)\mathchar
24635\relax\;D\left(\mathcal{A}\right)\right)\cap
C^{1}\left(\left(0,+\infty\right)\mathchar 24635\relax\;\mathcal{H}\right).$

The proof of Theorem 1 is based on the Hille-Yosida Theorem.

######  Theorem 2 (Hille-Yosida).

Brezis  Let $\mathcal{A}$ be a maximal monotone operator on a Hilbert space
$\mathcal{H}$, then, for any $u_{0}\in D(\mathcal{A})$ there exists a unique
function

$u\in C^{1}((0,+\infty)\mathchar 24635\relax\;\mathcal{H})\cap
C((0,+\infty)\mathchar 24635\relax\;D(\mathcal{A})),$

satisfying

$\left\\{\begin{array}[]{l}\dfrac{du}{dt}+\mathcal{A}u=0,\\\
u(0)=u_{0}.\end{array}\right.$

The proof of Theorem 1 will be established through several lemmas.

######  Lemma 2.

Let $T$ be the operator defined on $\mathcal{V}$ by

$T\eta=-\eta_{s},$

with domain

$D(T)=\left\\{\eta\in\mathcal{V}:\eta_{s}\in\mathcal{V},~{}\underset{s\rightarrow
0}{\lim}\|\eta_{x}\|=0\right\\}.$

Then, for every $\eta\in D(T)$, we have

$\langle
T\eta,\eta\rangle_{\mathcal{V}}=-\frac{1}{2}\int_{0}^{+\infty}\kappa^{\prime}(s)\left\|\eta_{x}(s)\right\|^{2}ds$
(30)

and

$\delta\left\|\eta\right\|_{\mathcal{V}}^{2}\leq-\int_{0}^{+\infty}\kappa^{\prime}(s)\left\|\eta_{x}(s)\right\|^{2}ds.$
(31)

Proof:

$\begin{array}[]{rl}2\langle
T\eta,\eta\rangle_{\mathcal{V}}&=\displaystyle-\int_{0}^{+\infty}\kappa(s)\frac{d}{ds}\|\eta_{x}(s)\|^{2}ds\\\
&=-\displaystyle\left.\kappa(s)\|\eta_{x}(s)\|^{2}\right|_{0}^{+\infty}+\int_{0}^{+\infty}\kappa^{\prime}(s)\|\eta_{x}(s)\|^{2}ds\end{array}$
$2\langle T\eta,\eta\rangle_{\mathcal{V}}=\displaystyle\lim_{s\rightarrow
0}\kappa(s)\|{\eta}_{x}(s)\|^{2}-\lim_{s\rightarrow\infty}\kappa(s)\|{\eta}_{x}(s)\|^{2}+\int_{0}^{+\infty}\kappa^{\prime}(s)\|\eta_{x}(s)\|^{2}ds.$
(32)

Since $\kappa\left(s\right)\|\eta_{x}\left(s\right)\|$ and
$\kappa\left(s\right)\|\eta_{xs}\left(s\right)\|$ are in
$L^{1}\left(\mathbb{R}^{+}\right)$ and $\eta_{x}(0)=0$, then, the first term
in the right hand side of (32) worth

$\displaystyle\lim_{s\rightarrow 0}\kappa(s)\|{\eta}_{x}(s)\|^{2}$
$\displaystyle=\lim_{s\rightarrow
0}\kappa(s)\left\|\int_{0}^{s}{\eta}_{xs}(\tau)d\tau\right\|^{2},$
$\displaystyle\leq\limsup_{s\rightarrow
0}\left(\int_{0}^{s}\kappa(s)^{1/2}\left\|{\eta}_{xs}(\tau)\right\|d\tau\right)^{2}.$

The use of Cauchy-Schwarz inequality, leads to

$\lim_{s\rightarrow 0}\kappa(s)\|{\eta}_{x}(s)\|^{2}\leq\limsup_{s\rightarrow
0}s\int_{0}^{s}\kappa(\tau)\left\|{\eta}_{xs}(\tau)\right\|^{2}d\tau=0.$

Therefore,

$2\langle
T\eta,\eta\rangle_{\mathcal{V}}=-\lim_{s\rightarrow\infty}\kappa(s)\|\eta_{x}(s)\|^{2}+\int_{0}^{+\infty}\kappa^{\prime}(s)\|\eta_{x}(s)\|^{2}ds.$

The left-hand side of the last equality is bounded, and from (h2) both terms
of the right-hand side are non-positive, we infer that the above limit exists
and is finite, and consequently equals zero. Thus

$\langle
T\eta,\eta\rangle_{\mathcal{V}}=\frac{1}{2}\int_{0}^{+\infty}\kappa^{\prime}(s)\|\eta_{x}(s)\|^{2}ds,$

Moreover, the use of the assumption (h4) on $\kappa$ leads to (31), which
completes the proof.

######  Remark 2.

According to (30) and (27), the energy $E(t)$, satisfies

$\frac{d}{dt}E(t)=\frac{1}{2}\int_{0}^{+\infty}\kappa^{\prime}(s)\|\eta_{x}(s)\|^{2}ds\leq
0.$ (33)

######  Lemma 3.

The operator $\mathcal{A}$ defined by (29) is monotone.

Proof: A direct calculation using integration by parts and boundary conditions
gives

$\displaystyle\left\langle\mathcal{A}U,U\right\rangle_{\mathcal{H}}=$
$\displaystyle-\int_{0}^{\pi}\Big{[}\left(\mu
u_{xx}+b\phi_{x}\right)\Big{]}vdx-\mu\int_{0}^{\pi}v_{x}u_{x}dx-\xi\int_{0}^{\pi}\psi\phi
dx-\alpha\int_{0}^{\pi}\psi_{x}\phi_{x}dx$
$\displaystyle-\int_{0}^{\pi}\Big{[}\alpha\phi_{xx}-\xi\phi-
bu_{x}-\beta\theta\Big{]}\psi dx-b\int_{0}^{\pi}\psi
u_{x}dx-b\int_{0}^{\pi}v_{x}\phi dx$
$\displaystyle-\int_{0}^{\pi}\Big{[}-\beta\psi+\int_{0}^{+\infty}\kappa(s)\eta_{xx}\left(s\right)ds\Big{]}\theta
dx$
$\displaystyle-\int_{0}^{+\infty}\kappa(s)\int_{0}^{\pi}\Big{[}\theta_{x}-\eta_{sx}(s)\Big{]}\eta_{x}\left(s\right)dx$
$\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{+\infty}\kappa(s)\eta_{x}\eta_{sx}ds$
$\displaystyle=$
$\displaystyle\dfrac{1}{2}\displaystyle\int_{0}^{+\infty}\kappa(s)\dfrac{d}{ds}\|\eta_{x}\left(s\right)\|^{2}ds.$

Therefore, the result follows from Lemma 2.

######  Lemma 4.

The operator $\mathcal{A}$ defined by (29) is maximal.

Proof: Let $F=(f^{1},f^{2},f^{3},f^{4},f^{5},f^{6})\in\mathcal{H}$, we will
find $U\in D(\mathcal{A})$ unique, such that $(I-\mathcal{A})U=F$, that is the
equation becomes the system

$\displaystyle u-v$ $\displaystyle=f^{1},$ (34) $\displaystyle\rho v-\mu
u_{xx}-b\phi_{x}$ $\displaystyle=\rho f^{2},$ (35) $\displaystyle\phi-\psi$
$\displaystyle=f^{3},$ (36) $\displaystyle
J\psi-\alpha\phi_{xx}+\xi\phi+bu_{x}+\beta\theta$ $\displaystyle=Jf^{4},$ (37)
$\displaystyle
c\theta+\beta\psi-\displaystyle\int_{0}^{+\infty}\kappa(s)\eta_{xx}(s)ds$
$\displaystyle=cf^{5},$ (38) $\displaystyle\eta(s)-\theta+\eta_{s}(s)$
$\displaystyle=f^{6}.$ (39)

By solving (39) subject to the initial condition $\eta(0)=0$, we get

$\eta(s)=(1-e^{s})\theta+\int_{0}^{s}e^{y-s}f^{6}(y)dy.$ (40)

Next, substituting $v,\psi$ and $\eta$ from (34), (36) and (40) into (35),
(37) and (39), we get

$\left\\{\begin{array}[]{l}au_{xx}+b\phi_{x}-\rho u=-\rho(f^{1}+f^{2}),\\\
\alpha\phi_{xx}-bu_{x}+\beta\theta-(J+\xi)\phi=-J\left(f^{3}-f^{4}\right)\\\
c_{\mu}\theta_{xx}-c\theta-\beta\phi=-\beta
f^{3}-cf^{5}-\displaystyle\int_{0}^{+\infty}\kappa(s)\int_{0}^{s}e^{y-s}f_{xx}^{6}(y)dy,\end{array}\right.$
(41)

where

$c_{\kappa}=\int_{0}^{+\infty}\kappa(s)(1-e^{-s})ds.$

The last term in the right-hand side of the third equation of (41) belongs to
$H^{-1}(0,\pi)$. Indeed, let $\varphi\in H_{0}^{1}(0,\pi)$ such that
$\|\varphi_{x}\|\leq 1,$ then

$\begin{array}[]{ll}\Big{|}\Big{\langle}\displaystyle\int_{0}^{+\infty}\kappa(s)\Big{(}\int_{0}^{s}e^{y-s}f_{xx}^{6}(y)dy\Big{)}ds,\varphi\Big{\rangle}\Big{|}=&\Big{|}\Big{\langle}\displaystyle\int_{0}^{+\infty}\kappa(s)\Big{(}\int_{0}^{s}e^{y-s}f_{x}^{6}(y)dy\Big{)}ds,\varphi_{x}\Big{\rangle}\Big{|}\\\
&\leq\displaystyle\int_{0}^{+\infty}\kappa(s)e^{-s}\Big{(}\int_{0}^{s}e^{y}\|f_{x}^{6}(y)\|dy\Big{)}ds\\\
&=\displaystyle\int_{0}^{+\infty}e^{y}\|f_{x}^{6}(y)\|\int_{y}^{+\infty}\kappa(s)e^{-s}dsdy\\\
&\leq\displaystyle\int_{0}^{+\infty}\kappa(y)e^{y}\|f_{x}^{6}(y)\|\int_{y}^{+\infty}e^{-s}dsdy\\\
&\leq\displaystyle\int_{0}^{+\infty}\kappa(y)\|f_{x}^{6}(y)\|<+\infty.\end{array}$

In view of the above, we multiply the equations (41)${}_{1},$(41)2 and (41)3
by $\widetilde{u},\widetilde{\phi}$ and $\widetilde{\theta}$ respectively,
integrating over $\left(0,\pi\right)$ and summing up, we obtain the following
variational formulation

$B((u,\phi,\theta),(u^{\ast},\phi^{\ast},\theta^{\ast}))=L(u^{\ast},\phi^{\ast},\theta^{\ast})$
(42)

where,

$\begin{array}[]{ll}B((u,\phi,\theta),(u^{\ast},\phi^{\ast},\theta^{\ast}))=&\mu\displaystyle\int_{0}^{\pi}u_{x}u_{x}^{\ast}dx-b\int_{0}^{\pi}\phi_{x}u^{\ast}dx+\rho\int_{0}^{\pi}uu^{\ast}dx\\\
&+\displaystyle\alpha\int_{0}^{\pi}\phi_{x}\phi_{x}^{\ast}dx+b\int_{0}^{\pi}u_{x}\phi^{\ast}dx-\beta\int_{0}^{\pi}\theta\phi^{\ast}dx\\\
&\displaystyle+(J+\xi)\int_{0}^{\pi}\phi\phi^{\ast}dx+c_{\kappa}\int_{0}^{\pi}\theta_{x}\theta_{x}^{\ast}dx\\\
&\displaystyle+c\int_{0}^{\pi}\theta\theta^{\ast}dx+\beta\int_{0}^{\pi}\phi\theta^{\ast}dx,\end{array}$

is the bilinear form over $\mathcal{W}=H_{0}^{1}\left(0,\pi\right)\times
H^{1}\left(0,\pi\right)\times L^{2}\left(0,\pi\right)$, and

$\begin{array}[]{ll}L(u^{\ast},\phi^{\ast},\theta^{\ast})=&\rho\displaystyle\int_{0}^{\pi}(f^{1}+f^{2})u^{\ast}dx+J\int_{0}^{\pi}\left[f^{3}+f^{4}\right]\phi^{\ast}dx+\int_{0}^{\pi}(\beta
f^{3}+cf^{5})\theta^{\ast}dx\\\
&+\displaystyle\int_{0}^{\pi}\theta^{\ast}\int_{0}^{+\infty}\kappa(s)\Big{(}\int_{0}^{s}e^{y-s}f_{xx}^{6}(y)dy\Big{)}dsdx\end{array}$

is a linear form. Clearly, $B$ and $L$ are continuous. Moreover, a
straightforward calculation shows that

$\begin{array}[]{ll}B(U,U)&\geq\displaystyle\dfrac{1}{2}(\mu-\dfrac{b^{2}}{\xi})\int_{0}^{\pi}u_{x}^{2}dx+c\int_{0}^{\pi}\theta^{2}dx+J\int_{0}^{\pi}\phi^{2}dx+\alpha\int_{0}^{\pi}\phi_{x}^{2}dx,\\\
&\geq C\|U\|_{\mathcal{W}}^{2},\end{array}$

Thus, $B$ is coercive. Consequently, Lax-Milgram theorem guarantees the
existence of a unique $(u,\phi,\theta)\in\mathcal{W}$ satisfying (42).

Next, we take $(u^{\ast},\phi^{\ast},\theta^{\ast})=(u^{\ast},0,0)$ in (42) to
get

$\displaystyle\mu\int_{0}^{\pi}u_{x}u_{x}^{\ast}dx=\int_{0}^{\pi}(b\phi_{x}-\rho
u+\rho\left(f^{1}+f^{2}\right))u^{\ast}dx.$

Standard arguments of elliptic equations infer tha that $u\in H^{2}(0,\pi)$
and

$u_{xx}=-\frac{1}{\mu}\left(b\phi_{x}-\rho u+\rho(f^{1}+f^{2})\right).$

Therefore,

$u\in H^{2}\left(0,\pi\right)\cap H_{0}^{1}\left(0,\pi\right).$

Similarly, by taking $(u^{\ast},\phi^{\ast},\theta^{\ast})=(0,\phi^{\ast},0)$,
we get

$\alpha\int_{0}^{\pi}\phi_{x}\phi_{x}^{\ast}dx=\int_{0}^{\pi}\left(-bu_{x}+\beta\theta-(J+\xi)\phi+J\left(f^{3}+f^{4}\right)\right)\phi^{\ast}dx,~{}\forall\phi^{\star}\in
H_{\star}^{1}(0,\pi).$ (43)

Here, we are not able to apply elliptic arguments. To do so, we proceed as
follows:
Let $\psi\in H^{1}_{0}(0,\pi)$ and set
$\phi^{\star}=\psi-\int_{0}^{\pi}\psi(x)dx$. Clearly $\phi^{\star}\in
H_{\star}^{1}(0,\pi)$. Plugging, $\phi^{\star}$ in (43) taking into account
that $-bu_{x}+\beta\theta-(J+\xi)\phi+J\left(f^{3}+f^{4}\right)\in
L_{\star}^{2}(0,\pi)$, we obtain

$\alpha\int_{0}^{\pi}\phi_{x}\psi_{x}dx=\int_{0}^{\pi}\left(-bu_{x}+\beta\theta-(J+\xi)\phi+J\left(f^{3}+f^{4}\right)\right)\psi
dx,~{}\forall\psi\in H_{0}^{1}(0,\pi).$

Thus,

$\phi\in H^{2}\left(0,\pi\right)\cap H_{\ast}^{1}\left(0,\pi\right),$

with

$\phi_{xx}=-\frac{1}{\alpha}(-bu_{x}+\beta\theta-(J+\xi)\phi+J\left(f^{3}+f^{4}\right)).$

Finally, we take $(u^{\ast},\phi^{\ast},\theta^{\ast})=(0,0,\theta^{\ast})$ to
get $\theta\in H_{0}^{1}\left(0,\pi\right).$
Back to (34),(36) and (40) taking into account the last results, we get

$v\in H_{0}^{1}\left(0,\pi\right),~{}~{}\psi\in
H_{\ast}^{1}\left(0,\pi\right)~{}~{}\mbox{and}~{}~{}\eta\in
H_{\kappa}^{1}((0,+\infty)\mathchar 24635\relax\;H_{0}^{1}).$

Furthermore, (38) yields

$\int_{0}^{+\infty}\kappa\left(s\right)\eta_{xx}^{t}\left(s\right)ds\in
L^{2}(0,\pi).$

Hence, the solution $U$ belongs to $D(\mathcal{A})$ and then $1$ is in the
resolvent set of $\mathcal{A}$, which completes the proof of the Lemma.

Proof: [Proof. (of Theorem 1)] From Lammas 3, 4 and by Hille-Yoside Theorem 2,
the problem (28) has a unique solution.

######  Remark 3.

The operator maximal and monotone $\mathcal{A}$ generates a
C${}_{0}-$semigroup of contractions $S(t)=e^{-\mathcal{A}t}$, and the solution
of (28) is given by

$U(x,t)=S(t)U_{0}(x),~{}\forall t\geq 0.$

Moreover, if $U_{0}\in\mathcal{H}$, then the solution $U$ is a weak solution
and satisfies

$U\in C((0,+\infty)\mathchar 24635\relax\;\mathcal{H}).$

##  4 Exponential stability

In this section we state and prove the stability result of our problem.
First, define

$\gamma_{g}=c\mu-\rho g\left(0\right)$ (44)

and for $\gamma_{g}\neq 0,$ we introduce the stability number

$\chi_{g}=\frac{\rho}{\mu}-\frac{J}{\alpha}+\frac{\rho\beta^{2}}{\alpha\gamma_{g}}.$
(45)

The main result reads as follow:

######  Theorem 3.

Let $(u,\phi,\theta,\eta)$ be the solution of (22) subjected to the initial
and boundary conditions (20),(23) respectively. Assume that $\gamma_{g}\neq 0$
and $\chi_{g}=0,$ then, the energy $E\left(t\right)$ associated to the
solution $(u,\phi,\theta,\eta)$ is exponentially stable, that there exist two
positive constants $\sigma,\omega$ such that

$E\left(t\right)\leq\sigma e^{-\omega t},\;\forall t\geq 0.$

The proof of Theorem 3 will be done by the multipliers method and will be
established through several lemmas.

######  Lemma 5.

Let $(u,\phi,\theta,\eta)$ be a solution of (22), then the functional

$F_{1}(t)=-\frac{2c}{g(0)}\int_{0}^{+\infty}\kappa(s)\left\langle\theta(t),\eta^{t}(s)\right\rangle
ds$

satisfies, for any $\varepsilon_{1}>0$, the estimate

$\frac{d}{dt}F_{1}(t)\leq-c\left\|\theta\right\|^{2}-\left\|\eta\right\|^{2}+\varepsilon_{1}\left\|\phi_{t}\right\|^{2}-M\left(1+\frac{1}{\varepsilon_{1}}\right)\int_{0}^{+\infty}\kappa^{\prime}\left(s\right)\|\eta_{x}(s)\|^{2}ds,$
(46)

where $M$ is a positive constant independent of $\varepsilon_{1}.$

Proof: Differentiating $F_{1}(t)$, then, using (22)3, (22)4 and integration by
parts, we infer

$\displaystyle\frac{d}{dt}F_{1}(t)=$
$\displaystyle-\frac{2c}{g(0)}\int_{0}^{+\infty}\kappa(s)\left\langle\theta_{t}(t),\eta(s)\right\rangle
ds-\frac{2c}{g(0)}\int_{0}^{+\infty}\kappa(s)\left\langle\theta(t),\eta_{t}(s)\right\rangle
ds,$ $\displaystyle=$
$\displaystyle-\frac{2}{g(0)}\int_{0}^{+\infty}\kappa(s)\left\langle-\beta\phi_{xt}+\int_{0}^{+\infty}\kappa(s)\eta_{xx}(x,s)ds,\eta(s)\right\rangle
ds$
$\displaystyle-\frac{2c}{g(0)}\int_{0}^{+\infty}\kappa(s)\left\langle\theta,\theta-\eta_{s}(s)\right\rangle
ds,$ $\displaystyle=$
$\displaystyle-\frac{2\beta}{g(0)}\int_{0}^{+\infty}\kappa(s)\left\langle\phi_{t},\eta_{x}(s)\right\rangle
ds+\frac{2}{g(0)}\left\|\int_{0}^{+\infty}\kappa(s)\eta_{x}(s)ds\right\|^{2}$
$\displaystyle-2c\left\|\theta\right\|^{2}+\frac{2c}{g(0)}\int_{0}^{+\infty}\kappa(s)\left\langle\theta,\eta_{s}(s)\right\rangle
ds.$ (47)

First, integrating by parts with respect to $s$, using Cauchy Schwarz, Young’s
and Poincaré’s inequalities we get

$\displaystyle\frac{2c}{g(0)}\int_{0}^{+\infty}\kappa(s)\left\langle\theta,\eta_{s}(s)\right\rangle
ds$
$\displaystyle=-\frac{2c}{g(0)}\left\langle\theta,\int_{0}^{+\infty}\kappa^{\prime}(s)\eta(s)ds\right\rangle$
$\displaystyle\leq
c\left\|\theta\right\|^{2}-M\int_{0}^{+\infty}\kappa^{\prime}\|\eta_{x}(s)\|^{2}ds.$
(48)

Next, Young’s inequality and (31) yield

$\displaystyle-\frac{2\beta}{g(0)}\int_{0}^{+\infty}\kappa(s)\left\langle\phi_{t},\eta_{x}(s)\right\rangle
ds$
$\displaystyle=-\frac{2\beta}{g(0)}\left\langle\phi_{t},\int_{0}^{+\infty}\kappa(s)\eta_{x}(s)ds\right\rangle$
$\displaystyle\leq\varepsilon_{1}\left\|\phi_{t}\right\|^{2}-\frac{M}{\varepsilon_{1}}\int_{0}^{+\infty}\kappa^{\prime}(s)\left\|\eta_{x}(s)\right\|^{2}ds.$
(49)

Cauchy-Schwarz’ inequality and (31) give

$\displaystyle\frac{2}{g(0)}\left\|\displaystyle\int_{0}^{+\infty}\kappa(s)\eta_{x}(s)ds\right\|^{2}$
$\displaystyle\leq\displaystyle
2\left\|\eta\right\|_{\mathcal{V}}^{2}\leq-\left\|\eta\right\|_{\mathcal{V}}^{2}-\frac{3}{\delta}\int_{0}^{+\infty}\kappa^{\prime}(s)\left\|\eta_{x}(s)\right\|^{2}ds,$
(50)
$\displaystyle\leq-\left\|\eta\right\|_{\mathcal{V}}^{2}-M\displaystyle\int_{0}^{+\infty}\kappa^{\prime}(s)\left\|\eta_{x}(s)\right\|^{2}ds.$

Substituting (48)-(50) into (47), (46) follows immediately.

######  Lemma 6.

Let $(u,\phi,\theta,\eta)$ be the solution of (22), the functional

$F_{2}(t)=-\frac{cJ}{\beta}\left\langle\theta,\int_{0}^{x}\phi_{t}(y)dy\right\rangle$

satisfies for any $\varepsilon_{2}>0,$ the estimate

$\displaystyle\frac{d}{dt}F_{2}(t)$
$\displaystyle=-\frac{J}{2}\left\|\phi_{t}\right\|^{2}-M\int_{0}^{+\infty}\kappa^{\prime}(s)\left\|\eta_{x}(s)\right\|^{2}ds+M\left(1+\frac{1}{\varepsilon_{2}}\right)\left\|\theta\right\|^{2}$
$\displaystyle+\varepsilon_{2}\left\|\sqrt{a}u_{x}+\frac{b}{\sqrt{a}}\phi\right\|^{2}+\varepsilon_{2}\left\|\phi\right\|^{2}+\varepsilon_{2}\left\|\phi_{x}\right\|^{2},$
(51)

where, $M$ is a positive constant independent of $\varepsilon_{2}.$

Proof: Differentiating $F_{2}(t)$, then using (22)2, (22)3 and integration by
parts, we get

$\displaystyle\frac{d}{dt}F_{2}(t)=$
$\displaystyle-\frac{cJ}{\beta}\left\langle\theta_{t},\int_{0}^{x}\phi_{t}(y)dy\right\rangle-\frac{cJ}{\beta}\left\langle\theta,\int_{0}^{x}\phi_{tt}(y)dy\right\rangle$
$\displaystyle=$
$\displaystyle-\frac{J}{\beta}\left\langle-\beta\phi_{xt}+\int_{0}^{+\infty}\kappa(s)\eta_{xx}^{t}(s)ds,\int_{0}^{x}\phi_{t}(y)dy\right\rangle$
$\displaystyle-\frac{c}{\beta}\left\langle\theta,\int_{0}^{x}\left(\alpha\phi_{xx}-bu_{x}-\xi\phi-\beta\theta_{x}\right)dy\right\rangle$
$\displaystyle=$
$\displaystyle-J\left\|\phi_{t}\right\|^{2}+\frac{J}{\beta}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{x}^{t}(s),\phi_{t}\right\rangle
ds-\frac{c\alpha}{\beta}\left\langle\theta,\phi_{x}\right\rangle$
$\displaystyle+c\left\|\theta\right\|^{2}+\frac{c}{\beta}\left\langle\theta,\int_{0}^{x}\left(bu_{x}+\xi\phi\right)dy\right\rangle.$

Notice that

$\displaystyle\frac{c}{\beta}\left\langle\theta,\int_{0}^{x}\left(bu_{x}+\xi\phi\right)\left(y\right)dy\right\rangle$
$\displaystyle=\frac{cb}{\beta\sqrt{\mu}}\left\langle\theta,\int_{0}^{x}\left(\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right)\left(y\right)dy\right\rangle$
$\displaystyle+\frac{c}{\beta}\left(\xi-\frac{b^{2}}{\mu}\right)\left\langle\theta,\int_{0}^{x}\phi(y)dy\right\rangle,$

we infer that

$\displaystyle\frac{d}{dt}F_{2}(t)$
$\displaystyle=-J\left\|\phi_{t}\right\|^{2}+\frac{J}{\beta}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{x}^{t}(s),\phi_{t}\right\rangle
ds-\frac{c\alpha}{\beta}\left\langle\theta,\phi_{x}\right\rangle+c\left\|\theta\right\|^{2}$
$\displaystyle+\frac{c}{\beta}\left(\xi-\frac{b^{2}}{\mu}\right)\left\langle\theta,\int_{0}^{x}\phi(y)dy\right\rangle+\frac{cb}{\beta\sqrt{\mu}}\left\langle\theta,\int_{0}^{x}\left(\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right)\left(y\right)dy\right\rangle.$
(52)

Young’s and Cauchy Schwarz inequalities and (31), give

$\displaystyle\frac{J}{\beta}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{x}(s),\phi_{t}\right\rangle
ds$
$\displaystyle\leq\frac{J}{\beta}\left\|\phi_{t}\right\|\int_{0}^{+\infty}\kappa(s)\left\|\eta_{x}\right\|ds$
$\displaystyle\leq\frac{J}{2}\left\|\phi_{t}\right\|^{2}+M\left\|\eta\right\|_{\mathcal{V}}^{2}$
$\displaystyle\leq\frac{J}{2}\left\|\phi_{t}\right\|^{2}-M\int_{0}^{+\infty}\kappa^{\prime}\left(s\right)\|\eta_{x}(s)\|^{2}ds.$
(53)

Similarly, for any $\varepsilon_{2}>0$ we have

$\frac{cb}{\beta\sqrt{\mu}}\left\langle\theta,\int_{0}^{x}\left(\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right)\left(y\right)dy\right\rangle\leq\frac{M}{\varepsilon_{2}}\left\|\theta\right\|^{2}+\varepsilon_{2}\left\|\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\|^{2},$
(54)
$\frac{c}{\beta}\left(\xi-\frac{b^{2}}{a}\right)\left\langle\theta,\int_{0}^{x}\phi(y)dy\right\rangle\leq\frac{M}{\varepsilon_{2}}\left\|\theta\right\|^{2}+\varepsilon_{2}\left\|\phi\right\|^{2}$
(55)

and

$c\alpha\left\langle\theta,\phi_{x}\right\rangle\leq\frac{M}{\varepsilon_{2}}\left\|\theta\right\|^{2}+\varepsilon_{2}\left\|\phi_{x}\right\|^{2}.$
(56)

The substitution of (53)-(56) into (4), yields (51).

######  Lemma 7.

Let

$I_{1}=\frac{\alpha\rho}{b}\left\langle\phi_{x},u_{t}\right\rangle+\frac{J\mu}{b}\left\langle\phi_{t},u_{x}\right\rangle+2J\left\langle\phi,\phi_{t}\right\rangle-\dfrac{\rho\xi}{b}\left\langle
u_{t},\int_{0}^{x}\phi(y)dy\right\rangle,$
$I_{2}(t)=-\frac{1}{\sqrt{\mu}}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{x}(s),\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\rangle
ds,$
$I_{3}\left(t\right)=-\frac{\beta\rho}{b}\left\langle\theta,u_{t}\right\rangle,$

and

$F_{3}\left(t\right)=I_{1}(t)+\frac{\rho\beta\mu}{b\gamma_{g}}I_{2}(t)+\frac{\mu
c}{\gamma_{g}}I_{3}\left(t\right).$ (57)

Suppose that $\chi_{g}=0$, then for all $\varepsilon>0$, the functional
$F_{3}$ satisfies,

$\displaystyle\frac{d}{dt}F_{3}\left(t\right)\leq$
$\displaystyle-\frac{1}{2}\left\|\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\|^{2}-\left(\xi-\frac{b^{2}}{a}\right)\left\|\phi\right\|^{2}-\frac{\alpha}{2}\left\|\phi_{x}\right\|^{2}+M\left(1+\dfrac{1}{\varepsilon_{3}}\right)\left\|\phi_{t}\right\|^{2}$
$\displaystyle+\varepsilon_{3}\left\|u_{t}\right\|^{2}+M\left\|\theta\right\|^{2}-M\left(1+\varepsilon_{3}\right)\int_{0}^{+\infty}\kappa^{\prime}(s)\|\eta_{x}\|^{2}ds.$
(58)

such that
$\chi=\frac{\rho}{\mu}-\frac{J}{\alpha}+\frac{\rho\beta^{2}}{\alpha\gamma_{g}}$
, $\gamma_{g}=c\mu-g\left(0\right)\rho\neq 0$ and

Proof: Direct differentiation, using (22)1 and integration by parts, yields

$\displaystyle\dfrac{d}{dt}\frac{\alpha\rho}{b}\left\langle\phi_{x},u_{t}\right\rangle$
$\displaystyle=\frac{\alpha\rho}{b}\left\langle\phi_{tx},u_{t}\right\rangle+\frac{\alpha\rho}{b}\left\langle\phi_{x},u_{tt}\right\rangle$
$\displaystyle=\frac{\alpha\rho}{b}\left\langle\phi_{tx},u_{t}\right\rangle+\frac{\alpha}{b}\left\langle\phi_{x},\mu
u_{xx}+b\phi_{x}\right\rangle$
$\displaystyle=\frac{\alpha\rho}{b}\left\langle\phi_{tx},u_{t}\right\rangle+\frac{\mu\alpha}{b}\left\langle\phi_{x},u_{xx}\right\rangle+\alpha\left\|\phi_{x}\right\|^{2},$
(59)

Similarly, using (22)2 we have

$\displaystyle\dfrac{d}{dt}\frac{J\mu}{b}\left\langle
u_{x},\phi_{t}\right\rangle$ $\displaystyle=\frac{J\mu}{b}\left\langle
u_{tx},\phi_{t}\right\rangle+\frac{\mu}{b}\left\langle
u_{x},J\phi_{tt}\right\rangle$ $\displaystyle=\frac{\mu}{b}\left\langle
u_{x},\alpha\phi_{xx}-bu_{x}-\xi\phi-\beta\theta_{x}\right\rangle+\frac{J\mu}{b}\left\langle
u_{tx},\phi_{t}\right\rangle$ $\displaystyle=-\frac{\mu\alpha}{b}\left\langle
u_{x},\phi_{xx}\right\rangle-\mu\left\|u_{x}\right\|^{2}-\frac{\mu\xi}{b}\left\langle
u_{x},\phi\right\rangle-\frac{\mu\beta}{b}\left\langle
u_{x},\theta_{x}\right\rangle+\frac{J\mu}{b}\left\langle
u_{tx},\phi_{t}\right\rangle$ (60)

and

$\displaystyle 2\dfrac{d}{dt}J\left\langle\phi,\phi_{t}\right\rangle$
$\displaystyle=2J\left\|\phi_{t}\right\|^{2}+2J\left\langle\phi,\phi_{tt}\right\rangle$
$\displaystyle=2J\left\|\phi_{t}\right\|^{2}+2\left\langle\phi,\alpha\phi_{xx}-bu_{x}-\xi\phi-\beta\theta_{x}\right\rangle$
$\displaystyle=2J\left\|\phi_{t}\right\|^{2}-2\alpha\left\|\phi_{x}\right\|^{2}-2\xi\left\|\phi\right\|^{2}-2b\left\langle\phi,u_{x}\right\rangle-2\beta\left\langle\phi,\theta_{x}\right\rangle.$
(61)

By exploiting (22)1 we infer that

$\displaystyle-\dfrac{d}{dt}\dfrac{\rho\xi}{b}\left\langle
u_{t},\int_{0}^{x}\phi(y)dy\right\rangle$
$\displaystyle=-\dfrac{\rho\xi}{b}\left\langle
u_{tt},\int_{0}^{x}\phi(y)dy\right\rangle-\dfrac{\rho\xi}{b}\left\langle
u_{t},\int_{0}^{x}\phi_{t}(y)dy\right\rangle$
$\displaystyle=-\dfrac{\xi}{b}\left\langle\mu
u_{xx}+b\phi_{x},\int_{0}^{x}\phi(y)dy\right\rangle-\dfrac{\rho\xi}{b}\left\langle
u_{t},\int_{0}^{x}\phi_{t}(y)dy\right\rangle$
$\displaystyle=\dfrac{\mu\xi}{b}\left\langle
u_{x},\phi\right\rangle+\xi\left\|\phi\right\|^{2}-\dfrac{\rho\xi}{b}\left\langle
u_{t},\int_{0}^{x}\phi_{t}(y)dy\right\rangle.$ (62)

The addition of (59)-(62) yields

$\displaystyle\frac{d}{dt}I_{1}\left(t\right)$
$\displaystyle=-\mu\left\|u_{x}\right\|^{2}-\xi\left\|\phi\right\|^{2}-2b\left\langle\phi,u_{x}\right\rangle-\alpha\left\|\phi_{x}\right\|^{2}+2J\left\|\phi_{t}\right\|^{2}$
$\displaystyle-2\beta\left\langle\phi,\theta_{x}\right\rangle-\frac{\mu\beta}{b}\left\langle\theta_{x},u_{x}\right\rangle-\dfrac{\rho\xi}{b}\left\langle
u_{t},\int_{0}^{x}\phi_{t}(y)dy\right\rangle+\left(\frac{J\mu}{b}-\frac{\alpha\rho}{b}\right)\left\langle
u_{tx},\phi_{t}\right\rangle.$

Therefore, (26) leads to

$\displaystyle\frac{d}{dt}I_{1}\left(t\right)$
$\displaystyle=-\left\|\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\|^{2}-\left(\xi-\frac{b^{2}}{\mu}\right)\left\|\phi\right\|^{2}-\alpha\left\|\phi_{x}\right\|^{2}+2J\left\|\phi_{t}\right\|^{2}$
$\displaystyle-2\beta\left\langle\phi,\theta_{x}\right\rangle-\frac{\mu\beta}{b}\left\langle\theta_{x},u_{x}\right\rangle-\dfrac{\rho\xi}{b}\left\langle
u_{t},\int_{0}^{x}\phi_{t}(y)dy\right\rangle+\left(\frac{Ja}{b}-\frac{\alpha\rho}{b}\right)\left\langle
u_{tx},\phi_{t}\right\rangle.$ (63)

The differentiation of $I_{2}\left(t\right)$ and integration by parts yield

$\displaystyle\frac{d}{dt}I_{2}(t)=$
$\displaystyle-\frac{1}{\sqrt{\mu}}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{tx}(s),\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\rangle
ds$
$\displaystyle-\displaystyle\frac{1}{\sqrt{\mu}}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{x}(s),\sqrt{\mu}u_{tx}+\frac{b}{\sqrt{\mu}}\phi_{t}\right\rangle
ds,$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\mu}}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{t}(s),\sqrt{\mu}u_{xx}+\frac{b}{\sqrt{\mu}}\phi_{x}\right\rangle
ds$
$\displaystyle+\frac{1}{\sqrt{\mu}}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{xx}(s),\sqrt{\mu}u_{t}\right\rangle
ds-\frac{1}{\sqrt{\mu}}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{x}(s),\frac{b}{\sqrt{\mu}}\phi_{t}\right\rangle
ds$

Now, using (22)3, (22)4 and integration by parts, we get

$\displaystyle\frac{d}{dt}I_{2}\left(t\right)=$
$\displaystyle\frac{g\left(0\right)}{\sqrt{\mu}}\left\langle\theta,\sqrt{\mu}u_{xx}+\frac{b}{\sqrt{\mu}}\phi_{x}\right\rangle-\frac{1}{\sqrt{\mu}}\int_{0}^{\infty}\kappa(s)\left\langle\eta_{s},\sqrt{\mu}u_{xx}+\frac{b}{\sqrt{\mu}}\phi_{x}\right\rangle
ds$ $\displaystyle+\left\langle
c\theta_{t}+\beta\phi_{xt},u_{t}\right\rangle-\frac{b}{\mu}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{x}(s),\phi_{t}\right\rangle
ds.$ $\displaystyle=$
$\displaystyle-g(0)\left\langle\theta_{x},u_{x}\right\rangle+\frac{b}{\mu}g(0)\left\langle\theta,\phi_{x}\right\rangle-\frac{1}{\sqrt{\mu}}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{s},\sqrt{\mu}u_{xx}+\frac{b}{\sqrt{\mu}}\phi_{x}\right\rangle
ds$ $\displaystyle+c\left\langle\theta_{t},u_{t}\right\rangle
ds-\beta\left\langle\phi_{t},u_{xt}\right\rangle-\frac{b}{\mu}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{x}(s),\phi_{t}\right\rangle
ds.$

The integration by parts gives

$\displaystyle-\frac{1}{\sqrt{\mu}}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{s},\sqrt{\mu}u_{xx}+\frac{b}{\sqrt{\mu}}\phi_{x}\right\rangle
ds=\frac{1}{\sqrt{\mu}}\int_{0}^{+\infty}\kappa^{\prime}(s)\left\langle\eta,\sqrt{\mu}u_{xx}+\frac{b}{\sqrt{\mu}}\phi_{x}\right\rangle
ds$
$\displaystyle=-\frac{1}{\sqrt{\mu}}\int_{0}^{+\infty}\kappa^{\prime}(s)\left\langle\eta_{x},\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\rangle
ds.$

Thus,

$\displaystyle\frac{d}{dt}I_{2}(t)$
$\displaystyle=-g(0)\left\langle\theta_{x},u_{x}\right\rangle+\frac{b}{\mu}g(0)\left\langle\theta,\phi_{x}\right\rangle-\frac{1}{\sqrt{\mu}}\int_{0}^{+\infty}\kappa^{\prime}(s)\left\langle\eta_{x},\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\rangle
ds$
$\displaystyle+c\left\langle\theta_{t},u_{t}\right\rangle+\beta\left\langle\phi_{xt},u_{t}\right\rangle-\frac{b}{\mu}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{x}(s),\phi_{t}\right\rangle
ds.$ (64)

Finally, using (22)1 we get

$\displaystyle\frac{d}{dt}I_{3}\left(t\right)$
$\displaystyle=-\frac{\beta\rho}{b}\left\langle\theta_{t},u_{t}\right\rangle-\frac{\beta\rho}{b}\left\langle\theta,u_{tt}\right\rangle$
$\displaystyle=-\frac{\beta\rho}{b}\left\langle\theta_{t},u_{t}\right\rangle-\frac{\beta}{b}\left\langle\theta,\mu
u_{xx}+b\phi_{x}\right\rangle$
$\displaystyle=-\frac{\beta\rho}{b}\left\langle\theta_{t},u_{t}\right\rangle+\frac{\beta\mu}{b}\left\langle\theta_{x},u_{x}\right\rangle-\beta\left\langle\theta,\phi_{x}\right\rangle.$
(65)

Plugging (63)-(65) into (57) we obtain

$\displaystyle\frac{d}{dt}F_{3}\left(t\right)$
$\displaystyle=-\left\|\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\|^{2}-\left(\xi-\frac{b^{2}}{\mu}\right)\left\|\phi\right\|^{2}-\alpha\left\|\phi_{x}\right\|^{2}+2J\left\|\phi_{t}\right\|^{2}$
$\displaystyle-2\beta\left\langle\phi,\theta_{x}\right\rangle-\frac{\mu\beta}{b}\left\langle\theta_{x},u_{x}\right\rangle-\dfrac{\rho\xi}{b}\left\langle
u_{t},\int_{0}^{x}\phi_{t}(y)dy\right\rangle+\left(\frac{J\mu}{b}-\frac{\alpha\rho}{b}\right)\left\langle
u_{tx},\phi_{t}\right\rangle$ $\displaystyle-\frac{\rho\beta\mu
g(0)}{b\gamma_{g}}\left\langle\theta_{x},u_{x}\right\rangle-\frac{\rho\beta\sqrt{\mu}}{b\gamma_{g}}\int_{0}^{+\infty}\kappa^{\prime}(s)\left\langle\eta_{x},\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\rangle
ds$ $\displaystyle+\frac{\rho
c\beta\mu}{b\gamma_{g}}\left\langle\theta_{t},u_{t}\right\rangle+\frac{\rho\beta^{2}\mu}{b\gamma_{g}}\left\langle\phi_{xt},u_{t}\right\rangle-\frac{\rho\beta}{\gamma_{g}}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{x}(s),\phi_{t}\right\rangle
ds.$ $\displaystyle+\frac{\rho\beta
g(0)}{\gamma_{g}}\left\langle\theta,\phi_{x}\right\rangle-\frac{\mu
c\beta\rho}{b\gamma_{g}}\left\langle\theta_{t},u_{t}\right\rangle+\frac{\beta\mu^{2}c}{b\gamma_{g}}\left\langle\theta_{x},u_{x}\right\rangle-\frac{\mu
c\beta}{\gamma_{g}}\left\langle\theta,\phi_{x}\right\rangle,$

then,

$\displaystyle\frac{d}{dt}F_{3}\left(t\right)=$
$\displaystyle-\left\|\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\|^{2}-\left(\xi-\frac{b^{2}}{\mu}\right)\left\|\phi\right\|^{2}-\alpha\left\|\phi_{x}\right\|^{2}+2J\left\|\phi_{t}\right\|^{2}$
$\displaystyle+\frac{\mu}{b}\left(\frac{\mu
c\beta}{\gamma_{g}}-\frac{\rho\beta
g(0)}{\gamma_{g}}-\beta\right)\left\langle\theta_{x},u_{x}\right\rangle-\dfrac{\rho\xi}{b}\left\langle
u_{t},\int_{0}^{x}\phi_{t}(y)dy\right\rangle$
$\displaystyle+\left(\frac{\rho\beta g(0)}{\gamma_{g}}-\frac{\mu
c\beta}{\gamma_{g}}+2\beta\right)\left\langle\theta,\phi_{x}\right\rangle-\frac{\mu\alpha}{b}\chi_{g}\left\langle\phi_{t},u_{tx}\right\rangle$
$\displaystyle-\frac{\rho\beta\sqrt{\mu}}{b\gamma_{g}}\int_{0}^{+\infty}\kappa^{\prime}(s)\left\langle\eta_{x},\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\rangle
ds-\frac{\rho\beta}{\gamma_{g}}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{x}(s),\phi_{t}\right\rangle
ds,$

Taking into account that $\chi_{g}=0,$ we obtain

$\displaystyle\frac{d}{dt}F_{3}\left(t\right)=$
$\displaystyle-\left\|\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\|^{2}-\left(\xi-\frac{b^{2}}{\mu}\right)\left\|\phi\right\|^{2}-\alpha\left\|\phi_{x}\right\|^{2}+2J\left\|\phi_{t}\right\|^{2}$
$\displaystyle-\dfrac{\rho\xi}{b}\left\langle
u_{t},\int_{0}^{x}\phi_{t}(y)dy\right\rangle-\frac{\rho\beta\sqrt{\mu}}{b\gamma_{g}}\int_{0}^{+\infty}\kappa^{\prime}(s)\left\langle\eta_{x},\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\rangle
ds$
$\displaystyle+\beta\left\langle\theta,\phi_{x}\right\rangle-\frac{\rho\beta}{\gamma_{g}}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{x}(s),\phi_{t}\right\rangle
ds.$ (66)

Cauchy-Schwarz and Young’s inequalities yield

$-\frac{\rho\beta\sqrt{\mu}}{b\gamma_{g}}\int_{0}^{+\infty}\kappa^{\prime}(s)\left\langle\eta_{x},\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\rangle
ds$
$\leq\frac{1}{2}\left\|\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\|^{2}-M\int_{0}^{+\infty}\kappa^{{}^{\prime}}(s)\left\|\eta_{x}\right\|^{2}ds.$
(67)

Similarly, for any $\varepsilon>0$,

$\dfrac{\rho\xi}{b}\left\langle
u_{t},\int_{0}^{x}\phi_{t}(y)dy\right\rangle\leq\varepsilon\left\|u_{t}\right\|^{2}+\dfrac{M}{\varepsilon}\left\|\phi_{t}\right\|^{2},$
(68)
$\beta\left\langle\theta,\phi_{x}\right\rangle\leq\frac{\alpha}{2}\left\|\phi_{x}\right\|^{2}+M\left\|\theta\right\|^{2}.$
(69)

and by the use of (31), for any $\varepsilon_{3}>0$ we have

$\displaystyle\frac{\rho\beta}{\gamma_{g}}\int_{0}^{+\infty}\kappa(s)\left\langle\eta_{x}(s),\phi_{t}\right\rangle
ds$
$\displaystyle\leq\frac{1}{\varepsilon_{3}}\left\|\phi_{t}\right\|^{2}+\varepsilon_{3}M\left\|\eta\right\|_{\mathcal{V}}^{2}$
$\displaystyle\leq\frac{1}{\varepsilon_{3}}\left\|\phi_{t}\right\|^{2}-\varepsilon_{3}M\int_{0}^{+\infty}\kappa^{{}^{\prime}}(s)\left\|\eta_{x}\right\|^{2}ds.$
(70)

Therefore, (58) follows from (66)-(70).

######  Lemma 8.

For $(u,\phi,\theta,\eta)$ solution of (22), the functional

$F_{4}(t):=-\rho\left\langle
u_{t},u\right\rangle-\dfrac{b\rho}{\mu}\left\langle
u_{t},\int_{0}^{x}\phi(y)dy\right\rangle$

satisfies

$\frac{d}{dt}F_{4}(t)\leq-\dfrac{\rho}{2}\left\|u_{t}\right\|^{2}+\left\|\sqrt{\mu}u_{x}+\dfrac{b}{\sqrt{\mu}}\phi\right\|^{2}+M\left\|\phi_{t}\right\|^{2},$
(71)

where $M$ is a positive constant.

Proof: By differentiating $F_{4}(t)$ and using (22), we arrive at

$\displaystyle\frac{d}{dt}F_{4}(t)=$
$\displaystyle-\rho\left\|u_{t}\right\|^{2}-\left\langle\mu
u_{xx}+b\phi_{x},u\right\rangle$ $\displaystyle-\dfrac{b}{\mu}\left\langle\mu
u_{xx}+b\phi_{x},\int_{0}^{x}\phi(y)dy\right\rangle-\dfrac{b\rho}{\mu}\left\langle
u_{t},\int_{0}^{x}\phi_{t}(y)dy\right\rangle,$ $\displaystyle=$
$\displaystyle-\rho\left\|u_{t}\right\|^{2}+\mu\left\|u_{x}\right\|^{2}+2b\left\langle
u_{x},\phi\right\rangle+\dfrac{b^{2}}{\mu}\left\|\phi\right\|^{2}-\dfrac{b\rho}{\mu}\left\langle
u_{t},\int_{0}^{x}\phi_{t}(y)dy\right\rangle,$ $\displaystyle=$
$\displaystyle-\rho\left\|u_{t}\right\|^{2}+\left\|\sqrt{\mu}u_{x}+\dfrac{b}{\sqrt{\mu}}\phi\right\|^{2}-\dfrac{b\rho}{\mu}\left\langle
u_{t},\int_{0}^{x}\phi_{t}(y)dy\right\rangle.$

Young and Cauchy Schwarz inequalities, give

$\dfrac{b\rho}{a}\left\langle
u_{t},\int_{0}^{x}\phi_{t}(y)dy\right\rangle\leq\dfrac{\rho}{2}\left\|u_{t}\right\|^{2}+M\left\|\phi_{t}\right\|^{2}$

and (71) follows immediately.

At this state, we introduce the Lyapunov functional

$\mathscr{L}(t)=NE\left(t\right)+N_{1}F_{1}\left(t\right)+\frac{1}{\varepsilon_{2}}F_{2}\left(t\right)+\frac{\rho}{4\varepsilon_{3}}F_{3}\left(t\right)+F_{4}(t),$

where, $N$ and $N_{1}$ are positive constants the will be determine later.

By differentiating $\mathscr{L}(t)$ and using (46),(51),(58),(71) we get

$\displaystyle\frac{d}{dt}\mathscr{L}(t)\leq$
$\displaystyle-\left(\frac{\rho}{8\varepsilon_{3}}-2\right)\left\|\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\|^{2}$
$\displaystyle-\left[\left(\xi-\frac{b^{2}}{\mu}\right)\frac{\rho}{4\varepsilon_{3}}-1\right]\left\|\phi\right\|^{2}-\frac{\rho}{4}\left\|u_{t}\right\|^{2}-\left(\frac{\rho\alpha}{8\varepsilon_{3}}-1\right)\left\|\phi_{x}\right\|^{2}$
$\displaystyle-\left(\frac{J}{2\varepsilon_{2}}-\varepsilon_{1}N_{1}-M\left(\frac{\rho}{4\varepsilon_{3}^{2}}+\frac{\rho}{4\varepsilon_{3}}+1\right)\right)\left\|\phi_{t}\right\|^{2}$
$\displaystyle-\left(cN_{1}-M\left(1+\frac{1}{\varepsilon_{2}}\right)\frac{1}{\varepsilon_{2}}-M\frac{\rho}{4\varepsilon_{3}}\right)\left\|\theta\right\|^{2}-N_{1}\left\|\eta\right\|_{\mathcal{V}}^{2}$
$\displaystyle+\left(N-MN_{1}\left(1+\frac{1}{\varepsilon_{1}}\right)-\frac{M}{\varepsilon_{2}}-\frac{M\rho}{4\varepsilon_{3}}\left(1+\varepsilon_{3}\right)\right)\int_{0}^{+\infty}\kappa^{\prime}(s)\left\|\eta_{x}\right\|^{2}ds$

First we choose $\varepsilon_{3}$ small enough such that

$\varepsilon_{3}\leq\min\left\\{\frac{\rho}{16},\frac{\rho\alpha}{8},\left(\xi-\frac{b^{2}}{\mu}\right)\frac{\rho}{4}\right\\}.$

Next, we choose

$\varepsilon_{2}=\frac{2J\varepsilon_{3}^{2}}{M\left[\rho\varepsilon_{3}+\rho+4\varepsilon_{3}^{2}\right]},$

to get

$\frac{J}{2\varepsilon_{2}}-M\left(\frac{\rho}{4\varepsilon_{3}^{2}}+\frac{\rho}{4\varepsilon_{3}}+1\right)=M\left(\frac{\rho}{4\varepsilon_{3}^{2}}+\frac{\rho}{4\varepsilon_{3}}+1\right).$

After that, we pick $N_{1}$ large such that

$cN_{1}>M\left[\left(1+\frac{1}{\varepsilon_{2}}\right)\frac{1}{\varepsilon_{2}}+\frac{\rho}{4\varepsilon_{3}}\right].$

Next, we choose $\varepsilon_{1}$ small enough such that

$M\left(\frac{\rho}{4\varepsilon_{3}^{2}}+\frac{\rho}{4\varepsilon_{3}}+1\right)>\varepsilon_{1}N_{1}.$

Thus, there exist $\varpi,\lambda>0,$ such that

$\displaystyle\frac{d}{dt}\mathscr{L}(t)$ $\displaystyle\leq$
$\displaystyle-\varpi\left(\left\|\phi\right\|^{2}+\left\|u_{t}\right\|^{2}+\left\|\phi_{x}\right\|^{2}+\left\|\phi_{t}\right\|^{2}+\left\|\theta\right\|^{2}+\left\|\eta\right\|^{2}\right)$
(72)
$\displaystyle-\varpi\left\|\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\|^{2}+\left(N-\lambda\right)\int_{0}^{\infty}\kappa^{{}^{\prime}}(s)\left\|\eta_{x}\right\|^{2}ds.$

On the other hand, let

$\mathscr{K}\left(t\right):=\mathscr{L}\left(t\right)-NE\left(t\right)=N_{1}F_{1}\left(t\right)+\frac{1}{\varepsilon_{2}}F_{2}\left(t\right)+\frac{\rho}{4\varepsilon_{3}}F_{3}\left(t\right)+F_{4}(t),$

then,

$\displaystyle\left|\mathscr{K}\left(t\right)\right|\leq$ $\displaystyle
C\left[\left|\int_{0}^{+\infty}\kappa(s)\left\langle\theta(t),\eta^{t}(s)\right\rangle
ds\right|+\left|\left\langle\theta,\int_{0}^{x}\phi_{t}(y)dy\right\rangle\right|\right]$
$\displaystyle+$ $\displaystyle
C\left[\left|\left\langle\phi_{x},u_{t}\right\rangle\right|+\left|\left\langle\phi_{t},u_{x}\right\rangle\right|+\left|\left\langle\phi,\phi_{t}\right\rangle\right|+\left|\left\langle
u_{t},u\right\rangle\right|+\left|\left\langle
u_{t},\int_{0}^{x}\phi(y)dy\right\rangle\right|\right]$ $\displaystyle+$
$\displaystyle
C\left[\left|\left\langle\int_{0}^{\infty}\kappa(s)\eta_{x}(s)ds,\sqrt{\mu}u_{x}+\frac{b}{\sqrt{\mu}}\phi\right\rangle\right|+\left|\left\langle\theta,u_{t}\right\rangle\right|\right]$

Therefore, Cauchy-Schwarz and Young’s inequalities yield

$\left|\mathscr{L}\left(t\right)-NE\left(t\right)\right|\leq
CE\left(t\right),$

that is

$\left(N-C\right)E\left(t\right)\leq\mathscr{L}\left(t\right)\leq\left(N-C\right)E\left(t\right).$

Now, we choose $N$ large enough such that $N>\max\left\\{C,\lambda\right\\}.$
Then, $\mathscr{L}\left(t\right)\sim E\left(t\right)$ and from (72) we infer
that there exist $\gamma>0$ such that

$\frac{d}{dt}\mathscr{L}(t)\leq-\gamma E\left(t\right),\;t\geq 0.$

Using the fact that $L\left(t\right)\sim E\left(t\right),$ we get

$\frac{d}{dt}\mathscr{L}(t)\leq-\omega L\left(t\right),\;t\geq 0,$

where, $\omega>0.$ Therefore,

$\mathscr{L}\left(t\right)\leq\mathscr{L}\left(0\right)e^{-\omega t},\;t\geq
0.$

Again, the fact that $\mathscr{L}\left(t\right)\sim E\left(t\right),$ infer
that

$E\left(t\right)\leq\sigma e^{-\omega t},\;t\geq 0,$

which completes the proof of Theorem 3.

##  5 Lack of exponential stability

In this section we examine the cases when $\gamma_{g}=0$ or $\chi_{g}\neq 0$
and prove that the solution $(u,\phi,\theta,\eta)$ loses its exponential
stability, that is:

######  Theorem 4.

Suppose that $\gamma_{g}=0,$ or $\gamma_{g}\neq 0$ and $\chi_{g}\neq 0,$ then
the solution $\left(u,\phi,\theta,\eta\right)$ can not be exponentially
stable.

The proof is based on the following theorem

######  Theorem 5.

Let $\mathcal{B}$ (unbounded operator) be the infinitesimal generator of a
semigroup of contractions $S(t)=e^{\mathcal{B}t}.$ Then $S\left(t\right)$ is
exponentially stable if and only if there exists a positive constant $m$ such
that

$\underset{\lambda\in\mathbb{R}}{\inf}\|(i\lambda I-\mathcal{B})U\|\geq
m\|U\|,~{}\forall U\in D(\mathcal{B}).$ (73)

Proof: [Proof of Theorem 4]

Taking $F_{n}=(0,\frac{\sin(n\pi)}{\rho},0,0,0,0)$, then

$\|F_{n}\|_{\mathscr{H}}=\sqrt{\frac{\pi}{2\rho}}.$

Let $U_{n}\in D(\mathscr{A})$ be the solution of
$(i\lambda_{n}I+\mathscr{A})U_{n}=F_{n}$, then

$\left\\{\begin{array}[]{rl}i\lambda_{n}u_{n}-v_{n}=&0\\\
i\rho\lambda_{n}v_{n}-\mu u_{nxx}-b\phi_{nx}=&\sin(n\pi),\\\
i\lambda_{n}\phi_{n}-\psi_{n}=&0,\\\
iJ\lambda_{n}\psi_{n}-\alpha\phi_{nxx}+\xi\phi_{n}+bu_{nx}+\beta\theta_{nx}=&0,\\\
ic\lambda_{n}\theta_{n}+\beta\psi_{nx}-\displaystyle\int_{0}^{+\infty}\kappa(s)\eta_{nxx}(s)ds=&0,\\\
i\lambda_{n}\eta_{n}-\theta_{n}+\eta_{ns}=&0\end{array}\right.$ (74)

Simplifying $v_{n},\psi_{n}$ from (74), we obtain

$\left\\{\begin{array}[]{rl}\rho\lambda_{n}^{2}u_{n}+\mu
u_{nxx}+b\phi_{nx}=&-\sin(n\pi),\\\
(J\lambda_{n}^{2}-\xi)\phi_{n}+\alpha\phi_{nxx}-bu_{nx}-\beta\theta_{nx}=&0,\\\
ic\lambda_{n}\theta_{n}+i\lambda_{n}\beta\phi_{nx}-\displaystyle\int_{0}^{+\infty}\kappa(s)\eta_{nxx}(s)ds=&0,\\\
i\lambda_{n}\eta_{n}+\theta_{n}-\eta_{ns}=&0\end{array}\right.$ (75)

Taking into account the boundary conditions (20), we set

$\left\\{\begin{array}[]{l}u_{n}=A_{n}\sin(nx),\\\ \phi_{n}=B_{n}\cos(nx),\\\
\theta_{n}=C_{n}\sin(nx),\\\ \eta_{n}=D_{n}(s)\sin(nx),\end{array}\right.$
(76)

where, $A_{n},B_{n},C_{n}\in\mathbb{C}$ and
$D_{n}:\mathbb{R}_{+}\rightarrow\mathbb{C}$ such that
$\displaystyle\int_{0}^{\infty}\kappa(s)\mbox{{}\rm\sf\leavevmode\hbox{}\hbox{}D\\_{n}(s)\/}^{2}ds<0$,and$D_{n}(0)=0.$Plugging(\ref{42'})into(\ref{42}),\
weinfer\begin{equation}\left\\{\begin{array}[]{rl}\left(\rho\lambda_{n}^{2}-\mu
n^{2}\right)A_{n}-bnB_{n}=&1,\\\ (J\lambda_{n}^{2}-\xi-\alpha
n^{2})B_{n}-bnA_{n}-\beta nC_{n}=&0,\\\ ic\lambda_{n}C_{n}-i\beta
n\lambda_{n}B_{n}+n^{2}\displaystyle\int_{0}^{+\infty}\kappa(s)D_{n}(s)ds=&0,\\\
i\lambda_{n}D_{n}\left(s\right)+C_{n}-D_{n}^{\prime}\left(s\right)=&0.\end{array}\right.\end{equation}Solving(\ref{43})$_{4}$,weget\begin{equation*}D_{n}\left(s\right)=\frac{C_{n}}{i\lambda_{n}}\left(1-e^{-i\lambda_{n}s}\right).\end{equation*}Thesubstitutionof$D_{n}\left(
s\right)
$into(\ref{43})$_{3},$entails\par\begin{equation*}\begin{array}[]{rl}\left(\rho\lambda_{n}^{2}-\mu
n^{2}\right)A_{n}-bnB_{n}&=-1\\\ (J\lambda_{n}^{2}-\xi-\alpha
n^{2})B_{n}-bnA_{n}-\beta nC_{n}&=0\\\ \beta
n\lambda_{n}^{2}B_{n}-\left[c\lambda_{n}^{2}-n^{2}\left(g\left(0\right)-\widehat{\kappa}\left(\lambda_{n}\right)\right)\right]C_{n}&=0\end{array}\end{equation*}whichcanbewrittenas\begin{equation}\left(\begin{array}[]{ccc}p_{1}\left(\lambda\right)&-bn&0\\\
-bn&p_{2}\left(\lambda\right)&-\beta n\\\ 0&\beta
n\lambda_{n}^{2}&p_{3}\left(\lambda\right)\end{array}\right)\left(\begin{array}[]{c}A_{n}\\\
B_{n}\\\ C_{n}\end{array}\right)=\left(\begin{array}[]{c}-1\\\ 0\\\
0\end{array}\right)\end{equation}where,\par\begin{equation*}p_{1}\left(\lambda\right)=\left(\rho\lambda_{n}^{2}-\mu
n^{2}\right),\text{ \ \
}p_{2}\left(\lambda\right)=(J\lambda_{n}^{2}-\xi-\alpha n^{2}),\text{ \ \
}p_{3}\left(\lambda\right)=-c\lambda_{n}^{2}+n^{2}\left(g\left(0\right)-\widehat{\kappa}\left(\lambda_{n}\right)\right).\end{equation*}Itfollowsthat\begin{equation*}A_{n}=\frac{-K_{1}}{p_{1}\left(\lambda_{n}\right)K_{1}-K_{2}},\end{equation*}where\begin{equation*}K_{1}=p_{2}\left(\lambda_{n}\right)p_{3}\left(\lambda_{n}\right)+\left(\beta
n\lambda_{n}\right)^{2},\;K_{2}=\left(bn\right)^{2}p_{3}\left(\lambda_{n}\right).\end{equation*}Taking$\lambda_{n}$suchthat$p_{1}\left(
\lambda_{n}\right)
=0,$thatis$n^{2}=\frac{\rho}{\mu}\lambda_{n}^{2},$then\par\begin{equation}A_{n}=\frac{K_{1}}{K_{2}}.\end{equation}where,\par\begin{equation*}K_{1}=\frac{\alpha}{\mu}\left(\frac{\rho}{\mu}-\frac{J}{\alpha}+\frac{\xi}{\alpha\lambda_{n}^{2}}\right)\left[c\mu-\rho
g\left(0\right)+\rho\widehat{\kappa}\left(\lambda_{n}\right)\right]\lambda_{n}^{4}+\frac{\rho\beta^{2}}{\mu}\lambda_{n}^{4},\end{equation*}as$\frac{\xi}{\alpha\lambda_{n}^{2}}\longrightarrow
0,$then\begin{equation}K_{1}\simeq\frac{\alpha}{\mu}\left[\left(\frac{\rho}{\mu}-\frac{J}{\alpha}\right)\gamma_{g}+\frac{\rho\beta^{2}}{\alpha}+\left(\frac{\rho}{\mu}-\frac{J}{\alpha}\right)\rho\widehat{\kappa}\left(\lambda_{n}\right)\right]\lambda_{n}^{4}\end{equation}and\@@eqnarray
K_{2}&=&\frac{b^{2}\rho}{\mu^{2}}\lambda_{n}^{4}\left(-c\mu+\rho
g\left(0\right)-\rho\widehat{\kappa}\left(\lambda_{n}\right)\right)\\\
&=&-\frac{b^{2}\rho}{\mu^{2}}\left(\gamma_{g}+\rho\widehat{\kappa}\left(\lambda_{n}\right)\right)\lambda_{n}^{4}.\cr
Substituting(\ref{47})and(\ref{48})into(\ref{46})weget\par\begin{equation*}A_{n}\simeq-\frac{\alpha\mu\left[\chi_{g}\gamma_{g}+\left(\frac{\rho}{\mu}-\frac{J}{\alpha}\right)\rho\widehat{\kappa}\left(\lambda_{n}\right)\right]}{b^{2}\rho\left(\gamma_{g}+\rho\widehat{\kappa}\left(\lambda_{n}\right)\right)}.\end{equation*}\begin{itemize}
\par\itemize@item@If $\dfrac{\rho}{\mu}-\dfrac{J}{\alpha}=0$ and
$\gamma_{g}\neq 0,$ since
$\underset{n\longrightarrow\infty}{\lim}\widehat{\kappa}\left(\lambda_{n}\right)=0,$
we have
\begin{equation*}A_{n}\simeq\frac{\beta^{2}\mu}{-b^{2}\left(\gamma_{g}+\rho\widehat{\kappa}\left(\lambda_{n}\right)\right)}\longrightarrow\frac{\beta^{2}\mu}{-b^{2}\gamma_{g}}=A\neq
0,\end{equation*}where $A$ is a constant. \par\par\itemize@item@If
$\dfrac{\rho}{\mu}-\dfrac{J}{\alpha}\neq 0$ and $\gamma_{g}=0,$
then\begin{equation*}A_{n}\simeq-\frac{\alpha\mu\left(\frac{\rho}{\mu}-\frac{J}{\alpha}\right)}{b^{2}\rho}=A\neq
0.\end{equation*}\par\itemize@item@If $\gamma_{g}=0$ and
$\dfrac{\rho}{\mu}-\dfrac{J}{\alpha}=0,$ then
\begin{equation*}K_{1}=\frac{\rho\xi}{\mu}\widehat{\kappa}\left(\lambda_{n}\right)\lambda_{n}^{2}+\frac{\rho\beta^{2}}{\mu}\lambda_{n}^{4},\;K_{2}=-\frac{b^{2}\rho^{2}}{\mu^{2}}\widehat{\kappa}\left(\lambda_{n}\right)\lambda_{n}^{4}\end{equation*}then\begin{equation*}A_{n}\simeq\frac{\mu\beta^{2}}{-b^{2}\rho\widehat{\kappa}\left(\lambda_{n}\right)}\longrightarrow\infty.\end{equation*},
since
$\underset{n\longrightarrow\infty}{\lim}\widehat{\kappa}\left(\lambda_{n}\right)=0.$
\par\itemize@item@If $\gamma_{g}\neq 0$, and $\chi_{g}\neq 0,$ then \
\begin{equation*}A_{n}\simeq-\frac{\alpha\mu\left[\chi_{g}+\left(\frac{\rho}{\mu}-\frac{J}{\alpha}\right)\rho\frac{\widehat{\kappa}\left(\lambda_{n}\right)}{\gamma_{g}}\right]}{b^{2}\rho\left(1+\rho\frac{\widehat{\kappa}\left(\lambda_{n}\right)}{\gamma_{g}}\right)},\end{equation*}and\begin{equation*}A_{n}\longrightarrow-\frac{\alpha\mu\chi_{g}}{b^{2}\rho}=A\neq
0.\end{equation*}\par\par\end{itemize}Therefore,if$A_{n}\longrightarrow
A,$wehave\begin{aligned}
\left\|U_{n}\right\|^{2}\geq&\rho\int_{0}^{\pi}\left|v_{n}\right|^{2}dx=\rho\int_{0}^{\pi}\left|\lambda_{n}u_{n}\right|^{2}dx=\rho\int_{0}^{\pi}\left|\lambda_{n}A\sin
nx\right|^{2}dx,\\\ \geq&\rho\left|\lambda_{n}A\right|\int_{0}^{\pi}\left|\sin
nx\right|^{2}dx=\frac{\rho\pi\left|\lambda_{n}A\right|}{2}\longrightarrow\infty,\end{aligned}andif$A_{n}\longrightarrow\infty,$wehave\begin{equation*}\left\|U_{n}\right\|^{2}\geq\mu\int_{0}^{\pi}\left|u_{n}\right|^{2}dx=\mu\int_{0}^{\pi}\left|A_{n}u_{n}\right|^{2}dx\longrightarrow\infty,\end{equation*}whichcompletestheproofofTheorem\ref{TH4}.\end@proof\par\@@unnumbered@section{section}{}{Declarations}
\par\par\begin{itemize} \par\itemize@item@{\bf Funding}: This work is funded
by DGRSDT Algeria PRFU N\textdegree C00L03UN390120220004.
\par\itemize@item@{\bf Conflict of interest}: There is no conflict of
interest. \par\itemize@item@{\bf Authors' contributions}: All the authors are
contributed equally to this work \par\itemize@item@{\bf Data Availability
Statements}: Data sharing not applicable to this article as no datasets were
generated or analysed during the current study since it is a theoretical
study. \par\par\end{itemize}
\par\thebibliography\@@lbibitem{Almeida}\NAT@@wrout{1}{}{}{}{(1)}{Almeida}\lx@bibnewblock
D.S. Almeida J\~{u}nior, M.L. Santos, J.E. Mu\~{n}oz Rivera, \emph{Stability
to 1-D thermoelastic Timoshenko beam acting on shear force}, Z. Angew. Math.
Phys. {\bf 65}, 1233--1249 (2014).
\par\@@lbibitem{Ammar}\NAT@@wrout{2}{}{}{}{(2)}{Ammar}\lx@bibnewblock F. Ammar
Khodja, A. Benabdallah, J.E. Mu\~{n}oz Rivera, R. Racke, \emph{Energy decay
for Timoshenko systems of memory type}, J. Differential Equations, {\bf 194},
82--115 (2003).
\@@lbibitem{Apalara}\NAT@@wrout{3}{}{}{}{(3)}{Apalara}\lx@bibnewblock T.
Apalara, \emph{Exponential decay in one dimensional porous dissipation
elasticity}, Quart. J. Mech. Appl. Math., {70} (4), 363--372 (2017).
\par\@@lbibitem{Apalara1}\NAT@@wrout{4}{}{}{}{(4)}{Apalara1}\lx@bibnewblock
T.A. Apalara, \emph{General decay of solutions in one-dimensional porous-
elastic system with memory}, J. Math. Anal. Appl., {\bf 469}, 457--471 (2019).
\par\@@lbibitem{Brezis}\NAT@@wrout{5}{}{}{}{(5)}{Brezis}\lx@bibnewblock H.
Brezis, {\em Functional Analysis, Sobolev Spaces and Partial Differential
Equations}, Universitext, Springer, New York, (2011).
\par\@@lbibitem{Casas1}\NAT@@wrout{6}{}{}{}{(6)}{Casas1}\lx@bibnewblock P.S.
Casas, R. Quintanilla, Exponential decay in one-dimensional porous-thermo-
elasticity, Mech. Res. Commun., 32, 652-658 (2005).
\par\@@lbibitem{Casas2}\NAT@@wrout{7}{}{}{}{(7)}{Casas2}\lx@bibnewblock P.S.
Casas,, R. Quintanilla, \emph{Exponential stability in thermoelasticity with
microtemperatures}, Inter. J. Engin. Sci., {\bf 43}, 33--47 (2005).
\par\@@lbibitem{Oro}\NAT@@wrout{8}{}{}{}{(8)}{Oro}\lx@bibnewblock F. Dell'Oro,
V. Pata, \emph{On the stability of Timoshenko systems with Gurtin-Pipkin
thermal law}, J. Diff. Equa., {\bf 257}, 523--548 (2014).
\par\@@lbibitem{Fareh5}\NAT@@wrout{9}{}{}{}{(9)}{Fareh5}\lx@bibnewblock A.
Fareh, S.A. Messaoudi, \emph{Energy decay for a porous thermoelastic system
with thermoelasticity of second sound and with a non-necessary positive
definite energy}, Appl. Math. Compu., {\bf 293}, 493--507 (2017).
\@@lbibitem{FarehRV}\NAT@@wrout{10}{}{}{}{(10)}{FarehRV}\lx@bibnewblock A.
Fareh, \emph{Exponential stability of a porous thermoelastic system with
Gurtin-Pipkin thermal law}, RACSAM {\bf 116} 6 (2022).
https://doi.org/10.1007/s13398-021-01132-1.
\@@lbibitem{Fatouri}\NAT@@wrout{11}{}{}{}{(11)}{Fatouri}\lx@bibnewblock L.H.
Fatori, J.E. Mu\~{n}oz Rivera, \emph{Energy decay for hyperbolic thermoelastic
systems of memory type}, Quart. Appl. Math. {\bf 59}, 441--458 (2001).
\par\@@lbibitem{Feng1}\NAT@@wrout{12}{}{}{}{(12)}{Feng1}\lx@bibnewblock B.
Feng, \emph{On the decay rates for a one-dimensional porous elasticity system
with past history}, Comm. Pur. Appl. Anal., {\bf 18} (6), 2905--2921 (2019).
\par\par\@@lbibitem{Gurtin}\NAT@@wrout{13}{}{}{}{(13)}{Gurtin}\lx@bibnewblock
M.E. Gurtin, A.C. Pipkin, \emph{A general theory of heat conduction with
finite wave speeds}, Arch. Ration. Mech. Anal., {\bf 31}, 113--126 (1968).
\par\@@lbibitem{Iesan86}\NAT@@wrout{14}{}{}{}{(14)}{Iesan86}\lx@bibnewblock D.
Ie\c{s}an, \emph{ A theory of thermoelastic materials with voids}, Acta
Meehanica, {\bf 60}, 67--89 (1986). doi.org/10.1007/BF01302942.
\@@lbibitem{Iesan}\NAT@@wrout{15}{}{}{}{(15)}{Iesan}\lx@bibnewblock D.
Ie\c{s}an, Thermoelastic Models of Continua. Springer, Berlin 2004.
\par\@@lbibitem{Kim}\NAT@@wrout{16}{}{}{}{(16)}{Kim}\lx@bibnewblock J.U. Kim,
Y. Renardy, \emph{Boundary control of the Timoshenko beam}, SIAM, J. Control
Optim. {\bf 25} (6), 1417--1429 (1987).
\par\@@lbibitem{Lord}\NAT@@wrout{17}{}{}{}{(17)}{Lord}\lx@bibnewblock H.W.
Lord, Y. Shulman, \emph{ A generalized dynamical theory of thermoelasticity},
J. Mech. Phys. Sol., {\bf 15}, 299--309 (1967).
\par\@@lbibitem{Magana}\NAT@@wrout{18}{}{}{}{(18)}{Magana}\lx@bibnewblock A.
Maga\~{n}a, R. Quintanilla, \emph{ On the time decay of solutions in one-
dimensional theories of porous materials}, Int. J. Sol. Struc. {\bf 43}, 3414
--3427 (2006).
\par\par\par\@@lbibitem{Mustafa}\NAT@@wrout{19}{}{}{}{(19)}{Mustafa}\lx@bibnewblock
S.A. Messaoudi, M.I. Mustafa, \emph{A stability result in a memory-type
Timoshenko system}, Dyn. Syst. Appl., {\bf 18}, 457--468 (2009).
\@@lbibitem{MunozQuintanilla}\NAT@@wrout{20}{}{}{}{(20)}{MunozQuintanilla}\lx@bibnewblock
J.E. Mu\~{n}oz Rivera, R. Quintanilla, \emph{On the time polynomial decay in
elastic solids with voids}, J. Math. Anal. Appl., {\bf 338}, 1296--1309
(2008). \@@lbibitem{Racke}\NAT@@wrout{21}{}{}{}{(21)}{Racke}\lx@bibnewblock
J.E. Mu\~{n}oz Rivera, R. Racke, \emph{Mildly dissipative nonlinear Timoshenko
systems-global existence and exponential stability}, J. Math. Anal. Appl. {\bf
276}, 248--278 (2002).
\par\@@lbibitem{Pata}\NAT@@wrout{22}{}{}{}{(22)}{Pata}\lx@bibnewblock V. Pata,
E. Vuk, \emph{ On the exponential stability of linear thermoelasticity},
Contin. Mech. Thermodyn. {\bf 12}, 121--130 (2000).
\par\@@lbibitem{Quin1}\NAT@@wrout{23}{}{}{}{(23)}{Quin1}\lx@bibnewblock R.
Quintanilla, \emph{ Slow decay for one-dimensional porous dissipation
elasticity}, Appl. Math. letters, {16}, 487-491 (2003).
\par\par\@@lbibitem{Raposo}\NAT@@wrout{24}{}{}{}{(24)}{Raposo}\lx@bibnewblock
C.A. Raposo, J. Ferreira, M.L. Santos, N.N.O. Castro, \emph{Exponential
stability for the Timoshenko system with two weak dampings}, Appl. Math.
Lett., {\bf 18}, 535--541 (2005).
\par\@@lbibitem{Santos}\NAT@@wrout{25}{}{}{}{(25)}{Santos}\lx@bibnewblock M.L.
Santos, D.S. Almeida J\~{u}nior, J.E. Mu\~{n}oz Rivera, \emph{The stability
number of the Timoshenko system with second sound}, J. Diff. Equa., {\bf 253},
2715--2733 (2012).
\@@lbibitem{Santos1}\NAT@@wrout{26}{}{}{}{(26)}{Santos1}\lx@bibnewblock M.L.
Santos, D.A. J\~{u}nior, \emph{On porous-elastic system with localized
damping}, Z. Angew. Math. Phys., {\bf 67}, 1--18 (2016).
\par\@@lbibitem{Shi}\NAT@@wrout{27}{}{}{}{(27)}{Shi}\lx@bibnewblock D.H. Shi,
D.X. Feng, \emph{Exponential decay of Timoshenko Beam with locally distributed
feedback}, in: Proceeding of the IFAC World Congress, Beijing, vol. F, (1999).
\par\@@lbibitem{Soufyane}\NAT@@wrout{28}{}{}{}{(28)}{Soufyane}\lx@bibnewblock
A. Soufyane, \emph{Stabilisation de la poutre de Timoshenko}, Compt. Rend.
Acad. Sci. Paris, Ser. I Math. 328 (1999), 731--734.
\par\@@lbibitem{Tatar}\NAT@@wrout{29}{}{}{}{(29)}{Tatar}\lx@bibnewblock
Tatar,N. e-, \emph{Stabilization of a viscoelastic Timoshenko beam}, Appl.
Ana., {\bf 92}, 27--43 (2013). \endthebibliography
\par\par\@add@PDF@RDFa@triples\par\end{document}$