# Theoretical and numerical study of the decay in a viscoelastic Bresse System

Jamilu Hashim Hassan<EMAIL_ADDRESS>Salim A. Messaoudi
<EMAIL_ADDRESS>Toufic El-Arwadi<EMAIL_ADDRESS>Mohamad El
Hindi<EMAIL_ADDRESS>
###### Abstract

In this paper, we consider a one-dimensional finite-memory Bresse system with
homogeneous Dirichlet-Neumann-Neumann boundary conditions. We prove some
general decay results for the energy associated with the system in the case of
equal and non-equal speeds of wave propagation under appropriate conditions on
the relaxation function. In addition, we show by giving an example that in the
case of equal speeds of wave propagation and for certain polynomially decaying
relaxation functions, our result gives an optimal decay rate in the sense that
the decay rate of the system is exactly the same as that of the relaxation
function considered.

##  1 Introduction

Bresse system is a mathematical model that describes the vibration of a
planar, linear shearable curved beam. The model was first derived by Bresse
[6] and it consists of three coupled wave equations given by

$\begin{array}[]{ll}\rho_{1}\varphi_{tt}-k_{1}(\varphi_{x}+\psi+lw)_{x}-lk_{3}(w_{x}-\varphi)+F_{1}=0&\mathrm{in\,\,\,}(0,L)\times(0,\infty),\\\
\\\
\rho_{2}\psi_{tt}-k_{2}\psi_{xx}+k_{1}(\varphi_{x}+\psi+lw)+F_{2}=0&\mathrm{in\,\,\,}(0,L)\times(0,\infty),\\\
\\\
\rho_{1}w_{tt}-k_{3}(w_{x}-\varphi)_{x}+lk_{1}(\varphi_{x}+\psi+lw)+F_{3}=0&\mathrm{in\,\,\,}(0,L)\times(0,\infty),\end{array}$
(1.1)

where $\varphi,\psi,w$ represent the vertical displacement, the shear angle,
and the longitudinal displacement, respectively;
$\rho_{1},\rho_{2},k_{1},k_{2},k_{3},l$ are positive parameters and
$F_{1},F_{2},F_{3}$ are external forces.

A lot of results dealing with well-posedness and asymptotic behaviour of the
above system have been published. We start with the work of Santos $et\ al.$
[28] from 2010, where they studied the Bresse system with Dirichlet-Dirichlet-
Dirichlet boundary conditions and linear frictional damping acting on each
equation, that is,

$(F_{1},F_{2},F_{3})=(\gamma_{1}\varphi_{t},\,\gamma_{2}\psi_{t},\,\gamma_{3}w_{t}),$
(1.2)

where $\gamma_{1},\gamma_{2},\gamma_{3}>0$. They established an exponential
decay rate for the system using spectral theory approach developed by Z. Liu
and S. Zheng in [18]. They also gave a numerical scheme using finite
difference method to illustrate their theoretical result. Soriano $et\ al.$
[29] used the method developed by Lasiecka and Tataru in [16] and proved a
uniform decay rate for the same system with a nonlinear frictional damping
acting on the second equation and locally distributed nonlinear damping acting
on the other equations. Precisely, the external forces are given by

$(F_{1},F_{2},F_{3})=(\alpha(x)g_{1}(\varphi_{t}),\,g_{2}(\psi_{t}),\,\gamma(x)g_{3}(w_{t}))$

with $\alpha,\gamma\in L^{\infty}(0,L)$ and the $g_{i}$’s are continuous and
monotone increasing functions. The results of [28] and [29] were established
without imposing any restriction on the speeds of wave propagation given by

$s_{1}=\sqrt{\frac{k_{1}}{\rho_{1}}},\qquad
s_{2}=\sqrt{\frac{k_{2}}{\rho_{2}}},\qquad\mathrm{and}\qquad
s_{3}=\sqrt{\frac{k_{3}}{\rho_{1}}}.$ (1.3)

Alves $et\ al.$ [4] used the semigroup and spectral theory to obtain the
exponential stability of the Bresse system with three controls at the
boundary.

In the presence of dissipating terms in only one or two of the equations in
system (1.1), the decay rates of the energy associated to the system depend
totally on the speeds of the wave propagation. As illustrated in [2], Alabau-
Boussouira $et\ al.$ studied (1.1) with linear frictional damping acting on
the second equation; that is, they used (1.2), with $\gamma_{1}=\gamma_{3}=0$
and $\gamma_{2}>0$ and showed that the system is exponentially stable if and
only if it has equal speeds of wave propagation,

$\frac{k_{1}}{\rho_{1}}=\frac{k_{2}}{\rho_{2}}=\frac{k_{3}}{\rho_{3}}.$ (1.4)

As mentioned by many authors [2, 3], relation (1.4) is physically unrealistic.
In the case of non-equal speeds of wave propagation, they proved polynomial
stability with rates which can be improved with the regularity of the initial
data. Fatori and Monteiro [8] improved this result in the case of non-equal
speeds of wave propagation by proving optimal decay rate. Soriano $et\ al.$
[30] established the same exponential stability result as in [2] by replacing
the frictional damping with indefinite one; that is, they replaced
$\gamma_{2}$ in [2] with a function $a:(0,L)\longrightarrow\mathbb{R}$ such
that $\displaystyle\bar{a}=\frac{1}{L}\int_{0}^{L}a(x)dx>0$ and
$\displaystyle\|a-\bar{a}\|_{L^{2}(0,L)}$ is small enough. Wehbe and Youcef
[31] inspected the situation of two locally distributed dampings acting on the
last two equations; that is,

$(F_{1},F_{2},F_{3})=(0,\,a_{1}(x)\psi_{t},\,a_{2}(x)w_{t}),$

where $a_{i}:(0,L)\longrightarrow\mathbb{R}$ are non-negative functions which
can take value zero on some part of the interval $(0,L)$. By using the
frequency domain and the multiplier method, they proved that the system is
exponentially stable if and only if $s_{1}=s_{2}$. When $s_{1}\neq s_{2}$ they
established a polynomial decay rate which can be improved with the regularity
of the initial data. The same result was established by Alves $et\ al.$ in
[3], in the case of non-equal speeds of wave propagation, they used the recent
result of Borichev and Tomilov in [5] to show that the solution is
polynomially stable with optimal decay rate.

Concerning the dissipation via heat effect, we mention the work of Liu and Rao
[17] where the following system

$\begin{array}[]{ll}\rho_{1}\varphi_{tt}-k_{1}(\varphi_{x}+\psi+lw)_{x}-lk_{3}(w_{x}-\varphi)+l\gamma\chi=0&\mathrm{in\,\,\,}(0,L)\times(0,\infty),\\\
\\\
\rho_{2}\psi_{tt}-k_{2}\psi_{xx}+k_{1}(\varphi_{x}+\psi+lw)+\gamma\theta_{x}=0&\mathrm{in\,\,\,}(0,L)\times(0,\infty),\\\
\\\
\rho_{1}w_{tt}-k_{3}(w_{x}-\varphi)_{x}+lk_{1}(\varphi_{x}+\psi+lw)+\gamma\chi_{t}=0&\mathrm{in\,\,\,}(0,L)\times(0,\infty),\\\
\\\
\rho_{3}\theta_{t}-\theta_{xx}+\gamma\psi_{xt}=0&\mathrm{in\,\,\,}(0,L)\times(0,\infty),\\\
\\\
\rho_{3}\chi_{t}-\chi_{xx}+\gamma(w_{x}-l\varphi)_{t}=0&\mathrm{in\,\,\,}(0,L)\times(0,\infty),\end{array}$
(1.5)

with boundary and initial conditions was considered. They showed that the
exponential stability of the system is equivalent to the validity of the
identity (1.4). In the case where (1.4) does not hold, they established a
polynomial-type decay rate. Fatori and Muñoz Rivera [9] obtained a similar
result as in [17] for the thermoelastic Bresse system (1.5) when the fifth
equation is omitted. They also showed that the polynomial decay rate is
optimal in the case of non-equal speeds of wave propagation. Filippo Dell’Oro
[7] gave a detail stability analysis of the thermoelastic Bresse-Gurtin-Pipkin
system of the form:

$\begin{array}[]{ll}\rho_{1}\varphi_{tt}-k(\varphi_{x}+\psi+lw)_{x}-lk_{0}(w_{x}-\varphi)=0&\mathrm{in\,\,\,}(0,L)\times(0,\infty),\\\
\\\
\rho_{2}\psi_{tt}-k_{2}\psi_{xx}+k(\varphi_{x}+\psi+lw)+\gamma\theta_{x}=0&\mathrm{in\,\,\,}(0,L)\times(0,\infty),\\\
\\\
\rho_{1}w_{tt}-k_{0}(w_{x}-\varphi)_{x}+lk(\varphi_{x}+\psi+lw)=0&\mathrm{in\,\,\,}(0,L)\times(0,\infty),\\\
\\\
\rho_{3}\theta_{t}-k_{1}\displaystyle\int_{0}^{\infty}g(s)\theta_{xx}(t-s)ds+\gamma\psi_{xt}=0&\mathrm{in\,\,\,}(0,L)\times(0,\infty),\end{array}$
(1.6)

where $g$ is a bounded convex integrable function on $[0,\infty)$ satisfying

$\int_{0}^{\infty}g(s)ds=1,$

and there exists a non-increasing absolutely continuous function
$\mu:(0,\infty)\longrightarrow[0,\infty)$ such that

$\mu(0)=\lim_{s\rightarrow 0}\mu(s)\in(0,\infty),\qquad
g(s)=\int_{s}^{\infty}\mu(\tau)d\tau,\qquad\forall\,s\in[0,\infty)$

and

$\mu^{\prime}(s)+\nu\mu(s)\leq 0\quad\mathrm{for\ \ some\ \
}\nu>0\quad\mathrm{and\quad}a.e.\ s\in(0,\infty).$

By introducing a new stability number of the form

$\chi_{g}=\left(\frac{\rho_{1}}{\rho_{3}k}-\frac{1}{g(0)k_{1}}\right)\left(\frac{\rho_{1}}{k}-\frac{\rho_{2}}{b}\right)-\frac{1}{g(0)k}\frac{\rho_{1}\gamma^{2}}{\rho_{3}bk},$

he proved that the semigroup generated by (1.6) is exponentially stable if and
only if

$\chi_{g}=0\qquad\mathrm{and\qquad}k=k_{0}.$

As a special case, he showed that his stability result gave the stability
characterization of Bresse systems with Fourier, Maxwell-Cataneo and Coleman-
Gurtin thermal dissipation. The reader is referred to [1, 10, 15, 22, 23, 24,
25, 26] and the references therein for more recent results on thermoelastic
Bresse system.

There are few results that dealt with stabilization of Bresse system via
infinite memory. We begin with the work of Guesmia and Kafini [12] in 2015.
They studied the following system

$\begin{cases}\rho_{1}\varphi_{tt}-k_{1}(\varphi_{x}+\psi+lw)_{x}-lk_{3}(w_{x}-l\varphi)+\displaystyle\int_{0}^{\infty}g_{1}(s)\varphi_{xx}(x,t-s)ds=0,\\\
\rho_{2}\psi_{tt}-k_{2}\psi_{xx}+k_{1}(\varphi_{x}+\psi+lw)+\displaystyle\int_{0}^{\infty}g_{2}(s)\psi_{xx}(x,t-s)ds=0,\\\
\rho_{1}w_{tt}-lk_{3}(w_{x}-l\varphi)_{x}+lk_{1}(\varphi_{x}+\psi+lw)+\displaystyle\int_{0}^{\infty}g_{3}(s)w_{xx}(x,t-s)ds=0,\\\
\varphi(0,t)=\psi(0,t)=w(0,t)=\varphi(L,t)=\psi(L,t)=w(L,t)=0,\\\
\varphi(x,-t)=\varphi_{0}(x,t),\ \varphi_{t}(x,0)=\varphi_{1}(x),\\\
\psi(x,-t)=\psi_{0}(x,t),\ \psi_{t}(x,0)=\psi_{1}(x),\\\ w(x,-t)=w_{0}(x,t),\
w_{t}(x,0)=w_{1}(x),\end{cases}$ (1.7)

where $(x,t)\in(0,L)\times\mathbb{R}_{+}$,
$g_{i}:\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}$ are differentiable non-
increasing and integrable functions, and $L,\,l_{i},\,\rho_{i},\,k_{i}$ are
positive constants. They proved the well-posedness and the asymptotic
stability of (1.7). Later, Guesmia and Kirane [13] used two infinite memories
to obtain the same stability result of [12] under the following conditions on
the speeds of wave propagation:

$\frac{k_{1}}{\rho_{1}}=\frac{k_{2}}{\rho_{2}}\quad\mathrm{in\ case\
}g_{1}=0,\quad\frac{k_{1}}{\rho_{1}}=\frac{k_{2}}{\rho_{2}}\quad\mathrm{in\
case\
}g_{2}=0,\quad\frac{k_{1}}{\rho_{1}}=\frac{k_{3}}{\rho_{3}}\quad\mathrm{in\
case\ }g_{3}=0.$

Santos $et\ al.$ [27] discussed the Bresse system with only one infinite
memory acting on the shear angle displacement equation. Precisely, they
studied problem (1.7) with

$g_{1}=g_{3}=0\qquad\mathrm{and\qquad}g_{2}\mathrm{\quad
satisfying:\quad}-\alpha_{1}g_{2}(t)\leq
g_{2}^{\prime}(t)\leq-\alpha_{2}g_{2}(t),\quad\forall\,t\geq 0,$

for some $\alpha_{1},\alpha_{2}>0.$ They showed that the solution of the
system decays exponentially to zero if and only if (1.4) holds, otherwise a
polynomial stability of the system with an optimal decay rate of type
$t^{-1/2}$ was obtained. Recently, Guesmia [11] analysed the asymptotic
stability of Bresse system with one infinite memory in the longitudinal
displacement.

To the best of our knowledge, there is no result in the literature that deals
with the stability of Bresse system via viscoelastic damping of finite memory-
type. In this paper we will discuss the decay property of the following finite
memory-type Bresse system:

$\begin{cases}\rho_{1}\varphi_{tt}-k_{1}(\varphi_{x}+\psi+lw)_{x}-lk_{3}(w_{x}-l\varphi)=0,&\mathrm{in}\,\,\,(0,L)\times(0,+\infty),\\\
\rho_{2}\psi_{tt}-k_{2}\psi_{xx}+k_{1}(\varphi_{x}+\psi+lw)+\displaystyle\int_{0}^{t}g(t-s)\psi_{xx}(s)ds=0,&\mathrm{in\,\,\,}(0,L)\times(0,+\infty),\\\
\rho_{1}w_{tt}-k_{3}(w_{x}-l\varphi)_{x}+lk_{1}(\varphi_{x}+\psi+lw)=0,&\mathrm{in\,\,\,}(0,L)\times(0,+\infty),\\\
\varphi(0,t)=\varphi(L,t)=\psi_{x}(0,t)=\psi_{x}(L,t)=w_{x}(0,t)=w_{x}(L,t)=0,&\mathrm{for\,\,\,}t\geq
0,\\\
\varphi(x,0)=\varphi_{0}(x),\,\,\,\,\varphi_{t}(x,0)=\varphi_{1}(x),&\mathrm{for}\,\,\,x\in(0,L),\\\
\psi(x,0)=\psi_{0}(x),\,\,\,\,\psi_{t}(x,0)=\psi_{1}(x),&\mathrm{for}\,\,\,x\in(0,L),\\\
w(x,0)=w_{0}(x),\,\,\,\,w_{t}(x,0)=w_{1}(x),&\mathrm{for}\,\,\,x\in(0,L),\end{cases}$
($P$)

where $l,\,\,k_{1},\,\,k_{2},\,\,k_{3},\,\,\rho_{1},\,\,\rho_{2}$ are positive
constants,
$\varphi_{0},\,\,\varphi_{1},\,\,\psi_{0},\,\,\psi_{1},\,\,w_{0},\,\,w_{1}$
are given data and $g$ is a relaxation function satisfying some conditions to
be specified in the next section. Our problem is motivated by the following
classical Bresse system

$\begin{array}[]{lll}\rho_{1}\varphi_{tt}-S_{x}-lN&=0&\mathrm{in}\quad(0,L)\times(0,\infty),\\\
\rho_{2}\psi_{tt}-M_{x}+S&=0&\mathrm{in}\quad(0,L)\times(0,\infty),\\\
\rho_{1}w_{tt}-N_{x}-lS&=0&\mathrm{in}\quad(0,L)\times(0,\infty),\end{array}$

where $t$ and $x$ represent the time and space variables, respectively, and
$N$, $S$ and $M$ denote the axial force, the shear force and the bending
moment given by

$S=k_{1}(\varphi_{x}+\psi+w),\quad
M=k_{2}\psi_{x}-\int_{0}^{t}g(t-s)\psi_{x}(\cdot,s)ds,\quad
N=k_{3}(w_{x}-\varphi).$

We will prove, under a smallness condition on $l$, generalized energy decay
results for the system in the case of equal and different speeds of wave
propagation. This paper is organized as follows: in Section 2, we state some
preliminary results. In Section 3, we state and prove some technical lemmas.
The statement and proof of our main results are given in Sections 4 and 5,
while in Section 6 we present some numerical illustrations to validate our
results. Through out this work we use $c$ to represent a generic positive
constant, independent of $t$ but may depend on the initial data.

##  2 Preliminaries

In this section, we introduce our assumptions, present some useful lemmas and
state the existence theorem.
Assumptions: We assume that the relaxation function $g$ satisfies the
following hypotheses:

  * (A1)

$g:[0,\infty)\longrightarrow[0,\infty)$ is a non-increasing differentiable
function such that

$g(0)>0\qquad\mathrm{and}\qquad k_{2}-\int_{0}^{+\infty}g(s)ds>0.$

  * (A2)

There exists a non-increasing differentiable function
$\xi\\!:[0,\infty)\longrightarrow(0,\infty)$ and a constant $p$, with $1\leq
p<\frac{3}{2}$, such that

$g^{\prime}(t)\leq-\xi(t)g^{p}(t),\qquad\forall t\geq 0.$

######  Lemma 2.1.

Assume that $g$ satisfies hypotheses $(A1)$ and $(A2)$. Then,

$\int_{0}^{+\infty}\xi(t)g^{1-\sigma}(t)dt<+\infty,\qquad\forall\,0<\sigma<2-p.$

###### Proof.

From (A1), we have

$\lim_{t\rightarrow+\infty}g(t)=0.$

Using (A2), we have

$\displaystyle\int_{0}^{+\infty}\xi(t)g^{1-\sigma}(t)dt$ $\displaystyle=$
$\displaystyle\int_{0}^{+\infty}\xi(t)g^{p}(t)g^{1-\sigma-p}(t)dt\leq-\int_{0}^{+\infty}g^{\prime}(t)g^{1-\sigma-p}(t)dt$
$\displaystyle=$
$\displaystyle-\left[\frac{1}{2-\sigma-p}g^{2-\sigma-p}(t)\right]_{t=0}^{t=+\infty}<+\infty,$

since $\sigma<2-p$. ∎

Now, integrating both sides of the second and third equations in $\eqref{p1}$
over $(0,L)$ and using the boundary conditions, we get

$\frac{d^{2}}{dt^{2}}\int_{0}^{L}\psi(x,t)dx+\frac{k_{1}}{\rho_{2}}\int_{0}^{L}\psi(x,t)dx+\frac{lk_{1}}{\rho_{2}}\int_{0}^{L}w(x,t)dx=0\quad\forall
t\geq 0$ (2.1)

and

$\frac{d^{2}}{dt^{2}}\int_{0}^{L}w(x,t)dx+\frac{l^{2}k_{1}}{\rho_{1}}\int_{0}^{L}w(x,t)dx+\frac{lk_{1}}{\rho_{1}}\int_{0}^{L}\psi(x,t)dx=0\quad\forall
t\geq 0.$ (2.2)

Solving these ODEs simultaneously yields

$\int_{0}^{L}\psi(x,t)dx=a_{1}\cos(a_{0}t)+a_{2}\sin(a_{0}t)+a_{3}t+a_{4}$
(2.3)

and

$\int_{0}^{L}w(x,t)dx=\frac{a_{1}}{l}\left(\frac{\rho_{2}a_{0}^{2}}{k_{1}}-1\right)\cos(a_{0}t)+\frac{a_{2}}{l}\left(\frac{\rho_{2}a_{0}^{2}}{k_{1}}-1\right)\sin(a_{0}t)-\frac{a_{3}}{l}t-\frac{a_{4}}{l},$
(2.4)

where

$\begin{cases}a_{0}=\displaystyle{\sqrt{\frac{k_{1}}{\rho_{2}}+\frac{l^{2}k_{1}}{\rho_{1}}}}\\\
\\\
a_{1}=\displaystyle{\frac{k_{1}}{\rho_{2}a_{0}^{2}}\int_{0}^{L}\psi_{0}(x)dx+\frac{lk_{1}}{\rho_{2}a_{0}^{2}}\int_{0}^{L}w_{0}(x)dx},\\\
\\\
a_{2}=\displaystyle{\frac{k_{1}}{\rho_{2}a_{0}^{3}}\int_{0}^{L}\psi_{1}(x)dx+\frac{lk_{1}}{\rho_{2}a_{0}^{3}}\int_{0}^{L}w_{1}(x)dx},\\\
\\\
a_{3}=\displaystyle{\left(1-\frac{k_{1}}{\rho_{2}a_{0}^{2}}\right)\int_{0}^{L}\psi_{1}(x)dx-\frac{lk_{1}}{\rho_{2}a_{0}^{2}}\int_{0}^{L}w_{1}(x)dx},\\\
\\\
a_{4}=\displaystyle{\left(1-\frac{k_{1}}{\rho_{2}a_{0}^{2}}\right)\int_{0}^{L}\psi_{0}(x)dx+\frac{lk_{1}}{\rho_{2}a_{0}^{2}}\int_{0}^{L}w_{0}(x)dx}.\end{cases}$

Therefore, we perform the following change of variables

$\begin{array}[]{l}\displaystyle{\widetilde{\psi}=\psi-\frac{1}{L}\big{(}a_{1}\cos(a_{0}t)+a_{2}\sin(a_{0}t)+a_{3}t+a_{4}\big{)}}\\\
\\\
\displaystyle{\widetilde{w}=w-\frac{1}{L}\left[\frac{a_{1}}{l}\left(\frac{\rho_{2}a_{0}^{2}}{k_{1}}-1\right)\cos(a_{0}t)+\frac{a_{2}}{l}\left(\frac{\rho_{2}a_{0}^{2}}{k_{1}}-1\right)\sin(a_{0}t)-\frac{a_{3}}{l}t-\frac{a_{4}}{l}\right]}\end{array}$

to get

$\int_{0}^{L}\widetilde{\psi}(x,t)dx=\int_{0}^{L}\widetilde{w}(x,t)dx=0,\qquad\forall\,t\geq
0.$

Furthermore, $(\varphi,\,\widetilde{\psi},\,\widetilde{w})$ satisfies the
equations and the boundary conditions in ($P$) with the initial data

$\begin{array}[]{ll}\displaystyle{\widetilde{\psi}_{0}=\psi_{0}-\frac{1}{L}(a_{1}+a_{4})},&{\displaystyle\widetilde{\psi}_{1}=\psi_{1}-\frac{1}{L}(a_{0}a_{2}+a_{3})}\\\
\\\
\displaystyle{\widetilde{w}_{0}=w_{0}-\frac{1}{L}\left[\frac{a_{1}}{l}\left(\frac{\rho_{2}a_{0}^{2}}{k_{1}}-1\right)-\frac{a_{4}}{l}\right]},&\displaystyle{\widetilde{w}_{1}=w_{1}-\frac{1}{L}\left[\frac{a_{2}a_{0}}{l}\left(\frac{\rho_{2}a_{0}^{2}}{k_{1}}-1\right)-\frac{a_{3}}{l}\right].}\end{array}$

From now on, we work with $\widetilde{\psi},\,\,\widetilde{w}$ and,
respectively, write $\psi,\,\,w$ for convenience. We also introduce the
following spaces,

$L^{2}_{*}(0,L):=\left\\{w\in L^{2}(0,L):\int_{0}^{L}w(x)dx=0\right\\},\qquad
H^{1}_{*}(0,L):=H^{1}(0,L)\cap L^{2}_{*}(0,L),$

and

$H^{2}_{*}(0,L):=\left\\{w\in H^{2}(0,L):w_{x}(0)=w_{x}(L)=0\right\\}.$

Then, Poincaré’s inequality is applicable to the elements of $H^{1}_{*}(0,L)$,
that is,

$\exists\,\,c_{0}>0\qquad\mathrm{such\,\,\,\,that}\qquad\int_{0}^{L}v^{2}dx\leq
c_{0}\int_{0}^{L}v_{x}^{2}dx\qquad\forall\,v\in H^{1}_{*}(0,L).$ (2.5)

For completeness, we state, without proof the global existence and regularity
result which can be established by repeating the steps of the proof of the
existence result in [21].

######  Theorem 2.1.

Let $(\varphi_{0},\varphi_{1})\in H^{1}_{0}(0,L)\times L^{2}(0,L)$ and
$(\psi_{0},\psi_{1}),\,(w_{0},w_{1})\in H^{1}_{*}(0,L)\times L^{2}_{*}(0,L)$
be given. Assume that $g$ satisfies hypothesis $(A1)$. Then, the problem ($P$)
has a unique global (weak) solution

$\varphi\in C(\mathbb{R}_{+};H^{1}_{0}(0,L))\cap
C^{1}(\mathbb{R}_{+};L^{2}(0,L)),\quad\psi,\,\,w\in
C(\mathbb{R}_{+};H^{1}_{*}(0,L))\cap C^{1}(\mathbb{R}_{+};L^{2}_{*}(0,L)).$

Moreover, if

$(\varphi_{0},\varphi_{1})\in(H^{2}(0,L)\cap H^{1}_{0}(0,L))\times
H^{1}_{0}(0,L)$

and

$(\psi_{0},\psi_{1}),\,\,(w_{0},w_{1})\in{(H^{2}_{*}(0,L)\cap
H^{1}_{*}(0,L))\times H^{1}_{*}(0,L)},$

then

$\varphi\in C(\mathbb{R}_{+};H^{2}(0,L)\cap H^{1}_{0}(0,L))\cap
C^{1}(\mathbb{R}_{+};H^{1}_{0}(0,L))\cap C^{2}(\mathbb{R}_{+};L^{2}(0,L)),$

and

$\psi,\,\,w\in C(\mathbb{R}_{+};H^{2}_{*}(0,L)\cap H^{1}_{*}(0,L))\cap
C^{1}(\mathbb{R}_{+};H^{1}_{*}(0,L))\cap C^{2}(\mathbb{R}_{+};L^{2}(0,L)).$

Now, we introduce the energy functional

$\begin{split}E(t):=\,\,\,&\frac{1}{2}\int_{0}^{L}\left[\rho_{1}\varphi_{t}^{2}+\rho_{2}\psi_{t}^{2}+\rho_{1}w_{t}^{2}+\left(k_{2}-\int_{0}^{t}g(s)ds\right)\psi_{x}^{2}\right.\\\
&+\left.\vphantom{\int_{0}^{t}}k_{3}(w_{x}-l\varphi)^{2}+k_{1}(\varphi_{x}+\psi+lw)^{2}\right]dx+\frac{1}{2}(g\circ\psi_{x})(t),\quad\forall\,t\geq
0,\end{split}$ (2.6)

where for any $v\in L^{2}_{loc}([0,+\infty);L^{2}(0,L))$,

$(g\circ
v)(t):=\int_{0}^{L}\int_{0}^{t}g(t-s)\big{(}v(t)-v(s)\big{)}^{2}dsdx.$

By multiplying the equations in ($P$) by $\varphi_{t},\,\psi_{t},\,w_{t}$,
respectively, integrating over $(0,L)$ and exploiting the boundary conditions
we have the following lemma.

######  Lemma 2.2.

Let $(\varphi,\psi,w)$ be the weak solution of ($P$). Then,

$E^{\prime}(t)=-\frac{1}{2}g(t)\int_{0}^{L}\psi_{x}^{2}dx+\frac{1}{2}(g^{\prime}\circ\psi_{x})(t)\leq
0,\qquad\forall t\geq 0.$ (2.7)

From the Cauchy-Schwarz and Poicaré’s inequalities we have the following
lemma.

######  Lemma 2.3 ([19]).

There exists a constant $c>0$ such that for any $v\in
L^{2}_{loc}(\mathbb{R}_{+};H^{1}_{*}(0,L))$, we have

$\int_{0}^{L}\left(\int_{0}^{t}g(t-s)(v(t)-v(s))ds\right)^{2}dx\leq c(g\circ
v_{x})(t),\quad\forall t\geq 0.$

######  Lemma 2.4 ([19]).

Assume that conditions $(A1)$ and $(A2)$ hold and let $(\varphi,\psi,w)$ be
the weak solution of ($P$). Then, for any $0<\sigma<1$, we have

$g\circ\psi_{x}\leq
c\left[\int_{0}^{t}g^{1-\sigma}(s)ds\right]^{\frac{p-1}{p+\sigma-1}}(g^{p}\circ\psi_{x})^{\frac{\sigma}{p+\sigma-1}}.$

For $\sigma=\tfrac{1}{2}$, we obtain the following inequality

$g\circ\psi_{x}\leq
c\left(\int_{0}^{t}g^{1/2}(s)ds\right)^{\frac{2p-2}{2p-1}}(g^{p}\circ\psi_{x})^{\frac{1}{2p-1}}.$
(2.8)

######  Corollary 2.1.

Assume that $g$ satisfies $(A1)$, $(A2)$ and $(\varphi,\psi,w)$ is the weak
solution of ($P$). Then,

$\xi(t)(g\circ\psi_{x})(t)\leq c(-E^{\prime}(t))^{\frac{1}{2p-1}},\quad\forall
t\geq 0.$

###### Proof.

Multiplying both sides of the inequality (2.8) by $\xi(t)$ and using Lemmas
2.1 and 2.2, we get

$\displaystyle\xi(t)(g\circ\psi_{x})(t)$ $\displaystyle\leq$ $\displaystyle
c\xi^{\frac{2p-2}{2p-1}}(t)\left(\int_{0}^{t}g^{1/2}(s)ds\right)^{\frac{2p-2}{2p-1}}(\xi
g^{p}\circ\psi_{x})^{\frac{1}{2p-1}}(t)$ $\displaystyle\leq$ $\displaystyle
c\left(\int_{0}^{t}\xi(s)g^{1/2}(s)ds\right)^{\frac{2p-2}{2p-1}}(-g^{\prime}\circ\psi_{x})^{\frac{1}{2p-1}}\leq
c(-E^{\prime}(t))^{\frac{1}{2p-1}}.$

∎

######  Lemma 2.5 (Jensen’s inequality).

Let $G:[a,b]\longrightarrow\mathbb{R}$ be a concave function. Assume that the
functions $f:\Omega\longrightarrow[a,b]$ and
$h:\Omega\longrightarrow\mathbb{R}$ are integrable such that $h(x)\geq 0$, for
any $x\in\Omega$ and $\displaystyle\int_{\Omega}h(x)dx=k>0$. Then,

$\frac{1}{k}\int_{\Omega}G(f(x))h(x)dx\leq
G\left(\frac{1}{k}\int_{\Omega}f(x)h(x)dx\right).$

In particular, for $\displaystyle G(y)=y^{\frac{1}{p}},\,\,y\geq 0,\,\,p>1$,
we have

$\frac{1}{k}\int_{\Omega}f^{1/p}(x)h(x)dx\leq\left(\frac{1}{k}\int_{\Omega}f(x)h(x)dx\right)^{1/p}.$

##  3 Technical Lemmas

In this section, we state and prove some lemmas needed to establish our main
results. All the computations are done for regular solutions but they still
hold for weak and strong solutions by a density argument.

######  Lemma 3.1.

Assume that conditions $(A1)$ and $(A2)$ hold. Then, the functional $I_{1}$
defined by

$I_{1}(t):=-\rho_{2}\int_{0}^{L}\psi_{t}\int_{0}^{t}g(t-s)(\psi(t)-\psi(s))dsdx$

satisfies, along the solution of ($P$), the estimates

$\displaystyle I_{1}^{\prime}(t)$ $\displaystyle\leq$
$\displaystyle-\rho_{2}\left(\int_{0}^{t}g(s)ds-\delta\right)\int_{0}^{L}\psi_{t}^{2}dx+\delta\int_{0}^{L}(\varphi_{x}+\psi+lw)^{2}dx$
(3.1)
$\displaystyle+c\delta\int_{0}^{L}\psi_{x}^{2}dx+\frac{c}{\delta}(g\circ\psi_{x}-g^{\prime}\circ\psi_{x}),\qquad\forall\,\delta>0.$

###### Proof.

Differentiating $I_{1}$, using equations in ($P$) and integrating by parts, we
get

$\displaystyle I_{1}^{\prime}(t)$ $\displaystyle=$
$\displaystyle-\rho_{2}\int_{0}^{L}\psi_{t}\int_{0}^{t}g^{\prime}(t-s)(\psi(t)-\psi(s))dsdx-\rho_{2}\left(\int_{0}^{t}g(s)ds\right)\int_{0}^{L}\psi_{t}^{2}dx$
$\displaystyle+k_{2}\int_{0}^{L}\psi_{x}\int_{0}^{t}g(t-s)(\psi_{x}(t)-\psi_{x}(s))dsdx$
$\displaystyle+k_{1}\int_{0}^{L}(\varphi_{x}+\psi+lw)\int_{0}^{t}g(t-s)(\psi(t)-\psi(s))dsdx$
$\displaystyle-\int_{0}^{L}\left(\int_{0}^{t}g(t-s)\psi_{x}(s)ds\right)\left(\int_{0}^{t}g(t-s)(\psi_{x}(t)-\psi_{x}(s))ds\right)dx.$

Next, we estimate the terms on the right-hand side of the above equation.

Using Young’s inequality and Lemma 2.3 for $(-g^{\prime})$, we obtain, for any
$\delta>0$,

$-\rho_{2}\int_{0}^{L}\psi_{t}\int_{0}^{t}g^{\prime}(t-s)(\psi(t)-\psi(s))dsdx\leq\delta\rho_{2}\int_{0}^{L}\psi_{t}^{2}dx-\frac{c}{\delta}(g^{\prime}\circ\psi_{x}).$

Similarly, we have

$k_{2}\int_{0}^{L}\psi_{x}\int_{0}^{t}g(t-s)(\psi_{x}(t)-\psi_{x}(s))dsdx\leq\delta\int_{0}^{L}\psi_{x}^{2}+\frac{c}{\delta}(g\circ\psi_{x}),$
$k_{1}\int_{0}^{L}(\varphi_{x}+\psi+lw)\int_{0}^{t}g(t-s)(\psi(t)-\psi(s))dsdx\leq
k_{1}\delta\int_{0}^{L}(\varphi_{x}+\psi+lw)^{2}dx+\frac{c}{\delta}(g\circ\psi_{x}),$

and

$\begin{split}-\int_{0}^{L}\left(\int_{0}^{t}g(t-s)\psi_{x}(s)ds\right)\left(\int_{0}^{t}g(t-s)(\psi_{x}(t)-\psi_{x}(s))ds\right)dx\leq&c\delta\int_{0}^{L}\psi_{x}^{2}dx\\\
&+c\left(\delta+\frac{1}{\delta}\right)(g\circ\psi_{x}).\end{split}$

A combination of these estimates gives the desired result. ∎

######  Lemma 3.2.

Assume that the hypotheses $(A1)$ and $(A2)$ hold. Then, for any
$\varepsilon_{0},\,\delta_{1}>0$, the functional $I_{2}$ defined by

$I_{2}(t):=-\rho_{1}k_{3}\int_{0}^{L}(w_{x}-l\varphi)\int_{0}^{x}w_{t}(y,t)dydx-\rho_{1}k_{1}\int_{0}^{L}\varphi_{t}\int_{0}^{x}(\varphi_{x}+\psi+lw)(y,t)dydx$

satisfies, along the solution of ($P$), the estimate

$\displaystyle I_{2}^{\prime}(t)$ $\displaystyle\leq$ $\displaystyle
k_{1}^{2}\int_{0}^{L}(\varphi_{x}+\psi+lw)^{2}dx-
k_{3}^{2}\int_{0}^{L}(w_{x}-l\varphi)^{2}dx+\frac{c}{\varepsilon_{0}}\int_{0}^{L}\psi_{t}^{2}dx$
$\displaystyle+\left(\varepsilon_{0}-\rho_{1}k_{1}+\frac{l\rho_{1}|k_{3}-k_{1}|\delta_{1}}{2}\right)\int_{0}^{L}\varphi_{t}^{2}dx$
$\displaystyle+\rho_{1}\left(k_{3}+\frac{c_{0}l|k_{3}-k_{1}|}{2\delta_{1}}\right)\int_{0}^{L}w_{t}^{2}dx.$

###### Proof.

Differentiation of $I_{2}$, using equations in ($P$) and integration by parts
yield

$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\rho_{1}k_{3}\int_{0}^{L}w_{t}^{2}dx+l\rho_{1}k_{3}\int_{0}^{L}\varphi_{t}\int_{0}^{x}w_{t}(y,t)dydx-
k_{3}^{2}\int_{0}^{L}(w_{x}-l\varphi)^{2}dx$
$\displaystyle+k_{1}^{2}\int_{0}^{L}(\varphi_{x}+\psi+lw)^{2}dx-\rho_{1}k_{1}\int_{0}^{L}\varphi_{t}^{2}dx-\rho_{1}k_{1}\int_{0}^{L}\varphi_{t}\int_{0}^{x}(\psi_{t}+lw_{t})(y,t)dydx.$

Using Young’s inequality, we get, for any $\varepsilon_{0},\,\delta_{1}>0$,

$\displaystyle I_{2}$ $\displaystyle\leq$ $\displaystyle
k_{1}^{2}\int_{0}^{L}(\varphi_{x}+\psi+lw)^{2}dx-
k_{3}^{2}\int_{0}^{L}(w_{x}-l\varphi)^{2}dx+\frac{c}{\varepsilon_{0}}\int_{0}^{L}\psi_{t}^{2}dx$
$\displaystyle+\left(\varepsilon_{0}-\rho_{1}k_{1}+\frac{l\rho_{1}|k_{3}-k_{1}|\delta_{1}}{2}\right)\int_{0}^{L}\varphi_{t}^{2}dx+\rho_{1}\left(k_{3}+\frac{c_{0}l|k_{3}-k_{1}|}{2\delta_{1}}\right)\int_{0}^{L}w_{t}^{2}dx.$

∎

######  Lemma 3.3.

Under the conditions $(A1)$ and $(A2)$, the functional $I_{3}$ defined by

$I_{3}(t):=-\rho_{1}\int_{0}^{L}(\varphi_{x}+\psi+lw)w_{t}dx-\frac{k_{3}\rho_{1}}{k_{1}}\int_{0}^{L}(w_{x}-l\varphi)\varphi_{t}dx$

satisfies, along the solution of ($P$) and for any $\varepsilon_{0}>0$, the
estimate

$\displaystyle I_{3}^{\prime}(t)$ $\displaystyle\leq$ $\displaystyle
lk_{1}\int_{0}^{L}(\varphi_{x}+\psi+lw)^{2}dx-\frac{lk_{3}^{2}}{k_{1}}\int_{0}^{L}(w_{x}-l\varphi)^{2}dx+\frac{c}{\varepsilon_{0}}\int_{0}^{L}\psi_{t}^{2}dx$
(3.3)
$\displaystyle+\frac{l\rho_{1}k_{3}}{k_{1}}\int_{0}^{L}\varphi_{t}^{2}dx+(\varepsilon_{0}-l\rho_{1})\int_{0}^{L}w_{t}^{2}dx+\rho_{1}\left(\frac{k_{3}}{k_{1}}-1\right)\int_{0}^{L}\varphi_{xt}w_{t}dx.$

###### Proof.

Differentiating $I_{3}$, using equations in ($P$) and integrating by parts, we
have

$\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle-\rho_{1}\int_{0}^{L}\psi_{t}w_{t}dx-l\rho_{1}\int_{0}^{L}w^{2}_{t}dx+lk_{1}\int_{0}^{L}(\varphi_{x}+\psi+lw)^{2}dx$
$\displaystyle+\frac{l\rho_{1}k_{3}}{k_{1}}\int_{0}^{L}\varphi_{t}^{2}dx-\frac{lk_{3}^{2}}{k_{1}}\int_{0}^{L}(w_{x}-l\varphi)^{2}dx+\rho_{1}\left(\frac{k_{3}}{k_{1}}-1\right)\int_{0}^{L}\varphi_{xt}w_{t}dx.$

Use of Young’s inequality for the first term in the right-hand side gives
(3.3). ∎

######  Lemma 3.4.

Assume that conditions $(A1)$ and $(A2)$ hold. Then for any $\delta>0$, the
functional $I_{4}$ defined by

$I_{4}(t):=-\int_{0}^{L}(\rho_{1}\varphi\varphi_{t}+\rho_{2}\psi\psi_{t}+\rho_{1}ww_{t})dx$

satisfies, along the solution of ($P$), the estimate

$\displaystyle I_{4}^{\prime}(t)$ $\displaystyle\leq$
$\displaystyle-\int_{0}^{L}(\rho_{1}\varphi_{t}^{2}+\rho_{2}\psi_{t}^{2}+\rho_{1}w_{t}^{2})dx+k_{1}\int_{0}^{L}(\varphi_{x}+\psi+lw)^{2}dx$
(3.4)
$\displaystyle+k_{3}\int_{0}^{L}(w_{x}-l\varphi)^{2}dx+\left(k_{2}+\delta-\int_{0}^{t}g(s)ds\right)\int_{0}^{L}\psi_{x}^{2}dx+\frac{c}{\delta}(g\circ\psi_{x}).$

###### Proof.

Differentiation of $I_{4}$, using equations of ($P$) gives

$\displaystyle I_{4}^{\prime}(t)$ $\displaystyle=$
$\displaystyle-\int_{0}^{L}(\rho_{1}\varphi_{t}^{2}+\rho_{2}\psi_{t}^{2}+\rho_{1}w_{t}^{2})dx+k_{1}\int_{0}^{L}(\varphi_{x}+\psi+lw)^{2}dx+k_{3}\int_{0}^{L}(w_{x}-l\varphi)^{2}dx$
$\displaystyle+\left(k_{2}-\int_{0}^{t}g(s)ds\right)\int_{0}^{L}\psi_{x}^{2}dx-\int_{0}^{L}\psi_{x}\int_{0}^{t}g(t-s)(\psi_{x}(t)-\psi_{x}(s))dsdx.$

Repeating the above computations yields the desired result. ∎

######  Lemma 3.5.

Assume that conditions $(A1)$ and $(A2)$ hold. Then for any
$\delta,\,\delta_{2}>0$, the functional $I_{5}$ defined by

$I_{5}(t):=-\rho_{2}\int_{0}^{L}\psi_{x}\int_{0}^{x}\psi_{t}(y,t)dydx$

satisfies, along the solution of ($P$), the estimate

$\displaystyle I_{5}^{\prime}(t)$ $\displaystyle\leq$
$\displaystyle\rho_{2}\int_{0}^{L}\psi_{t}^{2}dx+\left(\frac{k_{1}}{2\delta_{2}}+\int_{0}^{t}g(s)ds+\delta-
k_{2}\right)\int_{0}^{L}\psi_{x}^{2}dx$ (3.5)
$\displaystyle+\frac{c_{0}k_{1}\delta_{2}}{2}\int_{0}^{L}(\varphi_{x}+\psi+lw)^{2}dx+\frac{c}{\delta}(g\circ\psi_{x}).$

###### Proof.

Using equations of ($P$) and repeating similar computations as above, we
arrive at

$\displaystyle I_{5}^{\prime}(t)$ $\displaystyle=$
$\displaystyle\rho_{2}\int_{0}^{L}\psi_{t}^{2}dx-
k_{2}\int_{0}^{L}\psi_{x}^{2}dx+k_{1}\int_{0}^{L}\psi_{x}\int_{0}^{x}(\varphi_{x}+\psi+lw)(y,t)dydx$
$\displaystyle+\int_{0}^{L}\psi_{x}\int_{0}^{t}g(t-s)\psi_{x}(s)dsdx$
$\displaystyle\leq$
$\displaystyle\rho_{2}\int_{0}^{L}\psi_{t}^{2}dx+\left(\frac{k_{1}}{2\delta_{2}}+\int_{0}^{t}g(s)ds+\delta-
k_{2}\right)\int_{0}^{L}\psi_{x}^{2}dx$
$\displaystyle+\frac{k_{1}\delta_{2}}{2}\int_{0}^{L}\left(\int_{0}^{x}(\varphi_{x}+\psi+lw)(y,t)dy\right)^{2}dx+\frac{c}{\delta}(g\circ\psi_{x}).$

Poincaré’s inequality for the third term yields (3.5). ∎

######  Lemma 3.6.

Assume that the hypotheses $(A1)$ and $(A2)$ hold. Then, for any
$\varepsilon_{0},\,\varepsilon_{1},\,\varepsilon_{2},\,\delta>0$, the
functional $I_{6}$ defined by

$I_{6}(t):=\rho_{2}\int_{0}^{L}\psi_{t}(\varphi_{x}+\psi+lw)dx+\frac{b\rho_{1}}{k_{1}}\int_{0}^{L}\varphi_{t}\psi_{x}dx-\frac{\rho_{1}}{k_{1}}\int_{0}^{L}\varphi_{t}\int_{0}^{t}g(t-s)\psi_{x}(s)dsdx$

satisfies, along the solution of ($P$), the estimate

$\displaystyle I_{6}^{\prime}(t)$ $\displaystyle\leq$ $\displaystyle-
k_{1}\int_{0}^{L}(\varphi_{x}+\psi+lw)^{2}dx+\left(\frac{lk_{2}k_{3}\varepsilon_{1}}{2k_{1}}+\frac{lk_{3}\varepsilon_{2}}{2k_{1}}\int_{0}^{t}g(s)ds+\delta\right)\int_{0}^{L}(w_{x}-l\varphi)^{2}dx$
$\displaystyle+\delta\int_{0}^{L}\varphi^{2}_{t}dx+\left(\frac{lk_{2}k_{3}}{2k_{1}\varepsilon_{1}}+\frac{lk_{3}}{2k_{1}\varepsilon_{2}}\int_{0}^{t}g(s)ds+\frac{c}{\delta}g(t)\right)\int_{0}^{L}\psi_{x}^{2}dx$
$\displaystyle+\varepsilon_{0}\int_{0}^{L}w_{t}^{2}dx+\frac{c}{\varepsilon_{0}}\int_{0}^{L}\psi_{t}^{2}dx+\frac{c}{\delta}(g\circ\psi_{x}-g^{\prime}\circ\psi_{x})+\left(\frac{k_{2}\rho_{1}}{k_{1}}-\rho_{2}\right)\int_{0}^{L}\varphi_{t}\psi_{xt}dx.$

###### Proof.

Use of equations of ($P$) and integration by parts lead to

$\displaystyle I^{\prime}_{6}(t)$ $\displaystyle=$ $\displaystyle-
k_{1}\int_{0}^{L}(\varphi_{x}+\psi+lw)^{2}dx+\rho_{2}\int_{0}^{L}\psi_{t}^{2}dx+l\rho_{2}\int_{0}^{L}\psi_{t}w_{t}dx$
$\displaystyle+\frac{lk_{2}k_{3}}{k_{1}}\int_{0}^{L}(w_{x}-l\varphi)\psi_{x}dx-\frac{lk_{3}}{k_{1}}\int_{0}^{L}(w_{x}-l\varphi)\int_{0}^{t}g(t-s)\psi_{x}(s)dsdx$
$\displaystyle-\frac{\rho_{1}}{k_{1}}g(t)\int_{0}^{L}\varphi_{t}\psi_{x}dx+\frac{\rho_{1}}{k_{1}}\int_{0}^{L}\varphi_{t}\int_{0}^{t}g^{\prime}(t-s)(\psi_{x}(t)-\psi_{x}(s))dsdx$
$\displaystyle+\left(\frac{k_{2}\rho_{1}}{k_{1}}-\rho_{2}\right)\int_{0}^{L}\varphi_{x}\psi_{xt}dx.$

Next, we estimate the terms in the right-hand side of the above equation.

Exploiting Young’s inequality, we get

$l\rho_{2}\int_{0}^{L}\psi_{t}w_{t}dx\leq\varepsilon_{0}\int_{0}^{L}w^{2}_{t}dx+\frac{c}{\varepsilon_{0}}\int_{0}^{L}\psi_{t}^{2}dx,\qquad\forall\,\varepsilon_{0}>0.$

Using Young’s inequality and Lemma 2.8, we obtain, for any
$\varepsilon_{1},\,\varepsilon_{2},\,\delta>0$,

$\displaystyle\frac{lk_{2}k_{3}}{k_{1}}\int_{0}^{L}(w_{x}-l\varphi)\psi_{x}dx-\frac{lk_{3}}{k_{1}}\int_{0}^{L}(w_{x}-l\varphi)\int_{0}^{t}g(t-s)\psi_{x}(s)dsdx$
$\displaystyle=$
$\displaystyle\frac{lk_{3}}{k_{1}}\left(k_{2}-\int_{0}^{t}g(s)ds\right)\int_{0}^{L}(w_{x}-l\varphi)\psi_{x}dx$
$\displaystyle+\frac{lk_{3}}{k_{1}}\int_{0}^{L}(w_{x}-l\varphi)\int_{0}^{t}g(t-s)(\psi_{x}(t)-\psi_{x}(s))dsdx$
$\displaystyle\leq$
$\displaystyle\left(\frac{lk_{2}k_{3}\varepsilon_{1}}{2k_{1}}+\frac{lk_{3}\varepsilon_{2}}{2k_{1}}\int_{0}^{t}g(s)ds+\delta\right)\int_{0}^{L}(w_{x}-l\varphi)^{2}dx$
$\displaystyle+\left(\frac{lk_{2}k_{3}}{2k_{1}\varepsilon_{1}}+\frac{lk_{3}}{2k_{1}\varepsilon_{2}}\int_{0}^{t}g(s)ds\right)\int_{0}^{L}\psi_{x}^{2}dx+\frac{c}{\delta}(g\circ\psi_{x})$

and

$\displaystyle-\frac{\rho_{1}}{k_{1}}g(t)\int_{0}^{L}\varphi_{t}\psi_{x}dx+\frac{\rho_{1}}{k_{1}}\int_{0}^{L}\varphi_{t}\int_{0}^{t}g^{\prime}(t-s)(\psi_{x}(t)-\psi_{x}(s))dsdx$
$\displaystyle\leq$
$\displaystyle\delta\int_{0}^{L}\varphi_{t}^{2}dx+\frac{c}{\delta}g(t)\int_{0}^{L}\psi_{x}^{2}dx-\frac{c}{\delta}(g^{\prime}\circ\psi_{x}).$

A combination of these estimates gives the desired result.

∎

##  4 General Decay Rates for Equal Speeds of Wave Propagation

In this section, we state and prove a general decay result under equal speeds
of wave propagation condition. The exponential and polynomial decay results
are only special cases.

######  Theorem 4.1.

Let $(\varphi_{0},\varphi_{1})\in H^{1}_{0}(0,L)\times L^{2}(0,L)$ and
$(\psi_{0},\psi_{1}),\,(w_{0},w_{1})\in H^{1}_{*}(0,L)\times L^{2}_{*}(0,L)$.
Assume that $(A1)$ and $(A2)$ hold and that

$\frac{k_{1}}{\rho_{1}}=\frac{k_{2}}{\rho_{2}}\qquad\mathrm{and}\qquad
k_{1}=k_{3}.$ (4.1)

Then for $l$ small enough and for any $t_{0}>0$, the solution of ($P$)
satisfies, for $t>t_{0}$,

$E(t)\leq
C\exp\left(-\lambda\int_{t_{0}}^{t}\xi(s)ds\right),\qquad\mathrm{for\,\,\,\,}p=1,$
(4.2)

and

$E(t)\leq
C\left(\frac{1}{1+\int_{t_{0}}^{t}\xi^{2p-1}(s)ds}\right)^{\frac{1}{2p-2}},\qquad\mathrm{for\,\,\,\,}1<p<\frac{3}{2},$
(4.3)

where $C>0$ is a constant independent of $t$ but may depend on the initial
data and $\lambda>0$ is a constant independent of both $t$ and the initial
data. Moreover, if

$\int_{t_{0}}^{+\infty}\left(\frac{1}{1+\int_{t_{0}}^{t}\xi^{2p-1}(s)ds}\right)^{\frac{1}{2p-2}}dt<+\infty,\qquad\mathrm{for\,\,\,\,}1<p<\frac{3}{2},$
(4.4)

then

$E(t)\leq
C\left(\frac{1}{1+\int_{t_{0}}^{t}\xi^{p}(s)ds}\right)^{\frac{1}{p-1}},\qquad\mathrm{for\,\,\,\,}1<p<\frac{3}{2}.$
(4.5)

######  Remark 4.1.

Inequalities (4.3) and (4.4) together give

$\int_{0}^{+\infty}E(t)dt<+\infty.$

######  Remark 4.2.

The smallness condition on $l$ makes the Bresse system close to Timoshenko
system and, hence, inherits some of its stability properties.

###### Proof of Theorem 4.1.

Define a functional $\mathcal{L}$ by

$\mathcal{L}:=NE+\sum_{j=1}^{6}N_{j}I_{j},$

where $N,\,N_{j}>0$ for $j=1,2,\dots,6$ with $N_{3}=N_{6}=1$. Then from (3.1)
$-$ (3.6) we have

$\displaystyle\mathcal{L}^{\prime}(t)$ $\displaystyle\leq$
$\displaystyle\left[-\rho_{1}(k_{1}N_{2}+N_{4})+\frac{l\rho_{1}|k_{3}-k_{1}|\delta_{1}N_{2}}{2}+\frac{l\rho_{1}k_{3}}{k_{1}}+\varepsilon_{0}N_{2}+\delta\right]\int_{0}^{L}\varphi_{t}^{2}dx$
$\displaystyle+\left[-\rho_{2}\left(N_{1}\int_{0}^{t}g(s)ds+N_{4}-N_{5}\right)+\rho_{2}\delta
N_{1}+\frac{c}{\varepsilon_{0}}(1+N_{2})\right]\int_{0}^{L}\psi_{t}^{2}dx$
$\displaystyle+\left[-l\rho_{1}+\rho_{1}(k_{3}N_{2}-N_{4})+\frac{c_{0}l\rho_{1}|k_{3}-k_{1}|N_{2}}{2\delta_{1}}+\varepsilon_{0}\right]\int_{0}^{L}w_{t}^{2}dx$
$\displaystyle+\left[(N_{5}-N_{4})\int_{0}^{t}g(s)ds+k_{2}(N_{4}-N_{5})+\frac{k_{1}N_{5}}{2\delta_{2}}+\frac{lk_{2}k_{3}}{2k_{1}\varepsilon_{1}}\right.$
$\displaystyle\left.+\frac{lk_{3}}{2k_{1}\varepsilon_{2}}\int_{0}^{t}g(s)ds+\delta(cN_{1}+N_{4}+N_{5})+\frac{c}{\delta}g(t)\right]\int_{0}^{L}\psi_{x}^{2}dx$
$\displaystyle+\left[-\frac{lk_{3}^{2}}{k_{1}}-k_{3}(k_{3}N_{2}-N_{4})+\frac{lk_{2}k_{3}\varepsilon_{1}}{2k_{1}}+\frac{lk_{2}k_{3}\varepsilon_{2}}{2k_{1}}\int_{0}^{t}g(s)ds+\delta\right]\int_{0}^{L}(w_{x}-l\varphi)^{2}dx$
$\displaystyle+\left[-k_{1}\left(1-k_{1}N_{2}-l-N_{4}-\frac{c_{0}\delta_{2}N_{5}}{2}\right)+\delta
N_{1}\right]\int_{0}^{L}(\varphi_{x}+\psi+lw)^{2}dx$
$\displaystyle+\frac{c}{\delta}(1+N_{1}+N_{4}+N_{5})g\circ\psi_{x}-\frac{c}{\delta}(1+N_{1})g^{\prime}\circ\psi_{x}+NE^{\prime}(t)$
$\displaystyle+\left(\frac{k_{2}\rho_{1}}{k_{1}}-\rho_{2}\right)\int_{0}^{L}\varphi_{t}\psi_{xt}dx+\rho_{1}\left(\frac{k_{3}}{k_{1}}-1\right)\int_{0}^{L}\varphi_{xt}w_{t}dx.$

By setting
$\delta_{1}=1,\,\,\,N_{4}=k_{3}N_{2},\,\,\,N_{5}=4k_{3}N_{2},\,\,\,\delta_{2}=\frac{k_{1}}{k_{2}-g_{0}},\,\,\,\varepsilon_{1}=\frac{k_{3}}{k_{2}},\,\,$
and $\varepsilon_{2}=\frac{k_{3}}{2g_{0}},\,\,$ where ${\displaystyle
g_{0}=\int_{0}^{\infty}g(s)ds}$, we arrive at

$\displaystyle\mathcal{L}^{\prime}(t)$ $\displaystyle\leq$
$\displaystyle-\rho_{1}\left[(k_{1}+k_{3})N_{2}-l\left(\frac{|k_{3}-k_{1}|}{2}N_{2}+\frac{k_{3}}{k_{1}}\right)\right]\int_{0}^{L}\varphi_{t}^{2}dx$
$\displaystyle-\rho_{2}\left(N_{1}\int_{0}^{t}g(s)ds-3k_{3}N_{2}\right)\int_{0}^{L}\psi_{t}^{2}dx-l\rho_{1}\left(1-\frac{c_{0}|k_{3}-k_{1}|}{2}N_{2}\right)\int_{0}^{L}w_{t}^{2}dx$
$\displaystyle-\left[(k_{2}-g_{0})k_{3}N_{2}-\frac{l}{k_{1}}\left(\frac{k_{2}^{2}}{2}+g_{0}^{2}\right)\right]\int_{0}^{L}\psi_{x}^{2}dx-\frac{lk_{3}^{2}}{4k_{1}}\int_{0}^{L}(w_{x}-l\varphi)^{2}dx$
$\displaystyle-
k_{1}\left[1-\left(k_{1}+k_{3}+\frac{2c_{0}k_{1}k_{3}}{k_{2}-g_{0}}\right)N_{2}-l\right]\int_{0}^{L}(\varphi_{x}+\psi+lw)^{2}dx$
$\displaystyle+(1+N_{2})\varepsilon_{0}\int_{0}^{L}(\varphi_{t}^{2}+w_{t}^{2})dx+\frac{c}{\varepsilon_{0}}(1+N_{2})\int_{0}^{L}\psi_{t}^{2}dx+NE^{\prime}(t)$
$\displaystyle+\delta\int_{0}^{L}\Big{(}\varphi_{t}^{2}+\rho_{2}N_{1}\psi_{t}^{2}+c(N_{1}+5k_{3}N_{2})\psi_{x}^{2}+N_{1}(\varphi_{x}+\psi+lw)^{2}\Big{)}dx$
$\displaystyle+\frac{c}{\delta}(1+N_{1}+5k_{3}N_{2})g\circ\psi_{x}+\frac{c}{\delta}(1+N_{1})\left[g(t)\int_{0}^{L}\psi_{x}^{2}dx-g^{\prime}\circ\psi_{x}\right]$
$\displaystyle+\left(\frac{k_{2}\rho_{1}}{k_{1}}-\rho_{2}\right)\int_{0}^{L}\varphi_{t}\psi_{xt}dx+\rho_{1}\left(\frac{k_{3}}{k_{1}}-1\right)\int_{0}^{L}\varphi_{xt}w_{t}dx.$

Now, we set $\varepsilon_{0}=\frac{l\rho_{1}}{2(1+N_{2})}$, to get

$\displaystyle\mathcal{L}^{\prime}(t)$ $\displaystyle\leq$
$\displaystyle-\rho_{1}\left[(k_{1}+k_{3})N_{2}-l\left(\frac{1}{2}+\frac{k_{3}}{k_{1}}+\frac{|k_{3}-k_{1}|}{2}N_{2}\right)\right]\int_{0}^{L}\varphi_{t}^{2}dx$
$\displaystyle-\rho_{2}\left(N_{1}\int_{0}^{t}g(s)ds-3k_{3}N_{2}-\frac{c(1+N_{2})^{2}}{l\rho_{1}\rho_{2}}\right)\int_{0}^{L}\psi_{t}^{2}dx-\frac{lk_{3}^{2}}{4k_{1}}\int_{0}^{L}(w_{x}-l\varphi)^{2}dx$
$\displaystyle-\frac{l\rho_{1}}{2}\left(1-c_{0}|k_{3}-k_{1}|N_{2}\right)\int_{0}^{L}w_{t}^{2}dx-\left[(k_{2}-g_{0})k_{3}N_{2}-\frac{l}{k_{1}}\left(\frac{k_{2}^{2}}{2}+g_{0}^{2}\right)\right]\int_{0}^{L}\psi_{x}^{2}dx$
$\displaystyle-
k_{1}\left[1-\left(k_{1}+k_{3}+\frac{2c_{0}k_{1}k_{3}}{k_{2}-g_{0}}\right)N_{2}-l\right]\int_{0}^{L}(\varphi_{x}+\psi+lw)^{2}dx$
$\displaystyle+\delta
c_{N_{1},N_{2}}E(t)+\left[N-\frac{c}{\delta}(1+N_{1})\right]E^{\prime}(t)+\frac{c}{\delta}(1+N_{1}+5k_{3}N_{2})g\circ\psi_{x}$
$\displaystyle+\left(\frac{k_{2}\rho_{1}}{k_{1}}-\rho_{2}\right)\int_{0}^{L}\varphi_{t}\psi_{xt}dx+\rho_{1}\left(\frac{k_{3}}{k_{1}}-1\right)\int_{0}^{L}\varphi_{xt}w_{t}dx.$

Fix $t_{0}>0$ and choose $N_{2}$ so small that

$1-c_{0}|k_{3}-k_{1}|N_{2}>0\qquad\mathrm{and}\qquad
1-\left(k_{1}+k_{3}+\frac{2c_{0}k_{1}k_{3}}{k_{2}-g_{0}}\right)N_{2}>0.$

Next, we select $l$ small enough so that

$(k_{1}+k_{3})N_{2}-l\left(\frac{1}{2}+\frac{k_{3}}{k_{1}}+\frac{|k_{3}-k_{1}|}{2}N_{2}\right)>0,\qquad(k_{2}-g_{0})k_{3}N_{2}-\frac{l}{k_{1}}\left(\frac{k_{2}^{2}}{2}+g_{0}^{2}\right)>0,$

and

$1-\left(k_{1}+k_{3}+\frac{2c_{0}k_{1}k_{3}}{k_{2}-g_{0}}\right)N_{2}-l>0.$

After that, we pick $N_{1}$ very large so that

$N_{1}\int_{0}^{t_{0}}g(s)ds-3k_{3}N_{2}-\frac{c(1+N_{2})^{2}}{l\rho_{1}\rho_{2}}>0.$

Therefore, we have

$\displaystyle\mathcal{L}^{\prime}(t)$ $\displaystyle\leq$
$\displaystyle-(\beta-c\delta)E(t)+\left(N-\frac{c}{\delta}\right)E^{\prime}(t)+\frac{c}{\delta}g\circ\psi_{x}$
$\displaystyle+\left(\frac{k_{2}\rho_{1}}{k_{1}}-\rho_{2}\right)\int_{0}^{L}\varphi_{t}\psi_{xt}dx+\rho_{1}\left(\frac{k_{3}}{k_{1}}-1\right)\int_{0}^{L}\varphi_{xt}w_{t}dx,$

for some $\beta>0$. At this point, we take
$\displaystyle\delta<\frac{\beta}{c}$. Consequently, we obtain, for some
$k>0$,

$\displaystyle\mathcal{L}^{\prime}(t)$ $\displaystyle\leq$ $\displaystyle-
kE(t)+(N-c)E^{\prime}(t)+c(g\circ\psi_{x})$ (4.6)
$\displaystyle+\left(\frac{k_{2}\rho_{1}}{k_{1}}-\rho_{2}\right)\int_{0}^{L}\varphi_{t}\psi_{xt}dx+\rho_{1}\left(\frac{k_{3}}{k_{1}}-1\right)\int_{0}^{L}\varphi_{xt}w_{t}dx,\qquad\forall\,t\geq
t_{0}.$

Finally, we choose N so large that $N>c$ and $\mathcal{L}\sim E$, therefore we
have, $\forall\,t\geq t_{0}$,

$\mathcal{L}^{\prime}(t)\leq-
kE(t)+c(g\circ\psi_{x})+\left(\frac{k_{2}\rho_{1}}{k_{1}}-\rho_{2}\right)\int_{0}^{L}\varphi_{t}\psi_{xt}dx+\rho_{1}\left(\frac{k_{3}}{k_{1}}-1\right)\int_{0}^{L}\varphi_{xt}w_{t}dx.$
(4.7)

Note that from this point, the proof goes similarly as in [20]. But we will
continue for the sake of completeness.

By recalling (4.1) and multiplying both sides of (4.7) by $\xi(t)$ and using
Corollary 2.1, we arrive at

$\xi(t)\mathcal{L}^{\prime}(t)\leq-k\xi(t)E(t)+c\xi(t)(g\circ\psi_{x})(t)\leq-k\xi(t)E(t)+c(-E^{\prime}(t))^{\frac{1}{2p-1}},\qquad\forall\,t\geq
t_{0}.$ (4.8)

For $p=1$, it follows from non-increasing property of $\xi$ and (4.8) that

$\big{(}\xi(t)\mathcal{L}(t)+cE(t)\big{)}^{\prime}\leq\xi(t)\mathcal{L}^{\prime}(t)+cE^{\prime}(t)\leq-k\xi(t)E(t),\qquad\forall\,t\geq
t_{0}.$

Using the fact that $\mathcal{F}=\xi\mathcal{L}+cE\sim E$, there exists a
$\lambda>0$ such that

$\mathcal{F}^{\prime}(t)\leq-\lambda\xi(t)\mathcal{F}(t),\qquad\forall\,t\geq
t_{0}.$

A simple integration over $(t,t_{0})$ leads to

$E(t)\leq
C\exp\left(-\lambda\int_{t_{0}}^{t}\xi(s)ds\right),\qquad\forall\,t\geq
t_{0}.$

For $1<p<\frac{3}{2}$, we multiply both sides of (4.8) by $(\xi
E)^{\alpha}(t)$, with $\alpha=2p-2$, to obtain

$\xi^{\alpha+1}(t)E^{\alpha}(t)\mathcal{L}^{\prime}(t)\leq-k(\xi
E)^{\alpha+1}(t)+c(\xi E)^{\alpha}(t)(-E^{\prime}(t))^{\frac{1}{2p-1}}.$

Applying Young’s inequality with $\displaystyle q=\frac{\alpha+1}{\alpha}$ and
$q^{\prime}=\alpha+1$, we get

$\xi^{\alpha+1}(t)E^{\alpha}(t)\mathcal{L}^{\prime}(t)\leq-(k-c\gamma)(\xi
E)^{\alpha+1}(t)-c_{\gamma}E^{\prime}(t),\qquad\forall\gamma>0.$

We choose $\gamma$ such that $\lambda_{1}:=k-c\gamma>0$ and use the non-
increasing property of $\xi$ and $E$, to have

$(\xi^{\alpha+1}E^{\alpha}\mathcal{L})^{\prime}(t)\leq\xi^{\alpha+1}(t)E^{\alpha}(t)\mathcal{L}^{\prime}(t)\leq-\lambda_{1}(\xi
E)^{\alpha+1}(t)-cE^{\prime}(t),$

this entails that

$(\xi^{\alpha+1}E^{\alpha}\mathcal{L}+cE)^{\prime}(t)\leq-\lambda_{1}(\xi
E)^{\alpha+1}(t).$

Let $\mathcal{F}=\xi^{\alpha+1}E^{\alpha}\mathcal{L}+cE\sim E$, then

$\mathcal{F}^{\prime}(t)\leq-\lambda\xi^{\alpha+1}(t)\mathcal{F}^{\alpha+1}(t),$

for some $\lambda>0$. Integration over $(t_{0},t)$ gives

$E(t)\leq
C\left(\frac{1}{1+\int_{t_{0}}^{t}\xi^{2p-1}(s)ds}\right)^{\frac{1}{2p-2}},\qquad\forall
t\geq t_{0}.$

This establishes (4.3).

To prove (4.5), we treat (4.8) as follows

$\displaystyle\xi(t)\mathcal{L}^{\prime}(t)$ $\displaystyle\leq$
$\displaystyle-k\xi(t)E(t)+c\xi(t)(g\circ\psi_{x})(t)$ (4.9)
$\displaystyle\leq$
$\displaystyle-k\xi(t)E(t)+c\frac{\eta(t)}{\eta(t)}\int_{0}^{t}\big{(}\xi^{p}(s)g^{p}(s)\big{)}^{\frac{1}{p}}\|\psi_{x}(t)-\psi_{x}(t-s)\|^{2}_{2}ds,$

for any $t\geq t_{0}$, where

$\displaystyle\eta(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\|\psi_{x}(t)-\psi_{x}(t-s)\|^{2}_{2}ds\leq
2\int_{0}^{t}(\|\psi_{x}(t)\|^{2}_{2}+\|\psi_{x}(t-s)\|^{2}_{2})ds$
$\displaystyle\leq$ $\displaystyle 4\int_{0}^{t}(E(t)+E(t-s))ds\leq
8\int_{0}^{t}E(t-s)ds=8\int_{0}^{t}E(s)ds$ $\displaystyle\leq$ $\displaystyle
8\int_{0}^{+\infty}E(s)ds<+\infty,$

by Remark 4.1. Applying Jensen’s inequality to the second term in the right-
hand side of (4.9), with $\displaystyle G(y)=y^{\frac{1}{p}},$ $y>0$,
$f(s)=\xi^{p}(s)g^{p}(s)$ and $h(s)=\|\psi_{x}(t)-\psi_{x}(t-s)\|^{2}_{2}$, we
obtain

$\xi(t)\mathcal{L}^{\prime}(t)\leq-k\xi(t)E(t)+c\eta(t)\left(\frac{1}{\eta(t)}\int_{0}^{t}\xi^{p}(s)g^{p}(s)\|\psi_{x}(t)-\psi_{x}(t-s)\|^{2}_{2}ds\right)^{\frac{1}{p}},$

where we assume that $\eta(t)>0$, otherwise we get, from (4.7),

$E(t)\leq C\exp(-kt),\qquad\forall t\geq t_{0}.$

Therefore,

$\displaystyle\xi(t)\mathcal{L}^{\prime}(t)$ $\displaystyle\leq$
$\displaystyle-k\xi(t)E(t)+c\eta^{\frac{p-1}{p}}(t)\left(\xi^{p-1}(0)\int_{0}^{t}\xi(s)g^{p}(s)\|\psi_{x}(t)-\psi_{x}(t-s)\|^{2}_{2}ds\right)^{\frac{1}{p}}$
$\displaystyle\leq$
$\displaystyle-k\xi(t)E(t)+c(-g^{\prime}\circ\psi_{x})^{\frac{1}{p}}(t)\leq-k\xi(t)E(t)+c(-E^{\prime}(t))^{\frac{1}{p}}.$

Multiplying both sides of the above inequality by $(\xi E)^{\alpha}(t)$, for
$\alpha=p-1$, and repeating the above computations, we arrive at

$E(t)\leq
C\left(\frac{1}{1+\int_{t_{0}}^{t}\xi^{p}(s)ds}\right)^{\frac{1}{p-1}},\qquad\forall
t>t_{0},$

which establishes (4.5). ∎

######  Example 4.1.

Let $\displaystyle g(t)=\frac{a}{(1+t)^{q}}$ with $q>2$, and $a>0$ is to be
chosen so that $(A1)$ is satisfied. Then

$g^{\prime}(t)=-a_{0}\left(\frac{a}{(1+t)^{q}}\right)^{\frac{q+1}{q}}=-\xi(t)g^{p}(t),$

with $\displaystyle\xi(t)=a_{0}=\frac{q}{a^{1/q}}$ and $\displaystyle
p=\frac{q+1}{q}<\frac{3}{2}$, we have, for any fixed $t_{0}>0$,

$\int_{t_{0}}^{+\infty}\left(\frac{1}{1+\int_{t_{0}}^{t}\xi^{2p-1}(s)ds}\right)^{\frac{1}{2p-2}}dt=\int_{t_{0}}^{+\infty}\left(\frac{1}{1+c(t-t_{0})}\right)^{\frac{1}{2p-2}}dt<+\infty.$

Therefore, inequality (4.5) entails that there exists $C>0$ such that

$E(t)\leq
C\left(\frac{1}{1+\int_{t_{0}}^{t}\xi^{p}(s)ds}\right)^{\frac{1}{p-1}}=\frac{c}{(1+t)^{q}},$

with the optimal decay rate $q$. For more examples, see [20].

##  5 General Decay Rate for Different Speeds of Wave Propagation

In this section, we state and prove a generalized decay result in the case of
non-equal speeds of wave propagation. We start by differentiating both sides
of the differential equations in ($P$) with respect to $t$ and use the fact
that

$\displaystyle\frac{\partial}{\partial
t}\left[\int_{0}^{t}g(t-s)\psi_{xx}(s)ds\right]$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
t}\left[\int_{0}^{t}g(s)\psi_{xx}(t-s)ds\right]=g(t)\psi_{xx}(0)+\int_{0}^{t}g(s)\psi_{xxt}(t-s)ds$
$\displaystyle=$
$\displaystyle\int_{0}^{t}g(t-s)\psi_{xxt}(s)ds+g(t)\psi_{0xx},$

to obtain the following system

$\begin{cases}\rho_{1}\varphi_{ttt}-k_{1}(\varphi_{xt}+\psi_{t}+lw_{t})_{x}-lk_{3}(w_{xt}-l\varphi_{t})=0,\\\
\rho_{2}\psi_{ttt}-k_{2}\psi_{xxt}+k_{1}(\varphi_{xt}+\psi_{t}+lw_{t})+\displaystyle\int_{0}^{t}g(t-s)\psi_{xxt}(s)ds+g(t)\psi_{0xx}=0,\\\
\rho_{1}w_{ttt}-k_{3}(w_{xt}-l\varphi_{t})_{x}+lk_{1}(\varphi_{xt}+\psi_{t}+lw_{t})=0.\end{cases}$
($P_{*}$)

The energy functional associated to ($P_{*}$) is given by

$\begin{split}E_{*}(t):=\,\,\,&\frac{1}{2}\int_{0}^{L}\left[\rho_{1}\varphi_{tt}^{2}+\rho_{2}\psi_{tt}^{2}+\rho_{1}w_{tt}^{2}+\left(k_{2}-\int_{0}^{t}g(s)ds\right)\psi_{xt}^{2}\right.\\\
&+\left.\vphantom{\int_{0}^{t}}k_{3}(w_{xt}-l\varphi_{t})^{2}+k_{1}(\varphi_{xt}+\psi_{t}+lw_{t})^{2}\right]dx+\frac{1}{2}(g\circ\psi_{xt})(t),\quad\forall\,t\geq
0,\end{split}$ (5.1)

Using similar arguments as in [14, Lemma 3.11] we have the following result.

######  Lemma 5.1.

Let $(\varphi,\psi,w)$ be the strong solution of ($P$). Then, the energy of
($P_{*}$) satisfies, for all $t\geq 0$,

$E_{*}^{\prime}(t)=-\frac{1}{2}g(t)\int_{0}^{L}\psi_{xt}^{2}dx+\frac{1}{2}(g^{\prime}\circ\psi_{xt})-g(t)\int_{0}^{L}\psi_{tt}\psi_{0xx}dx$
(5.2)

and

$E_{*}(t)\leq c\left(E_{*}(0)+\int_{0}^{L}\psi_{0xx}^{2}dx\right).$ (5.3)

######  Lemma 5.2.

Assume that hypotheses $(A1)$ and $(A2)$ hold and let $(\varphi,\psi,w)$ be
the strong solution of ($P$). Then, for any $0<\sigma<1$, we have

$g\circ\psi_{xt}\leq\left[c\left(E_{*}(0)+\int_{0}^{L}\psi_{0xx}^{2}dx\right)\int_{0}^{t}g^{1-\sigma}(s)ds\right]^{\frac{p-1}{p+\sigma-1}}\big{(}g^{p}\circ\psi_{xt}\big{)}^{\frac{\sigma}{p+\sigma-1}}.$

In particular, for $\sigma=\frac{1}{2}$, we get the following inequality

$g\circ\psi_{xt}\leq
c\left(\int_{0}^{t}g^{1/2}(s)ds\right)^{\frac{2p-2}{2p-1}}\left(g^{p}\circ\psi_{xt}\right)^{\frac{1}{2p-1}}.$
(5.4)

###### Proof.

By setting $\displaystyle r=\frac{p+\sigma-1}{p-1}$ and $\displaystyle
q=\frac{(p-1)(1-\sigma)}{p+\sigma-1}$, we have
$\displaystyle\frac{r}{r-1}=\frac{p+\sigma-1}{\sigma}$ and $\displaystyle
1-q=\frac{\sigma p}{p+\sigma-1}$. Then exploiting Hölder’s inequality and
(5.3), we obtain

$\displaystyle g\circ\psi_{xt}$ $\displaystyle=$
$\displaystyle\int_{0}^{L}\int_{0}^{t}g(t-s)(\psi_{xt}(t)-\psi_{xt}(s))^{2}dsdx$
$\displaystyle=$
$\displaystyle\int_{0}^{L}\int_{0}^{t}\left[g^{q}(t-s)(\psi_{xt}(t)-\psi_{xt}(s))^{\frac{2}{r}}\right]\left[g^{1-q}(t-s)(\psi_{xt}(t)-\psi_{xt}(s))^{\frac{2r-2}{r}}\right]dsdx$
$\displaystyle\leq$
$\displaystyle\left[\int_{0}^{L}\int_{0}^{t}g^{qr}(t-s)(\psi_{xt}(t)-\psi_{xt}(s))^{2}dsdx\right]^{\frac{1}{r}}$
$\displaystyle\times\left[\int_{0}^{L}\int_{0}^{t}g^{\frac{(1-q)r}{r-1}}(t-s)(\psi_{xt}(t)-\psi_{xt}(s))^{2}dsdx\right]^{\frac{r-1}{r}}$
$\displaystyle\leq$
$\displaystyle\left[\int_{0}^{L}\int_{0}^{t}g^{1-\sigma}(t-s)(\psi_{xt}(t)-\psi_{xt}(s))^{2}dsdx\right]^{\frac{p-1}{p+\sigma-1}}\left(g^{p}\circ\psi_{xt}\right)^{\frac{\sigma}{p+\sigma-1}}$
$\displaystyle\leq$
$\displaystyle\left[2\int_{0}^{L}\int_{0}^{t}g^{1-\sigma}(s)(\psi_{xt}^{2}(t)+\psi_{xt}^{2}(t-s))dsdx\right]^{\frac{p-1}{p+\sigma-1}}\left(g^{p}\circ\psi_{xt}\right)^{\frac{\sigma}{p+\sigma-1}}$
$\displaystyle\leq$
$\displaystyle\left[\frac{4}{k_{2}-g_{0}}\int_{0}^{t}g^{1-\sigma}(s)(E_{*}(t)+E_{*}(t-s))ds\right]^{\frac{p-1}{p+\sigma-1}}\left(g^{p}\circ\psi_{xt}\right)^{\frac{\sigma}{p+\sigma-1}}$
$\displaystyle\leq$
$\displaystyle\left[c\left(E_{*}(0)+\int_{0}^{L}\psi_{0xx}^{2}dx\right)\int_{0}^{t}g^{1-\sigma}(s)ds\right]^{\frac{p-1}{p+\sigma-1}}\left(g^{p}\circ\psi_{xt}\right)^{\frac{\sigma}{p+\sigma-1}}.$

For $\sigma=\frac{1}{2}$, we get (5.4). This completes the proof. ∎

######  Corollary 5.1.

Assume that conditions $(A1)$ and $(A2)$ hold and let $(\varphi,\psi,w)$ be
the strong solution of ($P$). Then,

$\xi(t)(g\circ\psi_{xt})(t)\leq
c\big{(}-E_{*}^{\prime}(t)+c_{1}g(t)\big{)}^{\frac{1}{2p-1}},\qquad\forall
t\geq 0,$

for some positive constant $c_{1}$.

###### Proof.

From equation (5.2) and inequality (5.3) we have

$\displaystyle 0\leq-g^{\prime}\circ\psi_{xt}$ $\displaystyle=$
$\displaystyle-2E_{*}^{\prime}(t)-g(t)\int_{0}^{L}\psi_{xt}^{2}dx-2g(t)\int_{0}^{L}\psi_{tt}\psi_{0xx}dx$
(5.5) $\displaystyle\leq$
$\displaystyle-2E_{*}^{\prime}(t)-2g(t)\int_{0}^{L}\psi_{tt}\psi_{0xx}dx$
$\displaystyle\leq$
$\displaystyle-2E_{*}^{\prime}(t)+g(t)\int_{0}^{L}(\psi_{tt}^{2}+\psi_{0xx}^{2})dx$
$\displaystyle\leq$
$\displaystyle-2E_{*}^{\prime}(t)+g(t)\left(\frac{2}{\rho_{1}}E_{*}(t)+\int_{0}^{L}\psi_{0xx}^{2}dx\right)$
$\displaystyle\leq$ $\displaystyle
c\left(-E_{*}^{\prime}(t)+c_{1}g(t)\right),$

for some positive constant $c_{1}$. Multiplication of both sides of (5.4) by
$\xi(t)$ and use of Lemma 2.1 and inequality (5.5) give

$\displaystyle\xi(t)(g\circ\psi_{xt})(t)$ $\displaystyle\leq$ $\displaystyle
c\left(\xi(t)\int_{0}^{t}g^{1/2}(s)ds\right)^{\frac{2p-2}{2p-1}}\big{(}\xi
g^{p}\circ\psi_{xt}\big{)}^{\frac{1}{2p-1}}(t)$ $\displaystyle\leq$
$\displaystyle
c\left(\int_{0}^{t}\xi(s)g^{1/2}(s)ds\right)^{\frac{2p-2}{2p-1}}\big{(}-g^{\prime}\circ\psi_{xt}\big{)}^{\frac{1}{2p-1}}(t)$
$\displaystyle\leq$ $\displaystyle
c\big{(}-E_{*}^{\prime}(t)+c_{1}g(t)\big{)}^{\frac{1}{2p-1}}.$

∎

Now we estimate the third term in the right-hand side of (4.7) as in [14].

######  Lemma 5.3.

Let $(\varphi,\psi,w)$ be the strong solution of ($P$). Then, for any
$\varepsilon>0$, we have

$\left(\frac{\rho_{1}k_{2}}{k_{1}}-\rho_{2}\right)\int_{0}^{L}\varphi_{t}\psi_{xt}dx\leq\varepsilon
E(t)+\frac{c}{\varepsilon}(g\circ\psi_{xt}-E^{\prime}(t)+g(t)),\quad\forall
t\geq t_{0}.$ (5.6)

###### Proof.

$\displaystyle\left(\frac{\rho_{1}k_{2}}{k_{1}}-\rho_{2}\right)\int_{0}^{L}\varphi_{t}\psi_{xt}dx$
$\displaystyle=$
$\displaystyle\frac{\left(\frac{\rho_{1}k_{2}}{k_{2}}-\rho_{2}\right)}{\int_{0}^{t}g(s)ds}\int_{0}^{L}\varphi_{t}\int_{0}^{t}g(t-s)(\psi_{xt}(t)-\psi_{xt}(s))dsdx$
(5.7)
$\displaystyle+\frac{\left(\frac{\rho_{1}k_{2}}{k_{1}}-\rho_{2}\right)}{\int_{0}^{t}g(s)ds}\int_{0}^{L}\varphi_{t}\int_{0}^{t}g(t-s)\psi_{xt}(s)dsdx.$

By observing that $\displaystyle\int_{0}^{t}g(s)ds\geq\int_{0}^{t_{0}}g(s)ds$,
for all $t\geq t_{0}$ and exploiting Young’s inequality and

Lemma 2.3 (for $\psi_{xt}$), we get, for $\varepsilon>0$ and $t\geq t_{0}$,

$\frac{\left(\frac{\rho_{1}k_{2}}{k_{1}}-\rho_{2}\right)}{\int_{0}^{t}g(s)ds}\int_{0}^{L}\varphi_{t}\int_{0}^{t}g(t-s)(\psi_{xt}(t)-\psi_{xt}(s))dsdx\leq\frac{\varepsilon}{4}\rho_{1}\int_{0}^{L}\varphi_{t}^{2}dx+\frac{c}{\varepsilon}(g\circ\psi_{xt}).$

On the other hand, by integration by parts and using Lemma 2.3 (for
$-g^{\prime}$ and $\psi_{x}$) and the fact that $E$ is non-increasing, we get

$\displaystyle\frac{\left(\frac{\rho_{1}k_{2}}{k_{1}}-\rho_{2}\right)}{\int_{0}^{t}g(s)ds}\int_{0}^{L}\varphi_{t}\int_{0}^{t}g(t-s)\psi_{xt}(s)dsdx$
$\displaystyle=$
$\displaystyle\frac{\left(\frac{\rho_{1}k_{2}}{k_{1}}-\rho_{2}\right)}{\int_{0}^{t}g(s)ds}\int_{0}^{L}\varphi_{t}\left(g(0)\psi_{x}-g(t)\psi_{0x}+\int_{0}^{t}g^{\prime}(t-s)\psi_{x}(s)ds\right)dx$
$\displaystyle=$
$\displaystyle\frac{\left(\frac{\rho_{1}k_{2}}{k_{1}}-\rho_{2}\right)}{\int_{0}^{t}g(s)ds}\int_{0}^{L}\varphi_{t}\left(g(t)(\psi_{x}-\psi_{0x})-\int_{0}^{t}g^{\prime}(t-s)(\psi_{x}(t)-\psi_{x}(s))ds\right)dx$
$\displaystyle\leq$
$\displaystyle\frac{\varepsilon}{4}\rho_{1}\int_{0}^{L}\varphi_{t}^{2}dx+\frac{c}{\varepsilon}g(t)\int_{0}^{L}(\psi_{x}^{2}+\psi_{0x}^{2})dx-\frac{c}{\varepsilon}g^{\prime}\circ\psi_{x}$
$\displaystyle\leq$
$\displaystyle\frac{\varepsilon}{4}\rho_{1}\int_{0}^{L}\varphi_{t}^{2}dx+\frac{c}{\varepsilon}E(0)g(t)-\frac{c}{\varepsilon}g^{\prime}\circ\psi_{x}.$

Inserting the last two inequalities in (5.7), we get (5.6). ∎

######  Theorem 5.1.

Let

$(\varphi_{0},\varphi_{1})\in\left(H^{2}(0,L)\cap H^{1}_{0}(0,L)\right)\times
H^{1}_{0}(0,L)$

and

$(\psi_{0},\psi_{1}),\,\,(w_{0},w_{1})\in(H^{2}_{*}(0,L)\cap
H^{1}_{*}(0,L))\times H^{1}_{*}(0,L).$

Assume that conditions $(A1)$, $(A2)$ hold and that

$\frac{\rho_{1}}{k_{1}}\neq\frac{\rho_{2}}{k_{2}}\qquad\mathrm{and}\qquad
k_{1}=k_{3}.$

Then for $l$ small enough and for any $t_{0}>0$, there exists a positive
constant $C$ that may depend on the initial data but independent of $t$, for
which the strong solution of ($P$) satisfies, for $t>t_{0}$,

$E(t)\leq
C\left(\frac{1}{\int_{t_{0}}^{t}\xi^{2p-1}(s)ds}\right)^{\frac{1}{2p-1}},\qquad\mathrm{for\,\,\,\,}1\leq
p<\frac{3}{2}.$ (5.8)

###### Proof.

Repeating the steps of the proof of Theorem 4.1 up to inequality (4.6), then
inserting (5.6) into (4.6) we obtain

$\mathcal{L}^{\prime}(t)\leq-(k-\varepsilon)E(t)+\left[N-c\left(1+\frac{1}{\varepsilon}\right)\right]E^{\prime}(t)+cg\circ\psi_{x}+\frac{c}{\varepsilon}g\circ\psi_{xt}+\frac{c}{\varepsilon}g(t),\qquad\forall\,t\geq
t_{0}.$

Now we choose $\varepsilon$ so small that $k-\varepsilon>0$, and then pick
$N>c\left(1+\frac{1}{\varepsilon}\right)$ to get

$\mathcal{L}^{\prime}(t)\leq-
k_{0}E(t)+c(g\circ\psi_{x}+g\circ\psi_{xt})+cg(t),\qquad\forall\,t\geq t_{0},$

for some $k_{0}>0$. We then multiply both sides of the above inequality by
$\xi(t)$ and use Corollaries 2.1 and 5.1 to get

$\displaystyle\xi(t)\mathcal{L}^{\prime}(t)$ $\displaystyle\leq$
$\displaystyle-
k_{0}\xi(t)E(t)+c\xi(t)(g\circ\psi_{x}+g\circ\psi_{xt})+c\xi(t)g(t)$
$\displaystyle\leq$ $\displaystyle-
k_{0}\xi(t)E(t)+c\left[\big{(}-E^{\prime}(t)\big{)}^{\frac{1}{2p-1}}+\big{(}-E_{*}^{\prime}(t)+c_{1}g(t)\big{)}^{\frac{1}{2p-1}}\right]+c\xi(t)g(t).$

Next, we set $\alpha=2p-2$, then multiply both sides of the above inequality
by $(\xi E)^{\alpha}(t)$ and exploit Young’s inequality, with $\displaystyle
q=\frac{\alpha+1}{\alpha}$ and $q^{\prime}=\alpha+1$, to obtain

$\xi^{\alpha+1}(t)E^{\alpha}(t)\mathcal{L}^{\prime}(t)\leq-(k_{0}-c\gamma)(\xi
E)^{\alpha+1}(t)-cE^{\prime}(t)-cE_{*}^{\prime}(t)+c_{1}g(t)+c\xi^{\alpha+1}(t)E^{\alpha}(t)g(t),\,\,\,\,\forall\gamma>0.$

We choose $\gamma>0$ so small such that $\lambda_{2}:=k_{0}-c\gamma>0$ and use
the non-increasing property of $\xi$ and $g$ to get

$(\xi^{\alpha+1}E^{\alpha}\mathcal{L}+cE+cE_{*})^{\prime}(t)\leq-\lambda_{2}(\xi
E)^{\alpha+1}(t)+c\xi^{\alpha+1}(t)E^{\alpha}(t)g(t)+c_{1}g(t),$

which implies that

$\lambda_{2}(\xi
E)^{\alpha+1}(t)\leq-(\xi^{\alpha+1}E^{\alpha}\mathcal{L}+cE+cE_{*})^{\prime}(t)+c\xi^{\alpha+1}(t)E^{\alpha}(t)g(t)+c_{1}g(t).$

Then integration over $(t_{0},t)$ together with the non-increasing property of
$E$ and $\xi$, and the hypothesis $(A1)$ yield, for $t\geq t_{0}$,

$\displaystyle\lambda_{2}E^{\alpha+1}(t)\int_{t_{0}}^{t}\xi^{\alpha+1}(s)ds$
$\displaystyle\leq$ $\displaystyle\lambda_{2}\int_{t_{0}}^{t}(\xi
E)^{\alpha+1}(s)ds\leq-(\xi^{\alpha+1}E^{\alpha}\mathcal{L}+cE+cE_{*})(t)$
$\displaystyle+(\xi^{\alpha+1}E^{\alpha}\mathcal{L}+cE+cE_{*})(0)+\int_{0}^{L}\psi_{0xx}^{2}dx$
$\displaystyle+(c\xi^{\alpha+1}(0)E^{\alpha}(0)+c_{1})\int_{t_{0}}^{t}g(s)ds$
$\displaystyle\leq$
$\displaystyle(\xi^{\alpha+1}E^{\alpha}\mathcal{L}+cE+cE_{*})(0)+\int_{0}^{L}\psi_{0xx}^{2}dx$
$\displaystyle+(c\xi^{\alpha+1}(0)E^{\alpha}(0)+c_{1})\int_{0}^{\infty}g(s)ds.$

Therefore, we get

$E(t)\leq
C\left(\frac{1}{\int_{t_{0}}^{t}\xi^{2p-1}(s)ds}\right)^{\frac{1}{2p-1}},\qquad\forall
t>t_{0}.$

This completes the proof of the Theorem 5.1. ∎

######  Example 5.1.

Let $\displaystyle g(t)=e^{-at}$, where $a>0$. Then
$g^{\prime}(t)=-\xi(t)g(t)$ with $\xi(t)=a$. It follows from (5.8) that for
any fixed $t_{0}>0$, there exists $C>0$ such that

$E(t)\leq\frac{C}{t-t_{0}},\qquad\forall\,t>t_{0}.$

######  Example 5.2.

Consider the same function $g$ as in Example 4.1 and write $g^{\prime}$ as in
Example 4.1. Then it follows from (5.8) that for any fixed $t_{0}>0$, there
exists $C>0$ such that

$E(t)\leq
C\left(\frac{1}{\int_{t_{0}}^{t}\xi^{2p-1}(s)ds}\right)^{\frac{1}{2p-1}}=\frac{c}{(1+t)^{\frac{q}{q+2}}}\qquad\forall\,t>t_{0}.$

For more examples, see [14].

##  6 Full discrete problem

In this section, we introduce a scheme for the problem based on $P_{1}$-finite
element method in space and implicit Euler scheme for time discretization.
Then we draw graphs for the discredited energy showing it’s decay in both
cases, polynomial and exponential. Finally, we implement the approximation of
the solutions $\varphi,$ $\psi$ and $w$ in $3D$ and their cross section at
$x=0.5$.

###  6.1 Finite element setup

We denote by $(\Gamma_{h})_{h}$ a partition of $\Omega$ which fulfills the
following conditions:

  1. 1.

$\Gamma_{h}=\\{R\subset\bar{\Omega};$ $R$ is closed in $\Omega\\}$;

  2. 2.

$\forall(R,R^{\prime})\in\Gamma_{h}\times\Gamma_{h};\
\left|R\right|=\left|R^{\prime}\right|$, where their intersection is either
empty or an end point;

  3. 3.

$\bar{\Omega}=\bigcup\limits_{R\in\Gamma_{h}}R$.

We define the uniform partition of $\Omega$ as $0=x_{0}<x_{1}<\cdots<x_{s}$
and denote the length of $(x_{j},x_{j+1})$ as $h=\frac{L}{s}$. Now for time
discretization, denote by $\Delta t=\frac{T}{N}$ the step time, where $T$ is
the total time and $N$ is a positive integer. Finally we define the discrete
finite element space by

$s_{h}=\\{u_{h}\in H^{1}(0,L):\forall R\in\Gamma_{h};u_{h}|_{R}\in
P_{1}(R)\\},$

where $P_{k}(R)$ denotes the space of restrictions of $R$ of polynomials with
one variable and of order less than or equal to $k$.

Now we introduce the scheme and the discrete energy by using implicit Euler
scheme

$\begin{cases}\frac{\rho_{1}}{\Delta
t}(\Phi_{h}^{n}-\Phi_{h}^{n-1},\bar{\varphi}_{h})+k_{1}(\varphi_{h,x}^{n}+\psi_{h}^{n}+lw_{h}^{n},\bar{\varphi}_{h,x})-lk_{3}(w_{h,x}^{n}-l\varphi_{h}^{n},\bar{\varphi}_{h})=0\\\
\frac{\rho_{2}}{\Delta
t}(\Psi_{h}^{n}-\Psi_{h}^{n-1},\bar{\psi}_{h})+k_{2}(\psi_{h,x}^{n},\bar{\psi}_{x})+k_{1}(\varphi_{h,x}^{n}+\psi_{h}^{n}++lw_{h}^{n},\bar{\psi}_{h})\\\
\qquad\qquad\qquad\qquad-\Delta
t\sum\limits_{m=1}^{n}g(t_{n-m})(\psi_{h,x}^{m},\bar{\psi}_{h,x})=0\\\
\frac{\rho_{1}}{\Delta
t}(W_{h}^{n}-W_{h}^{n-1},\bar{w}_{h})+k_{3}(w_{h,x}^{n}-l\varphi_{h}^{n},\bar{w}_{h,x})+k_{1}l(\varphi_{h,x}^{n}+\psi_{h}^{n}++lw_{h}^{n},\bar{w}_{h})=0\end{cases}$

where $t_{j}=j\Delta t$ and

$\displaystyle E^{n}$ $\displaystyle=$
$\displaystyle\rho_{1}\left|\left|\Phi_{h}^{n}\right|\right|^{2}+\rho_{1}\left|\left|W_{h}^{n}\right|\right|^{2}+\rho_{2}\left|\left|\Psi_{h}^{n}\right|\right|^{2}+k_{1}\left|\left|\varphi_{h,x}^{n}+\psi_{h}^{n}+lw_{h}^{n}\right|\right|^{2}$
$\displaystyle+k_{3}\left|\left|w_{h,x}^{n}-l\varphi_{h}^{n}\right|\right|^{2}+k_{2}\left|\left|\psi_{h,x}^{n}\right|\right|^{2}-\left(\int\limits_{0}^{t_{n}}g(t)dt\right)\left|\left|\psi_{h,x}^{n}\right|\right|^{2}$
$\displaystyle+\frac{1}{2}\Delta
t\int\limits_{0}^{L}\sum\limits_{m=1}^{n}g(t_{n-m})(\psi_{h,x}^{n}-\psi_{h,x}^{m})^{2}dx$

###  6.2 Numerical Experiments

By using the following data

$k_{1}=k_{2}=k_{3}=1,\ \rho_{1}=\rho_{2}=0.1,\ \Delta t=0.012,\ h=0.024,\
T=7.4\ \ \mathrm{and}\ \ g(x)=e^{-3x};$

we draw the solutions $\varphi,$ $\psi,$ and $w$ in $3D$ (see Figures 1, 2 and
3, respectively) and their cross section at $x=0.5$ (see Figures 4, 5 and 6,
respectively).

Figure 1: The evolution in time and space of $\varphi$ Figure 2: The evolution
in time and space of $\psi$ Figure 3: The evolution in time and space of $w$
Figure 4: The evolution in time of $\varphi$ at $x=0.5$ Figure 5: The
evolution in time of $\psi$ at $x=0.5$ Figure 6: The evolution in time of $w$
at $x=0.5$

For the energy we have two cases, taking the conditions of equal and non-equl
speeds of wave propagation.

If $\frac{k_{1}}{k_{2}}=\frac{\rho_{1}}{\rho_{2}}$ and $k_{1}=k_{3}$ we obtain
an exponential decay by using the same data taken for the solutions as shown
in the following Figures 7 – 10.

Figure 7: The evolution in time of $E^{n}$ Figure 8: The evolution in time of
$ln(E^{n})$ that shows the exponential decay Figure 9: The evolution in time
of $ln(E^{n})$ with it’s regression line Figure 10: The evolution in time of
$ln(E^{n})/t$

If $\frac{k_{1}}{k_{2}}\neq\frac{\rho_{1}}{\rho_{2}}$ and $k_{1}\neq k_{3}$ we
obtain a polynomial decay by taking the following data $k_{1}=5$,
$k_{2}=k_{3}=1,$ $\rho_{1}=0.02$, $\rho_{2}=0.1,$ $\Delta t=0.03125,$
$h=0.0625,~{}$ and total time $T=16.4.$ with $g(x)=1/(x+1)^{2}$ as show in the
following Figures 11 – 13.

Figure 11: The evolution in time of $E^{n}$ Figure 12: The variation of
$-ln(E^{n})$ with respect to $ln(t)$ Figure 13: The variation of $-ln(E^{n})$
with respect to $ln(t)$ with it’s regression line

## Acknowledgement

The authors would like to express their gratitude to King Fahd University of
Petroleum and Minerals (KFUPM) for its continuous support. This work is
partially funded by KFUPM under Project SB181018.

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