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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 129, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/129\hfil Dynamic evolution of damage]
{Dynamic evolution of damage in elastic-thermo-viscoplastic materials}

\author[A. Merouani, F. Messelmi\hfil EJDE-2010/129\hfilneg]
{Abdelbaki Merouani, Farid Messelmi}  % in alphabetical order

\address{Abdelbaki Merouani \newline
Departement de Mathematiques, Univerisite de Bordj Bou Arreridj,
Bordj Bou Arreridj 34000, Algeria}
\email{badri\_merouani@yahoo.fr}

\address{Farid Messelmi \newline
Departement de Mathematiques, Univerisite Zian Achour de Djelfa,
Djelfa 17000, Algeria}
\email{foudimath@yahoo.fr}

\thanks{Submitted July 20, 2010. Published September 8, 2010.}
\subjclass[2000]{74H20, 74H25, 74M15, 74F05, 74R20}
\keywords{Damage field; temperature; elastic-thermo-viscoplastic;
\hfill \break \indent variational inequality}

\begin{abstract}
 We consider a mathematical model that describes the dynamic
 evolution  of damage in elastic-thermo-viscoplastic materials with
 displacement-traction, and Neumann and Fourier boundary conditions.
 We derive a weak formulation of the system consisting of a
 motion equation, an energy equation, and an evolution damage
 inclusion. This system has an integro-differential variational
 equation for the displacement and the stress fields, and a variational
 inequality for the damage field. We prove existence and uniqueness
 of the solution, and  the positivity of the temperature.
\end{abstract}

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

\section{Introduction}

The constitutive laws with internal variables has been used  in
various publications in order to model the effect of internal
variables in the behavior of real bodies like metals, rocks
polymers and so on, for which the rate of deformation depends on
the internal variables. Some of the internal state variables
considered by many authors are the spatial display of dislocation,
the work-hardening of materials, the absolute temperature and the
damage field, see for examples and details
\cite{b1,c1,n1,m1,m2,s3,s4}
and references therein for the case of hardening, temperature
and other internal state variables and the references
\cite{f2,f3,g1,m1,r1,s1} for the case of damage field.

The aim of this paper is to study the dynamic evolution of damage
in elastic-thermo-viscoplastic materials. For this, we consider a
rate-type constitutive equation with two internal variables of the
form
\begin{equation}
\begin{aligned}
\mathbf{\sigma }(t)
&=\mathcal{A}(\varepsilon (
\dot{\mathbf{u}}(t)))+\mathcal{E}(\varepsilon (\mathbf{u}(t)))\\
&\quad +\int_0^{t}\mathcal{G}\Big(\mathbf{\sigma }(s)
 -\mathcal{A}(\varepsilon (\dot{\mathbf{u}}(s)))
,\varepsilon (\mathbf{u}(s)),\theta ( s),\varsigma(s)\Big)ds,
\end{aligned} \label{e1.1}
\end{equation}
in which $\mathbf{u}$, $\mathbf{\sigma }$ represent, respectively,
the displacement field and the stress field where the dot above
denotes the derivative with respect to the time variable, $\theta
$ represents the absolute temperature, $\varsigma $ is the damage
field, $\mathcal{A}$ and $ \mathcal{E}$ are nonlinear operators
describing the purely viscous and the elastic properties of the
material, respectively, and $\mathcal{G}$ is a nonlinear
constitutive function which describes the visco-plastic behavior
of the material.

Examples and mechanical interpretation of elastic-viscoplastic can
be found in \cite{c2,i1}. Dynamic and quasistatic contact problems are
the topic of numerous papers, e.g. \cite{a1,a2,a4,f1,r2}, and the
comprehensive references \cite{h1,s2}. However, the mathematical
problem modelled the quasi-static evolution of damage in
thermo-viscoplastic materials has been studied in \cite{m1}.

The paper is organized as follows. In Section 2 we present the
mechanical problem of the dynamic evolution of damage in
elastic-thermo-viscoplastic materials. We introduce some notations
and preliminaries and we derive the variational formulation of the
problem. We prove in Section 3 the existence and uniqueness of the
solution as well as the positivity of the temperature.

\section{Statement of the Problem}

Let $\Omega \subset \mathbb{R} ^n$ $(n=2,3)$ be a bounded domain
with a Lipschitz boundary $\Gamma $, partitioned into two disjoint
measurable parts $\Gamma_1$ and $\Gamma _2$ such that
$\operatorname{meas}(\Gamma _1)>0$. We denote by $\mathbb{S}_{n}$
the space of symmetric tensors on $ \mathbb{R} ^n$. We define
the inner product and the Euclidean norm on $ \mathbb{R} ^n$ and
$\mathbb{S}_{n}$, respectively, by
\begin{gather*}
\mathbf{u}\cdot \mathbf{v}=u_{i}v_{i}\quad \forall  \mathbf{u},
\mathbf{v}\in \mathbb{R}^n, \quad
\mathbf{\sigma }\cdot \mathbf{\tau }=\sigma _{ij}\tau
_{ij}\quad \forall \mathbf{\sigma }, \mathbf{\tau }\in
\mathbb{S}_{n}, \\
| \mathbf{u}| =(\mathbf{u}\cdot \mathbf{u})^{1/2}\quad
\forall \mathbf{u}\in \mathbb{R}^n, \quad
| \mathbf{\sigma }| =(\mathbf{\sigma }\cdot \mathbf{\sigma })^{1/2}
\quad \forall \mathbf{\sigma }\in \mathbb{S}_{n}.
\end{gather*}

Here and below, the indices $i$ and $j$ run from $1$ to $n$ and the
summation convention over repeated indices is used.
We shall use the notation
\begin{gather*}
H=L^2(\Omega )^n=\{ \mathbf{u}=\{ u_{i}\}
: u_{i}\in L^2(\Omega )\} , \\
\mathcal{H}=\{ \mathbf{\sigma }=\{ \sigma _{ij}\} :
\sigma _{ij}=\sigma _{ji}\in L^2(\Omega )\} , \\
H_1=\{ \mathbf{u}\in H: \varepsilon (\mathbf{u}) \in
\mathcal{H}\} , \\
\mathcal{H}_1=\{ \mathbf{\sigma }\in \mathcal{H}:
\operatorname{Div}(\mathbf{\sigma })\in H\} , \\
V=H^{1}(\Omega ).
\end{gather*}
Here $\varepsilon :H_1\to \mathcal{H}$ and
$\operatorname{Div}: \mathcal{H}_1\to H$ are the deformation
and divergence operators, respectively, defined by
\[
\varepsilon (\mathbf{u})=(\varepsilon _{ij}(\mathbf{ u}) ) ,\quad
\varepsilon _{ij}( \mathbf{u})=
\frac{1}{2}(u_{i,j}+u_{j,i}), \quad
\operatorname{Div}(\mathbf{\sigma })=(\sigma _{ij,j}).
\]
The sets $H$, $\mathcal{H}$, $H_1$, $\mathcal{H}_1$ and $V$
are real Hilbert spaces endowed with the canonical inner products:
\begin{gather*}
(\mathbf{u},\mathbf{v})_{H}=\int_{\Omega }u_{i}v_{i}dx,\quad
(\mathbf{\sigma },\mathbf{\tau })_{\mathcal{H} }=\int_{\Omega
}\sigma _{ij}\tau _{ij}dx,
\\
(\mathbf{u},\mathbf{v})_{H_1}=( \mathbf{u},\mathbf{v} )
_{H}+(\varepsilon (\mathbf{u}),\varepsilon (
\mathbf{v}))_{\mathcal{H}}, \\
(\mathbf{\sigma },\mathbf{\tau })_{\mathcal{H}_1}=( \mathbf{\sigma
},\mathbf{\tau })_{\mathcal{H}}+( \operatorname{Div} (\mathbf{\sigma
}),\operatorname{Div}( \mathbf{\tau })
)_{H}, \\
(f,g)_{V}=(f,g)_{L^2(\Omega )}+( f_{x_{i}},g_{x_{i}})
_{L^2(\Omega )}\,.
\end{gather*}
The associated norms  are denoted by
$\|\cdot \|_{H}$, $\| \cdot\| _{\mathcal{H}}$,
$\|\cdot \| _{H_1}$, $ \| \cdot\| _{\mathcal{H}_1}$
and $\| \cdot\| _{V}$. Since the
boundary $\Gamma $ is Lipschitz continuous, the unit outward
normal vector field $\mathbf{\nu }$ on the boundary is defined
a.e. For every vector field $\mathbf{v}\in H_1$ we denote by
$v_{\nu }$ and $\mathbf{v}_{\tau }$ the normal and tangential
components of $\mathbf{v}$ on the boundary given by
\[
v_{\nu }=\mathbf{v}\cdot \mathbf{\nu },\quad
\mathbf{v}_{\tau }=\mathbf{ v-}v_{\nu }\mathbf{\nu }.
\]
Let $H_{\Gamma }=(H^{1/2}(\Gamma ))^n$ and $\gamma
:H_1\to H_{\Gamma }$ be the trace map. We denote by
$\mathcal{V}$ the closed subspace of $H_1$ defined by
\[
\mathcal{V}=\{ \mathbf{v}\in H_1: \gamma \mathbf{v}=0
\text{ on } \Gamma _1\} .
\]
We also denote by $H_{\Gamma }'$ the dual of
$H_{\Gamma}$. Moreover, since \textit{meas}$(\Gamma _1)>0$, Korn's
inequality holds and thus, there exists a positive constant
$C_0$ depending only on $\Omega $, $\Gamma _1$ such that
\[
\| \varepsilon (\mathbf{v})\| _{\mathcal{H} }\geq
C_0\| \mathbf{v}\| _{H_1}\quad \forall
\mathbf{v}\in \mathcal{V}.
\]
Furthermore, if $\mathbf{\sigma }\in \mathcal{H}_1$ there exists
an element $\mathbf{\sigma }\nu \in H_{\Gamma }'$ such
that the following Green formula holds
\[
(\mathbf{\sigma},\varepsilon (\mathbf{v}))_{
\mathcal{H}}+(\operatorname{Div}(\mathbf{\sigma }),\mathbf{v} )
_{H}=(\mathbf{\sigma }\nu ,\gamma \mathbf{v}) _{H_{\Gamma
}'\times H_{\Gamma }}\quad \forall \mathbf{v}\in H_1.
\]
In addition, if $\mathbf{\sigma }$ is sufficiently regular
(say $\mathcal{C}^{1}$), then
\[
(\mathbf{\sigma },\varepsilon (\mathbf{v}))_{
\mathcal{H}}+(\operatorname{Div}(\mathbf{\sigma }),\mathbf{v} )
_{H}=\int_{\Gamma }\mathbf{\sigma }\nu \cdot \gamma \mathbf{v}
d\gamma \quad  \forall \mathbf{v}\in H_1.
\]
where $d\gamma $ denotes the surface element. Similarly, for a
regular tensor field $\mathbf{\sigma }:\Omega \to
\mathbb{S}_{n}$ we
define its normal and tangential components on the boundary by
\[
\sigma _{\nu }=\mathbf{\sigma }\nu \cdot \mathbf{\nu },\quad
\mathbf{ \sigma }_{\tau }=\mathbf{\sigma }\nu -\sigma _{\nu
}\mathbf{\nu }.
\]
Moreover, we denote by $\mathcal{V}'$ and $V'$
the dual of the spaces $\mathcal{V}$ and $V$, respectively.
Identifying $H$, respectively $L^2(\Omega )$, with its own dual,
we have the inclusions
\[
\mathcal{V}\subset H\subset \mathcal{V}', \quad
V\subset L^2(\Omega )\subset V'.
\]
We use the notation $\langle \cdot,\cdot\rangle
_{\mathcal{V}'\times \mathcal{V}}$, $\langle
\cdot,\cdot\rangle _{V'\times V}$ to represent the duality
pairing between $\mathcal{V}',\mathcal{V}$ and $V'$, $V$, respectively.

For the rest of this article, we will denote by $c$ possibly different
positive constants depending only on the data of the problem.

The physical setting is the following. An
elastic-thermo-viscoplastic body occupies the domain $\Omega $. We
assume that the body is clamped on
$\Gamma _1\times ( 0,T)$, $(T>0)$ and therefore the displacement
field vanishes there.
Surface tractions of density $\mathbf{f} _0$ acts on $\Gamma
_2\times (0,T)$ and a volume forces of density $\mathbf{f}$ is
applied in $\Omega \times (0,T)$. In addition, we admit a possible
external heat source applied in $\Omega \times (0,T)$, given by
the function $q$.

The mechanical problem may be formulated as follows.

\subsection*{Problem (P)}
Find the displacement field $\mathbf{u}:\Omega \times
(0,T)\to \mathbb{R}^n$, the stress field
$\mathbf{\sigma }:\Omega \times (0,T)\to \mathbb{S}_{n}$,
the temperature $\theta :\Omega \times (0,T)\to
\mathbb{R}$ and the damage field $\varsigma :\Omega \times (0,T)
\to \mathbb{R}$ such that
\begin{gather}
\begin{aligned}
\mathbf{\sigma }(t)&=\mathcal{A}(\varepsilon (
\dot{\mathbf{u}}(t)))+\mathcal{E}(
\varepsilon (\mathbf{u}(t)))
 + \int_0^{t}\mathcal{G}\Big(\mathbf{\sigma }(s)\\
&\quad -
\mathcal{A}(\varepsilon (\dot{\mathbf{u}}(s)))
,\varepsilon (\mathbf{u}(s))
,\theta (s),\varsigma (s)\Big)ds
\quad \text{in  $\Omega$ a.e. }t\in (0,T),
\end{aligned}  \label{e2.1}
\\
\rho \ddot{\mathbf{u}}=\operatorname{Div}(\mathbf{\sigma }) +\mathbf{f}\quad
\text{in  }\Omega \times (0,T), \label{e2.2}
\\
\rho \dot{\theta}-k_0\Delta \theta =\psi (\mathbf{\sigma},
 \varepsilon (\dot{\mathbf{u}}),\theta ,
\varsigma )+q\quad \text{in }\Omega \times ( 0,T), \label{e2.3}
\\
\rho \dot{\varsigma}-k_1\Delta \varsigma +\partial _{K}\varphi (
\varsigma )\ni \phi (\mathbf{\sigma }, \varepsilon
(\mathbf{u}), \theta , \varsigma )\quad \text{in }\Omega \times (0,T),
\label{e2.4}
\\
\mathbf{u}=0\quad \text{on }\Gamma _1\times (0,T), \label{e2.5}
\\
\mathbf{\sigma }\nu =\mathbf{f}_0\quad \text{on }\Gamma _2\times (0,T),  \label{e2.6}
\\
k_0\frac{\partial \theta }{\partial \nu }+\beta \theta
=0\quad \text{on } \Gamma \times (0,T),  \label{e2.7}
\\
\frac{\partial \varsigma }{\partial \nu }=0\quad \text{on }
\Gamma \times (0,T),  \label{e2.8}
\\
\mathbf{u}(0)=\mathbf{u}_0,\quad
\dot{\mathbf{u}}( 0)=\mathbf{w},\quad
\theta (0)=\theta _0, \quad
\varsigma (0)=\varsigma _0\quad \text{in }\Omega . \label{e2.9}
\end{gather}


This problem represents the dynamic evolution of damage in
elastic-thermo-viscoplastic materials. Equation \eqref{e2.1} is the
elastic-thermo-viscoplastic constitutive law where $\mathcal{A}$
and $ \mathcal{E}$ are nonlinear operators describing the purely
viscous and the elastic properties of the material, respectively,
and $\mathcal{G}$ is a nonlinear constitutive function which
describes the viscoplastic behavior of the material. \eqref{e2.2}
represents the equation of motion in which the dot above denotes
the derivative with respect to the time variable and $\rho $ is
the density of mass. Equation \eqref{e2.3} represents the energy
conservation where $ \psi $ is a nonlinear constitutive function
which represents the heat generated by the work of internal forces
and $q$ is a given volume heat source. Inclusion \eqref{e2.4}
describes the evolution of damage field, governed by the source
damage function $\phi$, where $\partial _{K}\varphi ( \varsigma
)$ is the subdifferential of indicator function of the set $ K$ of
admissible damage functions given by
\[
K=\{ \xi \in V: 0\leq \xi (x)\leq 1\text{ \ a.e. }x\in
\Omega \} ,
\]
in such a way that the damage function $\varsigma $ varied between
$0$ and $1$. If $\varsigma =1$ there is no damage in the
material, if $\varsigma =0$ the material is completely damaged and
if $0<\varsigma <1$ the material is partially damaged.

Equalities \eqref{e2.5} and \eqref{e2.6} are the displacement-traction
boundary conditions, respectively. \eqref{e2.7}, \eqref{e2.8} represent, respectively on
$\Gamma $, a Fourier boundary condition for the temperature and an
homogeneous Neumann boundary condition for the damage field on
$\Gamma $. Finally the functions $ \mathbf{u}_0$, $\mathbf{w}$,
$\theta _0\ $and $\varsigma _0$ in \eqref{e2.9} are the initial data.

In the study of the mechanical problem (P), we consider the
following hypotheses

$\mathcal{A}:\Omega \times \mathbb{S}_{n}\to \mathbb{S}_{n}$
satisfies the following properties:
\begin{equation}
\parbox{10cm}{

(a) There exists an $L_{\mathcal{G}}>0$ such that
$|\mathcal{A}(x,\mathbf{\varepsilon }_1) -\mathcal{A}
(x,\mathbf{\varepsilon }_2)| \leq L_{\mathcal{A}
}| \mathbf{\varepsilon }_1-\mathbf{\varepsilon
}_2|$ for all $\mathbf{\varepsilon }_1,\mathbf{\varepsilon }
_2\in \mathbb{S}_{n}$  a.e. $x\in \Omega$;

(b) There exists an $m_{\mathcal{A}}$ such that
$(\mathcal{A}(x,\mathbf{\varepsilon }_1)-\mathcal{A}
(x,\mathbf{\varepsilon }_2)).(\mathbf{ \varepsilon
}_1-\mathbf{\varepsilon }_2)\geq m_{\mathcal{A} }|
\mathbf{\varepsilon }_1-\mathbf{\varepsilon }_2|
^2$ for all $\mathbf{\varepsilon }_1,\mathbf{\varepsilon }
_2\in \mathbb{S}_{n}$ a.e. $x\in \Omega$;

(c) The mapping $x\mapsto \mathcal{A}(x, \mathbf{\varepsilon })$
\ is Lebesgue measurable on $\Omega$ for all
$\mathbf{\varepsilon }\in \mathbb{S}_{n}$;

(d) The mapping $x\mapsto \mathcal{A}(x,0)\in
\mathcal{H}$.
}
  \label{e2.10}
\end{equation}

$\mathcal{E}:\Omega \times \mathbb{S}_{n}\to \mathbb{S}_{n}$
satisfies the following properties:
\begin{equation}
\parbox{10cm}{

(a) There exists an $L_{\mathcal{E}}>0$ such that
$| \mathcal{E}(x,\mathbf{\varepsilon }_1) -\mathcal{E}
(x,\mathbf{\varepsilon }_2)| \leq L_{\mathcal{E}
}| \mathbf{\varepsilon }_1-\mathbf{\varepsilon
}_2|$ for all $\mathbf{\varepsilon }_1, \mathbf{\varepsilon }
_2\in \mathbb{S}_{n}$ a.e. $x\in \Omega$;

(b) The mapping $x\mapsto \mathcal{E}(x,\mathbf{\varepsilon } )$
 is Lebesgue measurable on $\Omega$ for all
$\mathbf{\varepsilon }\in \mathbb{S}_{n}$;

(c) The mapping $x\mapsto \mathcal{E}(x,0)\in
\mathcal{H}$.
}  \label{e2.11}
\end{equation}

$\mathcal{G}:\Omega \times \mathbb{S}_{n}\times \mathbb{S}_{n}\times
\mathbb{R}\times\mathbb{R} \to \mathbb{S}_{n}$
satisifes the following properties:
\begin{equation}
\parbox{10cm}{

(a) There exists an $L_{\mathcal{G}}>0$  such that
$| \mathcal{G}(x,\mathbf{\sigma }_1,\mathbf{\varepsilon }
_1,\theta _1,\varsigma _1)-\mathcal{G}( x,\mathbf{\sigma }
_2,\mathbf{\varepsilon }_2,\theta _2,\varsigma _2) |
\leq L_{\mathcal{G}}(| \mathbf{\sigma }_1-\mathbf{\sigma }_2|
+| \mathbf{\varepsilon }_1-\mathbf{\varepsilon
}_2| +| \theta _1-\theta _2|
+| \varsigma _1-\varsigma _2| )$
for all $\mathbf{\sigma }_1, \mathbf{\sigma }_2\in
\mathbb{S}_{n}$,  for all $\mathbf{\varepsilon }_1,
\mathbf{\varepsilon } _2\in \mathbb{S}_{n}$, for all
$\theta _1, \theta _2\in \mathbb{R}$,
for all $\varsigma _1, \varsigma _2\in \mathbb{R}$ a.e.
$x\in \Omega$;

(b) The mapping $x\to \mathcal{G}(\mathbf{x},\mathbf{ \sigma },
\mathbf{\varepsilon },\theta ,\varsigma )$ is Lebesgue
measurable on $\Omega$ for all
$\mathbf{\sigma }, \mathbf{\varepsilon}\in \mathbb{S}_{n}$,
for all $\theta ,  \varsigma \in\mathbb{R}$;

(c) The mapping $x\to \mathcal{G}(x,0,0,0,0)\in \mathcal{H}$.
} \label{e2.12}
\end{equation}

$\psi :\Omega \times \mathbb{S}_{n}\times \mathbb{S}_{n}\times
\mathbb{R}\times\mathbb{R}\to \mathbb{R}$ satisfies
 the following properties:
\begin{equation}
\parbox{10cm}{

(a) There exists an $L_{\psi }>0$ such that
$| \psi (x,\mathbf{\sigma }_1,\varepsilon _1,\theta
_1,\varsigma _1)-\psi (x,\mathbf{\sigma }_2,\varepsilon
_2,\theta _2,\varsigma _2)|
\leq L_{\psi}(| \mathbf{\sigma }_1-\mathbf{\sigma }_2|
+| \varepsilon _1-\varepsilon _2|
+| \theta _1-\theta _2| +| \varsigma _1-\varsigma _2| )$
for all $\mathbf{\sigma }_1, \mathbf{\sigma }_2\in
\mathbb{S}_{n}$,  for all $\varepsilon _1, \varepsilon
_2\in \mathbb{S}_{n}$,  for all
$\theta _1, \theta_2\in \mathbb{R}$, for all
$\varsigma _1, \varsigma _2\in\mathbb{R}$ a.e.
$x\in \Omega$;

(b) The mapping $x\to \psi (x,\mathbf{\sigma }
,\varepsilon ,\theta ,\varsigma )$ is Lebesgue measurable
on $\Omega$ for all $\mathbf{\sigma }, \varepsilon \in
\mathbb{S}_{n}$, for all $\theta , \varsigma \in
\mathbb{R}$;

(c) The mapping $x\to \psi (x,0,0,0,0)\in L^2(\Omega )$.
}  \label{e2.13}
\end{equation}


$\phi :\Omega \times \mathbb{S}_{n}\times \mathbb{S}_{n}\times
\mathbb{R}\times \mathbb{R}\to\mathbb{R}$
satisfies the following properties:
\begin{equation}
\parbox{10cm}{

(a) There exists an $L_{\phi }>0$  such that
$| \phi (x,\mathbf{\sigma }_1,\mathbf{\varepsilon }
_1,\theta _1,\varsigma _1)-\phi ( x,\mathbf{\sigma }_2,
\mathbf{\varepsilon }_2,\theta _2,\varsigma _2) | \leq
L_{\phi }(| \mathbf{\sigma }_1-\mathbf{\sigma }_2|
+| \mathbf{ \varepsilon }_1-\mathbf{\varepsilon
}_2| +| \theta _1-\theta _2|
+| \varsigma _1-\varsigma _2| )$
for all $\mathbf{\sigma }_1, \mathbf{\sigma }_2\in
\mathbb{S}_{n}$,  for all
$\mathbf{\varepsilon }_1, \mathbf{\varepsilon } _2\in \mathbb{S}_{n}$,
for all $\theta _1, \theta _2\in\mathbb{R}$, for all
$\varsigma _1, \varsigma _2\in\mathbb{R}$ a.e. $x\in \Omega$;

(b) The mapping $x\mapsto \phi (x,\mathbf{\sigma
},\mathbf{\varepsilon },\theta ,\varsigma )$ is Lebesgue measurable
on $\Omega$ for all $\mathbf{\sigma }, \mathbf{ \varepsilon }
\in \mathbb{S}_{n}$, for all $\theta, \varsigma \in\mathbb{R}$;

(c) The mapping $x\mapsto \phi (x,0,0,0,0)\in L^2(\Omega )$.
}\label{e2.14}
\end{equation}

\begin{equation}
\begin{gathered}
\rho \in L^{\infty }(\Omega ),\quad
 \rho \geq \rho ^{\ast }>0. \\
\mathbf{f}\in L^2(0,T;H),\quad
\mathbf{f}_0\in L^2(0,T;L^2(\Gamma _2)^n). \\
q\in L^2(0,T;L^2(\Omega )).
\end{gathered} \label{e2.15}
\end{equation}

\begin{gather}
\mathbf{u}_0\in \mathcal{V},\quad \mathbf{w}_0\in H,\quad
\theta_0\in V,\quad \varsigma _0\in K.  \label{e2.16}
\\
k_{i}>0,\quad i=0, 1.  \label{e2.17}
\end{gather}
We denote by $\mathbf{F}(t)\in \mathcal{V}'$ the
following element
\begin{equation}
\langle \mathbf{F}(t),\mathbf{v}\rangle _{\mathcal{V}
'\times \mathcal{V}}=( \mathbf{f}( t),\mathbf{v} )
_{H}+(\mathbf{f}_0(t),\gamma \mathbf{v})
_{L^2(\Gamma _2)^n}\quad
\forall \mathbf{v}\in \mathcal{V},\ \ t\in (0,T).
\label{e2.18}
\end{equation}

The use of \eqref{e2.15} permits to verify that
\begin{equation}
\mathbf{F}\in L^2(0,T;\mathcal{V}').  \label{e2.19}
\end{equation}
We introduce the following continuous functionals
\begin{gather}
\mathfrak{a}_0:V\times V\to\mathbb{R},\quad
\mathfrak{a}_0(\zeta ,\xi )=k_0\int_{\Omega }\nabla
\zeta \cdot \nabla \xi dx+\beta \int_{\Gamma }\zeta \xi d\gamma ,\\
\mathfrak{a}_1:V\times V\to\mathbb{R},\quad
\mathfrak{a}_1(\zeta ,\xi )=k_1\int_{\Omega }\nabla \zeta
\cdot \nabla \xi dx.  \label{e2.22}
\end{gather}
Using the above notation and Green's formula, we derive the following
variational formulation of mechanical problem (P).

\subsection*{Problem PV}
Find the displacement field $\mathbf{u}:\Omega \times
(0,T)\to\mathbb{R}^n$, the stress field
$\mathbf{\sigma }:\Omega \times (0,T)\to \mathbb{S}_{n}$, the
temperature $\theta:\Omega \times (0,T)\to\mathbb{R}$ and the
damage field $\varsigma :\Omega \times (0,T)\to\mathbb{R}$ such that
\begin{gather}
\begin{aligned}
\mathbf{\sigma }(t)
&=\mathcal{A}(\varepsilon (
\dot{\mathbf{u}}(t)))+\mathcal{E}(
\varepsilon (\mathbf{u}(t)))\\
&\quad +  \int_0^{t}\mathcal{G}(\mathbf{\sigma }(s)-
\mathcal{A}(\varepsilon (\dot{\mathbf{u}}(s)))
,\varepsilon (\mathbf{u}(s)),\theta ( s),\varsigma
(s))ds\quad \text{a.e. } t\in (0,T),
\end{aligned}\label{e2.24}
\\
\langle \rho \ddot{\mathbf{u}}(t) ,\mathbf{v}\rangle
_{\mathcal{V}'\times \mathcal{V}}+(\mathbf{\sigma
}(t),\varepsilon ( \mathbf{v}))_{\mathcal{H} }=\langle
\mathbf{F}(t),\mathbf{v}\rangle _{\mathcal{
V}'\times \mathcal{V}}\quad
\forall \mathbf{v}\in \mathcal{V},\text{ a.e. }t\in (0,T) ,\label{e2.25}
\\
\begin{aligned}
&\langle \rho \dot{\theta}(t),\omega \rangle _{V'\times V}
+\mathfrak{a}_0(\theta (t),\omega)\\
&= \langle \psi (\mathbf{\sigma }(t),\varepsilon (
\dot{\mathbf{u}}(t)),\theta (t),\varsigma (t)),\omega
\rangle _{V'\times V}+(q(t),\omega ) _{L^2(\Omega)}\\
&\quad \forall \omega \in V,\text{ a.e. }t\in (0,T),
\end{aligned} \label{e2.26}
\\
\begin{aligned}
&\langle \rho \dot{\varsigma}
(t),\xi -\varsigma (t)\rangle _{V'\times
V}+\mathfrak{a}_1(\varsigma (t),\xi -\varsigma (t))\\
&\geq  \langle \phi (\mathbf{\sigma }(t),\varepsilon (
\mathbf{u}(t)),\theta (t),\varsigma ( t)),\xi -\varsigma (t)
\rangle _{V'\times V} \\
&  \quad \forall \xi \in K,\text{ a.e. }t\in (0,T), \varsigma (t)\in K,
\end{aligned} \label{e2.27}
\\
\mathbf{u}(0)=\mathbf{u}_0,\quad
\dot{\mathbf{u}}( 0) =\mathbf{w},\quad
\theta (0)=\theta _0, \quad
\varsigma(0)=\varsigma _0\quad \text{in }\Omega . \label{e2.28}
\end{gather}

\section{Main Results}

The main results are stated by the following theorems.

\begin{theorem}[Existence and uniqueness] \label{thm1}
Under  assumptions \eqref{e2.10}-\eqref{e2.17}, there exists a
unique solution $\{ \mathbf{u},\sigma ,\theta ,\varsigma \} $
to problem (PV). Moreover, the solution has the regularity
\begin{gather}
\mathbf{u}\in \mathcal{C}^{0}(0,T;\mathcal{V})\cap \mathcal{C}
^{1}(0,T;H),  \label{e3.1} \\
\dot{\mathbf{u}}\in L^2(0,T;\mathcal{V}),  \label{e3.2} \\
\ddot{\mathbf{u}}\in L^2(0,T;\mathcal{V}'),
\label{e3.3} \\
\mathbf{\sigma }\in L^2(0,T;\mathcal{H}),  \label{e3.4} \\
\theta \in L^2(0,T;V)\cap \mathcal{C}^{0}(
0,T;L^2(\Omega )),  \label{e3.5} \\
\dot{\theta}\in L^2(0,T;V'),  \label{e3.6} \\
\varsigma \in L^2(0,T;V)\cap \mathcal{C}^{0}(
0,T;L^2(\Omega )),  \label{e3.7} \\
\dot{\varsigma}\in L^2(0,T;V').  \label{e3.8}
\end{gather}
\end{theorem}

The proof will be done in several steps. Based on classical
arguments of functional analysis concerning variational problems,
and Banach fixed point theorem.


First step. Take an arbitrary element
\begin{equation}
(\mathbf{\eta },\lambda ,\mu )\in \mathbf{L}^2(0,T;
\mathcal{V}'\times V'\times V'),
\label{e3.9}
\end{equation}
and consider the  auxiliary problem.

\subsection*{Problem PV1$_{(\eta ,\lambda ,\mu )}$}
Find the displacement field $\mathbf{u}_{\eta }:\Omega \times
(0,T) \to \mathbb{R} ^n$, the temperature $\theta_{\lambda}:\Omega
\times (0,T)\to \mathbb{R}$ and the damage field
$\varsigma _{\mu }:\Omega \times(0,T)\to\mathbb{R}$ which are solutions
of the variational system
\begin{gather}
\begin{gathered}
\langle \rho \ddot{\mathbf{u}}_{\eta }(t),\mathbf{v}
\rangle _{\mathcal{V}'\times \mathcal{V}}+(
\mathcal{A} (\varepsilon (\dot{\mathbf{u}}_{\eta }(t) ) )
,\varepsilon (\mathbf{v}))_{\mathcal{H} }+\langle
\mathbf{\eta }(t),\mathbf{v}\rangle _{ \mathcal{V}'\times \mathcal{V}}=\langle \mathbf{F}(t)
,\mathbf{v}\rangle _{\mathcal{V}'\times
\mathcal{V}} \\
\forall \mathbf{v}\in \mathcal{V},\text{ a.e. }t\in (0,T) ,
\end{gathered} \label{e3.10}
\\
\begin{gathered}
\langle \rho \dot{\theta}_{\lambda }(t),\omega \rangle
_{V'\times V}+\mathfrak{a}_0(\theta _{\lambda
}(t),\omega )=\langle \lambda ( t)
+q(t),\omega \rangle _{V'\times V}   \\
\forall \omega \in V,\text{ a.e. }t\in (0,T),
\end{gathered} \label{e3.11}
\\
\begin{aligned}
&\langle \rho \dot{
\varsigma}_{\mu }(t),\xi  -\varsigma _{\mu }(t)
\rangle _{V'\times V}+\mathfrak{a}_1(\varsigma
_{\mu }(t),\xi  -\varsigma _{\mu }(t))\\
&\geq   \langle \mu  ,\xi  -\varsigma _{\mu }(t)
\rangle _{V'\times V}\quad \forall \xi \in
K,\text{ a.e. }t\in (0,T),\;
\varsigma _{\mu }(t)\in K,
\end{aligned} \label{e3.12}
\\
\mathbf{u}_{\eta }(0)=\mathbf{u}_0,\quad
\dot{\mathbf{u}}_{\eta }(0)=\mathbf{w},\quad
\theta _{\lambda }(0)=\theta _0, \quad
\varsigma _{\mu }( 0)=\varsigma _0\quad \text{in }\Omega. \label{e3.13}
\end{gather}

\begin{lemma} \label{lem2}
 For  all
$(\mathbf{\eta} ,\lambda ,\mu )\in L^2(0,T;\mathcal{V}'\times
V'\times V')$, there exists a unique solution
$\{ \mathbf{u}_{\eta },\theta _{\lambda },\varsigma _{\mu}\} $
 to the auxiliary problem PV1$_{(\mathbf{\eta },\lambda ,\mu )}$
satisfying
\eqref{e3.1}-\eqref{e3.3} and \eqref{e3.5}-\eqref{e3.8}.
\end{lemma}

\begin{proof}
Let us introduce the operator $A:\mathcal{V}\to \mathcal{V}'$,
\begin{equation}
\langle A\mathbf{ u},\mathbf{v}\rangle
_{\mathcal{V}'\times \mathcal{V}}=( \mathcal{A}(
\varepsilon (\mathbf{u})),\varepsilon ( \mathbf{v}))
_{\mathcal{H}}. \label{e3.14}
\end{equation}
It follows from  hypothesis \eqref{e2.10} that
\[
\| A\mathbf{u}-A\mathbf{v}\|
_{\mathcal{V}'}\leq L_{\mathcal{A}}\|
\mathbf{u}-\mathbf{v}\| _{\mathcal{V}} \quad \forall
\mathbf{u}, \mathbf{v}\in \mathcal{V}.
\]
Which proves that $A$ is bounded and hemi-continuous on
$\mathcal{V}$.

On the other hand, by  \eqref{e2.10} and Korn's inequality, we find
for every $\mathbf{v}\in \mathcal{V}$,
\[
\frac{\langle A\mathbf{v},\mathbf{v}\rangle
_{\mathcal{V} '\times \mathcal{V}}}{\|
\mathbf{v}\| _{\mathcal{V} }}\geq
C_0^2m_{\mathcal{A}}\| \mathbf{v}\|
_{\mathcal{V} }.
\]
The passage to the limit in this inequality when
$\|\mathbf{v} \| _{\mathcal{V}}\to +\infty $
implies that $A$ is coercive in $\mathcal{V}$.

Next, by definition of $A$, the use of  \eqref{e2.10} and Korn's
inequality permits also to obtain
\[
\langle
A\mathbf{u}-A\mathbf{v},\mathbf{u}-\mathbf{v}\rangle _{
\mathcal{V}'\times
\mathcal{V}}>C_0^2m_{\mathcal{A}}\|
\mathbf{u}-\mathbf{v}\| _{\mathcal{V}}\quad \text{if }\mathbf{u\neq v }.
\]
Then $A$ is strict monotone.
Therefore, \eqref{e3.10} can be rewritten, making use the
operator $A$, as follows
\begin{equation}
\rho \ddot{\mathbf{u}}_{\eta }(t)+A(\dot{\mathbf{u}} _{\eta
}(t))=\mathbf{F}_{\eta }(t)\quad\text{on }\mathcal{V}'\text{ a.e. }
t\in (0, T), \label{e3.15}
\end{equation}
where
\[
\mathbf{F}_{\eta }(t)=\mathbf{F}(t)-\mathbf{\eta } (t)\in
\mathcal{V}'.
\]

We recall that by \eqref{e2.19} we have
$\mathbf{F}_{\eta }\in L^2(0,T;\mathcal{V}')$.
Kipping in mind that the operator $A$ is strict monotone,
hemi-continuous, bounded and coercive, then by using classical
arguments of functional
analysis concerning parabolic equations \cite{b3,l1} we can
easily prove the existence and uniqueness of $\mathbf{w}_{\eta }$
satisfying
\begin{gather}
\mathbf{w}_{\eta }\in L^2(0,T;\mathcal{V})\cap \mathcal{C}
^{0}(0,T;H),  \label{e3.16} \\
\mathbf{\dot{w}}_{\eta }\in L^2(0,T;\mathcal{V}') ,
\label{e3.17} \\
\rho \mathbf{\dot{w}}_{\eta }(t)+A(\mathbf{w}_{\eta }( t) )
=\mathbf{F}_{\eta }(t)\text{ \ on \ }
\mathcal{V}'\text{ \ a.e. }t\in (0,T),  \label{e3.18} \\
\mathbf{w}_{\eta }(0)=\mathbf{w}_0.  \label{e3.19}
\end{gather}

Consider now the function $\mathbf{u}_{\eta }:(0,T)\to \mathcal{V}$
defined by
\begin{equation}
\mathbf{u}_{\eta }(t)=\int_0^{t}\mathbf{w}_{\eta
}(s)ds+\mathbf{u}_0\quad \forall t\in (0,T).  \label{e3.20}
\end{equation}
It follows from \eqref{e3.18} and \eqref{e3.19} that
$\mathbf{u}_{\eta }$ is a solution of the equation \eqref{e3.15}
and it satisfies \eqref{e3.1}-\eqref{e3.3}.

Furthermore, by an application of the Poincar\'{e}-Friedrichs
inequality, we can find a constant $\beta '>0$ such that
\[
\int_{\Omega }| \nabla \zeta |
^2dx+\frac{\beta }{ k_0}\int_{\Gamma }| \zeta
| ^2d\gamma \geq \beta '\int_{\Omega
}| \zeta | ^2dx\quad \forall \zeta \in V.
\]
Thus, we obtain
\begin{equation}
\mathfrak{a}_0(\zeta , \zeta )\geq c_1\| \zeta
\| _{V}^2\quad \forall \zeta \in V, \label{e3.21}
\end{equation}
where $c_1=k_0\min (1,\beta ')/2$, which
implies that $\mathfrak{a}_0$ is $V-$elliptic. Consequently,
based on classical arguments of functional analysis concerning
parabolic equations, the variational equation \eqref{e3.11} has a unique
solution $\theta _{\lambda }$ satisfies
\eqref{e3.5}-\eqref{e3.6}.

On the other hand, we know that the form $\mathfrak{a}_{1 }$ is not
$V$-elliptic. To solve this problem we introduce the functions
\[
\tilde{\varsigma}_{\mu }(t)=e^{-k_1t}\varsigma _{\mu}(t), \quad
\tilde{\xi}(t)=e^{-k_1t}\xi (t).
\]
We remark that if $\varsigma _{\mu }$, $\xi \in K$ then
$\tilde{\varsigma} _{\mu }$, $\tilde{\xi}\in K$. Consequently,
\eqref{e3.12} is equivalent to the inequality
\begin{equation}
\begin{aligned}
&\langle \rho \overset{\mathbf{ \cdot }}{\tilde{\varsigma}}_{\mu }(t)
,\tilde{\xi} - \tilde{\varsigma}_{\mu }(t) \rangle
_{V'\times V}+ \mathfrak{a}_1( \tilde{\varsigma}_{\mu
}(t),\tilde{\xi}  -\tilde{\varsigma}_{\mu }(t))
+k_1(\rho \tilde{\varsigma}_{\mu
},\tilde{\xi}-\tilde{\varsigma}_{\mu }(t))_{L^2(\Omega ) }\\
&\geq \langle e^{-k_1t}\mu ,\tilde{\xi} -\tilde{\varsigma}_{\mu }(t)
\rangle _{V'\times V}\quad \forall \tilde{\xi}\in K,\text{ a.e. }
t\in (0,T),\; \tilde{\varsigma}_{\mu }\in K.
\end{aligned} \label{e3.22}
\end{equation}
The fact that
\begin{equation}
\mathfrak{a}_1(\tilde{\xi},\tilde{\xi})+k_1(\rho
\tilde{\xi},\tilde{\xi} )_{L^2(\Omega )}\geq k_1\min
(\rho ^{\ast },1)\| \tilde{\xi}\| _{V}^2\quad
\forall \tilde{\xi}\in V, \label{e3.23}
\end{equation}
and using classical arguments of functional analysis
concerning parabolic inequalities \cite{b3,d1}, implies
that  \eqref{e3.22} has a unique solution $\tilde{\varsigma}_{\mu }$
having the regularity \eqref{e3.7}-\eqref{e3.8}.
This completes the proof .
\end{proof}

Let us consider now the auxiliary problem.

\subsection*{Problem PV2$_{(\eta ,\lambda ,\mu ) }$}
 Find the stress field $\mathbf{\sigma }_{\eta ,\lambda ,\mu
}:\Omega \times (0,T)\to \mathbb{S}_{n}$ which is a solution of
the  problem
\begin{equation}
\begin{aligned}
\mathbf{\sigma }_{\eta ,\lambda ,\mu }(t)
&=\mathcal{E}( \varepsilon (\mathbf{u}_{\eta }(t)))
 +\int_0^{t}\mathcal{G}\Big(\mathbf{\sigma }_{\eta ,\lambda
,\mu }(s)\\
&\quad -\mathcal{A}(\varepsilon (\dot{\mathbf{u}} _{\eta
}(s))),\varepsilon (\mathbf{ u}_{\eta }(s)),\theta_{\lambda }(s)
,\varsigma _{\mu }(s)\Big)ds  \quad
\text{a.e. }t\in (0,T),
\end{aligned} \label{e3.24}
\end{equation}

\begin{lemma} \label{lem3}
There exists a unique solution of Problem PV2$_{(\eta ,\lambda ,\mu )}$
and it satisfies  \eqref{e3.4}. Moreover, if
$\{\mathbf{u}_{\eta },\theta _{\lambda _{i}},\varsigma _{\mu
_{i}}\} $ and $\mathbf{\sigma }_{\eta ,\lambda ,\mu }$
represent the solutions of problems
PV1$_{(\eta_{i},\lambda _{i},\mu _{i})}$ and
PV2$_{( \eta_{i},\lambda _{i},\mu _{i})}$, respectively,
for $i=1,2$, then there exists $c>0$ such that
\begin{equation}
\begin{aligned}
&\| \mathbf{\sigma }_{\eta _1,\lambda _1,\mu
_1}(t)-\mathbf{\sigma }_{\eta _2,\lambda _2,\mu _2}(t)\|
_{\mathcal{H}}^2\\
&\leq c\int_0^{t}\Big(\| \dot{\mathbf{u}}_{\eta _1}(s)
-\dot{\mathbf{u}}_{\eta _2}(s)\| _{\mathcal{V}}^2
+\| \mathbf{u} _{\eta_1}( s) -\mathbf{u}_{\eta _2}(s)
\| _{\mathcal{V}}^2\\
&\quad +\| \theta _{\lambda _1}(s)-\theta _{\lambda
_2}(s)\| _{V}^2+\| \varsigma _{\mu_1}(s)
-\varsigma _{\mu _2}(s)\| _{V}^2\Big)ds.
\end{aligned}\label{e3.25}
\end{equation}
\end{lemma}

\begin{proof}
Let $\Sigma _{\eta ,\lambda ,\mu }:L^2(0,T;\mathcal{H})
\to L^2(0,T;\mathcal{H})$ be the mapping given by
\begin{equation}
\begin{aligned}
&\Sigma _{\eta ,\lambda ,\mu }\mathbf{\sigma }(t) \\
&=\mathcal{E}(\varepsilon (\mathbf{u}_{\eta }(t)))+
\int_0^{t}\mathcal{G}(\mathbf{\sigma }(s)-
\mathcal{A}(\varepsilon (\dot{\mathbf{u}}_{\eta }(s)) )
,\varepsilon (\mathbf{u}_{\eta }(s)),\theta _{\lambda
}(s),\varsigma _{\mu }(s))ds.
\end{aligned}  \label{e3.26}
\end{equation}
Let $\mathbf{\sigma }_{i}\in L^2(0,T;\mathcal{H})$, $i=1,2$ and
$t_1\in (0,T)$. We find be using hypothesis \eqref{e2.12}
and H\"older's inequality
\begin{equation}
\| \Sigma _{\eta ,\lambda ,\mu }\mathbf{\sigma }_1(
t_1)-\Sigma _{\eta ,\lambda ,\mu }\mathbf{\sigma
}_2(t_1)\| _{\mathcal{H}}^2\leq L_{\mathcal{G}
}^2T\int_0^{t_1}\| \mathbf{\sigma
}_1(s)-\mathbf{\sigma }_2(s)\| _{\mathcal{H}}^2ds.
\label{e3.27}
\end{equation}
Integration on the time interval  $(0,t_2)\subset (0,T) $, it
follows that
\[
\int_0^{t_2}\| \Sigma _{\eta ,\lambda ,\mu
}\mathbf{ \sigma }_1(t_1)-\Sigma _{\eta ,\lambda ,\mu
}\mathbf{\sigma }_2(t_1)\| _{\mathcal{H}}^2dt_1\leq L_{
\mathcal{G}}^2T\int_0^{t_2}\int_0^{t_1}\|
\mathbf{\sigma }_1(s)-\mathbf{\sigma }_2(s)\|
_{\mathcal{H}}^2dsdt_1.
\]
Using again \eqref{e3.27}, it follows that
\[
\| \Sigma _{\eta ,\lambda ,\mu }\mathbf{\sigma }_1(
t_2)-\Sigma _{\eta ,\lambda ,\mu }\mathbf{\sigma
}_2(t_2)\| _{\mathcal{H}}^2\leq L_{\mathcal{G}
}^{4}T^2\int_0^{t_2}\int_0^{t_1}\|
\mathbf{ \sigma }_1(s)-\mathbf{\sigma }_2( s) \|
_{\mathcal{H}}^2dsdt_1.
\]
For $t_1, t_2,\dots,t_{n}\in (0,T)$, we generalize the
procedure above by recurrence on $n$. We obtain the
inequality
\begin{align*}
&\| \Sigma _{\eta ,\lambda ,\mu }\mathbf{\sigma }_1(
t_{n})-\Sigma _{\eta ,\lambda ,\mu }\mathbf{\sigma }_2(
t_{n})\| _{\mathcal{H}}^2\\
&\leq L_{\mathcal{G}}^{2n}T^n\int_0^{tn}\dots \int_0^{t_2}
\int_0^{t_1}\| \mathbf{\sigma }_1( s) -
\mathbf{\sigma }_2(s)\| _{\mathcal{H}
}^2\,ds\,dt_1\dots dt_{n-1}.
\end{align*}
Which implies
\[
\| \Sigma _{\eta ,\lambda ,\mu }\mathbf{\sigma }_1(
t_{n})-\Sigma _{\eta ,\lambda ,\mu }\mathbf{\sigma
}_2(t_{n})\| _{\mathcal{H}}^2\leq \frac{L_{\mathcal{G}
}^{2n}T^{n+1}}{n!}\int_0^{T}\| \mathbf{\sigma
}_1(s)-\mathbf{\sigma }_2(s)\| _{\mathcal{H} }^2ds.
\]
Thus, we can infer, by integrating over the interval time $(0,T)$,
that
\[
\| \Sigma _{\eta ,\lambda ,\mu }\mathbf{\sigma }_1-\Sigma
_{\eta ,\lambda ,\mu }\mathbf{\sigma }_2\|
_{L^2(0,T;\mathcal{H} )}^2\leq
\frac{L_{\mathcal{G}}^{2n}T^{n+2}}{n!}\| \mathbf{ \sigma
}_1-\mathbf{\sigma }_2\| _{L^2( 0,T;\mathcal{H} )}^2.
\]

It follows from this inequality that for $n$ large enough, a power
$n$ of the mapping $\Sigma _{\eta ,\lambda ,\mu }$ is a
contraction on the space $ L^2(0,T;\mathcal{H})$ and, therefore,
from the Banach fixed point theorem, there exists a unique element
$\mathbf{\sigma }_{\eta ,\lambda ,\mu }\in L^2(0,T;\mathcal{H})$
such that $\Sigma _{\eta ,\lambda ,\mu }\mathbf{\sigma }_{\eta
,\lambda ,\mu }=\mathbf{\sigma } _{\eta ,\lambda ,\mu }$, which
represents the unique solution of the problem
PV2$_{(\eta,\lambda ,\mu )}$. Moreover, if
$\{ \mathbf{u}_{\eta },\theta
_{\lambda _{i}},\varsigma _{\mu _{i}}\} $ and
$\mathbf{\sigma }_{\eta ,\lambda ,\mu }$ represent the solutions
of problem PV1$_{(\eta _{i},\lambda _{i},\mu _{i}) }$ and
PV2$_{(\eta _{i},\lambda _{i},\mu _{i})}$, respectively,
for $i=1,2$, then we use \eqref{e2.10}, \eqref{e2.11}, \eqref{e2.12}
 and Young's inequality to obtain
\begin{align*}
&\| \mathbf{\sigma }_{\eta _1,\lambda _1,\mu
_1}(t)-\mathbf{\sigma }_{\eta _2,\lambda _2,\mu _2}(t)\|
_{\mathcal{H}}^2\\
&\leq c\| \mathbf{\sigma }_{\eta
_1,\lambda _1,\mu _1}(t) -\mathbf{\sigma }_{\eta
_2,\lambda _2,\mu _2}(t)\| _{\mathcal{H}}^2
+ c\int_0^{t}\Big(\| \dot{\mathbf{u}}_{\eta_1}(s)
-\dot{\mathbf{u}}_{\eta _2}(s)\| _{\mathcal{V}}^2\\
&\quad +\| \mathbf{u}_{\eta _1}( s)
 -\mathbf{u}_{\eta_2}(s)\| _{\mathcal{V}}^2
 +\| \theta _{\lambda_1}( s)-\theta _{\lambda _2}(s)\| _{V}^2
 +\| \varsigma _{\mu _1}(s)-\varsigma _{\mu_2}(s)\| _{V}^2\Big)ds.
\end{align*}
Which permits us to obtain, using Gronwall's lemma, the
inequality \eqref{e3.25}.

\noindent Second step. Let us consider the mapping
\[
\Lambda :L^2(0,T;\mathcal{V}'\times V'\times
V')\to L^2( 0,T;\mathcal{V}'\times
V'\times V'),
\]
defined by
\begin{equation}
\begin{aligned}
&\Lambda (\mathbf{\eta }(t),\lambda (t),\mu ( t))\\
&=\Big(\Lambda_0(\mathbf{\eta }(t),\lambda (t),\mu (t)),
\psi \big(\mathbf{\sigma }_{\eta ,\lambda ,\mu }(
t),\varepsilon (\dot{\mathbf{u}}_{\eta }(t)),\theta _{\lambda
}(t),\varsigma _{\mu }(t)\big),\\
&\quad \phi \big(\mathbf{\sigma }_{\eta
,\lambda ,\mu }( t),\varepsilon (\mathbf{u}_{\eta }(t) ) ,\theta
_{\lambda }(t),\varsigma _{\mu }(t)\big)\Big),
\end{aligned} \label{e3.28}
\end{equation}
where the mapping $\Lambda _0$ is given by
\begin{equation}
\begin{aligned}
&\langle \Lambda _0(\mathbf{\eta }(t),\lambda ( t) ,\mu
(t)),\mathbf{v}\rangle _{ \mathcal{V}'\times
\mathcal{V}}\\
&=\Big(\mathcal{E}( \varepsilon (\mathbf{u}_{\eta }(t)))
+  \int_0^{t}\mathcal{G}\Big(\mathbf{\sigma }_{\eta
,\lambda ,\mu }(s)\\
&\quad -\mathcal{A}(\varepsilon ( \dot{\mathbf{u}}
_{\eta }(s))),\varepsilon (\mathbf{ u}_{\eta }(s) )
,\theta _{\lambda }(s),\varsigma _{\mu }(s)\Big)ds, \varepsilon
(\mathbf{v})\Big)_{\mathcal{H}}.
\end{aligned} \label{e3.29}
\end{equation}
\end{proof}

\begin{lemma} \label{lem4}
 The mapping $\Lambda $ has a fixed point
\[
(\mathbf{\eta }^{\ast },\lambda ^{\ast },\mu ^{\ast })\in
L^2(0,T;\mathcal{V}'\times V'\times V').
\]
\end{lemma}

\begin{proof}
Let $t\in (0,T)$ and
\[
(\mathbf{\eta }_1,\lambda _1,\mu _1), ( \mathbf{\eta
}_2,\lambda _2,\mu _2)\in L^2( 0,T;\mathcal{ V}'\times
V'\times V').
\]
Let us start by using hypotheses \eqref{e2.10}, \eqref{e2.11}
and \eqref{e2.12} to obtain
\begin{equation}
\begin{aligned}
&\| \Lambda _0(\mathbf{\eta }_1(t),\lambda _1(t),\mu
_1(t))-\Lambda _0( \mathbf{\eta }_2(t),\lambda _2(t) ,\mu _2(
t))\| _{\mathcal{V}'}\\
&\leq L_{\mathcal{E}}\| \mathbf{u}_{\eta _1}(t)-\mathbf{u}
_{\eta _2}(t)\| _{\mathcal{V}}
+L_{\mathcal{G}}\int_0^{t}
\Big(\| \mathbf{\sigma}_{\eta _1,\lambda _1,\mu _1}(s)
-\mathbf{\sigma }_{\eta _2,\lambda_2,\mu _2}(s)\| _{\mathcal{H}}\\
&\quad +L_{ \mathcal{A}}\|
\dot{\mathbf{u}}_{\eta _1}(s) -\mathbf{ \dot{u}}_{\eta_2}(s)
\| _{\mathcal{V}}
 +  \| \mathbf{u}_{\eta _1}(s)
 -\mathbf{u}_{\eta_2}(s)\| _{\mathcal{V}}\\
&\quad +\| \theta _{\lambda_1}(s)-\theta _{\lambda _2}(s)\|
 _{L^2(\Omega)}
+\| \varsigma _{\mu _1}(s)-\varsigma _{\mu_2}(s)\| _{L^2(\Omega )}\Big)ds
    \quad \text{a.e. }t\in (0,T).
\end{aligned} \label{e3.30}
\end{equation}
On the other hand, we know that for a.e. $t\in (0,T)$,
\begin{equation}
\| \mathbf{u}_{\eta _1}(t)-\mathbf{u}_{\eta
_2}(t)\| _{\mathcal{V}}\leq \int_0^{t}\|
\dot{\mathbf{u}}_{\eta _1}( s)-\dot{\mathbf{u}}_{\eta
_2}(s)\| _{\mathcal{V}}ds.  \label{e3.31}
\end{equation}
Applying Young's and H\"older's inequalities, \eqref{e3.30} becomes,
via \eqref{e3.31},
\begin{equation}
\begin{aligned}
&\| \Lambda _0(\mathbf{\eta }_1(t),\lambda _1(t),\mu
_1(t))-\Lambda _0( \mathbf{\eta }_2(t),\lambda _2(t) ,\mu _2(
t))\| _{\mathcal{V}'}^2\\
&\leq c\int_0^{t}\Big(\| \mathbf{\sigma }_{\eta
_1,\lambda _1,\mu _1}(s)-\mathbf{\sigma }_{\eta _2,\lambda _2,\mu
_2}(s)\| _{\mathcal{H} }^2
+\|\dot{\mathbf{u}}_{\eta _1}(s)-\mathbf{\dot{u} }_{\eta
_2}(s)\| _{\mathcal{V}}^2\\
&\quad  + \| \mathbf{u}_{\eta _1}(s) -\mathbf{u}_{\eta
_2}(s)\| _{\mathcal{V}}^2+\| \theta _{\lambda
_1}( s)-\theta _{\lambda _2}(s)\| _{V}^2+\|
\varsigma _{\mu _1}(s)-\varsigma _{\mu _2}( s) \|
_{V}^2\Big)ds
\end{aligned} \label{e3.32}
\end{equation}
a.e. $t\in (0,T)$.
Furthermore, we find by taking the substitution $\mathbf{\eta
}=\mathbf{\eta }_1$, $\mathbf{\eta }=\mathbf{\eta }_2$ in \eqref{e3.10}
and choosing $\mathbf{v }=\dot{\mathbf{u}}_{\mathbf{\eta
}_1}-\dot{\mathbf{u}}_{\mathbf{\eta } _2} $ as test function
\begin{align*}
&\langle \rho (\ddot{\mathbf{u}}_{\eta _1}(t)-
\ddot{\mathbf{u}}_{\eta _2}(t))+A\dot{\mathbf{u}} _{\eta
_1}(t)-A\dot{\mathbf{u}}_{\eta _2}( t),  \dot{\mathbf{u}}_{\eta _1}(t)-\dot{\mathbf{u}}_{\eta
_2}(t)\rangle _{\mathcal{V}'\times \mathcal{V}}
\\
&=\langle \mathbf{\eta }_2(t)-\mathbf{\eta }_1(t)
,\dot{\mathbf{u}}_{\eta _1}(t)-\dot{\mathbf{u}} _{\eta
_2}(t)\rangle _{\mathcal{V}'\times \mathcal{V}}\quad
\text{a.e. }t\in (0,T).
\end{align*}
By virtue of  \eqref{e2.10}, this equation becomes
\begin{align*}
&\frac{(\rho ^{\ast })^2}{2}\frac{d}{dt}\| \mathbf{
\dot{u}}_{\eta _1}(t)-\dot{\mathbf{u}}_{\eta _2}(t) \|
_{H}^2+m_{\mathcal{A}}\| \dot{\mathbf{u}} _{\eta
_1}(t)-\dot{\mathbf{u}}_{\eta _2}(t)
\| _{\mathcal{V}}^2 \\
&\leq \| \mathbf{\eta }_2(t)-\mathbf{\eta }_1(t)
\| _{\mathcal{V}'}\| \dot{\mathbf{u}}
_{\eta _1}(t)-\dot{\mathbf{u}}_{\eta _2}(t)\|
_{\mathcal{V}}.
\end{align*}
Integrating this inequality over the interval time variable
$(0, t)$, Young inequality leads to
\begin{align*}
&(\rho ^{\ast })^2\| \dot{\mathbf{u}}_{\eta
_1}(t)-\dot{\mathbf{u}}_{\eta _2}(t)\|
_{H}^2+m_{\mathcal{A}}\int_0^{t}\| \mathbf{
\dot{u}}_{\eta _1}(s)-\dot{\mathbf{u}}_{\eta _2}(
s)\| _{\mathcal{V}}^2ds \\
&\leq \frac{2}{m_{\mathcal{A}}}\int_0^{t}\|
\mathbf{\eta } _1(s)-\mathbf{\eta }_2(s)\| _{
\mathcal{V}'}^2ds.
\end{align*}
Consequently,
\begin{equation}
\int_0^{t}\| \dot{\mathbf{u}}_{\eta _1}( s)-
\dot{\mathbf{u}}_{\eta _2}(s)\| _{\mathcal{V} }^2ds\leq
c\int_0^{t}\| \mathbf{\eta }_1(s)-\mathbf{\eta
}_2(s)\| _{\mathcal{V} '}^2ds\quad\text{a.e. }t\in (0,T).
\label{e3.33}
\end{equation}
which also implies, using a variant of \eqref{e3.31}, that
\begin{equation}
\| \mathbf{u}_{\eta _1}(s)-\mathbf{u}_{\eta
_2}(s)\| _{\mathcal{V}}^2\leq
c\int_0^{t}\| \mathbf{\eta }_1(s) -\mathbf{ \eta
}_2(s)\| _{\mathcal{V}'}^2ds\quad \text{a.e.}t\in (0,T),  \label{e3.34}
\end{equation}
Moreover, if we take the substitution $\lambda =\lambda _1$,
$\lambda =\lambda _2$ in \eqref{e3.11} and subtracting the two
obtained equations, we deduce by choosing $\omega =\theta
_{\lambda _1}-\theta _{\lambda _2}$ as test function
\begin{align*}
&\frac{(\rho ^{\ast })^2}{2}\| \theta _{\lambda
_1}(t)-\theta _{\lambda _2}(t)\| _{L^2(\Omega
)}^2+c_1\int_0^{t}\| \theta _{\lambda _1}(s)-\theta
_{\lambda _2}(s)
\| _{V}^2ds \\
&\leq \int_0^{t}\| \lambda _1(s)-\lambda _2(s)
\| _{V'}\| \theta _{\lambda
_1}(s)-\theta _{\lambda _2}(s)\| _{V}ds\quad \text{a.e. }t\in (0,T).
\end{align*}
Employing H\"older's and Young's inequalities, we deduce that
\begin{equation}
\begin{aligned}
\| \theta _{\lambda _1}(t)-\theta _{\lambda
_2}(t)\| _{L^2(\Omega ) }^2+\int_0^{t}\|
\theta _{\lambda _1}( s)-\theta
_{\lambda _2}(s)\| _{V}^2ds   \\
\leq c\int_0^{t}\| \lambda _1(s)-\lambda
_2(s)\| _{V'}^2ds\quad \text{a.e. }t\in (0,T).
\end{aligned}\label{e3.35}
\end{equation}
Substituting now $\{ \mu =\mu _1,\xi =\tilde{\varsigma}_{\mu
_1}\} $, $\{ \mu =\mu _2,\xi =\tilde{\varsigma}_{\mu
_2}\} $ in \eqref{e3.22} and subtracting the two
inequalities, we obtain
\begin{align*}
&\| \tilde{\varsigma}_{\mu _1}(t) -\tilde{\varsigma} _{\mu
_2}(t)\| _{L^2( \Omega ) }^2+\int_0^{t}\|
\tilde{\varsigma}_{\mu _1}(t)-
\tilde{\varsigma}_{\mu _2}(t)\| _{V}^2ds \\
&\leq c\int_0^{t}\| e^{-k_1t}(\mu _1(s)-\mu
_2(s))\| _{V'}^2ds\text{ \ a.e. } t\in (0,T),
\end{align*}
from which also follows that
\begin{equation}
\begin{aligned}
&\| \varsigma _{\mu _1}(t)-\varsigma _{\mu _2}(t)
\| _{L^2(\Omega )}^2+\int_0^{t}\|
\varsigma _{\mu _1}(s)
-\varsigma _{\mu _2}(s)\| _{V}^2ds   \\
&\leq c\int_0^{t}\| \mu _1(s)-\mu _2( s) \|
_{V'}^2ds\text{ \ a.e. }t\in (0,T), \label{e3.36}
\end{aligned}
\end{equation}
We can infer, using \eqref{e3.25}, \eqref{e3.32}, \eqref{e3.33},
\eqref{e3.35} and \eqref{e3.36}, that
\begin{align*}
&\int_0^{t}\| \Lambda _0(\mathbf{\eta
}_1(s),\lambda _1(s),\mu _1(s))-\Lambda _0(\mathbf{\eta
}_2(s),\lambda _2( s),\mu _2(s))\| _{\mathcal{V}'}^2ds
\\
&\leq c\int_0^{t}\int_0^{s}
\Big(\|\mathbf{ \dot{u}}_{\eta _1}(r)-\dot{\mathbf{u}}_{\eta
_2}(r)\| _{\mathcal{V}}^2+\| \theta _{\lambda
_1}(r)-\theta _{\lambda _2}(r)\| _{V}^2\\
&\quad +\|\mathbf{u}_{\eta _1}( r)-\mathbf{u}_{\eta _2}(r) \|
_{\mathcal{V}}^2
+\| \varsigma _{\mu _1}(r)-\varsigma _{\mu_2}(r)\| _{V}^2
 \Big)\,dr\,ds
\quad \text{a.e. }t\in (0,T)
\\
&\leq c\int_0^{T}\int_0^{T}
\Big(\|\mathbf{ \dot{u}}_{\eta _1}(r)-\dot{\mathbf{u}}_{\eta
_2}(r)\| _{\mathcal{V}}^2
+\| \theta _{\lambda_1}(r)-\theta _{\lambda _2}(r)\| _{V}^2\\
&\quad +\|\mathbf{u}_{\eta _1}( r)-\mathbf{u}_{\eta _2}(r) \|
_{\mathcal{V}}^2
+ \| \varsigma _{\mu _1}(r)-\varsigma _{\mu
_2}(r)\| _{V}^2\Big)\,dr\,ds
\\
&\leq c\int_0^{T}
\Big(\| \dot{\mathbf{u}}_{\eta _1}(s)-\dot{\mathbf{u}}_{\eta _2}(s)\|
_{\mathcal{V}}^2\\
&\quad +\| \mathbf{u}_{\eta _1}( s) -\mathbf{u}_{\eta _2}(s)\| _{\mathcal{V} }^2
+\| \theta _{\lambda _1}(s)-\theta _{\lambda _2}(s)\|_{V}^2
+ \| \varsigma _{\mu _1}(s)-\varsigma _{\mu_2}(s)\| _{V}^2\Big)ds
\\
&\leq c\int_0^{T}
\Big(\| \mathbf{\eta }_1(s)-\mathbf{\eta }_2(s)\| _{\mathcal{V} '}^2
+\| \lambda _1(s)-\lambda _2(s) \|_{V'}^2
+\| \mu _1(s)-\mu _2(s) \|_{V'}^2\\
&\quad +\| \mathbf{u}_{\eta _1}(s) -\mathbf{u}_{\eta_2}(s)
\| _{\mathcal{V}}^2\Big)ds
\end{align*}
Thus, by  \eqref{e3.34}, we find
\begin{equation}
\begin{aligned}
&\int_0^{T}\| \Lambda _0(\mathbf{\eta }_1( s) ,\lambda
_1(s),\mu _1(s))-\Lambda _0(\mathbf{\eta }_2(s),\lambda
_2(s),\mu _2(s))\| _{\mathcal{V}'}^2ds\\
&\leq c\int_0^{T}\Big(\| \mathbf{\eta }_1(s)- \mathbf{\eta
 }_2(s)\| _{\mathcal{V}'}^2+\| \lambda
 _1(s)-\lambda _2( s) \| _{V'}^2\
 +\| \mu_1(s)-\mu _2(s) \| _{V'}^2\Big)ds\, .
\end{aligned} \label{e3.37}
\end{equation}
Furthermore, hypothesis \eqref{e2.13} implies
\begin{align*}
&\int_0^{t}\| \psi \Big(\mathbf{\sigma }_{\eta
_1,\lambda _1,\mu _1}(s),\varepsilon ( \mathbf{\dot{u }}_{\eta
_1}(s)),\theta _{\lambda _1}( s)
,\varsigma _{\mu _1}(s)\Big)\\
&- \psi \Big(\mathbf{\sigma }_{\eta _2,\lambda _2,\mu
_2}(s),\varepsilon (\dot{\mathbf{u}}_{\eta _2}( s) ) ,\theta
_{\lambda _2}(s),\varsigma _{\mu _2}(s)\Big)\| _{V'}^2ds
 \\
&\leq 3L_{\psi }^2\int_0^{t}\Big(\| \mathbf{\sigma
} _{\eta _1,\lambda _1,\mu _1}(s)-\mathbf{\sigma }_{\eta
_2,\lambda _2,\mu _2}(s)\| _{\mathcal{H} }^2+\|
\dot{\mathbf{u}}_{\eta _1}( s) -\mathbf{\dot{u}
}_{\eta _2}(s)\| _{\mathcal{V}}^2\\
&\quad + \| \theta _{\lambda _1}(s)-\theta _{\lambda_2}(s)\| _{V}^2
+\| \varsigma _{\mu _1}(t)-\varsigma _{\mu _2}(t)\| _{V}^2\Big) ds\quad
\text{a.e. }t\in (0,T).
\end{align*}
This permits us to deduce, via \eqref{e3.25}, \eqref{e3.33}, \eqref{e3.35}
 and \eqref{e3.36}, that
\begin{equation}
\begin{aligned}
&\int_0^{T}\| \psi \Big(\mathbf{\sigma }_{\eta
_1,\lambda _1,\mu _1}(s),\varepsilon ( \mathbf{\dot{u }}_{\eta
_1}(s)),\theta _{\lambda _1}( s)
,\varsigma _{\mu _1}(s)\Big)\\
&- \psi \Big(\mathbf{\sigma }_{\eta _2,\lambda _2,\mu
_2}(s),\varepsilon (\dot{\mathbf{u}}_{\eta _2}( s) ) ,\theta
_{\lambda _2}(s),\varsigma _{\mu _2}(
s)\Big)\| _{V'}^2ds   \\
&\leq c\int_0^{T}\Big(\| \mathbf{\eta }_1( s)
-\mathbf{\eta }_2(s)\| _{\mathcal{V} '}^2+\| \lambda _1(s)-\lambda _2( s) \|
_{V'}^2+\| \mu _1(s)-\mu _2(s) \|
_{V'}^2\Big)ds
\end{aligned}\label{e3.38}
\end{equation}
Similarly,  using \eqref{e3.25}, \eqref{e3.34}, \eqref{e3.35} and
\eqref{e3.36}, we obtain the following
estimate for $\phi $,
\begin{equation}
\begin{aligned}
&\int_0^{T}\| \phi \Big(\mathbf{\sigma }_{\eta
_1,\lambda _1,\mu _1}(s),\varepsilon ( \mathbf{u} _{\eta
_1}(s)),\theta _{\lambda _1}( s)
,\varsigma _{\mu _1}(s)\Big)\\
&- \phi \Big(\mathbf{\sigma }_{\eta _2,\lambda _2,\mu
_2}(s), \varepsilon (\mathbf{u}_{\eta _2}( s) ) ,\theta
_{\lambda _2}(s),\varsigma _{\mu _2}(
s)\Big)\| _{V'}^2ds   \\
&\leq c\int_0^{T}\Big(\| \mathbf{\eta }_1( s)
-\mathbf{\eta }_2(s)\| _{\mathcal{V} '}^2+\| \lambda _1(s)-\lambda _2( s) \|
_{V'}^2+\| \mu _1(s)-\mu _2(s) \|
_{V'}^2\Big)ds.
\end{aligned} \label{e3.39}
\end{equation}

 From \eqref{e3.37}, \eqref{e3.38} and \eqref{e3.39}, we conclude
 that there exists a positive constant $C>0$ verifying
\begin{equation}
\begin{aligned}
&\| \Lambda (\mathbf{\eta }_1,\lambda _1,\mu _1)-\Lambda
(\mathbf{\eta }_2,\lambda _2,\mu _2) \|
_{L^2(0,T;\mathcal{V}'\times V'\times V')}   \\
&\leq C\| (\mathbf{\eta }_1-\mathbf{\eta }_2,
\lambda _1-\lambda _2, \mu _1-\mu _2) \|
_{L^2(0,T;\mathcal{V}'\times V'\times V')},
\end{aligned} \label{e3.40}
\end{equation}
and so, by reapplication of mapping $\mathbf{\Lambda }$, yields
\begin{align*}
&\| \Lambda ^2(\mathbf{\eta }_1,\lambda _1,\mu
_1)-\Lambda ^2(\mathbf{\eta }_2,\lambda _2,\mu _2) \|
_{L^2(0,T;\mathcal{V}'\times
V'\times V')} \\
&\leq \frac{C^2}{2!}\| (\mathbf{\eta }_1-\mathbf{\eta }
_2,\lambda _1-\lambda _2,\mu _1-\mu _2) \|
_{L^2(0,T;\mathcal{V}'\times V'\times
V')}.
\end{align*}
We generalize this procedure by recurrence on $n$. Then we obtain the
 formula
\begin{equation}
\begin{aligned}
&\| \Lambda ^n(\mathbf{\eta }_1, \lambda _1,
 \mu _1)-\Lambda ^n(\mathbf{\eta }_2,  \lambda
_2, \mu _2)\| _{L^2( 0,T;\mathcal{V
}'\times V'\times V')}   \\
&\leq \frac{C^n}{n!}\| (\mathbf{\eta }_1-\mathbf{\eta }
_2, \lambda _1-\lambda _2, \mu _1-\mu _2)\|
_{L^2(0,T;\mathcal{V}'\times V'\times
V')}.
\end{aligned}\label{e3.41}
\end{equation}
We know that the sequence $(C^n/n!)_{n}$ converges
to $0$. So, for $n$ sufficiently large $\frac{C^n}{n!}<1$. It
means that a large power $n$ of the operator $\Lambda $ is a
contraction on $ L^2( 0,T;\mathcal{V}'\times V'\times V')$.
Hence, Banach fixed point theorem shows
that $\Lambda $ admits a unique fixed point
$(\mathbf{\eta }^{\ast},\lambda ^{\ast },\mu ^{\ast })
\in L^2( 0,T;\mathcal{V}'\times V'\times V')$.

We can now prove the existence of a solution to problem
(PV). To this aim, it is sufficient to remark that for
a.e. $t\in (0,T)$,
\begin{gather*}
\begin{aligned}
&\mathcal{E}(\varepsilon (\mathbf{u}_{\eta ^{\ast }}(t)))
+ \int_0^{t}\mathcal{G}\Big(\mathbf{\sigma }_{\eta ^{\ast
},\lambda ^{\ast },\mu ^{\ast }}(s)-\mathcal{A}( \varepsilon
(\dot{\mathbf{u}}_{\eta ^{\ast }}(s))),\varepsilon
(\mathbf{u}_{\eta ^{\ast }}( s)),\theta _{\lambda ^{\ast
}}(s),\varsigma _{\mu ^{\ast }}(s)\Big)ds\\
&=\mathbf{\eta }^{\ast }( t),
\end{aligned}\\
\psi (\varepsilon (\mathbf{u}_{\eta ^{\ast }}(t)),\theta _{\lambda
^{\ast }}(t),\varsigma _{\mu ^{\ast
}}(t))=\lambda ^{\ast }(t), \\
\phi (\varepsilon (\mathbf{u}_{\eta ^{\ast }}(t)),\theta _{\lambda
^{\ast }}(t),\varsigma _{\mu ^{\ast }}(t))=\mu ^{\ast }(t),
\end{gather*}
which completes the proof.
\end{proof}

\begin{theorem}[Positivity of the temperature] \label{thm2}
Let the hypotheses of Theorem \ref{thm1} hold and suppose in addition that
\begin{gather}
\psi (\mathbf{\sigma },\varepsilon (\mathbf{u}),\theta ,\varsigma
)\geq 0\quad \text{a.e.  in  }\Omega \times (
0,T),  \label{e3.42} \\
q\geq 0\quad \text{  a.e.  in  }\Omega \times (0,T),  \label{e3.43} \\
\theta _0\geq 0\quad \text{  a.e.  in  }\Omega \times (0,T).
\label{e3.44}
\end{gather}
Then, the solution $\{ \mathbf{u},\mathbf{\sigma },\theta
,\varsigma\} $ to problem (PV) is such that
\begin{equation}
\theta (x,t)\geq 0\quad \text{for  a.e. }(x,t)\in \Omega \times
(0,T).  \label{e3.45}
\end{equation}
\end{theorem}

\begin{proof}
We use a maximum principle argument \cite{b1}. Thus, we test the
equation \eqref{e2.26} by the function $-\theta ^{-}$, where $f^{-}$
denoting the so-called negative part of a function $f$; i.e.,
$f^{-}=\max \{ 0,-f\} $, and integrate over $(0,T)$. We
can infer, using the hypothesis \eqref{e3.42}, \eqref{e3.43}
and \eqref{e3.44}, that
\begin{align*}
&\frac{1}{2}(\rho ^{\ast })^2\| \theta ^{-}\|
_{L^{\infty }(0,T;L^2(\Omega ))}^2+c_1\| \theta
^{-}\| _{L^2( 0,T;V) }^2\\
&\leq -\int_0^{t}\int_{\Omega }\psi (\varepsilon
(\mathbf{u}(x,s)),\theta (x,s),\varsigma
(x,s))\theta ^{-}(x,s)\,dx\,ds\\
&\quad - \int_0^{t}\int_{\Omega }q(x,s)\theta
^{-}(x,s)\,dx\,ds\leq 0\quad \text{a.e. }t\in (0,T).
\end{align*}
Consequently
\[
\| \theta ^{-}\| _{L^2(0,T;V)\cap L^{\infty}(0,T;L^2(\Omega ))}\leq 0,
\]
which eventually gives \eqref{e3.45}.
\end{proof}

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\end{document}