# Uniform stabilization for the semi-linear wave equation with nonlinear
Kelvin-Voigt damping

Kaïs Ammari  LR Analysis and Control of PDEs, LR 22ES03, Department of
Mathematics, Faculty of Sciences of Monastir, University of Monastir, Tunisia
<EMAIL_ADDRESS>,  Marcelo M. Cavalcanti  Department of Mathematics,
Universidade Estadual de Maringà, Maringà, PR87020-900, Brazil
<EMAIL_ADDRESS>and  Sabeur Mansouri  LR Analysis and Control of PDEs, LR
22ES03, Department of Mathematics, Faculty of Sciences of Monastir, University
of Monastir, Tunisia<EMAIL_ADDRESS>
###### Abstract.

This paper is concerned with the decay estimate of solutions to the semilinear
wave equation subject to two localized dampings in a bounded domain. The first
one is of the nonlinear Kelvin-Voigt type and is distributed around a
neighborhood of the boundary according to the Geometric Control Condition.
While the second one is a frictional damping and we consider it hurting the
geometric condition of control. We show uniform decay rate results of the
corresponding energy for all initial data taken in bounded sets of finite
energy phase-space. The proof is based on obtaining an observability
inequality which combines unique continuation properties and the tools of the
Microlocal Analysis Theory.

###### Key words and phrases:

uniform stabilization, semilinear wave equation, nonlinear Kelvin-Voigt
damping, viscoelatic feedback

###### 2010 Mathematics Subject Classification:

35B40, 35L20, 37L71, 37L15

###### Contents

  1. 1 Introduction
    1. 1.1 Description of the problem
    2. 1.2 Previous results and methodology
  2. 2 Well-posedness of the problem
  3. 3 Uniform decay for problem (1.1)
  4. 4 Appendix: Microlocal Analysis Background

##  1\. Introduction

###  1.1. Description of the problem

In this paper, we study the analysis of the uniform decay rates of solutions
to the wave equation subject to a non-linear Kelvin-Voigt damping. We consider
the following problem

(1.1) $\left\\{\begin{array}[]{lll}\partial_{t}^{2}u-\Delta u-\mathop{\rm
div}\big{(}a(x)g(\nabla\partial_{t}u)\big{)}+\eta(x)\partial_{t}u+f(u)=0,&\mbox{in}&\;\;\Omega\times(0,\infty),\\\
u=0,&\mbox{on}&\partial\Omega\times(0,\infty),\\\ \displaystyle
u(x,0)=u_{0}(x),\;\;\;\partial_{t}u(x,0)=u_{1}(x),&\mbox{in}&\;\;\Omega,\par\end{array}\right.$

where $\Omega$ is a bounded domain of $\mathbb{R}^{n}$, $n\geq 1$, with a
smooth boundary $\partial\Omega$. We denote by $\omega$ the intersection of
the domain $\Omega$ with a neighborhood of $\partial\Omega$ in
$\mathbb{R}^{n}$ and we call it a neighborhood of the boundary of $\Omega$.
Therewith, $f$ and $g$ are real valued functions satisfying Assumption 1.1 and
Assumption 1.2 below.

######  Assumption 1.1.

$f:\mathbb{R}\rightarrow\mathbb{R}$ is a $C^{2}(\mathbb{R})$ function which is
assumed to grow sub-critically satisfying the sign condition $f(s)s\geq 0$,
$\forall s\in\mathbb{R}$, and, furthermore:

(1.2) $f(0)=0,\;\;\;|f^{j}(s)|\leq k_{0}(1+|s|)^{p-j},\;\;j=1,2,\;\;\forall
s\in\mathbb{R},$

which implies, in particular, that for some $C>0$,

(1.3) $|f(s_{1})-f(s_{2})|\leq
C(1+|s_{1}|^{p-1}+|s_{2}|^{p-1})|s_{1}-s_{2}|,\;\;\forall
s_{1},\;s_{2}\in\mathbb{R},$

with

(1.4) $1\leq p<\frac{n}{n-2}\;\mbox{ if }\;n\geq 3\;\mbox{ or }\;p\geq
1\;\mbox{ if }\;n=1,2.$

In addition,

(1.5) $0\leq F(s)\leq f(s)s,\;\;\mbox{ for all }\;\;s\in\mathbb{R},$

where $F(s):=\int_{0}^{s}f(r)\;dr$.

######  Assumption 1.2.

The feedback function

$g:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}$
$s\longmapsto[g_{i}(s_{i})]_{i=1,...,n},$

where, for all $i=1,...,n$, the component
$g_{i}:\mathbb{R}\longrightarrow\mathbb{R}$ is a function such that

(1.6) $\left\\{\begin{array}[]{lll}g_{i}\mbox{ is continuous, monotone
increasing on }\mathbb{R}\mbox{ and }g_{i}(0)=0,\\\
g_{i}(s_{i})s_{i}>0,\;\;\forall s\neq 0,\\\ ms_{i}^{2}\leq
g_{i}(s_{i})s_{i}\leq Ms_{i}^{2}\mbox{ for }|s_{i}|\geq 1,\mbox{ for some
}m,\;M;0<m\leq M.\end{array}\right.$

The functions $a(\cdot)$ and $\eta(\cdot)$ responsible for the localized
dissipative effects are smooth functions satisfying:

######  Assumption 1.3.

$a(\cdot)\in L^{\infty}(\Omega)$ is a non-negative function. Further, there
exists a compact, connected subset $A\subset\Omega$ with smooth boundary and
no-empty interior such that

$A:=\left\\{x\in\Omega:\;a(x)=0\right\\}.$

We also suppose that $a(\cdot)\in C^{0}(\overline{\omega})$, where
$\omega:=\Omega/A.$

######  Assumption 1.4.

The frictional damping $\eta\in C^{0}(\overline{\Omega})$ is a nonnegative
function and is effective in a neighbourhood of the boundary of the set
$A=\left\\{x\in\Omega:\;a(x)=0\right\\}$, that is,

$\eta(x)\geq\eta_{0}>0$

in
$\mathcal{O}_{\varepsilon}=\left\\{x\in\Omega:\;d(x,y)<\varepsilon,\;y\in\partial
A\right\\}$ for $\varepsilon>0$ small enough.

Let us assume that the following assumptions are also made:

######  Assumption 1.5.

$\omega$ is a subset of $\Omega$ which controls geometrically $\Omega$, i.e.
there exists $T_{0}>0$ such that every ray of the geometric optics enters
$\omega$ in a time $t<T_{0}$.

######  Assumption 1.6.

For every $T>0$, the unique solution

$u\in C(0,T;L^{2}(\Omega))\cap C^{1}(0,T;H^{-1}(\Omega)),$

that satisfies

(1.7) $\left\\{\begin{array}[]{lll}\partial_{t}^{2}u-\Delta
u+V(x,t)u=0&\mbox{in}&\;\;\Omega\times(0,\infty),\\\
u=0&\mbox{in}&\;\;\omega\times(0,\infty),\par\par\end{array}\right.$

where $V(x,t)\in L^{\infty}\left(0,T;L^{\infty}(\Omega)\right)$, is the
trivial one.

###  1.2. Previous results and methodology

The stabilization of the wave equation have been extensively investigated in
literature by many authors. Some of them considered linear or nonlinear
viscous damping of the form $au_{t}$ or $ag(u_{t})$ in different setting. To
name few, we mention the following papers [37, 35, 4, 10, 13, 14, 15, 16, 19,
20, 23, 24, 25] and many references therein. In [37], E. Zuazua proved that
the energy of the linear wave equation decays exponentially if the damping
region, where $a(x)\geq a_{0}>0$, contains a neighbourhood of $\partial\Omega$
or at least a neighbourhood of
$\Gamma(x_{0})=\left\\{x\in\partial\Omega,\;(x-x_{0})\cdot\nu(x)\geq
0\right\\}$ where $\nu$ is the outward unit normal to $\Omega$ and
$x_{0}\in\mathbb{R}^{n}$.

Thereafter, Bardos et al. in [9] established, using microlocal analysis, sharp
sufficient conditions for the observation, control and stabilization of the
same linear equation on a compact Riemannian manifold $(M,g)$ with boundary.

In [31], using the piecewise multiplier method, P. Martinez weakened the usual
geometrical condition on the localization of the damping and eliminated the
polynomial growth of the feedback to prove the stability result with a precise
decay rate estimates. Later, Cavalcanti et al. in [12, 13, 14] studied the
wave equation where a localized nonlinear damping is considered. In the first
one, the authors consider compact surfaces, while the other two papers deal
with compact manifolds. Thanks to a damping effective in the complement of a
set satisfying a technical condition connected to the existence of
multipliers, they were able to show uniform and optimal decay rates of the
energy in [13], whereas in [14], the optimal decay rate was proved under
arbitrary small control regions based on the construction of a special
multiplier. In [1], Alabau-Boussouira considered a family of nonlinear
feedbacks, varying from very weak nonlinear dissipation to polynomial or
polynomial-logarithmic decaying feedbacks at the origin, for showing sharp or
quasi-optimal upper energy decay rates. Moreover, Alabau-Boussouira and Ammari
obtained in [3] sharp-energy decay rates for the nonlinearly damped abstract
infinite-dimensional systems, combining optimal geometric conditions as
provided by Bardos et al [9] with an optimal weight convexity method of
Alabau-Boussouira (see [1] and [2]). This approach has been later extended to
uniform discretization of nonlinearly damped evolution equations, see [4].

Furthermore, there are a numerous papers concerning the wave equation with the
linear viscoelastic damping of Kelvin-Voigt type in a bounded domain. In [28],
Liu an Liu investigated an one-dimensional linear wave equation with the
Kelvin-Voigt damping is effective in a subinterval which models an elastic
string with one segment made of viscoelastic material and the other of elastic
material. Since the discontinuity in the damping coefficient and the high
order of damping operator, the authors showed that the energy of that system
does not decay exponentially. This was partially confirmed in [29] where the
uniform stability of an one-dimensional wave equation is obtained for local
viscoelastic damping with differentiable coefficients, see also [7]. In the
same spirit, a result of uniform stability is established in [30] in the
higher dimensional case which present a more delicate problem due to the
unboundedness of viscoelastic damping and the lack of regularity of solution.
To overtake this difficulties the authors considered a special case where the
viscoelastic damping is localized in a neighborhood of the whole of the
boundary under some conditions. It also worth mentioning that this geometrical
condition has been treated in [5, 6] by considering a singular Kelvin-Voigt
damping which is localized far away from the boundary. In order to deal with
such a situation, Tebou has proposed a constructive frequency approach to
improve this result. He has weakened the hypothesis on the damping
coefficients and especially on the feedback control region, see [36]. More
recently, Tebou et al. studied in [16] the wave equation with two types of
linear locally distributed damping mechanisms: a frictional damping and a
Kelvin-Voigt type damping. Using a combination of the multiplier techniques
and the frequency domain approach, they established that a convenient
interaction of the two damping mechanisms is powerful enough for the
exponential stability of the dynamical system, provided that the coefficient
of the Kelvin-Voigt damping is smooth enough and satisfies a structural
condition. This latter result has been generalized in [17] to the semilinear
case in an inhomogeneous medium subject to a Kelvin-Voigt type damping acting
in a neighborhood of the boundary or in a mesh totally distributed in the
domain with measure arbitrarily small.

Motivated by all these works, the main concern of this paper is to consider a
wave equation, in the higher dimensional case, subject to a nonlinear locally
distributed Kelvin-Voigt damping to which we associate a linear frictional
one. What makes this problem interesting is the nature of the operator
defining the damping. Firstly, this type of damping generates a nonlinear and
unbounded operator. It is also of great interest to point that the domain
consists of two different materials, that is, there is an interaction between
the elastic component (the portion of domain where $a\equiv 0$) and the
nonlinear viscoelastic component (the portion of domain where $a>0$). Further,
the solution of the corresponding dynamical system does not have the requisite
smoothness to apply the usual multiplier method. All these points makes the
associated stabilization problem more difficult to study, especially in the
multidimensional setting. In turn, to the authors’ best knowledge, the wave
equation subject to nonlinear locally distributed Kelvin-Voigt damping was not
consider in literature (and a fortiori in the above references and the
references therein). Finally, it is important to remark that the problem under
consideration generalize many works in the literature, in particular, the
papers mentioned above.

To obtain the desired result, our strategy is based on microlocal analysis
techniques combined with unique continuation properties to establish an
observability inequality. In the following, we will give a briefly explication
for this strategy. The energy associated to problem (1.1) is given by

(1.8)
$E_{u}(t):=\frac{1}{2}\int_{\Omega}\left\\{|\partial_{t}u(x,t)|^{2}+|\nabla
u(x,t)|^{2}\right\\}\;dx+\int_{\Omega}F(u(x,t))\;dx,$

which satisfies the following identity

$E_{u}(t)+\sum_{i=1}^{n}\int_{0}^{t}\int_{\Omega}a(x)g_{i}(\partial_{x_{i}}\partial_{t}u(x,s))\partial_{x_{i}}\partial_{t}u(x,s)\;dxds$
(1.9)
$+\int_{0}^{t}\int_{\Omega}\eta(x)|\partial_{t}u(x,s)|^{2}\;dxds=E_{u}(0),\;\;\mbox{
for all }t\in[0,+\infty).$

The main objective of the present paper is to prove the existence and
uniqueness for weak solutions to problem (1.1) and, in addition, that those
solutions decay uniformly to zero, that is,

$E_{u}(t)\leq S\left(\frac{t}{T}-1\right),\;\;\mbox{ for all }t>T_{0},$

where the contraction semigroup $S(t)$ is the solution of the differential
equation

$\frac{d}{dt}S(t)+\mathcal{R}(S(t))=0,\;\;\;S(0)=E(0),$

where $\mathcal{R}$ is given in (3.29), for all mild solutions to problem
(1.1) provided that the initial data are taken in bounded sets of the
corresponding energy space. This result is a local stabilization result.
Inspired by Dehman, Gérard, Lebeau [21] or Dehman, Lebeau and Zuazua [20], we
give a direct proof of the inverse inequality to problem (1.1) namely, we
prove that there exist $T>0$ and a positive constant $C=C(T)$ such that

(1.10) $E_{u}(0)\leq
C\left(\int_{0}^{T}\int_{\Omega}a(x)\left(|\nabla\partial_{t}u|^{2}+|g(\nabla\partial_{t}u)|^{2}\right)dxdt+\int_{0}^{T}\int_{\Omega}\eta(x)|\partial_{t}u|^{2}\;dxdt\right),$

provided the initial data are taken in bounded sets of the corresponding
energy space and the nonlinarity $f$ is taken as in the Assumption 1.1.

Indeed, for a given $\delta>0$ we denote by $\mathcal{O}_{\delta}$ the
neighbourhood of $\partial A$ given by

$\mathcal{O}_{\delta}=\left\\{x\in\Omega:\;d(x,y)<\delta,\;y\in\partial
A\right\\}.$

First of all, it is worth mentioning that the basic damping to prove the
desired uniform result is the nonlinear viscoelstic one. Indeed, note that if
$a(\cdot)$ vanish in the whole of the domain $\Omega$, the frictional damping
$\eta(\cdot)$ is not strong enough to provide the exponential and uniform
decay of the energy since in this case the geometric control condition GCC is
violated, for instance see [9] or also [10]. However, the damping
$\eta(\cdot)$ plays an important role since the lack of convergence due to the
Kelvin-voigt damping which will be explained next. To prove (1.10) and
therefore the stability result, we will argue by contradiction and we will
obtain a sequence $\left\\{v_{k}\right\\}_{k\in\mathbb{N}}$ of solutions to
normalized problem associated to (1.1) such that $E_{v_{k}}(0)=1$. To do so,
we have to consider $\mathcal{V}=\left\\{x\in
A:\;d(x,y)>\varepsilon/2,\;y\in\partial A\right\\}$. In order to obtain a
contradiction we need to prove that $E_{v_{k}}(0)\rightarrow 0$ as
$k\rightarrow+\infty$. We shall show it, by exploiting the properties of the
function $a(\cdot)$, $\eta(\cdot)$, $f$ and $F$, and a unique continuation
principle, that

(1.11)
$\int_{0}^{T}\int_{\Omega\backslash\mathcal{V}}|\partial_{t}v_{k}|^{2}\;dxdt\rightarrow
0,\;\mbox{ as }m\rightarrow+\infty.$

Next we are going to propagate this convergence of $\partial_{t}v_{k}$ to $0$
from $L^{2}((\Omega/\mathcal{V})\times(0,T))$ to the whole space
$L^{2}(\Omega\times(0,T))$. For that, we will use microlocal analysis
arguments. Indeed, let us consider $\chi$ be the microlocal defect measure
associated with $\left\\{\partial_{t}v_{k}\right\\}_{k}$ in
$L^{2}_{loc}(\Omega\times(0,T))$. We are going to establish the convergence

(1.12)
$D(\partial_{t}v_{k}):=(\partial^{2}_{t}-\Delta)\partial_{t}v_{k}\rightarrow
0,\mbox{ strongly in }H^{-2}_{loc}(\Omega\times(0,T))\mbox{ as
}k\rightarrow\infty.$

In view of the convergence (1.12), we can deduce that the $\chi$ is contained
in the characteristic set of the wave equation. But, this convergence is not
enough to obtain the desired propagation. Indeed, we need the stronger
convergence

(1.13) $(\partial^{2}_{t}-\Delta)\partial_{t}v_{k}\rightarrow 0,\mbox{
strongly in }H^{-1}_{loc}(\Omega\times(0,T))\mbox{ as }k\rightarrow\infty.$

This problem of regularity due to the following weak convergence:

(1.14) $\partial_{t}\left(\mathop{\rm
div}\left[\frac{1}{\sigma_{k}}a(x)g(\nabla\partial_{t}u_{k})\right]\right)\rightarrow
0,\mbox{ strongly in }H^{-2}_{loc}(\Omega\times(0,T))\mbox{ as
}k\rightarrow\infty,$

where $\sigma_{k}=\sqrt{E_{u_{k}}(0)},u_{k}=\sigma_{k}\,v_{k}.$

Thus, the presence of the frictional damping in the neighborhood
$\mathcal{O}_{\epsilon}$ of the boundary of $\partial A$ will play a basic
role. In fact, we have $\mathop{\rm
div}\left(\frac{1}{\sigma_{k}}a(x)g(\nabla\partial_{t}u_{k})\right)=0$ in
$A\times(0,T)$ and, thus, we find that

(1.15)
$D\partial_{t}v_{k}=\partial_{t}\left(-\eta(x)\partial_{t}v_{k}-\frac{1}{\sigma_{k}}f(u_{k})\right)\rightarrow
0\mbox{ in }H^{-1}_{loc}(\mbox{int}(A)\times(0,T))\mbox{ as
}k\rightarrow\infty,$

which implies that $\chi$ propagates along the bicharacteristic flow of the
operator $D(\cdot)$. Particularly, if there is
$\mu_{0}=(t_{0},x_{0},\tau_{0},\xi_{0})$ does not belong to the supp$(\chi)$,
the whole bicharacteristic issued from $\mu_{0}$ is out of supp$(\chi)$.
Further, since supp$\chi\subset\mathcal{V}\times(0,T)$, and the frictional
damping effects in both sides of the boundary $\partial A$, we can propagate
the kinetic energy from $(\mathcal{O}_{\epsilon/2}\cap A)\times(0,T)$ towards
the set $\mathcal{V}\times(0,T)$.

Then, we find supp$(\chi)=\emptyset$ and $\partial_{t}v_{k}\rightarrow 0$ in
$L^{2}_{loc}(\Omega\times(0,T))$. Therewith, since some computations, we can
obtain

(1.16) $\int_{0}^{T}\int_{\Omega}|\partial_{t}v_{k}(x,t)|^{2}\;dxdt\rightarrow
0,\mbox{ as }k\rightarrow\infty.$

Finally, this convergence combined with an argument of equipartition of
energy, we can conclude the desired contradiction $E_{v_{k}}(0)\rightarrow 0$.

The rest of the paper is organized as follows: section 2 is devoted to the
well-posedness of the problem (1.1). In section 3, we will study the uniform
stability of our problem subject to strong damping. We prove the nonlinear
observability inequality associated to problem (1.1) by considering refined
arguments of microlocal analysis. An appendix is added as a last section.

##  2\. Well-posedness of the problem

To start the well-posedness study for considered problem, we recall the energy
space

$\mathcal{H}=H^{1}_{0}(\Omega)\times L^{2}(\Omega)$

which is endowed with the inner product

$\left\langle(u_{1},v_{1}),(u_{2},v_{2})\right\rangle_{\mathcal{H}}=\int_{\Omega}\left\\{\nabla
u_{1}\nabla u_{2}+v_{1}v_{2}\right\\}\;dx.$

Denoting $v=u_{t}$, we can rewrite the problem (1.1) in $\mathcal{H}$ as the
following abstract Cauchy problem

(2.17) $\left\\{\begin{array}[]{lll}\frac{\partial}{\partial
t}(u,v)+\mathcal{A}(u,v)=\mathcal{F}(u,v),\\\
(u,v)(0)=(u_{0},u_{1}),\end{array}\right.$

where the unbounded operator
$\mathcal{A}:D(\mathcal{A})\subset\mathcal{H}\rightarrow\mathcal{H}$ is given
by

(2.18) $\mathcal{A}(u,v)=\big{(}v,\mathop{\rm div}\big{(}\nabla u+a(x)g(\nabla
v)\big{)}+\eta(x)v\big{)},$

with domain

(2.19) $D(\mathcal{A})=\left\\{(u,v)\in\mathcal{H}:\;v\in
H^{1}_{0}(\Omega),\;\;\mathop{\rm div}\big{(}\nabla u+a(x)g(\nabla
v)\big{)}\in L^{2}(\Omega)\right\\},$

and $\mathcal{F}:\mathcal{H}\rightarrow\mathcal{H}$ is the operator

(2.20) $\mathcal{F}(u,v)=(0,-f(u)).$

The first point is to prove that the operator
$\mathcal{A}:D(\mathcal{A})\subset\mathcal{H}\rightarrow\mathcal{H}$ defined
above generates a $C_{0}$-semigroup of contractions $e^{t\mathcal{A}}$ on the
energy space $\mathcal{H}$ and the domain $D(\mathcal{A})$ is dense in
$\mathcal{H}$.

Consider the operators $A$ and $B$ defined as follows

$A:D(A)=H_{0}^{1}(\Omega)\subset H^{-1}(\Omega)\rightarrow H^{-1}(\Omega),$
$u\mapsto Au=-\Delta u,$

and

$B:H^{1}_{0}(\Omega)\rightarrow H^{-1}(\Omega),$ $v\mapsto Bv=-\mathop{\rm
div}\big{(}a(x)g(\nabla v)\big{)}+\eta(x)v.$

It is easy to verify that $A$ is a maximal monotone operator. In other hand,
we have

$\begin{array}[]{lll}&&\left\langle Bu-
Bv,u-v\right\rangle_{H^{-1}(\Omega),H_{0}^{1}(\Omega)}\\\
&=&\displaystyle\int_{\Omega}a(x)[g(\nabla u)-g(\nabla v)](\nabla u-\nabla
v)\;dx+\int_{\Omega}\eta(x)|u-v|^{2}\;dx\\\
&=&\displaystyle\sum_{i=1}^{n}\int_{\Omega}a(x)[g_{i}(\partial_{x_{i}}u)-g_{i}(\partial_{x_{i}}v)](\partial_{x_{i}}u-\partial_{x_{i}}v)+\int_{\Omega}\eta(x)|u-v|^{2}\;dx.\end{array}$

Since the fact that $a$ and $\eta$ are non-negative functions and every
component $g_{i}$ of $g$ is monotonic by hypothesis, we obtain that $B$ is
monotone.

We claim also that $B$ is hemicontinuous. In fact, consider $u$ and $v$ in
$D(B)$ and $t_{q}\rightarrow 0$ when $q\rightarrow+\infty$, then, for all
$w\in H_{0}^{1}(\Omega)$, we have

(2.21)
$\begin{array}[]{lll}&&\displaystyle\lim_{q\rightarrow\infty}\left\langle
B(u+t_{q}v),w\right\rangle_{H^{-1}(\Omega),H_{0}^{1}(\Omega)}\\\
&=&\displaystyle\lim_{q\rightarrow\infty}\int_{\Omega}a(x)g(\nabla
u+t_{q}\nabla v)\nabla w\;dx+\int_{\Omega}\eta(x)(u+t_{q}v)w\;dx\\\
&=&\displaystyle\sum_{i=1}^{n}\lim_{q\rightarrow\infty}\int_{\Omega}a(x)g_{i}(\partial_{x_{i}}u+t_{q}\partial_{x_{i}}v)\partial_{x_{i}}w\;dx+\int_{\Omega}\eta(x)uw\;dx.\end{array}$

Let
$\psi_{q}^{i}(x)=a(x)g_{i}(\partial_{x_{i}}u+t_{q}\partial_{x_{i}}v)\partial_{x_{i}}w$.
Taking in mind (1.6), if $|\partial_{x_{i}}u+t_{q}\partial_{x_{i}}v|\geq 1$,
we have

$\begin{array}[]{lll}|\psi_{q}^{i}(x)|&=&a(x)|g_{i}(\partial_{x_{i}}u(x)+t_{q}\partial_{x_{i}}v(x))||\partial_{x_{i}}w(x)|\\\
&\leq&K\left\|a\right\|_{\infty}|\partial_{x_{i}}u(x)+t_{q}\partial_{x_{i}}v(x)||\partial_{x_{i}}w(x)|\\\
&\leq&c_{1}|\partial_{x_{i}}u(x)||\partial_{x_{i}}w(x)|+c_{2}|\partial_{x_{i}}v(x)||\partial_{x_{i}}w(x)|\end{array}$

almost everywhere in $\Omega$, where $c_{1}$ and $c_{2}$ are two constants.
The second constant result of the convergence of $(t_{q})$. Then, for all
$1\leq i\leq n,\psi_{q}^{i}\in L^{1}(\Omega)$ since
$\partial_{x_{i}}u(x),\;\partial_{x_{i}}v(x),\;\partial_{x_{i}}w(x)\in
L^{2}(\Omega).$

In the case where $|\partial_{x_{i}}u+t_{q}\partial_{x_{i}}v|\leq 1$, using
the continuity of $g_{i}$, we obtain that $\psi_{q}^{i}\in L^{1}(\Omega)$ in
the same way.
Using once more the continuity of $g_{i}$ and thanks to Lebesgue’s dominated
convergence theorem, we find

$\lim_{q\rightarrow\infty}\int_{\Omega}a(x)g_{i}(\partial_{x_{i}}u(x)+t_{q}\partial_{x_{i}}v(x))\partial_{x_{i}}w(x)\;dx=\int_{\Omega}a(x)g_{i}(\partial_{x_{i}}u(x))\partial_{x_{i}}w(x)\;dx.$

Moreover, we have

(2.22)
$\begin{array}[]{lll}\displaystyle\sum_{i=1}^{n}\lim_{q\rightarrow\infty}\int_{\Omega}a(x)g_{i}(\partial_{x_{i}}u+t_{q}\partial_{x_{i}}v)\partial_{x_{i}}w\;dx&=&\displaystyle\displaystyle\sum_{i=1}^{n}\int_{\Omega}a(x)g_{i}(\partial_{x_{i}}u(x))\partial_{x_{i}}w(x)\;dx\\\
&=&\displaystyle\int_{\Omega}a(x)g\big{(}\nabla u\big{)}\nabla
w\;dx.\end{array}$

Thus, (2.21) and (2.22) yields

$\lim_{q\rightarrow\infty}\left\langle
B(u+t_{q}v),w\right\rangle_{H^{-1}(\Omega),H_{0}^{1}(\Omega)}=\left\langle
Bu,w\right\rangle_{H^{-1}(\Omega),H_{0}^{1}(\Omega)},$

which gives that $B$ is hemicontinuous.

Using (1.6), we get

(2.23) $\begin{array}[]{lll}\displaystyle\left\langle
u+Au+Bu,u\right\rangle_{H^{-1}(\Omega),H_{0}^{1}(\Omega)}=\\\
\displaystyle\left\langle
u,u\right\rangle_{H^{-1}(\Omega),H_{0}^{1}(\Omega)}+\displaystyle\left\langle
Au,u\right\rangle_{H^{-1}(\Omega),H_{0}^{1}(\Omega)}+\displaystyle\left\langle
Bu,u\right\rangle_{H^{-1}(\Omega),H_{0}^{1}(\Omega)}\\\
\geq\displaystyle|u|^{2}+\int_{\Omega}|\nabla
u|^{2}\;dx\geq\left\|u\right\|_{H^{1}_{0}(\Omega)}^{2},\end{array}$

therefore, we have that $I+A+B$ is coercive. Now, we rewrite the operator
$\mathcal{A}:D(\mathcal{A})\subset\mathcal{H}\rightarrow\mathcal{H}$ as
follows

$\mathcal{A}\left(\begin{array}[]{lll}u\\\
h\end{array}\right)=\left(\begin{array}[]{lll}-h\\\ Au+Bh\end{array}\right),$

and

$D(\mathcal{A})=\left\\{(u,h)\in\mathcal{H};\;\;h\in
H^{1}_{0}(\Omega),\;\;Au+Bh\in L^{2}(\Omega)\right\\}.$

Then, we have to prove that $\mathcal{A}$ is maximal monotone in
$\mathcal{H}$. Indeed, $\mathcal{A}$ is monotone, since

$\begin{array}[]{lll}&&\displaystyle\left(\mathcal{A}\left[\begin{array}[]{lll}u_{1}\\\
h_{1}\end{array}\right]-\mathcal{A}\left[\begin{array}[]{lll}u_{2}\\\
h_{2}\end{array}\right],\left[\begin{array}[]{lll}u_{1}\\\
h_{1}\end{array}\right]-\left[\begin{array}[]{lll}u_{2}\\\
h_{2}\end{array}\right]\right)_{\mathcal{H}}\\\
&=&\displaystyle\left(\left[\begin{array}[]{lll}-h_{1}+h_{2}\\\
A(u_{1}-u_{2})+Bh_{1}-Bh_{2}\end{array}\right],\left[\begin{array}[]{lll}u_{1}-u_{2}\\\
h_{1}-h_{2}\end{array}\right]\right)_{\mathcal{H}}\\\
&=&-(h_{1}-h_{2},u_{1}-u_{2})_{H^{1}_{0}(\Omega)}+(A(u_{1}-u_{2}),h_{1}-h_{2})_{L^{2}(\Omega)}+(Bh_{1}-Bh_{2},h_{1}-h_{2})_{L^{2}(\Omega)}\\\
&=&(Bh_{1}-Bh_{2},h_{1}-h_{2})_{L^{2}(\Omega)}\geq 0,\end{array}$

because $B$ is monotone.

To show the maximality of $\mathcal{A}$, we are going to show that
$R(I+\mathcal{A})=\mathcal{H}$. To do so, let $(u_{0},h_{0})\in\mathcal{H}$
and we look for an element $(u,h)\in D(\mathcal{A})$ such that

(2.24) $(I+\mathcal{A})\left[\begin{array}[]{lll}u\\\
h\end{array}\right]=\left[\begin{array}[]{lll}u_{0}\\\
h_{0}\end{array}\right].$

Equivalently, we consider the following system

$\left\\{\begin{array}[]{lll}u-h=u_{0},\\\ h+Au+Bh=h_{0}.\end{array}\right.$

Combining the above identities we deduce in the weak space

(2.25) $h+Ah+Bh=h_{0}-Au_{0}.$

Therefore, it sufficient to show that $I+A+B$ is maximal monotone. For that,
we have to prove that $R(I+A+B)=H^{-1}(\Omega)$. Since $B$ is monotone and
hemicontinuous, $I+A$ is maximal monotone and $I+A+B$ is coercive, we obtain
thanks to Corollary 1.3 in [8] that $I+A+B$ is maximal monotone which implies
that problem (2.25) admits a unique solution $h\in H^{1}_{0}(\Omega)$. Since
$u=u_{0}+h$ and $Au+Bh=u_{0}-h$, we obtain that $u\in H^{1}_{0}(\Omega)$ and
$Au+Bh\in L^{2}(\Omega)$.

Thus, the problem (2.24) has a unique solution $(u,h)\in D(\mathcal{A})$, and,
consequently, $\mathcal{A}$ is maximal monotone in $\mathcal{H}$.
Moreover, the nonlinear operator
$\mathcal{F}:\mathcal{H}\rightarrow\mathcal{H}$ given in (2.20) is a locally
Lipschitz continuous operator. Indeed, given a bounded set $\mathcal{S}$ in
$\mathcal{H}$ and $(u_{1},u_{2})$, $(v_{1},v_{2})$ in $\mathcal{S}.$ Then,
using Hölder’s inequality, the fact that $H^{1}$ is continuously embedded in
$L^{2q}(\Omega)$ for $2q=\frac{22^{*}}{2^{*}-2(p-1)}\leq 2^{*}$ and by
Assumption (1.3), we get

$\begin{array}[]{lll}\left\|\mathcal{F}(u_{1},u_{2})-\mathcal{F}(v_{1},v_{2})\right\|_{\mathcal{H}}^{2}&\leq&\left\|f(u_{1})-f(v_{1})\right\|_{L^{2}(\Omega)}\\\
&\leq&C\left(1+\left\|u_{1}\right\|^{2(p-1)}_{L^{2^{*}}}+\left\|v_{1}\right\|^{2(p-1)}_{L^{2^{*}}}\right)\left\|u_{1}-v_{1}\right\|^{2}_{L^{2q}}\\\
&\leq&C\left\|u_{1}-v_{1}\right\|^{2}_{H^{1}}\\\
&\leq&C\left\|(u_{1}-u_{2})-(v_{1}-v_{2})\right\|^{2}_{\mathcal{H}}.\end{array}$

Therefore, thanks to Theorems 6.1.4 and 6.1.5 in Pazy’s book [32], see also
Theorem 7.2 in [18], the problem (2.17) has a unique mild solution on the
interval $[0,T_{\max})$. This solution is given by

$(u(t),v(t))=e^{t\mathcal{A}}(u_{0},u_{1})+\int_{0}^{t}e^{(t-s)\mathcal{A}}\mathcal{F}(u(s),v(s))\;ds,\;\;\;\forall
t\in[0,T_{\max}).$

Furthermore, if $(u_{0},u_{1})\in D(\mathcal{A})$ the solution is regular. We
claim that $T_{\max}=\infty$.

Indeed, by a derivative of the energy functional defined in (1.8), it follows
that

(2.26)
$\frac{dE_{u}}{dt}(t)=-\int_{\Omega}a(x)g(\nabla\partial_{t}u)\nabla\partial_{t}u+\eta(x)|\partial_{t}u|^{2}\;dx,$

which shows that the energy $E_{u}(t)$ is non-increasing and thus
$E_{u}(t)\leq E_{u}(0)$ for all $t\in[0,T_{\max})$. Therefore, taking (1.5) in
mind, we have

$E_{u}(t)\geq\frac{1}{2}\int_{\Omega}|\partial_{t}u(x,t)|^{2}\;dx+\frac{1}{2}\int_{\Omega}|\nabla
u(x,t)|\;dx=\frac{1}{2}\left\|(u,u_{t})\right\|_{\mathcal{H}}^{2}.$

Therefore

$\frac{1}{2}\left\|(u,u_{t})\right\|_{\mathcal{H}}^{2}\leq E_{u}(t)\leq
E_{u}(0),\;\;\;\forall t\in[0,T_{\max}),$

and, thus, the local solutions cannot blow-up in finite time which gives that
$T_{\max}=\infty.$ Finally, we have to give the following well-posedness
theorem.

######  Theorem 2.1.

Suppose that the conditions on the nonlinear term $f$ specified in Assumption
1.1 are satisfied in the case $1\leq p\leq\frac{n}{n-2}$ if $n\geq 3$, and the
initial data $(u_{0},u_{1})$ is in the space $\mathcal{H}$. Then problem
(2.17) admits a unique mild solution

$u\in C(0,+\infty;H_{0}^{1}(\Omega))\cap C(0,+\infty;L^{2}(\Omega)).$

Further, if $(u_{0},u_{1})$ is in $D(\mathcal{A})$, then the solution is
regular, i.e. $C(0,+\infty;H^{2}(\Omega)\cap H_{0}^{1}(\Omega))\cap
C(0,+\infty;H^{1}_{0}(\Omega))$.

##  3\. Uniform decay for problem (1.1)

This section is devoted to prove the uniform decay result for the system (1.1)
under suitable growth hypothesis on the feedback and potential functions.
Before stating our goal, we have to define some needed functions. We will
adapt ideas firstly introduced in the literature by Lasiecka and Tataru [25]
for attractive forces and expanded by Cavalcanti et al. [13] for repulsive
forces (sources). Let consider $h_{i}$ be a concave, strictly increasing
functions, with $h_{i}(0)=0$, satisfying

(3.27) $h_{i}(s_{i}h_{i}(s_{i})\geq s_{i}^{2}+g_{i}^{2}(s_{i}),\;\mbox{ for
}|s_{i}|<1,$

for $i=1,..,n$. Note that such function can be straightforwardly constructed,
given the hypothesis on function $g_{i}$ given in (1.6). We define, also, for
$i=1,..,n$, the function $\hat{h}_{i}$ by
$\hat{h}_{i}(\cdot)=h_{i}\left(\frac{\cdot}{meas(Q_{T})}\right)$, where
$Q_{T}=\Omega\times(0,T)$. With these later functions, let us take the
function $\hat{h}$ defined as

$\hat{h}(\cdot)=\sum_{i=1}^{n}\hat{h}_{i}(\cdot).$

As $\hat{h}$ is monotone increasing, $\mu I+\hat{h}$ is invertible for all
$\mu\geq 0$. For $L$ a positive constant, we then set, respectively,

(3.28) $z(x)=(\mu
I+\hat{h})^{-1}(Lx),\;\;\;L:=(C(1+\left\|a\right\|_{\infty})\,\mbox{meas}(Q_{T}))^{-1},$

where $C$ and $\mu$ are a positive constants that will be determined later.

The function $z$ is easily seen to be positive, continuous, and strictly
increasing with $z(0)=0.$ Finally, let

(3.29) $\mathcal{R}=x-(I-z)^{-1}(x).$

Let us consider $T_{0}>0$ associated to the geometric control condition
introduced in Bardos, Lebeau and Rauch [9] which means that every ray of the
geometric optics enters $\omega:=\Omega/A$ in a time $T^{*}<T_{0}$.

Our goal is to show that the solutions of problem (1.1) decay uniformly to
zero which reads as follows:

######  Theorem 3.1.

Suppose that the Assumptions 1.1-1.6 are fulfilled. Let $u$ be the mild
solution of the problem (1.1) (according to Theorem 2.1). Then, the following
uniform decay rate estimate, for bounded initial data, holds

$E_{u}(t)\leq S\left(\frac{t}{T_{0}}-1\right),\;\;\mbox{ for all }t>T_{0},$

where the contraction semigroup $S(t)$ is the solution of the differential
equation

$\frac{d}{dt}S(t)+\mathcal{R}(S(t))=0,\;\;\;S(0)=E(0),$

where $\mathcal{R}$ is given in (3.29).

The principal tool of the proof of this theorem consists to obtain an
observability inequality for the system (1.1). To find such a desired result
some micro-local analysis techniques will be developed.

Integrating the identity (2.26), we get for all $0\leq t_{1}\leq t_{2}<\infty$
the following one

(3.30)
$E_{u}(t_{2})-E_{u}(t_{1})=-\int_{t_{1}}^{t_{2}}\int_{\Omega}\left\\{a(x)g(\nabla\partial_{t}u)\nabla\partial_{t}u+\eta(x)|\partial_{t}u|^{2}\right\\}\;dxdt.$

Thus, we have to prove the following lemma.

######  Lemma 3.2.

For every $T\geq T_{0}$ and for $R>0$, there exists a real $c>0$ depending
only in initial data such that the associated solution of (1.1) verifies that

(3.31) $E_{u}(0)\leq
c\left(\int_{0}^{T}\int_{\Omega}a(x)\left(|\nabla\partial_{t}u|^{2}+|g(\nabla\partial_{t}u)|^{2}\right)dxdt+\int_{0}^{T}\int_{\Omega}\eta(x)|\partial_{t}u|^{2}\;dxdt\right),$

where the energy-norm of the initial data is in $(0,2R)$.

###### Proof.

The proof of this lemma will be conducted by arguing by contradiction. Suppose
(3.31) is not true and hence there is $T\geq T_{0}$ and $R>0$ such that for
all real $c>0$, there exists a solution $u_{c}$ for (1.1) satisfying
$E_{u_{c}}(0)<R$ and such that $u_{c}$ violates the inequality (3.31).
Particularly, for every $c=k\in\mathbb{N}$, we obtain the existence of an
initial datum $(u_{0,k},u_{1,k})$ such that
$\left\|(u_{0,k},u_{1,k})\right\|_{\mathcal{H}}\leq 2R$, and whose
corresponding solution $u_{k}$ satisfies

$E_{u_{k}}(0)\geq
k\left(\int_{0}^{T}\int_{\Omega}a(x)\left(|\nabla\partial_{t}u_{k}|^{2}+|g(\nabla\partial_{t}u_{k})|^{2}\right)dxdt+\int_{0}^{T}\int_{\Omega}\eta(x)|\partial_{t}u_{k}|^{2}\;dxdt\right).$

Further, we find a sequence of solution $(u_{k})_{k\in\mathbb{N}}$ to problem
(1.1) such that

$\lim_{k\rightarrow\infty}\frac{E_{u_{k}}(0)}{\int_{0}^{T}\int_{\Omega}a(x)\left(|\nabla\partial_{t}u_{k}|^{2}+|g(\nabla\partial_{t}u_{k})|^{2}\right)dxdt+\int_{0}^{T}\int_{\Omega}\eta(x)|\partial_{t}u_{k}|^{2}\;dxdt}=+\infty,$

which is equivalent to the following limit to zero

(3.32)
$\lim_{k\rightarrow\infty}\frac{\int_{0}^{T}\int_{\Omega}a(x)\left(|\nabla\partial_{t}u_{k}|^{2}+|g(\nabla\partial_{t}u_{k})|^{2}\right)dxdt+\int_{0}^{T}\int_{\Omega}\eta(x)|\partial_{t}u_{k}|^{2}\;dxdt}{E_{u_{k}}(0)}=0.$

Bearing in mind the boundedness of the sequence
$(E_{u_{k}}(0))_{k\in\mathbb{N}}$, the convergence (3.32) gives

(3.33)
$\lim_{k\rightarrow\infty}\int_{0}^{T}\int_{\Omega}a(x)\left(|\nabla\partial_{t}u_{k}|^{2}+|g(\nabla\partial_{t}u_{k})|^{2}\right)dxdt+\int_{0}^{T}\int_{\Omega}\eta(x)|\partial_{t}u_{k}|^{2}\;dxdt=0.$

It is worth mentioning, also, that there exists a subsequence of
$(u_{k})_{k\in\mathbb{N}}$, which will be denoted by the same notation, such
that

(3.34) $u_{k}\rightharpoonup u\mbox{ weakly}^{\ast}\;\mbox{ in
}L^{\infty}(0,T;H^{1}_{0}(\Omega)),\;\;\mbox{ as }k\rightarrow+\infty,$ (3.35)
$\partial_{t}u_{k}\rightharpoonup\partial_{t}u\mbox{ weakly}^{\ast}\;\mbox{ in
}L^{\infty}(0,T;L^{2}(\Omega)),\;\;\mbox{ as }k\rightarrow+\infty,$

and, thanks to Aubin-Lions-Simon Theorem (see [33]), we have

(3.36) $u_{k}\rightarrow u\mbox{ strongly }\;\mbox{ in
}L^{\infty}(0,T;L^{q}(\Omega)),\;\;\mbox{ as }k\rightarrow+\infty,\;\mbox{ for
all }q\in\left[2,\frac{2n}{n-2}\right).$

Accordingly, we can deduce that

(3.37) $f(u_{k})\rightharpoonup f(u)\mbox{ weakly}^{\ast}\;\mbox{ in
}L^{2}(0,T;L^{2}(\Omega)),\;\;\mbox{ as }k\rightarrow+\infty.$

Now, we are in position to distinguish two cases where $u\neq 0$ and $u=0$.
First of all, let us take the subset $\mathcal{V}$ of $A$ defined as

$\mathcal{V}=\left\\{x\in A:\;d(x,y)>\frac{\epsilon}{2},\;\;y\in\partial
A\right\\}.$

Case (i): $u\neq 0$.
We recall that for all $k\in\mathbb{N}$, $u_{k}$ solve the problem

(3.38) $\left\\{\begin{array}[]{lll}\partial_{t}^{2}u_{k}-\Delta
u_{k}-\mathop{\rm
div}\big{(}a(x)g(\nabla\partial_{t}u_{k})\big{)}+\eta(x)\partial_{t}u_{k}+f(u_{k})=0,&\mbox{in}&\;\;\Omega\times(0,\infty),\\\
u_{k}=0,&\mbox{on}&\partial\Omega\times(0,\infty),\\\ \displaystyle
u_{k}(x,0)=u_{0,k}(x),\;\;\;\partial_{t}u_{k}(x,0)=u_{1,k}(x),&\mbox{in}&\;\;\Omega.\par\end{array}\right.$

Passing to the limit in (3.38) and exploring (3.33)-(3.37), it follows

(3.39) $\left\\{\begin{array}[]{lll}\partial_{t}^{2}u-\Delta
u+f(u)=0,&\mbox{in}&\;\;\Omega\times(0,\infty),\\\
u=0,&\mbox{on}&\partial\Omega\times(0,\infty),\\\
\partial_{t}u=0,&\mbox{in}&\;\;\Omega\backslash\mathcal{V}.\par\end{array}\right.$

Denoting $w=\partial_{t}u$, the problem (3.39) implies

(3.40) $\left\\{\begin{array}[]{lll}\partial_{t}^{2}w-\Delta
w+V(x,t)w=0,&\mbox{in}&\;\;\Omega\times(0,\infty),\\\
w=0,&\mbox{on}&\partial\Omega\times(0,\infty),\\\
w=0,&\mbox{in}&\;\;\Omega\backslash\mathcal{V},\par\end{array}\right.$

with $V(x,t)=f^{\prime}(u(x,t)).$

Consequently, since Assumption 1.6, we deduce that $w=\partial_{t}u=0$ and,
thus, returning to (3.39) we deduce that $u=0$ which presents the desired
contradiction as well.
Case (ii): $u=0$.
Now, we define

(3.41)
$\sigma_{k}:=\sqrt{E_{u_{k}}(0)},\;\;\;\mbox{and}\;\;\;v_{k}=\frac{u_{k}}{\sigma_{k}},$

and, thus, let us consider the following normalized problem

(3.42) $\left\\{\begin{array}[]{lll}\displaystyle\partial_{t}^{2}v_{k}-\Delta
v_{k}-\mathop{\rm
div}\left(\frac{1}{\sigma_{k}}a(x)g(\nabla\partial_{t}u_{k})\right)+\eta(x)\partial_{t}v_{k}+\frac{1}{\sigma_{k}}f(u_{k})=0,&\mbox{in}&\;\Omega\times(0,\infty),\\\
v_{k}=0,&\mbox{on}&\partial\Omega\times(0,\infty),\\\ \displaystyle
v_{k}(x,0)=\frac{u_{0,k}}{\sigma_{k}},\;\;\;\partial_{t}v_{k}(x,0)=\frac{u_{1,k}}{\sigma_{k}},&\mbox{in}&\Omega.\end{array}\right.$

We define the associated energy by

(3.43)
$E_{v_{k}}(t)=\frac{1}{2}\int_{\Omega}\left\\{|\partial_{t}v_{k}(x,t)|^{2}+|\nabla
v_{k}(x,t)|^{2}\right\\}\;dx+\frac{1}{\alpha_{k}^{2}}\int_{\Omega}F(u_{k}(x,t))\;dx.$

It is easy to verify that

(3.44) $E_{v_{k}}(t)=\frac{1}{\alpha_{k}^{2}}E_{u_{k}}(t),\;\;\forall t\geq
0.$

Note that, by the definition of $\sigma_{k}$, (3.44) gives in particular, for
$t=0$, that $E_{v_{k}}(0)=1$. To conclude the contradiction, our strategy is
to prove that $E_{v_{k}}(0)$ converges to zero. Indeed, first, taking (3.33)
and (3.41) into account we infer

(3.45)
$\lim_{k\rightarrow\infty}\int_{0}^{T}\int_{\Omega}a(x)\left(|\nabla\partial_{t}v_{k}|^{2}+\frac{1}{\sigma_{k}^{2}}|g(\nabla\partial_{t}u_{k})|^{2}\right)dxdt+\int_{0}^{T}\int_{\Omega}\eta(x)|\partial_{t}v_{k}|^{2}\;dxdt=0.$

On the other hand, from the boundedness of $E_{v_{k}}(0)$, we also deduce that
for an eventual subsequence of $(v_{k})_{k\in\mathbb{N}}$, still denoted by
the same way, satisfying

(3.46) $v_{k}\rightharpoonup v\mbox{ weakly}^{\ast}\;\mbox{ in
}L^{\infty}(0,T;H^{1}_{0}(\Omega)),\;\;\mbox{ as }k\rightarrow+\infty,$ (3.47)
$\partial_{t}v_{k}\rightharpoonup\partial_{t}v\mbox{ weakly}^{\ast}\;\mbox{ in
}L^{\infty}(0,T;L^{2}(\Omega)),\;\;\mbox{ as }k\rightarrow+\infty.$

From standard compactness arguments, we deduce, for an eventual subsequence,
that will be denoted by the same notation, that

(3.48) $v_{k}\rightarrow v\mbox{ strongly }\;\mbox{ in
}L^{\infty}(0,T;L^{q}(\Omega)),\;\;\mbox{ as }k\rightarrow+\infty,\;\mbox{ for
all }q\in\left[2,\frac{2n}{n-2}\right).$

Further, note that, for an eventual subsequence,
$\sigma_{k}\rightarrow\sigma\in[0,\infty)$.

Subcase (a): $\sigma>0$. Recalling that we are in the case where $u=0$, using
(3.37) and the fact that $f(0)=0$, it follows

(3.49) $\frac{1}{\sigma_{k}}f(u_{k})\rightharpoonup 0,\mbox{ weakly in
}L^{2}(0,T,L^{2}(\Omega)).$

Then, passing to the limit in (3.42) and using the convergences (3.45)-(3.49),
we infer that

(3.50) $\left\\{\begin{array}[]{lll}\partial_{t}^{2}v-\Delta
v=0,&\mbox{in}&\;\;\Omega\times(0,\infty),\\\
v=0,&\mbox{on}&\partial\Omega\times(0,\infty),\\\
\partial_{t}v=0,&\mbox{in}&\;\;\Omega\backslash\mathcal{V}.\par\end{array}\right.$

Therefore, taking $y=\partial_{t}v,$ we obtain the following system

(3.51) $\left\\{\begin{array}[]{lll}\partial_{t}^{2}y-\Delta
y=0,&\mbox{in}&\;\;\Omega\times(0,\infty),\\\
y=0,&\mbox{on}&\partial\Omega\times(0,\infty),\\\
y=0,&\mbox{in}&\;\;\Omega\backslash\mathcal{V}.\par\end{array}\right.$

Once more, thanks to Assumption 1.6, it follows that $y=\partial_{t}v=0$ which
gives, since (3.50), that $v=0$.

Subcase (b): $\sigma=0$. In this case, we are going to show that

(3.52) $\frac{1}{\sigma_{k}}f(u_{k})\rightharpoonup f^{\prime}(0)v,\mbox{
weakly in }L^{2}(0,T,L^{2}(\Omega)).$

First of all, the fact that $f\in C^{2}(\mathbb{R})$ and, thus, Taylor’s
formula allows us to get

(3.53) $f(s)=f^{\prime}(0)s+R(s),\mbox{ where }|R(s)|\leq
c(|s|^{2}+|s|^{p+1}).$

This is obtain by using the condition (1.2) for the semi-linearity $f$.

Thereafter, we have to write

(3.54)
$\frac{1}{\sigma_{k}}f(\sigma_{k}v_{k})=f^{\prime}(0)v_{k}+\frac{1}{\sigma_{k}}R(\sigma_{k}v_{k}).$

Now, to prove (3.52) it suffices to show that

(3.55) $\frac{1}{\sigma_{k}}R(\sigma_{k}v_{k})\rightharpoonup 0\mbox{ weakly
in }L^{2}(0,T,L^{2}(\Omega)).$

To do so, according (3.53) and (1.3), we get

(3.56)
$\begin{array}[]{lll}\displaystyle\left\|\frac{1}{\sigma_{k}}R(\sigma_{k}v_{k})\right\|^{2}_{L^{2}(0,T,L^{2}(\Omega))}&\leq&\displaystyle
C\left\\{\int_{0}^{T}\int_{\Omega}\left|\frac{f(\sigma_{k}v_{k})}{\sigma_{k}}\right|^{2}\;dxdt+\int_{0}^{T}\int_{\Omega}|f^{\prime}(0)v_{k}|^{2}\;dxdt\right\\}\\\
\\\ &\leq&\displaystyle
C\left\\{\frac{1}{\alpha_{k}^{2}}\int_{0}^{T}\int_{\Omega}\left(|\sigma_{k}v_{k}|^{2}+|\sigma_{k}v_{k}|^{p}\right)^{2}\;dxdt+\int_{0}^{T}\int_{\Omega}|v_{k}|^{2}\;dxdt\right\\}\\\
\\\ &\leq&\displaystyle
C\left\\{\sigma_{k}^{2(p-1)}\left\|v_{k}\right\|^{2p}_{L^{2p}(0,T;L^{2p}(\Omega))}+\left\|v_{k}\right\|^{2}_{L^{2}(0,T;L^{2}(\Omega))}\right\\}.\end{array}$

Using the two facts that $H^{1}(\Omega)$ is continuously embedded in
$L^{2p}(\Omega)$ and the boundedness of the sequence $(v_{k})$, we can
immediately conclude from (3.56) that
$\left\\{\frac{1}{\sigma_{k}}R(\sigma_{k}v_{k})\right\\}_{k}$ is bounded in
the space $L^{2}(0,T,L^{2}(\Omega))$.

Whereupon, this sequence admits a subsequence which converges in the
associated space, that is, there exists $\delta\in L^{2}(0,T,L^{2}(\Omega))$
such that

(3.57) $\frac{1}{\sigma_{k}}R(\sigma_{k}v_{k})\rightharpoonup\delta\mbox{
weakly in }L^{2}(0,T,L^{2}(\Omega)).$

Since (3.53), we deduce that

(3.58)
$\left|\frac{1}{\sigma_{k}}R(\sigma_{k}v_{k})\right|\leq\sigma_{k}|v_{k}|^{2}+|\sigma_{k}|^{p}|v_{k}|^{p+1}\;\mbox{
in the case }\;1\leq\frac{n}{n-2},$

which implies that

(3.59)
$\left\|\frac{1}{\sigma_{k}}R(\sigma_{k}v_{k})\right\|_{L^{1}(0,T,L^{1}(\Omega))}\leq
C\left(\sigma_{k}\left\|v_{k}\right\|^{2}_{L^{2}(0,T,L^{2}(\Omega))}+(\sigma_{k})^{p}\left\|v_{k}\right\|^{p+1}_{L^{p+1}(0,T,L^{p+1}(\Omega))}\right).$

Therefore

(3.60) $\frac{1}{\sigma_{k}}R(\sigma_{k}v_{k})\rightharpoonup 0\mbox{ weakly
in }L^{1}(0,T,L^{1}(\Omega))\mbox{ as }k\rightarrow\infty.$

Combining (3.57) and (3.60) together with the embedding
$L^{2}(0,T,L^{2}(\Omega))$ in $L^{1}(0,T,L^{1}(\Omega))$, it result the
desired convergence in (3.55). Consequently, (3.52) follows from (3.46),
(3.54) and (3.55). Next, observing all the convergences (3.45)-(3.47) and
(3.52) and passing to the limit in (3.42), we obtain in the distributional
sense

(3.61) $\left\\{\begin{array}[]{lll}\partial_{t}^{2}v-\Delta
v+f^{\prime}(0)v=0,&\mbox{in}&\;\;\Omega\times(0,\infty),\\\
v=0,&\mbox{on}&\partial\Omega\times(0,\infty),\\\
\partial_{t}v=0,&\mbox{in}&\;\;\Omega\backslash\mathcal{V},\par\end{array}\right.$

and, for $z=\partial_{t}v$, we infer

(3.62) $\left\\{\begin{array}[]{lll}\partial_{t}^{2}z-\Delta
z+f^{\prime}(0)z=0,&\mbox{in}&\;\;\Omega\times(0,\infty),\\\
z=0,&\mbox{on}&\partial\Omega\times(0,\infty),\\\
z=0,&\mbox{in}&\;\;\Omega\backslash\mathcal{V}.\par\end{array}\right.$

As a consequence, via the Assumption 1.6 once again, it follows that
$w=\partial_{t}u=0$. Returning to (3.39), we immediately find that $v=0$.

Thereby, in any of both cases $\sigma>0$ and $\sigma=0$, we find that $v=0$.
As a consequence, $v=0$ in all convergences (3.46)-(3.48). Moreover, by virtue
of (3.48), (3.54) and (3.55) together with the nullity of $v$, it follows that

(3.63) $\frac{1}{\sigma_{k}}f(u_{k})\rightarrow 0,\mbox{ in
}L^{2}(0,T,L^{2}(\Omega)).$

Recall that our objective is to show that $E_{v_{k}}(0)$ converges to zero,
where $E_{v_{k}}(t)$ is defined by (3.43) as well. To do so, let us consider
the D’Alembert operator given by

$D:=\partial^{2}_{t}-\Delta.$

By the first equation in (3.42), we have

(3.64) $Dv_{k}=\mathop{\rm
div}\left(\frac{1}{\sigma_{k}}a(x)g(\nabla\partial_{t}u_{k})\right)-\eta(x)\partial_{t}v_{k}-\frac{1}{\sigma_{k}}f(u_{k}),$

which gives, after applying the operator $\partial_{t}$, that

(3.65) $\partial_{t}Dv_{k}=\partial_{t}\left(\mathop{\rm
div}\left(\frac{1}{\sigma_{k}}a(x)g(\nabla\partial_{t}u_{k})\right)-\eta(x)\partial_{t}v_{k}-\frac{1}{\sigma_{k}}f(u_{k})\right).$

Our immediate task is to seek an appropriate limit to $\partial_{t}Dv_{k}$,
so, we have to determine the convergences of each term in the right hand side
of (3.65). Firstly, observing (3.63), we deduce that

(3.66) $\frac{1}{\sigma_{k}}f(u_{k})\rightarrow 0,\mbox{ strongly in
}L^{2}(\Omega\times(0,T))\mbox{ as }k\rightarrow\infty.$

Secondly, thanks to (3.45), we have obviously that

$\frac{1}{\sigma_{k}}a(x)g(\nabla\partial_{t}u_{k})\rightarrow 0\mbox{
strongly in }L^{2}(\Omega\times(0,T))\mbox{ as }k\rightarrow\infty,$

that is,

(3.67) $\mathop{\rm
div}\left(\frac{1}{\sigma_{k}}a(x)g(\nabla\partial_{t}v_{k})\right)\rightarrow
0,\mbox{ strongly in }H^{-1}_{loc}(\Omega\times(0,T))\mbox{ as
}k\rightarrow\infty.$

Also, by (3.45) once more, we deduce that

(3.68) $\eta(x)\partial_{t}v_{k}\rightarrow 0,\mbox{ strongly in
}L^{2}(\Omega\times(0,T))\mbox{ as }k\rightarrow\infty.$

Amalgamating (3.65) with (3.66)-(3.68), it follows that

(3.69) $D\partial_{t}v_{k}\rightarrow 0,\mbox{ strongly in
}H^{-2}_{loc}(\Omega\times(0,T))\mbox{ as }k\rightarrow\infty.$

Let $\chi$ be the microlocal defect measure associated with
$(\partial_{t}v_{k})_{k}$ in $L^{2}_{loc}(\Omega\times(0,T))$. In view of
(3.69), we deduce that the supp($\chi$) is contained in the characteristic set
of the wave operator $\left\\{\tau^{2}=\left\|\xi\right\|^{2}\right\\}$, for
that see Theorem 4.4 in Appendix or also Proposition 2.1 and Corollary 2.2 due
to Gérard [22].

Additionally, the existence of the dissipative effects yields that

(3.70) $\partial_{t}v_{k}\rightarrow 0\;\;\mbox{ in
}\;(\Omega\backslash\mathcal{V})\times(0,T).$

Therewith, the aim is to propagate the convergence of $\partial_{t}v_{k}$ to
$0$ from $L^{2}((\Omega\backslash\mathcal{V})\times(0,T))$ to the whole space
$L^{2}(\Omega\times(0,T))$. It is worth mentioning that the convergence (3.69)
is not regular enough to guarantee the desired propagation. This problem of
regularity due to the following weak convergence:

(3.71) $\partial_{t}\left(\mathop{\rm
div}\left[\frac{1}{\sigma_{k}}a(x)g(\nabla\partial_{t}u_{k})\right]\right)\rightarrow
0,\mbox{ strongly in }H^{-2}_{loc}(\Omega\times(0,T))\mbox{ as
}k\rightarrow\infty.$

Whereupon, to obtain the propagation to the whole space, the presence of the
frictional damping in the neighborhood $\mathcal{O}_{\epsilon}$ of the
boundary of $\partial A$ plays a basic role. In fact, we have $\mathop{\rm
div}\left(\frac{1}{\sigma_{k}}a(x)g(\nabla\partial_{t}u_{k})\right)=0$ in
$A\times(0,T)$ and, thus, taking in mind (3.65), we find that

(3.72)
$D\partial_{t}v_{k}=\partial_{t}\left(-\eta(x)\partial_{t}v_{k}-\frac{1}{\sigma_{k}}f(u_{k})\right)\rightarrow
0\mbox{ in }H^{-1}_{loc}(\mbox{int}(A)\times(0,T))\mbox{ as
}k\rightarrow\infty.$

In addition, thanks to general microlocal results given by Proposition 4.6 and
Theorem 4.7 in Appendix, we can deduce that supp$(\chi)$ in
$(\mbox{int}(A)\times(0,T))\times S^{d}$ presents a union of curves like

(3.73) $t\in I\cap(0,\infty)\mapsto m_{\pm}(t)=\left(t,x(t),\frac{\pm
1}{\sqrt{1+|G(x)\dot{x}|^{2}}},\frac{\mp
G(x)\dot{x}}{\sqrt{1+|G(x)\dot{x}|^{2}}}\right),$

where $t\in I\mapsto x(t)\in\Omega$ is a geodesic associated to the metric.

Therefore, The above convergence yields that $\chi$ propagates along the
bicharacteristic flow of the operator $D(\cdot)$, which signifies,
particularly, that if some point $\mu_{0}=(t_{0},x_{0},\tau_{0},\xi_{0})$ does
not belong to the supp$(\chi)$, the whole bicharacteristic issued from
$\mu_{0}$ is out of supp$(\chi)$. Further, since
supp$\chi\subset\mathcal{V}\times(0,T)$, and the frictional damping effects in
both sides of the boundary $\partial A$, we can propagate the kinetic energy
from $(\mathcal{O}_{\epsilon/2}\cap A)\times(0,T)$ towards the set
$\mathcal{V}\times(0,T)$. Indeed, consider $t_{0}\in(0,+\infty)$ and $x$ a
geodesic of $G=I_{d}$ defined near $t_{0}$. Once the geodesics inside
$\mathcal{V}$, enter necessarily in $\omega$, they again intersect the set
$A\backslash\mathcal{V}$ , consequently, for any geodesic of the metric $G$,
with $0\in I$ there exists $t>0$ such that $m_{\pm}(t)$ does not belong to
supp$(\chi)$, so that $m_{\pm}(t_{0})$ does not belong as well. Once the time
$t_{0}$ and the geodesic $x$ were taken arbitrary, we conclude that
supp$(\chi)$ is empty. Whereupon, we have

$\partial_{t}v_{k}\rightarrow 0,\mbox{ in }L^{2}(\omega\times(0,T)),$

and considering (3.70), we can deduce the following convergence in the the
whole space

(3.74) $\int_{0}^{T}\int_{\Omega}|\partial_{t}v_{k}(x,t)|^{2}\;dxdt\rightarrow
0,\mbox{ as }k\rightarrow\infty.$

The task ahead is to show that $E_{v_{k}}(0)\rightarrow 0$ as
$k\rightarrow+\infty$. Let us consider the following cut-off function

$\displaystyle\theta\in C^{\infty}(0,T),\quad 0\leq\theta(t)\leq
1,\quad\theta(t)=1\ \mbox{in}\ (\varepsilon,T-\varepsilon),\;\;\epsilon>0.$

Multiplying the first equation in (3.42) by $v_{k}\theta$ and integrating by
parts, we get

(3.75)
$\begin{array}[]{lll}&&\displaystyle-\int_{0}^{T}\theta(t)\int_{\Omega}|\partial_{t}v_{k}|^{2}\,dxdt-\int_{0}^{T}\theta^{\prime}(t)\int_{\Omega}\partial_{t}v_{k}v_{k}\,dxdt+\int_{0}^{T}\theta(t)\int_{\Omega}|\nabla
v_{k}|^{2}\,dxdt\\\
&&\displaystyle+\frac{1}{\sigma_{k}}\int_{0}^{T}\theta(t)\int_{\Omega}f(\sigma_{k}v_{k})v_{k}\,dxdt+\frac{1}{\sigma_{k}}\int_{0}^{T}\theta(t)\int_{\Omega}a(x)g(\nabla\partial_{t}u_{k})\nabla
v_{k}\,dxdt\\\
&&\displaystyle+\int_{0}^{T}\theta(t)\int_{\Omega}\eta(x)\partial_{t}v_{k}v_{k}\,dxdt=0.\end{array}$

Employing the convergences (3.45)-(3.48) and (3.74), and, having $v=0$, the
identity (3.75) implies that

(3.76) $\lim_{k\rightarrow+\infty}\int_{0}^{T}\theta(t)\int_{\Omega}|\nabla
v_{k}|^{2}+\frac{1}{\sigma_{k}}f(\sigma_{k}u_{k})v_{k}\;dxdt=0,$

that is,

(3.77)
$\lim_{k\rightarrow+\infty}\int_{\epsilon}^{T-\epsilon}\int_{\Omega}|\nabla
v_{k}|^{2}+\frac{1}{\sigma_{k}}f(\sigma_{k}u_{k})v_{k}\;dxdt=0.$

Furthermore, by the definition of functions $f$ and $F$ and the property as
well, the convergence in (3.77) yields

(3.78)
$\lim_{k\rightarrow+\infty}\frac{1}{\sigma_{k}^{2}}\int_{\epsilon}^{T-\epsilon}\int_{\Omega}F(u_{k})\;dxdt=0.$

Thereafter, combining (3.74), (3.77) and (3.78), it follows that

(3.79) $\int_{\epsilon}^{T-\epsilon}E_{v_{k}}(t)\,dt\rightarrow 0\mbox{ as
}k\rightarrow+\infty.$

Thus, by the decrease of the energy, it result

(3.80) $(T-2\epsilon)E_{v_{k}}(T-\epsilon)\rightarrow 0,\mbox{ as
}k\rightarrow+\infty.$

The energy identity implies that

$E_{v_{k}}(T-\epsilon)-E_{v_{k}}(\epsilon)=-\int_{\epsilon}^{T-\epsilon}\int_{\Omega}\frac{1}{\sigma_{k}}a(x)g(\nabla\partial_{t}u_{k})\nabla\partial_{t}v_{k}+\eta(x)|\partial_{t}v_{k}|^{2}\;dxdt.$

This combining with (3.45) and (3.80) and the arbitrariness of $\epsilon$, we
achieve the desired contradiction

$E_{v_{k}}(0)\rightarrow 0,\mbox{ as }k\rightarrow+\infty,$

and the proof of Lemma 3.2 is, thereby, finished. ∎

###### Proof.

of Theorem 3.1 In order to obtain the decay rate of energy to the problem
(1.1), it sufficient to apply the Lemma 3.3. Indeed, Since (3.30), (3.31) we
have

(3.81) $\begin{array}[]{lll}E_{u}(T)&\leq&\displaystyle
C\sum_{i=1}^{n}\int_{0}^{T}\int_{\Omega}a(x)\left(|\partial_{x_{i}}\partial_{t}u|^{2}+|g_{i}(\partial_{x_{i}}\partial_{t}u)|^{2}\right)dxdt+C\int_{0}^{T}\int_{\Omega}\eta(x)|\partial_{t}u|^{2}\;dxdt,\\\
&\leq&\displaystyle
C\sum_{i=1}^{n}\int_{\Sigma_{A}^{i}\cup\Sigma_{B}^{i}}a(x)\left(|\partial_{x_{i}}\partial_{t}u|^{2}+|g_{i}(\partial_{x_{i}}\partial_{t}u)|^{2}\right)dxdt+C\int_{0}^{T}\int_{\Omega}\eta(x)|\partial_{t}u|^{2}\;dxdt,\end{array}$

where $\Sigma_{A}^{i}=\left\\{(t,x)\in
Q_{T};\;\;|\partial_{x_{i}}\partial_{t}u|>1\mbox{ a.e.}\right\\}$,
$\Sigma_{B}^{i}=Q_{T}\backslash\Sigma_{A}^{i}$ and $Q_{T}$ is already defined
in the begin of this section.

On other hand, taking in mind the hypothesis (1.6), we find

(3.82)
$\int_{\Sigma_{A}^{i}}a(x)\left(|\partial_{x_{i}}\partial_{t}u|^{2}+|g_{i}(\partial_{x_{i}}\partial_{t}u)|^{2}\right)\;dxdt\leq(m^{-1}+M)\int_{\Sigma_{A}^{i}}a(x)g_{i}(\partial_{x_{i}}\partial_{t}u)\partial_{x_{i}}\partial_{t}u\;dxdt,$

and, since (3.27), the facts that, for $i=1,..,n$, $h_{i}$ is a concave,
increasing function and that $\frac{a(x)}{1+\left\|a\right\|_{\infty}}\leq
a(x)$, it follows

(3.83)
$\begin{array}[]{lll}\displaystyle\int_{\Sigma_{B}^{i}}a(x)\left(|\partial_{x_{i}}\partial_{t}u|^{2}+|g_{i}(\partial_{x_{i}}\partial_{t}u)|^{2}\right)dxdt\\\
\leq\displaystyle\int_{\Sigma_{B}^{i}}a(x)h_{i}\left(g_{i}(\partial_{x_{i}}\partial_{t}u)\partial_{x_{i}}\partial_{t}u\right)\;dxdt\\\
\leq\displaystyle\int_{\Sigma_{B}^{i}}(1+\left\|a\right\|_{\infty})h_{i}\left(\frac{a(x)}{1+\left\|a\right\|_{\infty}}g_{i}(\partial_{x_{i}}\partial_{t}u)\partial_{x_{i}}\partial_{t}u\right)\;dxdt\\\
\leq\displaystyle\int_{\Sigma_{B}^{i}}(1+\left\|a\right\|_{\infty})h_{i}\left(a(x)g_{i}(\partial_{x_{i}}\partial_{t}u)\partial_{x_{i}}\partial_{t}u\right)\;dxdt.\end{array}$

Thereafter, using Jensen’s inequality in (3.83), we have

(3.84)
$\begin{array}[]{lll}\displaystyle\int_{\Sigma_{B}^{i}}a(x)\left(|\partial_{x_{i}}\partial_{t}u|^{2}+|g_{i}(\partial_{x_{i}}\partial_{t}u)|^{2}\right)dxdt\\\
\leq\displaystyle(1+\left\|a\right\|_{\infty})\,\mbox{meas}(Q_{T})h_{i}\left(\frac{1}{\mbox{meas}(Q_{T})}\int_{Q_{T}}a(x)g_{i}(\partial_{x_{i}}\partial_{t}u)\partial_{x_{i}}\partial_{t}u\;dxdt\right)\\\
\leq\displaystyle(1+\left\|a\right\|_{\infty})\,\mbox{meas}(Q_{T})\hat{h}_{i}\left(\int_{Q_{T}}a(x)g_{i}(\partial_{x_{i}}\partial_{t}u)\partial_{x_{i}}\partial_{t}u\;dxdt\right).\end{array}$

Therewith, substituting (3.82) and (3.84) in (3.81), we can deduce

(3.85) $\begin{array}[]{lll}E_{u}(T)&\leq&\displaystyle
C(m^{-1}+M)\int_{Q_{T}}a(x)g(\nabla\partial_{t}u)\nabla\partial_{t}u\;dxdt+C\int_{Q_{T}}\eta(x)|\partial_{t}u|^{2}\;dxdt\\\
&+&\displaystyle
C(1+\left\|a\right\|_{\infty})\,\mbox{meas}(Q_{T})\sum_{i=1}^{n}\hat{h}_{i}\left(\int_{Q_{T}}a(x)g_{i}(\partial_{x_{i}}\partial_{t}u)\partial_{x_{i}}\partial_{t}u\;dxdt\right)\\\
&\leq&\displaystyle
C(m^{-1}+M)\int_{Q_{T}}a(x)g(\nabla\partial_{t}u)\nabla\partial_{t}u\;dxdt+\int_{Q_{T}}\eta(x)|\partial_{t}u|^{2}\;dxdt\\\
&+&\displaystyle
C(1+\left\|a\right\|_{\infty})\mbox{meas}(Q_{T})\hat{h}\left(\int_{Q_{T}}a(x)g(\nabla\partial_{t}u)\nabla\partial_{t}u\;dxdt\right).\end{array}$

Therefore, since the increasing of $\hat{h}$, it follows that

(3.86)
$\begin{array}[]{lll}LE_{u}(T)&\leq&\displaystyle\mu\left\\{\int_{Q_{T}}a(x)g(\nabla\partial_{t}u)\nabla\partial_{t}u\;dxdt+\int_{Q_{T}}\eta(x)|\partial_{t}u|^{2}\;dxdt\right\\}\\\
&+&\displaystyle\hat{h}\left(\int_{Q_{T}}a(x)g(\nabla\partial_{t}u)\nabla\partial_{t}u\;dxdt+\int_{Q_{T}}\eta(x)|\partial_{t}u|^{2}\;dxdt\right)\\\
&\leq&\displaystyle\left(\mu
I+\hat{h}\right)\left(\int_{Q_{T}}a(x)g(\nabla\partial_{t}u)\nabla\partial_{t}u\;dxdt+\int_{Q_{T}}\eta(x)|\partial_{t}u|^{2}\;dxdt\right),\end{array}$

where

$L=\frac{1}{C(1+\left\|a\right\|_{\infty})\mbox{meas}(Q_{T})}\;\;\mbox{ and
}\;\;\mu=\frac{1+m^{-1}+M}{(1+\left\|a\right\|_{\infty})\mbox{meas}(Q_{T})}.$

Furthermore, taking the function $z$ in (3.28), we obtain

(3.87)
$z(E_{u}(T))\leq\int_{Q_{T}}a(x)g(\nabla\partial_{t}u)\nabla\partial_{t}u\;dxdt+\int_{Q_{T}}\eta(x)|\partial_{t}u|^{2}\;dxdt,$

which implies, thanks to (3.30), that

(3.88) $z(E_{u}(T))+E_{u}(T)\leq E(0).$

Whereupon, we replace $T$ (resp. $0$) in (3.88) with $(m+1)T$ (resp. $mT$) to
obtain

(3.89) $z(E_{u}((m+1)T))+E_{u}((m+1)T)\leq E(mT),\;\;\;\mbox{for
}\;m=0,1,2,...$

Now, to deduce the proof of Theorem 3.1, it sufficient to use the following
lemma due to Lasiecka and Tataru [25].

######  Lemma 3.3.

Let $z$ be a positive, increasing function such that $z(0)=0$. Since $z$ is
increasing, we can define an increasing function $\mathcal{R}$,
$\mathcal{R}=x-(I+z)^{-1}(x)$. Consider a sequence $s_{m}$ of positive numbers
which satisfies

$s_{m+1}+z(s_{m+1})\leq s_{m}.$

Then, $s_{m}\leq S(m)$ where $S(t)$ is a solution of the differential equation

$\frac{d}{dt}S(t)+\mathcal{R}S(t)=0,\;\;\;S(0)=s_{0}.$

Moreover, if $z(x)>0$ for $x>0$, then
$\displaystyle\lim_{t\rightarrow+\infty}S(t)=0$.

Applying Lemma 3.3 with $s_{m}=E(mT)$, thus results in

(3.90) $E_{u}(mT)\leq S(m),\;\;\mbox{ for }\;m=0,1,2...$

Lastly, using the dissipativity of $E(t)$ and S(t), we have for $t=mT+\tau$
and $0\leq\tau\leq T$,

(3.91) $E_{u}(t)\leq E_{u}(mT)\leq S(m)=S\left(\frac{t-\tau}{T}\right)\leq
S\left(\frac{t}{T}-1\right),\;\;\mbox{ for }\;t>T.$

The proof of decay for $E_{u}$, and thus of the Theorem 3.1, is now complete.
∎

##  4\. Appendix: Microlocal Analysis Background

For the readers comprehension, we will announce some results which can be
found in Burq and Gérard [11] and in Gérard [22] and were used in the proof of
the uniform stability result.

######  Theorem 4.1.

Let $\\{u_{n}\\}_{n\in\mathbb{N}}$ be a bounded sequence in
$L_{loc}^{2}(\mathcal{O})$ such that it converges weakly to zero in
$L_{loc}^{2}(\mathcal{O})$. Then, there exists a subsequence
$\\{u_{\varphi(n)}\\}$ and a positive Radon measure $\mu$ on
$T^{1}\mathcal{O}:=\mathcal{O}\times S^{d-1}$ such that for all pseudo-
differential operator $A$ of order $0$ on $\Omega$ which admits a principal
symbol $\sigma_{0}(A)$ and for all $\chi\in C_{0}^{\infty}(\mathcal{O})$ such
that $\chi\sigma_{0}(A)=\sigma_{0}(A)$, one has

(4.92) $\displaystyle\left(A(\chi u_{\varphi(n)}),\chi
u_{\varphi_{n}}\right)_{L^{2}}\underset{n\rightarrow+\infty}{\longrightarrow}\int_{\mathcal{O}\times
S^{n-1}}\sigma_{0}(A)(x,\xi)\,d\mu(x,\xi).$

######  Definition 4.2.

Under the circumstances of Theorem 4.1 $\mu$ is called the microlocal defect
measure of the sequence $\\{u_{\varphi(n)}\\}_{n\in\mathbb{N}}$.

######  Remark 4.3.

Theorem 4.1 assures that for all bounded sequence
$\\{u_{n}\\}_{n\in\mathbb{N}}$ of $L_{loc}^{2}(\mathcal{O})$ which converges
weakly to zero, the existence of a subsequence admitting a microlocal defect
measure. We observe that from (4.92) in the particular case when $A=f\in
C_{0}^{\infty}(\mathcal{O})$, it follows that

(4.93)
$\displaystyle\int_{\Omega}f(x)|u_{\varphi(n)}(x)|^{2}\,dx\rightarrow\int_{\mathcal{O}\times
S^{d-1}}f(x)\,d\mu(x,\xi),$

so that $u_{\varphi(n)}$ converges to $0$ strongly if and only if $\mu=0$.

The second important result reads as follows.

######  Theorem 4.4.

Let $P$ be a differential operator of order $m$ on $\Omega$ and let
$\\{u_{n}\\}$ a bounded sequence of $L_{loc}^{2}(\mathcal{O})$ which converges
weakly to $0$ and admits a m.d.m. $\mu$. The following statement are
equivalents:

$\displaystyle(i)$ $\displaystyle
Pu_{n}\underset{n\rightarrow+\infty}{\longrightarrow}0\hbox{ strongly in
}H_{loc}^{-m}(\mathcal{O})\leavevmode\nobreak\ (m>0).$ $\displaystyle(ii)$
$\displaystyle\hbox{supp}(\mu)\subset\\{(x,\xi)\in\mathcal{O}\times
S^{n-1}:\sigma_{m}(P)(x,\xi)=0\\}.$

######  Theorem 4.5.

Let $P$ be a differential operator of order $m$ on $\Omega$, verifying
$P^{\ast}=P$, and let $\\{u_{n}\\}$ be a bounded sequence in
$L_{loc}^{2}(\mathcal{O})$ which converges weakly to $0$ and it admits a
m.d.m. $\mu$. Let us assume that
$Pu_{n}\underset{n\rightarrow+\infty}{\longrightarrow}0$ strongly in
$H_{loc}^{1-m}(\mathcal{O})$. Then, for all function $a\in
C^{\infty}(\mathcal{O}\times\mathbb{R}^{n}\backslash\\{0\\})$ homogeneous of
degree $1-m$ in the second variable and with compact support in the first one,

(4.94) $\displaystyle\int_{\Omega\times
s^{n-1}}\\{a,p\\}(x,\xi)\,d\mu(x,\xi)=0.$

Next, Let us consider the wave operator in the following general setting:

$\displaystyle\rho(x)\partial_{t}^{2}-\sum_{i,j=1}^{n}\partial_{x_{i}}\left[K(x)\partial_{x_{j}}\right].$

Using the notation $D_{j}=\frac{1}{i}\partial_{j}$ we can write

$\displaystyle
P(t,x,D_{t},D_{x})=-\rho(x)D_{t}^{2}+\sum_{i,j=1}^{n}D_{x_{i}}[K(x)D_{x_{j}}],\quad
D_{x}=(D_{x_{1}},\cdots,D_{x_{n}}),$

whose principal symbol $p(t,x,\tau,\xi)$ is given by

(4.95) $\displaystyle
p(t,x,\tau,\xi)=-\rho(x)\tau^{2}+K(x)\,\xi\cdot\xi,\quad\xi=(\xi_{1},\cdots,\xi_{n}),$

where $t\in\mathbb{R}$, $x\in\Omega\subset\mathbb{R}^{d}$,
$(\tau,\xi)\in\mathbb{R}\times\mathbb{R}^{d}$.

######  Proposition 4.6.

Unless a change of variables, the bicharacteristics of (4.95) are curves of
the form

$\displaystyle
t\mapsto\left(t,x(t),\tau,-\tau\left(\frac{K(x(t))}{\rho(x(t))}\right)^{-1}\dot{x}(t)\right),$

where $t\mapsto x(t)$ is a geodesic of the metric $G=\left(K/\rho\right)^{-1}$
on $\Omega$, parameterized by the curvilinear abscissa.

The main result is the following:

######  Theorem 4.7.

Let $P$ be a self-adjoint differential operator of order $m$ on $\mathcal{O}$
which admits a principal symbol $p$. Let $\\{u_{n}\\}_{n}$ be a bounded
sequence in $L_{loc}^{2}(\mathcal{O})$ which converges weakly to zero, with a
microlocal defect measure $\mu$. Let us assume that $Pu_{n}$ converges to $0$
in $H_{loc}^{-(m-1)}(\mathcal{O})$. Then the support of $\mu$,
$\hbox{supp}(\mu)$, is a union of curves like $s\in
I\mapsto\left(x(s),\frac{\xi(s)}{|\xi(s)|}\right)$, where $s\in
I\mapsto(x(s),\xi(s))$ is a bicharacteristic of $p$.

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