A thermoviscoelastic contact problem with friction, damage and wear diffusion

Abstract

In this paper we present a model for quasistatic frictional contact between a thermoviscoelastic body and a moving foundation that involves wear of contacting surface and diffusion of wear debris. The damage effect is taken into account in the thermoviscoelastic constitutive law, its evolution is described by a parabolic inclusion with the homogeneous Neumann boundary condition. Contact is modeled with a normal compliance condition and is associated to a dry friction. The wear takes place on a part of the contact surface, when the wear debris surface density diffuse on the whole of the contact surface and is accompanied by frictional heat exchange. We derive a variational formulation of the problem and state that, under a smallness assumption on the problem data, there exists a unique weak solution for the model. The proof is based on elliptic variational inequalities, parabolic variational inequalities, first order evolution equations and fixed point arguments.

1 Introduction

This work studies a quasistatic model for the process of frictional contact between a thermoviscoelastic body and a moving foundation when wear debris is generated and diffuses on the contact surface. The damage effect is included in the thermoviscoelastic constitutive law. The contact is described with a normal compliance condition and the associated a version of Coulomb law of dry friction in which the coefficient of friction is assumed to depend on the density of the wear particles and on the slip rate. The motion is accompanied by wear diffusion and frictional heat generation. The wear takes place on a part of the contact surface and its rate is described by the Archard differential condition. So, our interest is to describe a physical process in which thermal effect, damage effect, friction, wear diffusion and frictional heat generation are involved, and to show that the resulting model leads to well-posed mathematical problem. Then we present the result on the existence and uniqueness of a weak solution to the system. The model is set as a system of an evolutionary variational inequality for the displacements, a parabolic variational equation for the density of the wear particles, a parabolic variational inequality for the damage and an evolution equation for the temperature.

Frictional contact arise in structural and mechanical systems, a considerable progress has been achieved in modeling and mathematical analysis. Models of frictional contact problems are investigated in [13, 15, 18, 20, 21]. Frictional contact problems with wear, both in the dynamic and the quasistatic case, can be found in [16, 17, 19, 20]. Mathematical models for frictional contact with wear under thermodynamic considerations have been considered in [1, 5]. General dynamic thermoelastic models, which were derived from thermodynamical principles, can be found in [12, 23]. Quasistatic or dynamic thermoviscoelastic frictional contact problems can be found in [4, 5, 14]. A quasistatic thermoviscoelastic problem for a beam can be found in [10, 11], where the wear of the contacting surface is included. Quasistatic thermoviscoelastic problem with normal compliance, multivalued friction and wear diffusion can be found in [9].

Following [6, 7], the evolution of the microscopic cracks responsible for the damage is determined by a parabolic inclusion with a constitutive function describing the source of damage in the system which results from tension or compression. Using the subdifferential of indicator function of the interval [0, 1] guarantees that the damage function $\beta$ which measures the decrease in the load-bearing capacity of the material, varies between 0 and 1. When $\beta =1$ there is no damage in the material, when $\beta =0$ the material is completely damaged, when $0<\beta <1$ there is partial damage and the system has a reduced load carrying capacity. Contact problems with damage have been investigated in [8, 17, 20] and the monograph [22].

The rest of the manuscript is structured as follows. In Sect. 2 we present the notation we shall use as well as some preliminary material. In Sect. 3 we present the physical setting, describe the mechanical problem, list the assumptions on the data and give the variational formulation of the problem. In Sect. 4 we state our main existence and uniqueness result based on arguments of elliptic variational inequalities, parabolic variational inequalities, first order evolution equations and fixed point.

2 Notations and preliminaries

In this section we present some notations and preliminary material we shall use later in this paper. For further details, we refer the reader to [3]. Let $\varOmega \subset \mathbb {R}^{3}$ be a bounded domain with a Lipschitz boundary $\varGamma$ and $\varvec{\nu }$ denote the unit outward normal on $\varOmega$. We denote by $S_{3}$ the space of second order symmetric tensors on $\mathbb {R}^{3}$ while "." and $\mid .\mid$ will represent the inner product and the Euclidean norm on the spaces $\mathbb {R}^{2}$ and $\mathbb {R}^{3}$. Everywhere in the sequel the index i and j run from 1 to 3. The summation convention over repeated indices is adopted and the index that follows a comma indicates a partial derivative with respect to the corresponding component of the independent variable. We introduce the following spaces

$$\begin{aligned} H&=\left\{ \varvec{u}=(u_i)~ | ~ u_i\in L^{2}(\varOmega ) \right\} =L^{2}(\varOmega )^{3},\\ \mathcal {H}&=\left\{ {\varvec{\sigma }=(\sigma _{ij})~ | ~\sigma _{ij}=\sigma _{ji} \in L^{2}(\varOmega )}\right\} =L^{2}(\varOmega )_{s}^{3\times 3},\\ H_1&=\left\{ {\varvec{u}=(u_i)~ | ~ \varvec{\varepsilon } (\varvec{u})\in \mathcal {H} }\right\} = H^{1}(\varOmega )^{3},\\ \mathcal {H}_1&=\left\{ {\varvec{\sigma }\in \mathcal {H} ~ | ~ Div \varvec{\sigma }\in H}\right\} . \end{aligned}$$

Here $\varvec{\varepsilon }$ and Div are the deformation and divergence operators, respectively, defined by $\varvec{\varepsilon }(\varvec{u})=(\varepsilon _{ij}(\varvec{u}))$, $\varepsilon _{ij}(\varvec{u})=\frac{1}{2}(u_{i,j}+u_{j,i})$, $Div\varvec{\sigma }=(\sigma _{ij,\,j})$. The spaces H, $\mathcal {H}$, $H_{1}$ and $\mathcal {H}_{1}$ are real Hilbert spaces endowed with the canonical inner products given by

$$\begin{aligned} (\varvec{u,v})_{H}= & {} \int \limits _{\varOmega }u_{i}v_{i}\text { }dx,\qquad (\varvec{\sigma ,\tau })_{\mathcal {H}}=\int \limits _{\varOmega }\sigma _{ij}\tau _{ij}\text { }dx,\\ (\varvec{u,v})_{H_{1}}= & {} (\varvec{u,v})_{H}+(\varvec{\varepsilon }(\varvec{u}), \varvec{\varepsilon }(\varvec{v}))_{\mathcal {H}}, \qquad (\varvec{\sigma ,\tau })_{\mathcal {H}_{1}}=(\varvec{\sigma ,\tau })_{\mathcal {H}}+(Div\text { }\varvec{\sigma },Div\text { }\varvec{\tau })_{H}. \end{aligned}$$

The associated norms on the spaces H, $\mathcal {H}$, $H_{1}$ and $\mathcal {H}_{1}$ are denoted by $\mid \cdot \mid _{H},$ $\mid \cdot \mid _{\mathcal {H}}$, $\mid \cdot \mid _{H_{1}}$ and $\mid \cdot \mid _{\mathcal {H}_{1}}$, respectively. For an element $\varvec{v}\in H_{1}$ we denote by $\varvec{v}$ its trace on $\varGamma$ and by $v_{\nu }=\varvec{v} \cdot \varvec{\nu }$, $\varvec{v}_{\tau }=\varvec{v}-v_{\nu }\varvec{\nu }$ its normal and tangential components on the boundary. For an element $\varvec{\sigma }\in \mathcal {H}_{1}$, by $\sigma _{\nu }=(\varvec{\sigma \nu }) \cdot \varvec{\nu }$ and $\varvec{\sigma }_{\tau } =\varvec{\sigma \nu }-\sigma _{\nu }\varvec{\nu }$ we denote the normal and the tangential traces of $\varvec{\sigma }$. The following two Green formulas hold

$$\begin{aligned}&(div~\varvec{v},u)_{L^2(\varOmega )}+(\varvec{v},\nabla u )_H=\int \limits _{\varGamma }u(\varvec{v} \cdot \varvec{\nu })~da\quad \text { for all } u\in H^1(\varOmega ) \text { and } \varvec{v}\in H_1, \end{aligned}$$

(1)

$$\begin{aligned}&(\varvec{\sigma },\varvec{\varepsilon }(\varvec{v}))_{\mathcal {H}}+(Div\text { }\varvec{\sigma },\varvec{v})_{H}=\int \limits _{\varGamma }\varvec{\sigma \nu }.\varvec{v}~da\,~\qquad \forall \,\varvec{v}\in H_{1} \text { and } \varvec{\sigma } \in \mathcal {H}_1. \end{aligned}$$

(2)

Let $T>0$. For every real Banach space X we denote by C(0, T; X) and $C^{1}(0,T;X)$ the spaces of continuous and continuously differentiable functions from $\left[ 0,T\right]$ to X, with norms

$$\begin{aligned} \mid \varvec{f}\mid _{C(0,T;X)}=\underset{t\in \left[ 0,T\right] }{\max }\ \mid \varvec{f}(t)\mid _{X},\qquad \mid \varvec{f}\mid _{C^{1}(0,T;X)}=\underset{ t\in \left[ 0,T\right] }{\max }\mid \varvec{f}(t)\mid _{X}+\underset{t\in \left[ 0,T\right] }{\max }\mid \varvec{\dot{f}}(t)\mid _{X}. \end{aligned}$$

For $k\in \mathbb {N}$ and $p\in \left[ 1,\infty \right]$, we use the standard notation for the Lebesgue spaces $L^{p}(0,T;X)$ and for the Sobolev spaces $W^{k,p}(0,T;X)$. Moreover, if $X_1$ and $X_2$ are two real Hilbert spaces, then $X_1 \times X_2$ denotes the product space endowed with the canonical inner product $(\cdot , \cdot )_{X _1\times X_2}$ and norm $|. |_{X _1\times X_2}$.

3 Problem statement and variational formulation

A thermoviscoelastic body occupies a bounded domain $\varOmega \subset \mathbb {R} ^{3}$ with a Lipschitz surface $\varGamma$ that is divided into three disjoint measurable parts $\varGamma _{1}$, $\varGamma _{2}$ and $\varGamma _{3}$ such that meas $(\varGamma _{1})>0$ and $meas(\varGamma _{3})>0$. Let $\left[ 0,T\right]$ be the time interval of interest, for $T>0$. The body is clamped on $\varGamma _{1}\times \left( 0,T\right)$, so the displacement field vanishes there. Surface tractions of density $\varvec{f}_{2}$ act on $\varGamma _{2}\times \left( 0,T\right)$ and a body force of density $\varvec{f}_{0}$ is applied in $\varOmega \times \left( 0,T\right)$. An initial gap g exists between the potential contact surface $\varGamma _3$ and the foundation, and it is measured along the outward normal $\varvec{\nu }$. We assume that the coordinate system is such that $\varGamma _{3}$ occupies a regular domain in the $Ox_{1}x_{2}$ plane and the foundation is moving with velocity $\varvec{v}^{*}$ in the $Ox_{1}x_{2}$ plane. Furthermore, $\varGamma _{3}$ is divided into two subdomains $D_{d}$ and $D_{\omega }$ by a smooth curve $\gamma ^{*}$. The wear takes place only on $D_{\omega }$, while the wear particles diffuse on the whole of the contact surface $\varGamma _{3}$. The boundary $\partial \varGamma _{3}$ of $\varGamma _{3}$ is assumed to be Lipschitz and is composed of two parts $\gamma _{d}$ and $\gamma _{\omega }$. Then $\partial D_{\omega }=\gamma _{\omega }\cup$ $\gamma ^{*}$ and $\partial D_{d}=\gamma _{d}\cup$ $\gamma ^{*}$.

The wear function $\omega =\omega (\varvec{x},t)$ is defined on $D_{\omega }$ and the wear particle surface density function $\zeta =\zeta (\varvec{x},t)$ is defined on $\varGamma _{3}$. The function $\zeta$ measures the surface density of the diffusing wear particles and the wear function $\omega$ measures the depth of the wear i.e., the amount of material per unit surface that has been removed, then $\omega =\lambda \zeta$ in $D_{\omega }$, where $\lambda$ is a conversion factor from wear debris surface density to wear depth, which we assume to be a positive constant. For the sake of convenience we extend $\omega$ by zero to the whole of $\varGamma _{3}$, and below when confusion is unlikely we use the same symbol for the function and its extension. Thus,

$$\begin{aligned} \omega =\lambda \zeta \chi _{[D_{w}]}\text { on }\varGamma _{3}\times (0,T), \end{aligned}$$

(3)

where $\chi _{[D_{w}]}$ is the characteristic function of the set $D_{\omega }$ (i.e., $\chi _{[D_{w}]}(\textbf{x})=1$ when $\varvec{x}\in D_{\omega }$ and $\chi _{[D_{w}]}(\varvec{x})=0$ if $\varvec{x}\notin$ $D_{\omega }$). The wear diffusion coefficient k is given by

$$\begin{aligned} k=k(\varvec{x})=\left\{ \begin{array}{c} k_{w}\text { in }D_{\omega }\text {,} \\ k_{d}\text { in }D_{d}\text {.} \end{array} \right. \end{aligned}$$

Here, wear diffusion is described by the following nonlinear diffusion equation

$$\begin{aligned} \dot{\zeta }-div(k\nabla \zeta )={\kappa }\left| \varvec{\sigma }_{\tau }\right| R^{*}(\left| \varvec{\dot{u}}_{\tau }-\varvec{v}^{*}\right| )\chi _{[D_{\omega }]}\text { \qquad in }\varGamma _{3}\times (0,T)\text {,} \end{aligned}$$

(4)

where $R^{*}:\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ is the truncation operator

$$\begin{aligned} R^{*}(r)=\left\{ \begin{array}{lll} r &{}\text { if } &{}r\le R, \\ R &{} \text { if } &{}r>R, \end{array} \right. \end{aligned}$$

(5)

R is a fixed positive constant and ${\kappa }$ is the wear rate coefficient. We need this operator in order to avoid some mathematical difficulties, however, from the physical point of view the use of $R^*$ is not restrictive since, in practice, the slip velocity is bounded and no smallness assumption will be made on R.

Then, the classical model for the above process is as follows:

Problem P. Find a displacement field $\varvec{u}:\varOmega \times \left[ 0,T\right] \rightarrow \mathbb {R}^{3}$, a stress field $\varvec{\sigma }:\varOmega \times \left[ 0,T\right] \rightarrow S_{3}$, a temperature field $\theta :\varOmega \times \left[ 0,T\right] \rightarrow \mathbb {R},$ a damage field $\beta :\varOmega \times \left[ 0,T\right] \rightarrow \mathbb {R}$ and a surface particle density field $\zeta :\varGamma _{3}\times \left[ 0,T\right] \rightarrow \mathbb {R}$ such that

$$\begin{aligned}&\varvec{\sigma }=\mathcal {A}\varvec{\varepsilon }(\dot{\varvec{u}})+ \mathcal {G}(\varvec{\varepsilon }(\varvec{u}),\beta )-C(\theta ,\beta )&\quad \text { in }\varOmega \times \left( 0,T\right) , \end{aligned}$$

(6)

$$\begin{aligned}&\dot{\beta }-k_{1}\bigtriangleup \beta +\partial \varphi _{Y}(\beta )\ni \phi (\varvec{\varepsilon } (\varvec{u}),\theta ,\beta )&\quad \text { in }\varOmega \times \left( 0,T\right) , \end{aligned}$$

(7)

$$\begin{aligned}&Div\varvec{\sigma }+\varvec{f}_{0}=0&\quad \text { in }\varOmega \times \left( 0,T\right) , \end{aligned}$$

(8)

$$\begin{aligned}&\varvec{u}=\varvec{0}&\quad \text { on } \varGamma _{1}\times \left( 0,T\right) , \end{aligned}$$

(9)

$$\begin{aligned}&\varvec{\sigma \nu }=\varvec{f}_{2}&\quad \text { on } \varGamma _{2}\times \left( 0,T\right) , \end{aligned}$$

(10)

$$\begin{aligned}&\left\{ \begin{array}{l} -\sigma _{\nu }=p_{\nu }, \quad \mid \varvec{\sigma }_{\tau }\mid \le \mu p_{\nu },\\ \varvec{\sigma }_{\tau }=-\mu p_{\nu }\frac{\dot{\varvec{u}}_{\tau }- \varvec{v}^{*}}{\mid \dot{\varvec{u}}_{\tau }-\varvec{v}^{*}\mid } ~~ \text { if }\dot{\varvec{u}}_{\tau }\ne \varvec{v}^{*} \end{array} \right.&\quad \text { on }\varGamma _{3}\times (0,T), \end{aligned}$$

(11)

$$\begin{aligned}&\dot{\zeta }-div(k\nabla \zeta )={\kappa }\mu p_{\nu }R^{*}(\mid \overset{.}{\varvec{u}}_{\tau }-\varvec{v}^{*}\mid )\chi _{[D_{\omega }]}&\quad \text { on }\varGamma _{3}\times (0,T), \end{aligned}$$

(12)

$$\begin{aligned}&\zeta =0&\quad \text { on } \partial \varGamma _{3}\times (0,T), \end{aligned}$$

(13)

$$\begin{aligned}&\dot{\theta }-div(K_c\nabla \theta )=\psi (\dot{\varvec{u}}, \theta , \beta )+q&\quad \text { in }\varOmega \times (0,T), \end{aligned}$$

(14)

$$\begin{aligned}&-k_{ij}\theta _{,i}\eta _{j}=k_{e}(\theta -\theta _{R})&\quad \text { on }\varGamma _{3}\times (0,T), \end{aligned}$$

(15)

$$\begin{aligned}&\theta =0&\quad \text { on }\varGamma _{1}\cup \varGamma _{2}\times (0,T), \end{aligned}$$

(16)

$$\begin{aligned}&\frac{\partial \beta }{\partial \nu }=0&\quad \text { on }\varGamma \times \left( 0,T\right) , \end{aligned}$$

(17)

$$\begin{aligned}&\varvec{u}(0)=\varvec{u}_{0},\quad \theta (0)=\theta _{0},\quad \beta (0)=\beta _{0}&\quad \text { in }\varOmega , \end{aligned}$$

(18)

$$\begin{aligned}&\zeta (0)=\zeta _{0}&\quad \text { in }\varGamma _{3}. \end{aligned}$$

(19)

Here $p_{\nu }=p_{\nu }(u_{\nu }-\lambda \zeta \chi _{[D_{\omega }]}-g)$ and $\mu =\mu (\zeta ,\mid \dot{\varvec{u}}_{\tau }-\varvec{v}^{*}\mid )$ is the coefficient of friction which depends on the density of the wear particles and on the slip rate. Equation (6) represents the thermoviscoelastic constitutive law, where $\varvec{\sigma }$ denotes the stress tensor, $\varvec{u}$ represents the displacement field, $\dot{\varvec{u}}$ the velocity, $\theta$ is the temperature field and $\varvec{\varepsilon }(\varvec{u})$ is the small strain tensor. Here $\mathcal {A}$ and $\mathcal {G}$ are nonlinear operators describing the purely viscous and the elastic properties of the material, respectively and C represents the thermal expansion tensor. Equation (7) represents the inclusion used for the evolution of the damage field, where the set of admissible damage functions defined by

$$\begin{aligned} Y=\{\xi \in H^{1}(\varOmega )\text { }/\text { }0\le \xi \le 1~~~\text { a.e. }\varOmega \}, \end{aligned}$$

$k_{1}$ is a positive coefficient, $\partial \varphi _{Y}$ is the subdifferential of the indicator function $\varphi _{Y}$ and $\phi$ is a given constitutive function which describes the sources of the damage in the system. Equation (8) represents the equilibrium equation, since the process is assumed to be quasistatic. Equations (9)–(10) are the displacement-traction conditions. Equation (12) represents the nonlinear diffusion equation, Eq. (13) is the absorbing boundary condition. In (18) $\varvec{u}_{0}$ is the given initial displacement field, $\theta _{0}$ is the initial temperature and $\beta _0$ is the given initial damage field. In (19), $\zeta _{0}$ is the given initial surface particle density field. To simplify the notation, we do not indicate explicitly the dependence of various functions on the variables $\varvec{x} \in \varOmega \cup \varGamma$ and $t\in \left[ 0,T\right] .$

The evolution of the temperature field $\theta$ is governed by the heat equation (see [1, 13]), obtained from the conservation of energy, and defined by the differential equation for the temperature given in (14), where $K_c=(k_{ij})$ represents the thermal conductivity tensor, $div(K_c\nabla \theta )=(k_{ij}\theta _{,i})_{,i}$ and q(t) the density of volume heat sources. The associated temperature boundary condition is given by (15), where $\theta _{R}$ is the temperature of the foundation and $k_{e}$ is the heat exchange coefficient between the body and the obstacle. Condition (11) represents the normal compliance condition with wear and the associated general law of dry friction on the contact surface $\varGamma _{3}$. In (16) the temperature vanishes on $\varGamma _1\cup \varGamma _2$. Equation (17) represents the Neumann boundary condition. To obtain a variational formulation of the problem (6)–(19) we need additional notation. Let V be the closed subspace of $H_{1}$ defined by

$$\begin{aligned} V=\left\{ \varvec{v}\in H_{1}\text { / }\varvec{v}=\varvec{0} \text { on }\varGamma _{1}\text { }\right\} , \end{aligned}$$

and let E be the closed subspace of $H^{1}(\varOmega )$ given by

$$\begin{aligned} E=\left\{ y\in H^{1}(\varOmega )\text { / }y=0\text { on }\varGamma _{1}\cup \varGamma _{2}\right\} . \end{aligned}$$

Since $\varGamma$ is Lipschitz continuous and $meas(\varGamma _{1})>0$, Korn’s and Poincare’s inequalities hold true

$$\begin{aligned}&\mid \varvec{\varepsilon }(\varvec{v})\mid _{\mathcal {H}}\ge C\mid \varvec{v }\mid _{H_{1}}\forall \,\varvec{v}\in V, \end{aligned}$$

(20)

$$\begin{aligned}&\mid \nabla y\mid _{H}\ge C\mid y\mid _{H^{1}(\varOmega )}\qquad \forall \,y\in E, \end{aligned}$$

(21)

where here and below C is a positive constant depending on the problem data but is independent of the solutions, its value may change from line to line. We define the inner products on V and on E by

$$\begin{aligned}&(\varvec{u,v})_{V}=(\varvec{\varepsilon }(\varvec{u}),\varvec{\varepsilon }(\varvec{v}))_{\mathcal {H}}\text { }\forall \varvec{u,v}\in V, \end{aligned}$$

(22)

$$\begin{aligned}&(y,z)_{E}=(\nabla y,\nabla z)_{H}\text { }\forall y,z\in E, \end{aligned}$$

(23)

respectively. It follows from (20) and (22) that $\mid \cdot \mid _{H_{1}}$ and $\mid \cdot \mid _{V}$ are equivalent norms on V and from (21) and (23), it follows that $\mid \cdot \mid _{H^{1}(\varOmega )}$ and $\mid \cdot \mid _{E}$ are equivalent norms on E. Therefore $(V,\mid \cdot \mid _{V})$ and $(E,\mid \cdot \mid _{E})$ are real Hilbert spaces. By the Sobolev’s trace theorem, there exists a constant $C_{\varGamma }>0$ which depends only on $\varOmega$, $\varGamma _{1}$ and $\varGamma _{3}$ such that

$$\begin{aligned} \mid \varvec{v}\mid _{L^{2}(\varGamma _{3})^3}\le C_{\varGamma }\mid \varvec{v} \mid _{V}\qquad \forall \,\varvec{v}\in V. \end{aligned}$$

(24)

There exists $\hat{C}_{\varGamma }>0$ depending on $\varOmega$, $\varGamma _{1}$, $\varGamma _2$ and $\varGamma _{3}$ such that

$$\begin{aligned} \mid \theta \mid _{L^{2}(\varGamma _{3})}\le \hat{C}_{\varGamma }\mid \theta \mid _{E}\qquad \forall \theta \in E. \end{aligned}$$

(25)

$E'$ is the dual of the space E. Identifying $L^{2}(\varOmega )$ with its own dual we can write $E\subset L^{2}(\varOmega )\subset E'$. Below $\left\langle \cdot , \cdot \right\rangle$ represents the duality pairing between $E'$ and E, and $\mid \cdot \mid _{E'}$ denotes the norm on $E'$. Also, $\left\langle \theta ,\eta \right\rangle =\left( \theta ,\eta \right) _{L^{2}(\varOmega )}$ for $\theta \in L^{2}(\varOmega )$ and $\eta \in E$.

Recall that $\varGamma _{3}$ is assumed to be a regular domain in the $Ox_{1}x_{2}$ plane with Lipschitz boundary $\partial \varGamma _3$. Keeping in mind the boundary condition (13), for the surface particle density function, we shall use the space

$$\begin{aligned} H_{0}^{1}(\varGamma _{3})=\left\{ \xi \in H^{1}(\varGamma _{3})\text { / }\xi =0 \text { on }\partial \varGamma _3 \text { }\right\} . \end{aligned}$$

This is a real Hilbert space endowed with the inner product

$$\begin{aligned} \left( \zeta ,\xi \right) _{H_{0}^{1}(\varGamma _{3})}=\left( \nabla \zeta ,\nabla \xi \right) _{L^{2}(\varGamma _{3})^2}, \end{aligned}$$

where $\nabla :H_{0}^{1}(\varGamma _{3})\rightarrow L^{2}(\varGamma _{3})^{2}$ denotes the gradient operator, that is $\nabla \xi =\left( \xi _{,x_{1}},\xi _{,x_{2}}\right)$. By the Friedrichs–Poincaré inequality there exists a constant $\widetilde{C}_{\varGamma }>0,$ which depends on $\varGamma _{3}$, such that

$$\begin{aligned} \mid \zeta \mid _{L^{2}(\varGamma _{3})}\le \widetilde{C}_{\varGamma }\mid \zeta \mid _{H_{0}^{1}(\varGamma _{3})}\quad \forall \zeta \in H_{0}^{1}(\varGamma _{3}). \end{aligned}$$

(26)

We use the notation $H^{-1}(\varGamma _{3})$ for the dual of the space $H_{0}^{1}(\varGamma _{3})$. Identifying $L^{2}(\varGamma _{3})$ with its own dual we can write $H_{0}^{1}(\varGamma _{3})\subset L^{2}(\varGamma _{3})\subset H^{-1}(\varGamma _{3})$. Below $\left\langle \cdot , \cdot \right\rangle$ represents the duality pairing between $H^{-1}(\varGamma _{3})$ and $H_{0}^{1}(\varGamma _{3})$, and $\mid \cdot \mid _{H^{-1}(\varGamma _{3})}$denotes the norm on $H^{-1}(\varGamma _{3})$. Also, $\left\langle \zeta ,\xi \right\rangle =\left( \zeta ,\xi \right) _{L^{2}(\varGamma _{3})}$ for $\zeta \in L^{2}(\varGamma _{3})$ and $\xi \in H_{0}^{1}(\varGamma _{3})$.

For our existence and uniqueness result we will need the following hypotheses.

The viscosity operator $\mathcal {A}$: $\varOmega \times S_{3}\rightarrow S_{3}$ satisfies

$$\begin{aligned} \left\{ \begin{array}{l} (a)\text { There exists }L_{\mathcal {A}}>0\text { such that} \\ \quad \mid \mathcal {A(}\varvec{x},\varvec{\varepsilon }_{1})-\mathcal {A(}\varvec{x} ,\varvec{\varepsilon }_{2})\mid \le L_{\mathcal {A}}\mid \varvec{\varepsilon }_{1}-\varvec{\varepsilon }_{2}\mid ~\forall \varvec{\varepsilon } _{1},\varvec{\varepsilon }_{2}\in S_{3},\text { a.e. }\varvec{x}\in \varOmega . \\ (b)\text { There exists }m_{\mathcal {A}}>0\text { such that} \\ \quad (\mathcal {A}(\varvec{x},\varvec{\varepsilon }_{1})-\mathcal {A}(\varvec{x}, \varvec{\varepsilon }_{2})).(\varvec{\varepsilon }_{1}-\varvec{\varepsilon } _{2})\ge m_{\mathcal {A}}\mid \varvec{\varepsilon }_{1}-\varvec{\varepsilon } _{2}\mid ^{2}~\forall \varvec{\varepsilon }_{1},\varvec{\varepsilon } _{2}\in S_{3},\text { a.e. }\varvec{x}\in \varOmega . \\ (c)\text { The mapping }\varvec{x}\rightarrow \mathcal {A}(\varvec{x},\varvec{\varepsilon })\text { is Lebesgue measurable on }\varOmega \text { for any } \varvec{\varepsilon }\in S_{3}. \\ (d)\text { The mapping }\varvec{x}\rightarrow \mathcal {A}(\varvec{x,0})\text { }\in \mathcal {H}. \end{array} \right. \end{aligned}$$

(27)

The elasticity operator $\mathcal {G}: \varOmega \times S_{3}\times \mathbb {R}\rightarrow S_{3}$ satisfies

$$\begin{aligned} \left\{ \begin{array}{l} (a)\text { There exists }L_{\mathcal {G}}>0\text { such that} \\ \mid \mathcal {G}(\varvec{x},\varvec{\varepsilon }_{1},\beta _{1})-\mathcal {G} (\varvec{x},\varvec{\varepsilon }_{2},\beta _{2})\mid \le L_{\mathcal {G} }(\mid \varvec{\varepsilon }_{1}-\varvec{\varepsilon }_{2}\mid +\mid \beta _{1}-\beta _{2}\mid ) \\ \quad ~\forall \varvec{\varepsilon }_{1},\varvec{\varepsilon }_{2}\in S_{3},~\forall \beta _{1},\beta _{2}\in \mathbb {R},\text { a.e. }\varvec{x}\in \varOmega . \\ (b)\text { The mapping }\varvec{x}\rightarrow \mathcal {G}(\varvec{x},\varvec{\varepsilon },\beta )\text { is Lebesgue measurable on }\varOmega \text { for any }\varvec{\varepsilon }\in S_{3}\text { and }\beta \in \mathbb {R}. \\ (c)\text { The mapping }\varvec{x}\rightarrow \mathcal {G}(\varvec{x},\varvec{0},0)\in \mathcal {H}. \end{array} \right. \end{aligned}$$

(28)

The damage source function $\phi : \varOmega \times S_{3}\times \mathbb {R\times R}\rightarrow \mathbb {R}$ satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} (a)\text { There exists }L_{\phi }>0\text { such that} \\ \quad \mid \phi (\varvec{x},\varvec{\varepsilon }_{1},\theta _{1},\beta _{1})-\phi ( \varvec{x},\varvec{\varepsilon }_{2},\theta _{2},\beta _{2})\mid \le L_{\phi }(\mid \varvec{\varepsilon }_{1}-\varvec{\varepsilon }_{2}\mid +\mid \theta _{1}-\theta _{2}\mid +\mid \beta _{1}-\beta _{2}\mid )\text { } \\ \quad \forall \varvec{\varepsilon }_{1},\varvec{\varepsilon }_{2},\in S_{3}\text { and } \theta _{1},\theta _{2},\beta _{1},\beta _{2}\in \mathbb {R},\text { a.e. } \varvec{x}\in \varOmega . \\ (b)\text { The mapping }\varvec{x}\rightarrow \phi (\varvec{x},\varvec{\varepsilon },\theta ,\beta )\text { is Lebesgue measurable on }\varOmega \text { for any }\varvec{\varepsilon }\in S_{3}\text { and } \theta ,\beta \in \mathbb {R}. &{} \\ (c)\text { The mapping }\varvec{x}\rightarrow \phi (\varvec{x},\varvec{0},0,0) \in L^{2}(\varOmega ). &{} \end{array} \right. \end{aligned}$$

(29)

The thermal expansion operator C: $\varOmega \times \mathbb {R\times R}\rightarrow S_{3}$ satisfies

$$\begin{aligned} \left\{ \begin{array}{l} (a)\text { There exists }L_{C}>0\text { such that} \\ \quad \mid C(\varvec{x},\theta _{1},\beta _{1})-C(\varvec{x},\theta _{2},\beta _{2})\mid \le L_{C}(\mid \theta _{1}-\theta _{2}\mid +\mid \beta _{1}-\beta _{2}\mid ) ~\forall \theta _{1},\theta _{2},\beta _{1},\beta _{2}\in \mathbb {R},\text { a.e. }\varvec{x}\in \varOmega .\text { \quad } \\ (b)\text { The mapping }\varvec{x}\rightarrow C(\varvec{x},\theta ,\beta )\text { is Lebesgue measurable on } \varOmega \text { for any } \theta ,\beta \in \mathbb {R}. \\ (c)\text { The mapping }\varvec{x}\rightarrow C(\varvec{x},0,0)\in \mathcal {H}. \end{array} \right. \end{aligned}$$

(30)

The normal compliance functions $p_{\nu }:\varGamma _{3}\times \mathbb {R}\rightarrow \mathbb {R}_{+}$ satisfy

$$\begin{aligned} \left\{ \begin{array}{l} (a)\text { There exists }L_{\nu }>0\text { such that} \\ \mid p_{\nu }(\varvec{x},u_{1})-p_{\nu }(\varvec{x},u_{2})\mid \le L_{\nu }\mid u_{1}-u_{2}\mid ~\forall u_{1},u_{2}\in \mathbb {R},\text { a.e. } \varvec{x}\in \varGamma _{3}. \\ (b)\text { the mapping }\varvec{x}\rightarrow p_{\nu }(\varvec{x},u)\text { is measurable on }\varGamma _{3}\text { for any } u\in \mathbb {R}. \\ (c)\text { }p_{\nu }(\varvec{x},u)=0\text { for all } u\le 0,\text { a.e. } \varvec{x}\in \varGamma _{3}. \\ (d)\text { There exists } p_{\nu }^*>0\text { such that } p_{\nu }(\varvec{x},u)\le p_{\nu }^{*}~\forall u\in \mathbb {R},\text { a.e. }\varvec{x}\in \varGamma _{3}. \end{array} \right. \end{aligned}$$

(31)

The coefficient of friction $\mu :\varGamma _{3}\times \mathbb {R}^{2}\rightarrow \mathbb {R}_{+}$ satisfies

$$\begin{aligned} \left\{ \begin{array}{l} (a)\text { There exists }L_{\mu }>0 \text { such that} \\ \mid \mu (\varvec{x},a_{1},b_{1})-\mu (\varvec{x},a_{1},b_{1})\mid \le L_{\mu }(\mid a_{1}-a_{2} \mid +\mid b_{1}-b_{2} \mid ) \\ \forall a_{1},a_{2},b_{1},b_{2}\in \mathbb {R},\text { a.e. }\varvec{x}\in \varGamma _{3}. \\ (b)\text { The mapping }\varvec{x}\rightarrow \mu (\varvec{x},a,b,c)\text { is Lebesgue measurable on }\varGamma _{3}\text {, }\forall a,b,c\in \mathbb {R}. \\ (c)\text { There exists } \mu ^*>0 \text { such that }\mu (\varvec{x},a,b)\le \mu ^{*}\text { }\forall a,b\in \mathbb {R}, \text { a.e. }\varvec{x}\in \varGamma _{3}. \end{array} \right. \end{aligned}$$

(32)

The operator in the heat equation $\psi :\varOmega \times \mathbb {R}^{3}\rightarrow \mathbb {R}$ satisfies

$$\begin{aligned} \left\{ \begin{array}{l} (a)\text { There exists }L_{\psi }>0\text { such that} \\ \mid \psi (\varvec{x},\varvec{\varepsilon }_{1}, \theta _1, \beta _1)-\psi (\varvec{x},\varvec{\varepsilon }_{2},\theta _2, \beta _2)\mid \le L_{\psi }(\mid \varvec{\varepsilon }_{1}-\varvec{\varepsilon }_{1}\mid + \mid \theta _1 -\theta _2 \mid + \mid \beta _1 -\beta _2\mid ) \\ \forall \varvec{\varepsilon }_{1},\varvec{\varepsilon }_{2}\in \mathbb {R}^{3} ~~\forall \theta _1, \theta _2, \beta _1,\beta _2 \in \mathbb {R} ,\text { a.e. } \varvec{x}\in \varOmega . \\ (b)\text { The mapping } \varvec{x}\rightarrow \psi (\varvec{x},\varvec{\varepsilon },\theta ,\beta )\text { is Lebesgue measurable on }\varOmega , ~\forall \varvec{\varepsilon }\in \mathbb {R}^{3} ~\forall \theta , \beta \in \mathbb {R}. \\ (c)\text { }\psi (\varvec{x},\varvec{\varepsilon },\theta ,\beta )\in L^{2}(\mathcal {H}). \end{array} \right. \end{aligned}$$

(33)

For some $c_{k}>0$, for all $(\xi _{i})\in \mathbb {R}^{3}$

$$\begin{aligned} K_c=(k_{ij}),~~k_{ij}=k_{ji}\in L^{\infty }(\varOmega ),~~k_{ij}\xi _{j}\xi _{i}\ge c_{k}\xi _{i}\xi _{i}. \end{aligned}$$

(34)

For the initial gap function, wear diffusion coefficient, wear rate coefficient, velocity of the foundation, heat source density, body forces and surface traction we make the following assumptions

$$\begin{aligned}&g\in L^{2}(\varGamma _{3}),\quad g\ge 0\text { a.e. on }\varGamma _{3}. \end{aligned}$$

(35)

$$\begin{aligned}&k\in L^{\infty }(\varGamma _{3}),\quad k\ge k^{*}>0\text { a.e. on }\varGamma _{3}. \end{aligned}$$

(36)

$$\begin{aligned}&\kappa \in L^\infty (\varGamma _3),\quad \kappa > 0 \text { a.e. on } \varGamma _3. \end{aligned}$$

(37)

$$\begin{aligned}&\varvec{v}^{*}:\varGamma _{3}\times \left[ 0,T\right] \rightarrow \mathbb {R}^{3} \text { is a continuous function}. \end{aligned}$$

(38)

$$\begin{aligned}&q\in L^{2}(0,T;L^{2}(\varOmega )). \end{aligned}$$

(39)

$$\begin{aligned}&\varvec{f}_{0}\in C(0,T;H),~\varvec{f}_{2}\in C(0,T;L^{2}(\varGamma _{2})^{3}). \end{aligned}$$

(40)

The boundary and initial data satisfy

$$\begin{aligned} \varvec{u}_{0}\in V,~~\zeta _{0}\in L^{2}(\varGamma _{3}), ~~\beta _{0}\in Y,\theta _{0}\in L^{2}(\varOmega ),~~\theta _{R}\in L^{2}(0,T;L^{2}(\varGamma _{3})),~~k_{e}\in L^{\infty }(\varOmega ,\mathbb {R}_{+}). \end{aligned}$$

(41)

We define the vector valued function $\varvec{f}:\left[ 0,T\right] \rightarrow V$ and the bilinear forms $a: H_{0}^{1}(\varGamma _{3})\times H_{0}^{1}(\varGamma _{3})\rightarrow \mathbb {R}$ and $b:H^{1}(\varOmega )\times H^{1}(\varOmega )\rightarrow \mathbb {R}$ by

$$\begin{aligned} (\varvec{f}(t),\varvec{v})_{V}&=\int \limits _{\varOmega }\varvec{f}_{0}(t) \cdot \varvec{v} dx+\int \limits _{\varGamma _{2}}\varvec{f}_{2}(t) \cdot \varvec{v}~da\text { \quad }\forall \varvec{v}\in V. \end{aligned}$$

(42)

$$\begin{aligned} a(\zeta ,\xi )&=\int \limits _{\varGamma _{3}}k\nabla \zeta \cdot \nabla \xi ~ da. \end{aligned}$$

(43)

$$\begin{aligned} b(\xi ,\pmb {\varphi })&=k_{1}\int \limits _{\varOmega }\nabla \xi \cdot \nabla \pmb {\varphi }~dx. \end{aligned}$$

(44)

Finally, the functional $j:L^{2}(\varGamma _{3})\times V^{3}\rightarrow \mathbb {R}$ and the operator $F:H_{0}^{1}(\varGamma _{3})\times V^{3}\rightarrow H^{-1}(\varGamma _{3})$ are given by

$$\begin{aligned} j(\zeta , \varvec{u},\varvec{v},\varvec{w})&=\int \limits _{\varGamma _{3}}p_{\nu }(u_{\nu }-\lambda \zeta \chi _{[D_{\omega }]}-g)w_{\nu }da \nonumber \\&\quad +\int \limits _{\varGamma _{3}}\mu (\zeta , \mid \varvec{v}_{\tau }-\varvec{v}^{*}\mid )p_{\nu }(u_{\nu }-\lambda \zeta \chi _{[D_{\omega }]}-g)\mid \varvec{w}_{\tau } \nonumber \\&\quad - \varvec{v}^{*}\mid da \quad \forall \zeta \in L^{2}(\varGamma _{3}),\forall \varvec{u},\varvec{v}, \varvec{w}\in V. \end{aligned}$$

(45)

$$\begin{aligned} (F(\zeta ,\varvec{u},\varvec{v},\varvec{w}),\xi )_{H^{-1}(\varGamma _3)\times H^1_0(\varGamma _3)}&=\int \limits _{\varGamma _{3}}{ \kappa }\mu (\zeta , \mid \varvec{v}_{\tau }-\varvec{v}^{*}\mid )p_{\nu }(u_{\nu }-\lambda \zeta \chi _{[D_{\omega }]}-g)\nonumber \\&\qquad R^{*}(\mid \varvec{w}_{\tau }- \varvec{v}^{*}\mid )\xi da \quad \forall \zeta ,\xi \in H^1_0(\varGamma _3), \forall \varvec{u,v,w}\in V. \end{aligned}$$

(46)

Using standard arguments based on Green formulas given on (1) and (2), we obtain the following formulation of the mechanical problem (6)–(19).

Problem PV. Find a displacement field $\varvec{u}:[0,T]\rightarrow V,$ a stress field $\varvec{\sigma }:[0,T]\rightarrow \mathcal {H},$ a temperature field $\theta :\left[ 0,T\right] \rightarrow E$, a damage field $\beta :\left[ 0,T\right] \rightarrow H^{1}(\varOmega )$ and a surface particle density field $\zeta : [0,T]\rightarrow H_{0}^{1}(\varGamma _{3})$ such that

$$\begin{aligned}&\varvec{\sigma }(t)=\mathcal {A}\varvec{\varepsilon }(\dot{\varvec{u}}(t))+\mathcal {G}(\varvec{\varepsilon }(\varvec{u}(t)),\beta (t))-C(\theta (t),\beta (t)) \end{aligned}$$

(47)

$$\begin{aligned}&(\varvec{\sigma } (t),\varvec{\varepsilon }(\varvec{v}-\dot{\varvec{u}}(t))_{\mathcal {H}}+j(\zeta (t),\varvec{u}(t),\dot{\varvec{u}}(t),\varvec{v})-j(\zeta (t),\varvec{u}(t),\dot{\varvec{u}}(t),\dot{\varvec{u}}(t)) \nonumber \\&\qquad \ge (\varvec{f}(t),\varvec{v}-\dot{\varvec{u}}(t))_{V}~~\quad \forall \varvec{v}\in V, \end{aligned}$$

(48)

$$\begin{aligned}&\beta (t)\in Y, (\dot{\beta }(t),\xi -\beta (t))_{L^{2}(\varOmega )}+b(\beta (t),\xi -\beta (t))\nonumber \\&\qquad \ge (\phi (\varvec{\varepsilon }(\varvec{u}(t)),\theta (t),\beta (t)),\xi -\beta (t))_{L^{2}(\varOmega )}~~~\forall \xi \in Y\text { \quad a.e. } t\in \left( 0,T\right) , \end{aligned}$$

(49)

$$\begin{aligned}&\dot{\theta }(t)+K\theta (t)=S(\dot{\varvec{u}}(t), \theta (t), \beta (t))+Q(t) ~~\text { in } E' ~~\text { a.e. } t\in (0,T) , \end{aligned}$$

(50)

$$\begin{aligned}&(\dot{\zeta }(t),\xi )_{H^{-1}(\varGamma _3) \times H^1_0(\varGamma _3)}+a(\zeta (t),\xi )=(F(\zeta (t),\varvec{u}(t),\dot{\varvec{u}}(t),\overset{.}{\varvec{u}}(t)),\xi )_{H^{-1}(\varGamma _3) \times H^1_0(\varGamma _3)}\nonumber \\&\qquad \qquad \qquad \forall \xi \in H_{0}^{1}(\varGamma _{3})~~\text { a.e. }t\in \left( 0,T\right) , \end{aligned}$$

(51)

$$\begin{aligned}&\varvec{u}(0)=\varvec{u}_{0},~~\beta (0)=\beta _0, ~~\zeta (0)=\zeta _{0},~~\theta (0)=\theta _{0}, \end{aligned}$$

(52)

where $Q:[0,T]\rightarrow E^{\prime }$, $K:E\rightarrow E^{\prime }$ and $S:V\times E \times L^2(\varOmega ) \rightarrow E^{\prime }$ are given by

$$\begin{aligned}&(Q(t),\eta )_{E^{\prime }\times E}=\int \limits _{\varGamma _{3}}k_{e}\theta _{R}(t)\eta da+\int \limits _{\varOmega }q(t)\eta dx, \end{aligned}$$

(53)

$$\begin{aligned}&(K\tau ,\eta )_{E^{\prime }\times E}=\underset{i,j=1}{\overset{3}{\sum }} \int \limits _{\varOmega }k_{ij}\frac{\partial \tau }{\partial x_{j}}\frac{\partial \eta }{\partial x_{i}}dx+\int \limits _{\varGamma _{3}}k_{e}\tau \cdot \eta da, \end{aligned}$$

(54)

$$\begin{aligned}&(S(\varvec{u}, \theta , \beta ),\eta )_{E^{\prime }\times E}=\int \limits _{\varOmega }\psi (\varvec{u}, \theta , \beta )\eta dx, \end{aligned}$$

(55)

for all $\varvec{u}, \varvec{v}\in V$, $\theta ,\eta ,\tau \in E$, $\beta \in L^2(\varOmega )$ and $\zeta \in L^2(\varGamma _3)$. Below in this section $\varvec{u}_{1}\varvec{,u}_{2}$, $\varvec{v}_{1}$ and $\varvec{v}_{2}$ represent elements of V, $\theta _1$, $\theta _2$ are elements of E and $\zeta _1$, $\zeta _2$ are in $L^2(\varGamma _3)$. Finally, we use (45), the assumption (31) on $p_\nu$, the assumption (32) on $\mu$, (24) and (25) to see that

$$\begin{aligned}&j(\zeta _1,\varvec{u}_{1},\varvec{v}_{1},\varvec{v}_{2})-j(\zeta _1,\varvec{u}_{1},\varvec{v}_{1},\varvec{v}_{1})+j(\zeta _2,\varvec{u}_{2},\varvec{v}_{2},\varvec{v}_{1})-j(\zeta _2,\varvec{u}_{2},\varvec{v}_{2},\varvec{v}_{2})\nonumber \\&\qquad \le C^2_\varGamma (L_\nu + \mu ^{*}L_\nu )|\varvec{u}_1-\varvec{u}_2|_V |\varvec{v}_1-\varvec{v}_2|_V + C_\varGamma (L_\nu \lambda + \mu ^{*}L_\nu \lambda +p_\nu ^{*} L_\mu )|\zeta _1 \nonumber \\&\quad \qquad -\zeta _2|_{L^2(\varGamma _3)} |\varvec{v}_1-\varvec{v}_2|_V+ p_\nu ^{*}L _\mu C^2_\varGamma |\varvec{v}_1-\varvec{v}_2|_V^2 \end{aligned}$$

(56)

This inequality will be used in the following section.

4 An existence and uniqueness result

Our main existence and uniqueness result is the following.

Theorem 1

Let the assumptions (27)–(41) hold. There exists a constant $C^{*}>0$, depending on $C_{\varGamma }$, $m_\mathcal {A}$ and $L_\mu$ such that if $p_{\nu }^{*}<C^{*}$ then problem PV has a unique solution $\left\{ \varvec{u},\varvec{\sigma },\beta ,\theta ,\zeta \right\}$ satisfying

$$\begin{aligned}&\varvec{u}\in C^{1}(0,T;V), ~~\varvec{\sigma }\in C(0,T;\mathcal {H}_{1}), \end{aligned}$$

(57)

$$\begin{aligned}&\beta \in H^{1}(0,T;L^{2}(\varOmega ))\cap L^{2}(0,T;H^{1}(\varOmega )), \end{aligned}$$

(58)

$$\begin{aligned}&\theta \in C(0,T;L^{2}(\varOmega ))\cap L^{2}(0,T;E)\cap W^{1,2}(0,T;E^{\prime }), \end{aligned}$$

(59)

$$\begin{aligned}&\zeta \in L^{2}(0,T;H_{0}^{1}(\varGamma _{3}))\cap C(0,T;L^{2}(\varGamma _{3})),~~\dot{\zeta }\in L^{2}(0,T;H^{-1}(\varGamma _{3})). \end{aligned}$$

(60)

We conclude that, under assumptions (27)–(41), the mechanical problem (6)–(19) has a unique weak solution with the regularities (57)–(60).

The proof of this theorem will be carried out in several steps. We assume in what follows that assumptions (27)–(41) are satisfied and moreover

$$\begin{aligned} C_{\varGamma }^{2}p_{\nu }^{*}L_{\mu }<m_{\mathcal {A}}. \end{aligned}$$

(61)

First let $\alpha \in L^{2}(0,T;H^{-1}(\varGamma _{3})),$ we solve the following parabolic equation.

Problem $PV_{\alpha }$. Find $\zeta _{\alpha }:\left[ 0,T\right] \rightarrow H_{0}^{1}(\varGamma _{3})$ such that

$$\begin{aligned}&(\dot{\zeta }_{\alpha }(t),\xi )_{H^{-1}(\varGamma _3) \times H^1_0(\varGamma _3)}+a(\zeta _{\alpha }(t),\xi )=(\alpha (t),\xi )_{H^{-1}(\varGamma _3) \times H^1_0(\varGamma _3)}\nonumber \\&\quad \forall \xi \in H_{0}^{1}(\varGamma _{3}),\text { a.e. } t\in \left( 0,T\right) , \end{aligned}$$

(62)

$$\begin{aligned}&\zeta _{\alpha }(0)=\zeta _{0}. \end{aligned}$$

(63)

Lemma 1

There exists a unique solution of problem (62)–(63) satisfying

$$\begin{aligned} \zeta _{\alpha }\in L^{2}(0,T;H_{0}^{1}(\varGamma _{3}))\cap C(0,T;L^{2}(\varGamma _{3})),\text { }\overset{.}{\zeta }_{\alpha }\in L^{2}(0,T;H^{-1}(\varGamma _{3})). \end{aligned}$$

(64)

Moreover, if $\zeta _i$ is the solution to Problem $PV_{\alpha }$ corresponding to $\alpha = \alpha _i \in L^2(0,T; H^{-1}(\varGamma _3))$, for $i = 1, 2$, then

$$\begin{aligned}&|\zeta _1(t)- \zeta _2(t)|_{L^2(\varGamma _3)}^2+ \int \limits _{0}^{t}|\nabla \zeta _1(s)- \nabla \zeta _2(s)|^2_{L^2(\varGamma _3)^2}ds\nonumber \\&\quad \le C\int \limits _{0}^{t} |\alpha _1(s)-\alpha _2(s)|_{H^{-1}(\varGamma _3)}^2 ds \qquad \forall t\in [0,T]. \end{aligned}$$

(65)

Proof

The proof follows from an evolution equation result with linear continuous operators, see for example [24].

Now let $\rho \in L^{2}(0,T;E^{\prime })$, we solve the following evolution equation.

Problem $PV_{\rho }$. Find $\theta _{\rho }:\left[ 0,T\right] \rightarrow E$ such that

$$\begin{aligned}&\dot{\theta }_{\rho }(t)+K\theta _{\rho }(t) =\rho (t) \quad \text { in } E', \text { a.e. } t\in \left( 0,T\right) , \end{aligned}$$

(66)

$$\begin{aligned}&\theta _{\rho }(0)=\theta _{0}. \end{aligned}$$

(67)

We have the following result.

Lemma 2

Problem $PV_{\rho }$ has a unique solution satisfying the regularity (59). Moreover, $\exists C>0$ such that $\forall \rho _{i}\in L^2(0,T;E')$, denote $\theta _{\rho _{i}}=\theta _{i}$, $i=1,2$,

$$\begin{aligned} \mid \theta _{1}(t)-\theta _{2}(t)\mid _{L^{2}(\varOmega )}^{2}+\int \limits _{0}^{t}\mid \theta _{1}(s)-\theta _{2}(s)\mid _E^2 ds\le C\int \limits _{0}^{t}\mid \rho _{1}(s)-\rho _{2}(s)\mid _{E'}^{2}ds\quad \forall t\in [0,T]. \end{aligned}$$

(68)

Proof

The proof follows from classical first order evolution equation given in [2, 22]. Here the Gelfand triple is given by $E\subset \varvec{\ }L^{2}(\varOmega )\subset E^{\prime }$. The operator K is linear and coercive. By Korn’s inequality, we have $(K\tau ,\tau )_{E^{\prime }\times E}\ge C\mid \tau \mid _{E}^{2}$.

Next, for $\forall \rho _{1}, \rho _{2}\in L^2(0,T;E')$ we have for a.e. $s\in (0,T)$

$$\begin{aligned}&(\dot{\theta }_{1}(s)-\dot{\theta }_{2}(s), \theta _{1}(s)-\theta _{2}(s))_{L^2(\varOmega )}+(K\theta _{1}(s)-K\theta _{2}(s),\theta _{1}(s)-\theta _{2}(s))_{E'\times E}\\&\qquad \qquad \qquad =(\rho _1 (s)-\rho _2(s), \theta _{1}(s)-\theta _{2}(s))_{E'\times E}, \end{aligned}$$

then by integrating over (0, t) for $t \in [0, T ]$, (68) follows by using (34) and (54).

In the third step, we let $\gamma \in L^{2}(0,T;L^{2}(\varOmega ))$ be given and consider the following variational problem for the damage field.

Problem $PV_{\gamma }$. Find a damage field $\beta _{\gamma }:\left[ 0,T\right] \rightarrow H^{1}(\varOmega )$ such that

$$\begin{aligned}&\beta _{\gamma }(t)\in Y,\text { }(\ \overset{.}{\beta }_{\gamma }(t),\xi -\beta _{\gamma }(t))_{L^{2}(\varOmega )}+b(\beta _{\gamma }(t),\xi -\beta _{\gamma }(t))\nonumber \\&\quad \ge (\gamma (t),\xi -\beta _{\gamma }(t))_{L^{2}(\varOmega )} \quad \forall \xi \in Y\text { a.e. } t\in \left( 0,T\right) , \end{aligned}$$

(69)

$$\begin{aligned}&\beta _{\gamma }(0)=\beta _{0}. \end{aligned}$$

(70)

Lemma 3

Problem $PV_{\gamma }$ has a unique solution $\beta _{\gamma }$ such that

$$\begin{aligned} \beta _{\gamma }\in H^{1}(0,T;L^{2}(\varOmega ))\cap L^{2}(0,T;H^{1}(\varOmega )). \end{aligned}$$

(71)

Moreover, if $\beta _i$ is the solution to Problem $PV_{\gamma }$ corresponding to $\gamma = \gamma _i \in L^2(0,T; L^2(\varOmega ))$, for $i = 1, 2$, then

$$\begin{aligned} |\beta _1(t)- \beta _2(t)|_{L^{2}(\varOmega )}^2+\int \limits _{0}^{t}|\beta _1(s)- \beta _2(s)|_{L^{2}(\varOmega )}^2 ds \leqslant C\int \limits _{0}^{t} |\gamma _1(s)-\gamma _2(s)|_{L^{2}(\varOmega )}^2 ds \qquad \forall t\in [0,T]. \end{aligned}$$

(72)

Proof

We use (44), $\beta _0$ in (41) and a classical existence and uniqueness result on parabolic inequalities (see for example [2, 22]).

Now we substitute (47) in (48) and we consider the obtained variational inequality with $\zeta =\zeta _{\alpha },\theta =\theta _{\rho }$ and $\beta =\beta _{\gamma }$. Let $(\varvec{z},\varvec{h})\in C(0,T;V)^{2}$ be given, we consider the following variational problem.

Problem $PV_{\alpha \rho \gamma zh}$. Find a displacement field $\varvec{v}_{\alpha \rho \gamma zh}:\left[ 0,T\right] \rightarrow V$ such that

$$\begin{aligned}&(\mathcal {A}\varvec{\varepsilon }(\varvec{v}_{\alpha \rho \gamma zh}(t)), \varvec{\varepsilon }(\varvec{v}-\varvec{v}_{\alpha \rho \gamma zh}(t)))_{\mathcal {H}} + j(\zeta _{\alpha }(t),\varvec{z}(t),\varvec{h}(t),\varvec{v})-j(\zeta _{\alpha }(t),\varvec{z}(t),\varvec{h}(t),\varvec{v}_{\alpha \rho \gamma zh}(t))\nonumber \\&\ge (\varvec{f}(t),\varvec{v}-\varvec{v}_{\alpha \rho \gamma zh}(t))_{V}-(\mathcal {G}(\varvec{\varepsilon } (\varvec{z}(t)),\beta _{\gamma }(t)), \varvec{\varepsilon }(\varvec{v}-\varvec{v}_{\alpha \rho \gamma zh}(t)))_{\mathcal {H}}\nonumber \\&\quad +(C(\theta _{\rho }(t),\beta _{\gamma }(t)),\varvec{\varepsilon }(\varvec{v-v}_{\alpha \rho \gamma zh}(t)))_{\mathcal {H}} \quad \qquad \forall \varvec{v}\in V, t\in \left( 0,T\right) . \end{aligned}$$

(73)

Lemma 4

There exists a unique solution of problem (73) satisfying $\varvec{v}_{\alpha \rho \gamma zh}\in C(0,T;V)$.

Proof

By using variational inequalities results (see for example [3]), we conclude that there exists a unique solution $\varvec{v}_{\alpha \rho \gamma zh}(t)$ of problem (73) for $t\in \left( 0,T\right)$. Now we show that $\varvec{v} _{\alpha \rho \gamma zh}:\left[ 0,T\right] \rightarrow V$ is continuous. Let $t_{1},t_{2}\in \left( 0,T\right)$, we denote $\varvec{v}_{\alpha \rho \gamma zh}(t_{i})=\varvec{v}_{i}$, $\zeta _{\alpha }(t_{i})=\zeta _{i}$, $\theta _{\rho }(t_{i})=\theta _{i}$, $\beta _{\gamma }(t_{i})=\beta _{i}$, $\varvec{z}(t_{i})=\varvec{z}_{i}$, $\varvec{h}(t_{i})=\varvec{h}_{i}$ and $\varvec{f}(t_{i})=\varvec{f}_{i}$ for $i=1,2$. We use (73) to find

$$\begin{aligned} (\mathcal {A}\varvec{\varepsilon }(\varvec{v}_{1})-\mathcal {A}\varvec{\varepsilon }(\varvec{v}_{2}),\varvec{\varepsilon }(\varvec{v}_{1}-\varvec{v}_{2}))_{\mathcal {H}}\,&\le j(\zeta _{1},\varvec{z}_{1},\varvec{h}_{1} \varvec{,v}_{2})-j(\zeta _{1},\varvec{z}_{1},\varvec{h}_{1}\varvec{,v}_{1}) \\&+j(\zeta _{2},\varvec{z}_{2},\varvec{h}_{2}\varvec{,v}_{1})-j(\zeta _{2},\varvec{z}_{2},\varvec{h}_{2}\varvec{,v}_{2})\\&+(\mathcal {G}(\varvec{\varepsilon } (\varvec{z}_{1}),\beta _{1})-\mathcal {G}(\varvec{\varepsilon } (\varvec{z}_{2}),\beta _{2}),\varvec{\varepsilon }(\varvec{v}_{1}\varvec{-v}_{2}))_{\mathcal {H}} \\&+(C(\theta _{1},\beta _{1})-C(\theta _{2},\beta _{2}),\varvec{\varepsilon }(\varvec{v}_{1}\varvec{-v}_{2}))_{\mathcal {H}}\\&+(\varvec{f}_{1}-\varvec{f}_{2},\varvec{v}_{1}\varvec{-v}_{2})_{V}. \end{aligned}$$

Condition (22), the estimate (56) and assumptions (27), (28) and (30) give us

$$\begin{aligned} m_{\mathcal {A}}\mid \varvec{v}_{1}\varvec{-v}_{2}\mid _{V}&\le (L_{\mathcal {G }}+C_{\varGamma }^{2}(L_{\nu } +\mu ^{*}L_{\nu }))\mid \varvec{z}_{1}-\varvec{z} _{2}\mid _{V}\nonumber \\&\quad +C_{\varGamma }^{2}p_{\nu }^{*}L_{\mu }\mid \varvec{h}_{1}-\varvec{h}_{2}\mid _{V}+(L_\mathcal {G}+L_C) \mid \beta _1-\beta _2\mid _{L^2(\varOmega )}\nonumber \\&\quad +C_{\varGamma }(\lambda L_{\nu } +\mu ^{*}L_{\nu }\lambda +p_{\nu }^{*}L_{\mu })\mid \zeta _{1}-\zeta _{2}\mid _{L^{2}(\varGamma _{3})}\nonumber \\&\ \quad +\mid \varvec{f}_{1}- \varvec{f}_{2}\mid _{V}+L_C\mid \theta _{1}-\theta _{2}\mid _{E}, \end{aligned}$$

(74)

which implies that $\varvec{v}_{\alpha \rho \gamma zh}:\left[ 0,T\right] \rightarrow V$ is a continuous function.

We now consider the operator $\varLambda _{\alpha \rho \gamma z}:C(0,T;V)\rightarrow C(0,T;V)$ defined by

$$\begin{aligned} \varLambda _{\alpha \rho \gamma z}\varvec{h}=\varvec{v}_{\alpha \rho \gamma zh}. \end{aligned}$$

(75)

We have the following result.

Lemma 5

The operator $\varLambda _{\alpha \rho \gamma z}$ has a unique fixed point $\varvec{h}_{\alpha \rho \gamma z}$ $\in C(0,T;V).$

Proof

Let $\varvec{h}_{1},\varvec{h}_{2}\in C(0,T;V)$ and let $\varvec{v}_{i}$ denote the solution of (73) for $\varvec{h=h}_{i}$, i.e. $\varvec{v}_{i}=\varvec{v}_{\alpha \rho \gamma z h_{i}}$, $i=1,2$. Using the definition of the operator $\varLambda _{\alpha \rho \gamma z}$ given in (75) we find that

$$\begin{aligned} \mid \varLambda _{\alpha \rho \gamma z}\varvec{h}_{1}(t)-\varLambda _{\alpha \rho \gamma z}\varvec{h}_{2}(t)\mid _{V}=\mid \varvec{v}_{1}(t)-\varvec{v} _{2}(t)\mid _{V}\quad \forall t\in \left[ 0,T\right] . \end{aligned}$$

We use arguments like those used for the estimate (74) to obtain

$$\begin{aligned} m_{\mathcal {A}}\mid \varvec{v}_{1}(t)\varvec{-v}_{2}(t)\mid _{V}\le C_{\varGamma }^{2}p_{\nu }^{*}L_{\mu }\mid \varvec{h}_{1}(t)-\varvec{h} _{2}(t)\mid _{V}\forall t\in \left[ 0,T\right] . \end{aligned}$$

The two previous inequalities and the assumption (61) imply that the operator $\varLambda _{\alpha \rho \gamma z}$ is a contraction on the Banach space C(0, T; V).

In the sequel let $\varvec{h}_{\alpha \rho \gamma z}$ be the fixed point obtained in Lemma 5 and let $\varvec{v}_{\alpha \rho \gamma z} \in C(0,T;V)$ be the function defined by

$$\begin{aligned} \varvec{v}_{\alpha \rho \gamma z}=\varvec{v}_{\alpha \rho \gamma zh_{\alpha \rho \gamma z}}. \end{aligned}$$

(76)

We have $\varLambda _{\alpha \rho \gamma z}\varvec{h}_{\alpha \rho \gamma z}=\varvec{h}_{\alpha \rho \gamma z}$ and

$$\begin{aligned} \varvec{v}_{\alpha \rho \gamma z}=\varvec{h}_{\alpha \rho \gamma z}. \end{aligned}$$

(77)

We take $\varvec{h}=\varvec{h}_{\alpha \rho \gamma z}$ in (73) and we use (76) and (77), to see that $\varvec{v}_{\alpha \rho \gamma z}$ satisfies

$$\begin{aligned}&(\mathcal {A}\varvec{\varepsilon }(\varvec{v}_{\alpha \rho \gamma z}(t)), \varvec{\varepsilon }(\varvec{v}-\varvec{v}_{\alpha \rho \gamma z}(t)))_{\mathcal {H}}+j(\zeta _{\alpha }(t),\varvec{z}(t),\varvec{v}_{\alpha \rho \gamma z}(t),\varvec{v})\nonumber \\&\qquad -j(\zeta _{\alpha }(t),\varvec{z}(t),\varvec{v}_{\alpha \rho \gamma z}(t),\varvec{v}_{\alpha \rho \gamma z}(t)) \nonumber \\&\qquad \ge (\varvec{f}(t),\varvec{v}-\varvec{v}_{\alpha \rho \gamma z}(t))_{V}- (\mathcal {G}(\varvec{\varepsilon } (\varvec{z}(t)),\beta _{\gamma }(t)), \varvec{\varepsilon }(\varvec{v}-\varvec{v}_{\alpha \rho \gamma zh}(t)))_{\mathcal {H}} \nonumber \\&\qquad +(C(\theta _{\rho }(t),\beta _{\gamma }(t)),\varvec{\varepsilon }(\varvec{v}-\varvec{v}_{\alpha \rho \gamma zh}(t)))_{\mathcal {H}}\qquad \qquad \qquad \forall \varvec{v}\in V,~~ t\in \left( 0,T\right) . \end{aligned}$$

(78)

Let now $\varvec{u}_{\alpha \rho \gamma z}\in C^{1}(0,T;V)$ be the function defined by

$$\begin{aligned} \varvec{u}_{\alpha \rho \gamma z}(t)=\int \limits _{0}^{t}\varvec{v}_{\alpha \rho \gamma z}(s)ds+\varvec{u}_{0} \quad \forall t\in [0,T]. \end{aligned}$$

(79)

We define the operator $\varLambda _{\alpha \rho \gamma }:C(0,T;V)\rightarrow C(0,T;V)$ by

$$\begin{aligned} \varLambda _{\alpha \rho \gamma }\varvec{z}=\varvec{u}_{\alpha \rho \gamma z}. \end{aligned}$$

(80)

We have the following result.

Lemma 6

The operator $\varLambda _{\alpha \rho \gamma }$ has a unique fixed point $\varvec{z}_{\alpha \rho \gamma }\in C(0,T;V)$.

Proof

Let $\varvec{z}_{1}\varvec{,z}_{2}\in C(0,T;V)$ and denote $\varvec{v}_{i}=\varvec{v}_{\alpha \rho \gamma z_{i}}$, $\varvec{u}_{i}=\varvec{u}_{\alpha \rho \gamma z_{i}}$ for $i=1,2$. We use (78) and arguments like those used for the estimate (74) in the proof of Lemma 4 to have

$$\begin{aligned} (m_{\mathcal {A}}-C_{\varGamma }^{2}p_{\nu }^{*}L_{\mu })\left| \varvec{v} _{1}(s)-\varvec{v}_{2}(s)\right| _{V}\le (L_{\mathcal {G}}+C_{\varGamma }^{2}(L_{\nu }+\mu ^{*}L_{\nu }))\left| \varvec{z}_{1}(s)-\varvec{z} _{2}(s)\right| _{V}, \end{aligned}$$

(81)

for all $s\in [0,T]$. Using now (80)–(81) we obtain

$$\begin{aligned} \left| \varLambda _{\alpha \rho \gamma }\varvec{z}_{1}(t)-\varLambda _{\alpha \rho \gamma }\varvec{z}_{2}(t)\right| _{V}\le C\int \limits _{0}^{t}\left| \varvec{z}_{1}(s)-\varvec{z}_{2}(s)\right| _{V}ds, \end{aligned}$$

for all $t\in [0,T]$ and C is a positive constant. By reiterating this inequality we obtain that a power of $\varLambda _{\alpha \rho \gamma }$ is a contraction mapping on C(0, T; V), which concludes the proof.

We are now ready to prove the unique solvability of the variational problem.

Problem $PV_{\alpha \rho \gamma }$. Find a displacement field $\varvec{u}_{\alpha \rho \gamma }:\left[ 0,T\right] \rightarrow V$ such that

$$\begin{aligned}&(\mathcal {A}\varvec{\varepsilon }(\dot{\varvec{u}}_{\alpha \rho \gamma }(t)),\varvec{\varepsilon }(\varvec{v}-\dot{\varvec{u}} _{\alpha \rho \gamma }(t)))_{\mathcal {H}}+j(\zeta _{\alpha }(t),\varvec{u}_{\alpha \rho \gamma }(t),\dot{\varvec{u}}_{\alpha \rho \gamma }(t),\varvec{v}) \nonumber \\&-j(\zeta _{\alpha }(t),\varvec{u}_{\alpha \rho \gamma }(t),\dot{\varvec{u}}_{\alpha \rho \gamma }(t),\dot{\varvec{u}}_{\alpha \rho \gamma }(t))\ge (\varvec{f}(t),\varvec{v}-\dot{\varvec{u}}_{\alpha \rho \gamma }(t))_{V}\nonumber \\&-(\mathcal {G}(\varvec{\varepsilon } (\varvec{u}_{\alpha \rho \gamma }(t)),\beta _{\gamma }(t)),\varvec{\varepsilon }(\varvec{v}-\dot{\varvec{u}}_{\alpha \rho \gamma }(t)))_{\mathcal {H}}+(C(\theta _{\rho }(t),\beta _{\gamma }(t)),\varvec{\varepsilon }(\varvec{v}-\dot{\varvec{u}}_{\alpha \rho \gamma }(t)))_{\mathcal {H}}\nonumber \\&\forall \varvec{v}\in V,\quad t\in \left( 0,T\right) , \end{aligned}$$

(82)

$$\begin{aligned}&\varvec{u}_{\alpha \rho \gamma }(0)=\varvec{u}_{0}. \end{aligned}$$

(83)

Lemma 7

There exists a unique function $\varvec{u}_{\alpha \rho \gamma }\in C^{1}(0,T;V)$ satisfying (82)–(83).

Proof

Let $\varvec{z}_{\alpha \rho \gamma }\in C(0,T;V)$ be the fixed point guaranteed by Lemma 6 and let $\varvec{u}_{\alpha \rho \gamma }\in C^{1}(0,T;V)$ be defined by (79), for $\varvec{z}=\varvec{z}_{\alpha \rho \gamma }$. We have $\dot{\varvec{u}}_{\alpha \rho \gamma }=\varvec{v}_{\alpha \rho \gamma z_{\alpha \rho \gamma }}$ and, writing (78) for $\varvec{z}=\varvec{z}_{\alpha \rho \gamma }$, we find

$$\begin{aligned}&(\mathcal {A}\varvec{\varepsilon }(\dot{\varvec{u}}_{\alpha \rho \gamma }(t)),\varvec{\varepsilon }(\varvec{v}-\dot{\varvec{u}}_{\alpha \rho \gamma }(t)))_{\mathcal {H}}+j(\zeta _{\alpha }(t),\varvec{z}_{\alpha \rho \gamma }(t),\dot{\varvec{u}}_{\alpha \rho \gamma }(t),\varvec{v}) \nonumber \\&\qquad \quad -j(\zeta _{\alpha }(t),\varvec{z}_{\alpha \rho \gamma }(t),\dot{\varvec{u}}_{\alpha \rho \gamma }(t),\dot{\varvec{u}} _{\alpha \rho \gamma }(t)) \nonumber \\&\qquad \quad \ge (\varvec{f}(t),\varvec{v}-\dot{\varvec{u}}_{\alpha \rho \gamma }(t))_{V}-(\mathcal {G}(\varvec{\varepsilon } (\varvec{z}_{\alpha \rho \gamma }(t)),\beta _{\gamma }(t)),\varvec{\varepsilon }(\varvec{v}-\dot{\varvec{u}}_{\alpha \rho \gamma }(t)))_{\mathcal {H}} \nonumber \\&\qquad \quad +(C(\theta _{\rho }(t),\beta _{\gamma }(t)),\varvec{\varepsilon }(\varvec{v}-\dot{\varvec{u}}_{\alpha \rho \gamma }(t)))_{\mathcal {H}} \qquad \qquad \qquad \forall \varvec{v}\in V,\quad t\in \left( 0,T\right) . \end{aligned}$$

(84)

Inequality (82) follows now from (84) and (75) since $\varvec{u} _{\alpha \rho \gamma }=\varvec{z}_{\alpha \rho \gamma }$. Moreover, (83) results from (79). We conclude that $\varvec{u}_{\alpha \rho \gamma }$ is a solution of (82)–(83). For the uniqueness, let $\varvec{u}_{\alpha \rho \gamma }$ be the solution of (82)–(83) and let $\varvec{u}_{\alpha \rho \gamma }^{*}$ be any other solution such that $\varvec{u}_{\alpha \rho \gamma }^{*}\in C^{1}(0,T;V)$. Let $\varvec{v}_{\alpha \rho \gamma }^{*}=\dot{\varvec{u}}_{\alpha \rho \gamma }^{*}$. Using (82) we obtain that $\varvec{v}_{\alpha \rho \gamma }^{*}$ satisfies

$$\begin{aligned}&(\mathcal {A}\varvec{\varepsilon }(\varvec{v}_{\alpha \rho \gamma }^{*}(t)), \varvec{\varepsilon }(\varvec{v}-\varvec{v}_{\alpha \rho \gamma }^{*}(t)))_{\mathcal {H}}+j(\zeta _{\alpha }(t),\varvec{u}_{\alpha \rho \gamma }^{*}(t),\varvec{v}_{\alpha \rho \gamma }^{*}(t),\varvec{v})\nonumber \\&\qquad \quad -j(\zeta _{\alpha }(t),\varvec{u}_{\alpha \rho \gamma }^{*}(t),\varvec{v}_{\alpha \rho \gamma }^{*}(t),\varvec{v}_{\alpha \rho \gamma }^{*}(t)) \nonumber \\&\quad \ge (\varvec{f}(t),\varvec{v}-\varvec{v}_{\alpha \rho \gamma }^{*}(t))_{V} -(\mathcal {G}(\varvec{\varepsilon } (\varvec{u}_{\alpha \rho \gamma }^*(t)),\beta _{\gamma }(t)),\varvec{\varepsilon }(\varvec{v}-\varvec{v}^*_{\alpha \rho \gamma }(t)))_{\mathcal {H}} \nonumber \\&\qquad \quad +(C(\theta _{\rho }(t),\beta _{\gamma }(t)),\varvec{\varepsilon }(\varvec{v}-\varvec{v}^*_{\alpha \rho \gamma }(t)))_{\mathcal {H}} \qquad \forall \varvec{v}\in V,\quad t\in \left( 0,T\right) . \end{aligned}$$

(85)

This inequality has a form of (78) with $\varvec{z}=\varvec{u}_{\alpha \rho \gamma }^{*}$ and, therefore, it follows from (73) that it has a unique solution, already denoted by $\varvec{v}_{\alpha \rho \gamma u_{\alpha \rho \gamma }^{*}}$. We conclude that $\varvec{v}_{\alpha \rho \gamma }^{*}=\varvec{v}_{\alpha \rho \gamma u_{\alpha \rho \gamma }^{*}}$. Since $\varvec{v}_{\alpha \rho \gamma }^{*}= \dot{\varvec{u}}_{\alpha \rho \gamma }^{*}$ it follows from (79) that

$$\begin{aligned} \varvec{u}_{\alpha \rho \gamma }^{*}(t)=\int \limits _{0}^{t}\varvec{v}_{\alpha \rho \gamma u_{\alpha \rho \gamma }^{*}}(s)ds+\varvec{u}_{0}\quad \forall t\in [0,T]. \end{aligned}$$

(86)

From (79) and (86) we obtain $\varvec{u}_{\alpha \rho \gamma }^{*}=\varvec{u}_{\alpha \rho \gamma u_{\alpha \rho \gamma }^{*}}$, which shows that $\varvec{u}_{\alpha \rho \gamma }^{*}$ is a fixed point of the operator $\varLambda _{\alpha \rho \gamma }$ defined by (80). Using now Lemma 5 we deduce that

$$\begin{aligned} \varvec{u}_{\alpha \rho \gamma }^{*}=\varvec{z}_{\alpha \rho \gamma }^{*}. \end{aligned}$$

(87)

The uniqueness of the problem (82)–(83) is now a consequence of the fact that $\varvec{u}_{\alpha \rho \gamma }=\varvec{z}_{\alpha \rho \gamma }$ and equality (87).

Next, we need to investigate the properties of the operator $F:H_{0}^{1}(\varGamma _{3})\times E\times V^{3}\rightarrow H^{-1}(\varGamma _{3})$ given by (46).

Lemma 8

The following inequality holds

$$\begin{aligned}&\left| F(\zeta _{1},\varvec{u}_{1},\varvec{v}_{1},\varvec{w}_{1})-F(\zeta _{2},\varvec{u}_{2},\varvec{v}_{2},\varvec{w}_{2})\right| _{H^{-1}(\varGamma _{3})} \nonumber \\&\quad \le L_{F}(\mid \zeta _{1}-\zeta _{2}\mid _{H_{0}^{1}(\varGamma _{3})} +\mid \varvec{u}_{1}-\varvec{u}_{2}\mid _{V}+\mid \varvec{v}_{1}-\varvec{v} _{2}\mid _{V}+\mid \varvec{w}_{1}-\varvec{w}_{2}\mid _{V}) \nonumber \\&\qquad \forall \zeta _{1},\zeta _{2}\in H_{0}^{1}(\varGamma _{3}),~\forall \varvec{u}_{1},\varvec{u}_{2},\varvec{v}_{1},\varvec{v}_{2},\varvec{w}_{1},\varvec{w}_{2}\in V, \end{aligned}$$

(88)

where

$$\begin{aligned} L_{F}=\mid \kappa \mid _{L^\infty (D_\omega )}C_\varGamma \max \left\{ \mu ^{*}p_{\nu }^{*}C_{\varGamma },~\mu ^{*}L_{\nu }RC_{\varGamma },~ (L_{\mu }p_{\nu }^{*}+\lambda \mu ^{*} L_{\nu })R\tilde{C}_{\varGamma },~ p_{\nu }^{*}L_{\mu }RC_{\varGamma }\right\} . \end{aligned}$$

Proof

The estimate (88) follows from the definition (46) on F, the assumptions (31) on $p_\nu$, (32) on $\mu$, (24), (25), (26) and the definition of $R^{*}$.

Now we define the operator $\varLambda :L^{2}(0,T;H^{-1}(\varGamma _{3})\times E^{\prime }\times L^{2}(\varOmega ))\rightarrow L^{2}(0,T;H^{-1}(\varGamma _{3})\times E^{\prime }\times L^{2}(\varOmega ))$ by

$$\begin{aligned} \varLambda (\alpha ,\rho ,\gamma )=(\varLambda _{1}(\alpha ,\rho ,\gamma ),\varLambda _{2}(\alpha ,\rho ,\gamma ),\varLambda _{3}(\alpha ,\rho ,\gamma )), \end{aligned}$$

(89)

such that

$$\begin{aligned}&\varLambda _{1}(\alpha (t),\rho (t),\gamma (t))=F(\zeta _{\alpha }(t),\varvec{u}(t),\dot{\varvec{u}}(t),\dot{\varvec{u}}(t)), \end{aligned}$$

(90)

$$\begin{aligned}&\varLambda _{2}(\alpha (t),\rho (t),\gamma (t))=S(\dot{\varvec{u}}(t) , \theta _\rho (t), \beta _\gamma )+Q(t), \end{aligned}$$

(91)

$$\begin{aligned}&\varLambda _{3}(\alpha (t),\rho (t),\gamma (t))=\phi (\varvec{\varepsilon }(\varvec{u}(t)),\theta _{\rho }(t),\beta _{\gamma }(t)). \end{aligned}$$

(92)

Lemma 9

The operator $\varLambda$ has a unique fixed point $(\alpha ,\rho ,\gamma )\in L^{2}(0,T;H^{-1}(\varGamma _{3})\times E^{\prime }\times L^{2}(\varOmega ))$ such that $\varLambda (\alpha ,\rho ,\gamma )=(\alpha ,\rho ,\gamma )$.

Proof

Let $(\alpha _{i},\rho _{i},\gamma _{i})\in L^{2}(0,T;H^{-1}(\varGamma _{3})\times E^{\prime }\times L^{2}(\varOmega ))$ $i=1,2$. Denote $\varvec{u}_{\alpha _{i}\rho _{i}\gamma _{i}}= \varvec{u}_{i}$, $\dot{\varvec{u}}_{\alpha _{i}\rho _{i}\gamma _{i}}= \varvec{v}_{i}$, $\zeta _{\alpha _{i}}=\zeta _{i}$, $\theta _{\rho _{i}}=\theta _{i}$ and $\beta _{\gamma _{i}}=\beta _{i}$ for $i=1,2$. Using (90)–(92), (29), (30), (31), (32), (33), the definition of $R^{*}$ and Lemma 8 we deduce that

$$\begin{aligned}&\mid ( \varLambda _{1}(\alpha _1 ,\rho _1 ,\gamma _1 )-\varLambda _{1}(\alpha _2 ,\rho _2 ,\gamma _2 ),\xi )\mid _{H^{-1}(\varGamma _{3})\times H^1_0(\varGamma _3)}^2\\&\quad \le \int \limits _{\varGamma _3}\left| F(\zeta _{1},\varvec{u}_{1},\dot{\varvec{u}}_{1},\dot{\varvec{u}}_{1})-F(\zeta _{2},\varvec{u}_{2},\dot{\varvec{u}}_{2},\dot{\varvec{u}}_{2}) \right| ^2 _{H^{-1}(\varGamma _{3})} \mid \xi \mid ^2_{H_0^1(\varGamma _3)}da \\&\quad \le C(\mid \zeta _{1}-\zeta _{2}\mid ^2 _{L^{2}(\varGamma _{3})}+\mid \varvec{u}_{1}-\varvec{u}_{2}\mid ^2 _{V}+\mid \dot{\varvec{u}}_{1}-\dot{\varvec{u}}_{2}\mid ^2 _{V})\mid \xi \mid ^2_{L^2(\varGamma _3)}.\\&\mid (\varLambda _{2}(\alpha _1 ,\rho _1 ,\gamma _1 )-\varLambda _{2}(\alpha _2 ,\rho _2,\gamma _2 ),\eta )\mid _{E^{\prime }\times E}^2 \\&\quad \le \int \limits _{\varOmega }|\big (\psi (\dot{\varvec{u}}_1,\theta _1 ,\beta _1 )-\psi (\dot{\varvec{u}}_2,\theta _2 ,\beta _2 )\big )\eta |^2 dx \\&\quad \le C(\mid \dot{\varvec{u}}_{1}-\dot{\varvec{u}}_{2}\mid ^2 _{V}+ \mid \theta _1-\theta _2\mid ^2_{L^2(\varOmega )}+|\beta _1-\beta _2|^2_{L^2(\varOmega )}) \mid \eta \mid ^2_{L^2(\varOmega )}.\\&\mid \varLambda _{3}(\alpha _1 ,\rho _1 ,\gamma _1 )-\varLambda _{3}(\alpha _2,\rho _2,\gamma _2 )\mid _{L^2(\varOmega )}^2= \mid \phi (\varvec{\varepsilon }(\varvec{u}_1), \theta _1,\beta _1)-\phi (\varvec{\varepsilon }(\varvec{u}_2), \theta _2,\beta _2) \mid ^2_{L^2(\varOmega )}\\&\quad \le C( |\varvec{u}_1-\varvec{u}_2|^2_V+|\theta _1-\theta _2|^2_{L^2(\varOmega )}+|\beta _1-\beta _2|^2_{L^2(\varOmega )} ). \end{aligned}$$

Hence

$$\begin{aligned}&\mid \varLambda (\alpha _1 ,\rho _1 ,\gamma _1 )-\varLambda (\alpha _2 ,\rho _2,\gamma _2 )\mid _{H^{-1}(\varGamma _3)\times E^{\prime }\times L^2(\varOmega )}^2 \nonumber \\&\le C(|\zeta _1-\zeta _2|^2_{L^2(\varGamma _3)}+|\varvec{u}_1-\varvec{u}_2|^2_V+ \mid \dot{\varvec{u}}_{1}-\dot{\varvec{u}}_{2}\mid ^2 _{V}+\mid \theta _1-\theta _2\mid ^2_{L^2(\varOmega )}+|\beta _1-\beta _2|^2_{L^2(\varOmega )}). \end{aligned}$$

(93)

Since $\varvec{u}_{1}(0)=\varvec{u}_{2}(0)=\varvec{u}_{0}$ we have

$$\begin{aligned} \mid \varvec{u}_{1}(t)-\varvec{u}_{2}(t)\mid _{V}^{2}\le \int \limits _{0}^{t}\mid \varvec{v}_{1}(s)-\varvec{v}_{2}(s)\mid _{V}^{2}ds. \end{aligned}$$

(94)

Moreover, from (82) we obtain that

$$\begin{aligned} (\mathcal {A}\varvec{\varepsilon }(\varvec{v}_{1})-\mathcal {A}\varvec{\varepsilon }(\varvec{v}_{2}),\varvec{\varepsilon }(\varvec{v}_{1}-\varvec{v} _{2}))_{\mathcal {H}}&\le j(\zeta _1 ,\varvec{u}_{1},\varvec{v}_{1},\varvec{v}_{2})-j(\zeta _1 ,\varvec{u}_{1},\varvec{v}_{1},\varvec{v}_{1}) \nonumber \\&\quad +j(\zeta _2,\varvec{u}_{2},\varvec{v}_{2},\varvec{v}_{1})-j(\zeta _2,\varvec{u} _{2},\varvec{v}_{2},\varvec{v}_{2})\nonumber \\&\quad +(C(\theta _1,\beta _1)-C(\theta _2,\beta _2),\varvec{\varepsilon }(\varvec{v}_1-\varvec{v}_2))_{\mathcal {H}}\nonumber \\&\quad +(\mathcal {G}(\varvec{\varepsilon } (\varvec{u}_1),\beta _1)-\mathcal {G}(\varvec{\varepsilon } (\varvec{u}_2),\beta _2),\varvec{\varepsilon }(\varvec{v}_1-\varvec{v}_2))_{\mathcal {H}}. \end{aligned}$$

(95)

The assumption (27) on $\mathcal {A}$, the assumption (61) and the estimate (56) on j give us

$$\begin{aligned} \mid \varvec{v}_{1}-\varvec{v}_{2}\mid ^2 _{V}\le C\left( \mid \varvec{u}_{1}- \varvec{u}_{2}\mid _{V}^2+\mid \zeta _{1}-\zeta _{2}\mid _{L^2(\varGamma _3)}^2+\mid \theta _{1}-\theta _{2}\mid _{E}^2+|\beta _1-\beta _2|_{L^2(\varOmega )}^2\right) . \end{aligned}$$

(96)

From (94), (96) and by using Gronwall inequality we see that

$$\begin{aligned} \mid \varvec{v}_{1}-\varvec{v}_{2}\mid _{V}^2\le C\left( \mid \zeta _{1}-\zeta _{2}\mid _{L^2(\varGamma _3)}^2+\mid \theta _{1}-\theta _{2}\mid _E^2+|\beta _1-\beta _2|^2_{L^2(\varOmega )}\right) . \end{aligned}$$

(97)

From (93), (94) and (97) we find that

$$\begin{aligned}&\mid \varLambda (\alpha _1(t) ,\rho _1(t) ,\gamma _1(t) )-\varLambda (\alpha _2 (t),\rho _2(t),\gamma _2(t) )\mid _{H^{-1}(\varGamma _3)\times E^{\prime }\times L^2(\varOmega )}^2 \\&\quad \le C\bigg (|\zeta _1(t)-\zeta _2(t)|^2_{L^2(\varGamma _3)}+\mid \theta _1(t)-\theta _2(t)\mid ^2_{E}+\mid \theta _1(t)-\theta _2(t)\mid ^2_{L^2(\varOmega )}+|\beta _1(t)-\beta _2(t)|^2_{L^2(\varOmega )}\bigg )\\&\qquad +C\bigg (\int \limits _{0}^{t}|\nabla \zeta _1(s)-\nabla \zeta _2(s)|^2_{L^2(\varGamma _3)^2}ds+ \int \limits _{0}^{t}|\theta _1(s)-\theta _2(s)|^2_E ds+\int \limits _{0}^{t}|\beta _1(s)-\beta _2(s)|^2_{L^2(\varOmega )}ds\bigg ) \\&\quad \le C \bigg (|\zeta _1(t)-\zeta _2(t)|^2_{L^2(\varGamma _3)}+ \int \limits _{0}^{t}|\nabla \zeta _1(s)-\nabla \zeta _2(s)|^2_{L^2(\varGamma _3)^2}ds+|\theta _1(t)-\theta _2(t)|^2_{L^2(\varOmega )}\\&\qquad + \int \limits _{0}^{t}|\theta _1(s)-\theta _2(s)|^2_E ds +|\beta _1(t)-\beta _2(t)|^2_{L^2(\varOmega )}+\int \limits _{0}^{t}|\beta _1(s)-\beta _2(s)|^2_{L^2(\varOmega )}ds\bigg ). \end{aligned}$$

Using estimates (64), (68) and (72), we obtain

$$\begin{aligned}&\mid \varLambda (\alpha _{1}(t),\rho _1(t), \gamma _1(t))-\varLambda (\alpha _{2}(t),\rho _2(t), \gamma _2(t)) \mid ^{2}_{H^1_0(\varGamma _3)\times E' \times L^2(\varOmega ) }\\&\quad \le C\int \limits _{0}^{t}\big (\mid \alpha _{1}(s)-\alpha _{2}(s) \mid _{H^{-1}(\varGamma _3)}^{2}+\mid \rho _{1}(s)-\rho _{2}(s)\mid _{E^{\prime }}^{2}+\mid \gamma _{1}(s)-\gamma _{2}(s) \mid _{L^2(\varOmega )}^{2}\big )ds.\\&\quad \le C\int \limits _{0}^{t} \mid (\alpha _1(s), \rho _1(s),\gamma _1(s))-(\alpha _2(s), \rho _2(s),\gamma _2(s))\mid ^2_{H^{-1}(\varGamma _3)\times E' \times L^2(\varOmega )}ds. \end{aligned}$$

Reiterating this inequality n times leads to

$$\begin{aligned}&\mid \varLambda ^n (\alpha _{1},\rho _1, \gamma _1)-\varLambda ^n (\alpha _{2},\rho _2, \gamma _2)\mid _{L^2(0,T;H^{-1}(\varGamma _3)\times E' \times L^2(\varOmega ) )}^2 \\&\quad \le \frac{(CT)^{n}}{n!} \mid (\alpha _1, \rho _1,\gamma _1)-(\alpha _2, \rho _2,\gamma _2)\mid ^2_{L^2(0,T;H^{-1}(\varGamma _3)\times E' \times L^2(\varOmega ))}. \end{aligned}$$

Thus, for n sufficiently large, $\varLambda ^{n}$ is a contraction on the Banach space $L^2(0,T;H^{-1}(\varGamma _3)\times E^{\prime }\times L^2(\varOmega ))$, and so $\varLambda$ has a unique fixed point.

Next, we show the existence of a solution to problem PV.

Let $(\alpha ^{*},\rho ^{*},\gamma ^{*})$ be the fixed point of the operator $\varLambda$ defined by (89)–(92) and obtained in lemma 9.

Let $\zeta _{\alpha ^{*}}$ be the solution to problem $PV_\alpha$ for $\alpha =\alpha ^*$ (see Lemma 1), let $\theta _{\rho ^*}$ be the solution to problem $PV_\rho$ for $\rho =\rho ^{*}$ (see Lemma 2) and let $\beta _{\gamma ^{*}}$ be the solution to problem $PV_{\gamma }$ for $\gamma =\gamma ^*$ (see Lemma 3). Denote $\varvec{u}^*=\varvec{u}_{\alpha ^*\rho ^*\gamma ^*}$, $\dot{\varvec{u}}^*=\dot{\varvec{u}}_{\alpha ^*\rho ^*\gamma ^*}$, $\zeta ^*=\zeta _{\alpha ^*}$, $\theta ^*=\theta _{\rho ^*}$ and $\beta ^*=\beta _{\gamma ^*}$.

$$\begin{aligned}&\varLambda _1(\alpha ^*,\rho ^*, \gamma ^*)=\alpha ^*= F(\zeta ^*, \varvec{u}^*,\dot{\varvec{u}}^*,\dot{\varvec{u}}^*),\\&\varLambda _2(\alpha ^*,\rho ^*, \gamma ^*)=\rho ^*=S(\dot{\varvec{u}}^*, \theta ^*,\beta ^*)+Q,\\&\varLambda _3(\alpha ^*,\rho ^*, \gamma ^*)=\gamma ^*=\phi (\varvec{\varepsilon }(\varvec{u}^*),\theta ^*,\beta ^*). \end{aligned}$$

$\varvec{u}^*=\varvec{u}_{\alpha ^*\rho ^*\gamma ^*}$ is a solution to the problem $PV_{\alpha \rho \gamma }$ for $\alpha =\alpha ^*$, $\rho =\rho ^*$ and $\gamma =\gamma ^*$

$$\begin{aligned}&(\mathcal {A}\varvec{\varepsilon }(\dot{\varvec{u}}^*(t)),\varvec{\varepsilon }(\varvec{v}-\dot{\varvec{u}}^*(t)))_{\mathcal {H}}+j(\zeta ^*(t),\varvec{u}^* (t),\dot{\varvec{u}}^*(t),\varvec{v}) \nonumber \\&\quad -j(\zeta ^*(t),\varvec{u}^*(t),\dot{\varvec{u}}^*(t),\dot{\varvec{u}}^*(t))+(\mathcal {G}(\varvec{\varepsilon } (\varvec{u}^*(t)),\beta ^*(t)),\varvec{\varepsilon }(\varvec{v}-\dot{\varvec{u}}^*(t)))_{\mathcal {H}}\nonumber \\&\quad -(C(\theta ^*(t),\beta ^*(t)),\varvec{\varepsilon }(\varvec{v}-\dot{\varvec{u}}^*(t)))_{\mathcal {H}}\ge (\varvec{f}(t),\varvec{v}-\dot{\varvec{u}}^*(t))_{V}\quad \forall \varvec{v}\in V,\quad t\in \left( 0,T\right) ,\end{aligned}$$

(98)

$$\begin{aligned}&\quad \varvec{u}^*(0)=\varvec{u}_{0}, \end{aligned}$$

(99)

and $\varvec{\sigma }^*=\mathcal {A}\varvec{\varepsilon }(\dot{\varvec{u}}^*)+\mathcal {G}(\varvec{\varepsilon }(\varvec{u}^*),\beta ^*)-C(\theta ^*,\beta ^*)$. The uniqueness of the solution is a consequence of the uniqueness of the solution of problems $PV_{\alpha \rho \gamma }$, $PV_\alpha$, $PV_\rho$, $PV_\gamma$ and the uniqueness of the operator $\varLambda$.