Int. J. Anal. Appl. (2023), 21:56

Frictional Contact Problem With Wear for Thermo-Viscoelastic Materials With

Damage

Safa Gherian1, Abdelaziz Azeb Ahmed1, Fares Yazid2,∗, Fatima Siham Djeradi2

1Laboratory of Operator Theory and PDE: Foundations and Applications, Faculty of Exact Sciences,

University of El Oued, El Oued 39000, Algeria
2Laboratory of Pure and Applied Mathematics, Amar Telidji University of Laghouat, Laghouat

03000, Algeria

∗Corresponding author: f.yazid@lagh-univ.dz

Abstract. We consider a mathematical model which describes a dynamic frictional contact problem for

thermo-viscoelastic materials with long memory and damage. The contact is modeled by the normal

compliance condition and wear between surfaces are taken into account. We establish a variational

formulation for the model and prove the existence and uniqueness of the weak solution. The proof

is based on arguments of hyperbolic nonlinear differential equations, parabolic variational inequalities

and Banach fixed point.

1. Introduction

Contact problems involving deformable bodies arise naturally in many situations and industrial pro-

cesses as well as in everyday life and play an important role in mechanical and structural systems.

That is way have been widely studied in the last years. The aim of this paper is to model and establish

the variational analysis of a frictional contact problem for a dynamic thermo-viscoelastic body with

wear and damage. The contact is modelled with normal compliance and wear. Thermoviscoelas-

tic contact problems by taking into account the evolution of the temperature parameter could be

found in [5,6,13]. General models for thermoelastic frictional contact, derived from thermodynamical

principles, have been obtained in [14, 23, 24]. Quasistatic contact problems with normal compliance

and friction have been considered in [1] and [16]. Dynamic problems with normal compliance were

Received: Mar. 8, 2023.

2020 Mathematics Subject Classification. 74M15, 74M10, 74F15, 49J40.

Key words and phrases. thermoviscoelastic; damage; wear; dynamic contact problem.

https://doi.org/10.28924/2291-8639-21-2023-56
ISSN: 2291-8639

© 2023 the author(s).

https://doi.org/10.28924/2291-8639-21-2023-56


2 Int. J. Anal. Appl. (2023), 21:56

first considered in [17]. The existence of weak solutions to dynamic thermoelastic contact problems

with frictional heat generation have been proven in [2] and when wear is taken into account in [3,9].

Recently contact problems with wear were studied in [6, 8, 10, 18, 19]. The damage of the material

caused by growth, temperature and various other external factors. 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 ς, which

measures the decrease in the load bearing capacity of the material, varies between 0 and 1; when

ς = 1 the material has its full capacity; wvhen ς = 0 it is completely damaged, and if ς = 1, the

material is partially damaged. Because of the importance of the subject, three-dimensional problems

which include this approach to material damage have been investigated recently in [7,12,13]

This paper is organized as follows. In Section 2, we present the notation and some preliminaries. In

section 3 we present the original model and list the assumptions on the problem’s data and we derive

the variational formulation. In section 4 we present our main result stated in Theorem 4.1 and its

proof which is based on arguments of time-dependent variational inequalities, parabolic inequalities,

differential equations and fixed point.

2. Notations and Preliminaries

First we will introduce some notations and preliminaries that we will use later. We denote by Sd

the space of second order symmetric tensors on Rd(d = 1, 2, 3). Let ” : ” and ” · ” represent the
inner product on Sd and Rd, respectively, and ‖ ·‖ denotes the Euclidean norm on Sd and Rd. Thus,
for all u,v ∈ Rd, u · v = uivi,‖v‖ = (v · v)

1
2 and for all σ,ζ ∈ Sd, σ : ζ = σijζij,‖ζ‖ = (ζ : ζ)

1
2 ,

1 ≤ i, j ≤ d. Also, an index that follows a comma represents the partial derivative with respect
to the corresponding component of the spatial variable, e.g. ui;j =

∂ui
∂xj

. We denote by t the time

variable and a dot superscript represents the time derivative with respect to the time variable t, e.g.

u̇ = ∂u
∂t
, ü = ∂

2u
∂t2

.

In what follows, we use the standard notation for Lebesgue and Sobolev spaces associated to Ω and

Γ introduce the spaces

H = {u = (ui ) : ui ∈ L2(Ω)},

H = {σ = (σij) : σij = σji ∈ L2(Ω)},

H1 = {u = (ui ) : ε(u) ∈ H},

H1 = {σ ∈H : Divœ ∈ H}.

Here ε and Div are the deformation and divergence operators, respectively, defined by

ε(u) = (εij(u)), εij(u) =
1

2
(ui,j + uj,i ), Div σ = (σij,j).



Int. J. Anal. Appl. (2023), 21:56 3

The spaces H, H, H1 and H1 are real Hilbert spaces endowed with the canonical inner products given
by

(u,v)H =

∫
Ω

u.vdx ∀u,v∈ H, (ζ,σ)H =
∫

Ω

ζ : σdx ∀ζ,σ ∈H,

(u,v)H1 = (u,v)H + (ε(u),ε(v))H ∀u,v∈ H1,

(ζ,σ)H1 = (ζ,σ)H + (Divı, Divœ)H ∀ı, œ ∈H1.

The associated norms on the spaces H, H, H1 and H1 are denoted by ‖ · ‖H, ‖ · ‖H, ‖ · ‖H1 and
‖ · ‖H1, respectively. For every element v ∈ H1 we also use the notation v for the trace of v on Γ
and we denote by vν and vτ the normal and the tangential components of v on Γ given by vν = v ·ν,
vτ = v−vνν. We also denote by σν and στ the normal and the tangential traces of a function σ ∈H1,
we recall that when σ is a regular function then σν = σν ·ν, στ = σν−σνν, and the following Green’s
formula holds

(σ,ε(v))H + (Divœ,v)H =

∫
Γ

œ˚.˚da ∀v ∈ H1. (2.1)

For a real Banach space (X,‖ · ‖X) we use the usual notation for the spaces Lp(0,T ; X) and
Wk,p(0,T ; X) where k ∈ N and 1 ≤ p ≤ ∞; we also denote by C(0,T ; X) and C1([0,T ]; X)
the spaces of continuous and continuously differentiable functions on [0,T ] with values in X, with the

respective norms

‖x‖C([0,T ];X) = max
t∈[0,T ]

‖x(t)‖X,

‖x‖C1([0,T ];X) = max
t∈[0,T ]

‖x(t)‖X + max
t∈[0,T ]

‖ẋ(t)‖X.

We end this section by giving an existence, uniqueness and regularity result concerning evolution

problems, taken from [4, p.268].

Theorem 2.1. Let V and H be two real Hilbert spaces such that V ⊆ H and the inclusion mapping of
V into H is continuous and densely defined. We suppose that V is endowed with the norm ‖·‖ induced
by the inner product (·, ·〉) and H is endowed with the norm | · |. We denote by V ′ the dual space of
V , by 〈·, ·〉V ′×V the duality pairing between an element of V and an element of V ′, and H is identified
with its own dual H′. We assume that M is a maximal monotone set in V ′ × V and A is a linear,
continuous and symmetric operator from V to V ′ satisfying the following coerciveness condition:

〈Av,v〉V ′×V + λ|v|2 ≥ ω‖v‖2 ∀v∈ V (2.2)

where λ ∈R and ω > 0. Let f ∈ W 1,1(0,T ; H) be given in W 1,1(0,T ; H) and u0,v0 be given with

u0 ∈ V, v0 ∈ D(M), {Au0 + Mv0}∩H 6= ∅. (2.3)



4 Int. J. Anal. Appl. (2023), 21:56

Then there exists a unique solution u to the following problem:

d2u

dt2
+ Au + M(

du

dt
) 3 f(t) a.e on (0,T ),

u(0) = u0,
du

dt
(0) = v0.

which satisfies

u∈ W 1,∞(0,T ; V ) ∩W 2,∞(0,T ; H).

3. Mechanical and variational formulations

The physical setting is the following. A thermo-viscoelastic body occupies a bounded domain

Ω ⊂ Rd (d = 2, 3) with outer surface Γ = ∂Ω, assumed to be sufficiently smooth and decomposed
into three disjoint measurable parts Γ1, Γ2, and Γ3, such that meas(Γ1) > 0. Let us denote by [0,T ],

T > 0 the time interval of interest. The body is clamped on Γ1, so the displacement field vanishes

there. A surface traction of density f2 act on Γ2. Moreover, the body is submitted to the action of

body forces of density f0 and a heat source of constant strength q.

The body could come in sliding frictional contact with a moving obstacle made of a hard perfectly rigid

material, and assume that the contact surface of the body Γ3 is covered by a layer of soft material.

This layer is deformable and the foundation may penetrate it, and could deteriorate over time as a

result of frictional contact with the foundation.

Problem P. Find a displacement field u : Ω × [0,T ] −→ Rd, a stress field σ : Ω × [0,T ] −→ Sd,
a damage field ς : Ω × [0,T ] −→ R, a temperature field θ : Ω × [0,T ] −→ R and a wear field
w : Γ3 × [0,T ] −→R such that

σ = Aε(u̇) + Gε(u) +
∫ t

0

B(t − s,ε(u(s)),ς(s),θ(s))ds in Ω × (0,T ), (3.1)

ς̇ − ∆ς + ∂Ψk(ς) 3S(ε(u),ς,θ) in Ω × (0,T ), (3.2)

θ̇−µ0∆θ = Φ(ε(u),ς,θ) + q in Ω × (0,T ), (3.3)

Div σ + f0 = ρü in Ω × (0,T ), (3.4)

u = 0 on Γ1 × (0,T ), (3.5)

σν = f2 on Γ2 × (0,T ), (3.6)

−σν = pν(uν −w),‖στ‖≤ pτ (uν −w),
‖στ‖ < pτ (uν −w) =⇒ u̇τ = v∗,
‖στ‖ = pτ (uν −w) =⇒ u̇τ = v∗ −λστ, λ > 0,
ẇ = kω‖v∗‖pν(uν −w),

on Γ3 × (0,T ), (3.7)



Int. J. Anal. Appl. (2023), 21:56 5

µ0
∂θ

∂ν
+ µ1θ = 0 on Γ × (0,T ), (3.8)

∂ς

∂ν
= 0 on Γ × (0,T ), (3.9)

w(0) = 0 on Γ3, (3.10)

u(0) = u0, u̇(0) = u̇0, θ(0) = θ0, ς(0) = ς0 on Ω. (3.11)

We now describe problem (3.1)−(3.11). Equation (3.1) represents the thermo-viscoelastic constitutive
with long memory and damage, A and G denote the linear viscosity operator and the elastic operator,
respectively and B is the relaxation tensor depending on the damage ς and the temperature θ. Equation
(3.2) describes the evolution of the damage field, governed by the source damage function S and ∂ΨK
is the subdifferential of indicator function of the set of admissible damage functions. Equation (3.3)

represents the evolution of the temperature field θ where Φ is a nonlinear constitutive function which

represents the heat generated by the work of internal forces, q represents the density of volume heat

sources and µ0 is a strictly positive constant. Equation (3.4) represents the equilibrium equation for

the stress displacement fields. Equations (3.5) and (3.6) are the displacement and traction boundary

conditions, respectively. Equation (3.7) describes the condition with normal compliance, wear and the

Coulomb’s friction law. The wear function w which measures the wear accumulated of the surface.

The evolution of the wear of the contacting surface is governed by the differential form of Archard’s

law (see, eg., [2,21,23,24]), where v∗ is a constant vector which represents the displacement of the

foundation, kw > 0 is a wear coefficient, pν and pτ are prescribed functions of the normal compliance

and friction bound, respectively. (3.8) is a Fourier boundary condition for the temperature θ where

µ1 > 0 and it represents a conduction coefficient of Γ. (3.9) is a homogeneous Niemann boundary

condition for the damage ς, where ∂ς
∂ν

is the normal derivative of ς. (3.10) represents the initial

condition for the wear function, which shows that at the initial moment the foundation is new. Finally

the functions u0, u̇0, θ0 and ς0 in (3.11) are the initial data.

We now turn to the variational formulation of Problem P.

We introduce the following space for the temperature field denoted by

E = H1(Ω).

The following Friedrichs-Poincaré inequality holds on E is

‖∇ϑ‖L2(Ω)d ≥ CF ‖ϑ‖E, ∀ϑ ∈ E. (3.12)

L2(Ω) is identified with its dual and with a subspace of the dual E′ of E, i.e., E ⊂ L2(Ω) ⊂ E′, and
we say that the inclusions above define a Gelfand triple. We use the notation 〈., .〉E′×E to represent
the duality pairing between E′ and E.

〈ω,ϑ〉E′×E = (ω,ϑ)L2(Ω), ∀ω,ϑ ∈ L
2(Ω). (3.13)



6 Int. J. Anal. Appl. (2023), 21:56

We define the admissible space

V = {v∈ (H1(Ω))d : v = 0 on Γ1},

also, the admissible damage set

K = {ς ∈ H1(Ω) : 0 ≤ ς ≤ 1 a.e in Ω}.

Since meas(Γ1) > 0, Korn’s inequality holds and there exists a constant C0 > 0, that depends only

on Ω and Γ1, such that

C0‖v ‖H≤‖ε(v) ‖H, ∀ v ∈ V.

On the space V , we consider the inner product and the associated norm given by

(u,v)V = (ε(u),ε(v))H, ‖u‖V = ‖ε(u)‖H, ∀u,v ∈ V. (3.14)

It follows that ‖ · ‖H1 and ‖ · ‖V are equivalent norms on V and therefore (V,‖ · ‖V ) is a real Hilbert
space. Moreover, by the Sobolev trace theorem and (3.14), there exists a constant C1 > 0, depending

only on Ω, Γ1 and Γ3, such that

‖v‖L2(Γ3)d ≤ C1‖v‖V , ∀ v ∈ V. (3.15)

In the study of the mechanical problem (3.1)-(3.11), we still need to assume that the operators

A,G,B and the functions S, Φ,pr (for r = ν, τ) satisfy the following conditions




(1) A : Ω ×Sd →Sd;
(2) There exists mA > 0 such that

(A(x,ε1) −A(x,ε2)).(ε1 −ε2) ≥ mA‖ε1 −ε2‖2,
for any ε1,ε2 ∈Sd a.e x ∈ Ω;

(3) There exists M1,M2 > 0 such that

‖(A(x,ε)‖≤ M1‖ε‖ + M2, for any ε ∈Sd a.e x ∈ Ω;
(4) The mapping x 7→A(x,ε) is continuous on Sd, a.e. x ∈ Ω;
(5) The mapping x 7→A(x,ε)is Lebesgue measurable on Ω

for any ε ∈Sd.

(3.16)




(1) G : Ω ×Sd →Sd;
(2) There exists mG > 0 such that

G(x,ε) ·ε ≥ mG‖ε‖2, for any ε ∈Sd a.e x ∈ Ω;
(3) G(x,ε1) ·ε2 = ε1 · G(x,ε2) for all ε1,ε2 ∈Sd, a.e. x ∈ Ω;
(4) Gijkl ∈ L∞(Ω),∀i, j,k, l = 1, ..,d

(3.17)



Int. J. Anal. Appl. (2023), 21:56 7


(1) B : Ω × [0,T ] ×Sd ×R×R→ Sd;
(2) There exists LB > 0 such that

‖B(x,t,ε1,ζ1,θ1) −B(x,ε2,ζ2,θ2)‖≤ LB(‖ε1 −ε2‖ + |ζ1 −ζ2| + |θ1 −θ2|)
for all t ∈ [0,T ],ε1,ε2 ∈Sd,ζ1,ζ2,θ1,θ2 ∈R, a.e. x ∈ Ω;

(3) The mapping x 7→B(x,t,ε,ζ,θ) is Lebesgue measurable on Ω
for any t ∈ [0,T ],ε ∈Sd,ζ,θ ∈R;

(4) The mapping t 7→B(x,t,ε,ζ,θ) is continuous measurable on [0,T ]
for any ε ∈Sd,ζ ∈,θ ∈R, a.e. x ∈ Ω;

(5) The mapping x 7→B(x,t,0, 0, 0) ∈H.

(3.18)




(1) S : Ω ×Sd ×R×R−→R;
(2) There exists LS > 0 such that

|S(x,ε1,ζ1,θ1) −S(x,ε2,ζ2,θ2)| ≤ LS(‖ε1 −ε2‖ + |ζ1 −ζ2| + |θ1 −θ2|)
∀x ∈ Ω,∀ε1,ε2 ∈Sd, ∀ ζ1,ζ2,θ1,θ2 ∈ R;

(3) The function x −→S(x,ε,ζ,θ) is Lebesgue measurable on Ω
∀ ε ∈Sd,∀ζ ∈R

(4) The function x −→S(x,0, 0, 0) ∈ L2(Ω).

(3.19)




(1) Φ : Ω ×Sd ×R×R−→R;
(2) There exists LΦ > 0 such that

|Φ(x,ε1,ς1,θ1) − Φ(x,ε2,ς2,θ2)| ≤ LΦ(‖ε1 −ε2‖ + |ς1 − ς2| + |θ1 −θ2|)
∀x ∈ Ω,∀ε1,ε2 ∈Sd ∀ ς1,ς2,θ1,θ2 ∈ R;

(3) The mappingx −→ Φ(x,ε, ∀ς,θ) is Lebesgue mesurable on Ω
∀ ε ∈Sd,ς,θ ∈R;

(4) The function x −→ Φ(x,0, 0, 0) ∈ L2(Ω).

(3.20)




(1) pr : Γ3 ×R−→R+, (r = ν,τ);
(2) There exists Lr > 0 such that ‖pr (x,a1) −pr (x,a2)‖≤ Lr|a1 −a2|

∀x ∈ Ω,∀a1,a2 ∈R,a.e. x ∈ Γ3;
(3) pr (x,a) ≤ 0 ∀a ≤ 0, a.e. x ∈ Γ3;
(4) The mapping x −→ pr (x,a) is Lebesgue measurable on Γ3 ∀a ∈R.

(3.21)

We suppose that the mass density satisfies, for ρ∗ > 0

ρ ∈ L∞(Ω), ρ(x) ≥ ρ∗, a.e.x ∈ Ω. (3.22)

We also suppose the mechanical and heat forces satisfy

q ∈ L2(0,T ; L2(Ω)), f0 ∈ W 1,1(0,T ; H), f2 ∈ W 1,1(0,T ; L2(Γ2)d). (3.23)



8 Int. J. Anal. Appl. (2023), 21:56

Next, we define the elements f (t) ∈ V by

(f(t),v)V =

∫
Ω

f0 ·vdx +
∫

Γ2

f2 ·vda, ∀v∈ V.

Let us define j : V ×V ×L2(Γ3) →R be the functional

j(u,v,w) =

∫
Γ3

pν(uν −w)vνda +
∫

Γ3

pτ (uν −w)‖vτ‖da, (3.24)

the functional j satisfies

for all u∈ V and w ∈ L2(Γ3)

j : v→ j(u,v,w) is proper, convex and lower semicontinuous on V. (3.25)

Taking into account assumptions (3.21) combined with (3.15), we get

j(u1,v2,w) − j(u1,v1,w) + j(u2,v1,w) − j(u2,v2,w)

≤ C21 (Lν + Lτ )‖u1 −u2‖V‖v1 −v2‖V ,

∀u1,u2,v1,v2 ∈ V, ∀w ∈ L2(Γ3).

(3.26)

We note that condition (3.23) implies

f ∈ W 1,1(0,T ; V ) (3.27)

We suppose that the initial data satisfy

θ0 ∈ L2(Ω), ς0 ∈ K, w0 ∈ L2(Γ3), u0 ∈ V, u̇0 ∈ D(∂2j), (3.28)

where ∂2j denotes the partial subdifferential with respect to the second argument of the operator j

and D(∂2j) represent its domain.

There exists g∈ H such that(
Aε(u̇0) + Bε(u0),ε(v) −ε(u̇0)

)
H + j(u0,v, 0) − j(u0, u̇0, 0)

≥ (g,v− u̇0), ∀v∈ V.
(3.29)

We introduce the following continuous functionals A : V → V
′
and B : V → V

′
defined by ∀u ∈ V ,

∀v∈ V

〈Au,v〉
V
′×V = (Gε(u),ε(v))H, (3.30)

〈Bu,v〉
V
′×V = (Aε(u),ε(v))H. (3.31)

We define the bilinear forms a : E ×E →R and b : H1(Ω) ×H1(Ω) →R defined by

a(ξ,κ) = µ0

∫
Ω

∇ξ ·∇κdx + µ1
∫

Γ

ξκda, ∀ ξ, κ ∈ E, (3.32)

b(ς,ζ) =

∫
Ω

∇ς ·∇ζdx, ∀ ς, ζ ∈ H1(Ω). (3.33)



Int. J. Anal. Appl. (2023), 21:56 9

We will use a modified inner product on the Hilbert space H given by

((u,v))H = (ρu,v)H, ∀u,v∈ H, (3.34)

that is, it is weighted with ρ, and we let ||| · |||H be the associated norm, i.e.,

|||v|||H = (ρv,v)
1
2
H
, ∀v∈ H. (3.35)

It follows from assumptions (3.22) that ||| · |||H and ‖ · ‖H are equivalent norms on H, and also the
inclusion mapping of (V,‖ · ‖H) into (H, ||| · |||H) is continuous and dense. We denote by V

′
the dual

space of V , and by identifying H with its own dual, write

V ⊂ H = H ⊂ V
′
.

We use the notation 〈·, ·〉
V
′×V to represent the duality pairing between V

′
and V and recall that

〈u,v〉
V
′×V = ((u,v))H, ∀u,v∈ H. (3.36)

Using the above notation and a standard procedure based on integrals by parts, we have the following

variational formulation of the problem thermo-mechanical (3.1)-(3.11).

Problem PV. Find a displacement field u : Ω×[0,T ] −→ V , a damage field ς : Ω×[0,T ] −→ L2(Ω),
a temperature field θ : Ω × [0,T ] −→ E and a wear field w : Γ3 × [0,T ] −→ L2(Γ3) such that

(Aε(u̇) + Gε(u) +
∫ t

0

B
(
ε(u(s)),ς(s),θ(s)

)
ds,ε(v) −ε(u̇))H

+ ((ü,v− u̇))H + j(u,v,w) − j(u, u̇,w) ≥ (f(t),v− u̇)V , ∀ v ∈ V,
(3.37)

(ς̇,β − ς(t))L2(Ω) + b(ς(t),β − ς(t))

≥ (S(ε(u(t)),ς(t),θ(t)),β − ς(t))L2(Ω), ∀ β ∈ K,
(3.38)

〈θ̇(t),α〉
E
′×E + a(θ(t),α) = 〈Φ(ς(t),ε(u(t)),θ(t)),α〉E′×E

+ (q(t),α)L2(Ω), ∀ α ∈ E, a.e t ∈ (0,T ),
(3.39)

ẇ = kω‖v∗‖pν(uν −w), a.e. t ∈ (0,T ). (3.40)

To study Problem (3.37)-(3.40), we need the following smallness assumption

Lν + Lτ <
2mA

C21 + C1
, (3.41)

where mA, C1 and Lr, (r = ν,τ) are given in (3.16), (3.15) and (3.17), respectively.

Our main existence and uniqueness result is stated and proved in the next section.



10 Int. J. Anal. Appl. (2023), 21:56

4. Existence and uniqueness result

Theorem 4.1. Let the assumptions (3.16)-(3.29) and (3.41). Then there exists an unique solution

(u,ς,θ,w) to problem PV . Moreover, the solution satisfies

u∈ W 1,∞(0,T ; V ) ∩W 2,∞(0,T ; H), (4.1)

ς ∈ H1(0,T ; L2(Ω)) ∩L2(0,T ; H1(Ω)), (4.2)

θ ∈ C(0,T ; L2(Ω)) ∩L2(0,T ; E), θ̇ ∈ L1(0,T ; E
′
) (4.3)

w ∈ C1(0,T ; L2(Γ3)). (4.4)

The functions u,σ,ς,θ and w which satisfy (3.1) and (3.37)-(3.40) are called a weak solution of

the contact problem P.

We conclude that, under the assumptions (3.16)-(3.29) and (3.41), the mechanical problem (3.1)-

(3.11) has a unique weak solution satisfying (4.1)-(4.4). The regularity of the weak solution is given

by (4.1)-(4.4) and, in term of stress,

σ ∈ W 1,∞(0,T ;H1). (4.5)

Indeed, it follows from (3.1), (3.16)(3), (3.17)(3), (3.17)(4), (3.18)(2) and the regularity (4.1)-(4.3)

implies σ ∈ W 1,∞(0,T ;H). Let t ∈ [0,T ] and we choose as a test function v = u̇(t) ± z where
z∈D(Ω)d in (3.37) and using (2.1), (3.34) and (3.36) to obtain

Div σ + f0 = ρü in H.

It now follows from (3.23) and (4.1) that Div σ ∈ W 1,∞(0,T ; H) which show (4.5).
The proof of Theorem 4.1 is carried out in several steps that we prove in what follows, everywhere in

this section we suppose that assumptions of Theorem 4.1 hold, and we consider that C is a generic

positive constant which may depend on the problem’s data but it is independent on time, and whose

value may change from place to place.

Let w ∈ C(0,T ; L2(Γ3)) and η ∈ L2(0,T ;H). In the first step we consider the following variational
problem

Problem PVwη. Find uwη : [0,T ] → V such that

((üwη(t),v− u̇wη(t)))H + (Auwη(t),v− u̇wη(t))

+ (Bu̇wη(t),v− u̇wη(t)) + j(uwη(t),v,w(t))

− j(uwη(t), u̇wη(t),w(t)) ≥ (F(t),v− u̇wη(t))V , ∀ v ∈ V,

uwη(0) = u0, u̇wη(0) = u̇0,

(4.6)

where (F (t),v)V = (f (t),v)V − (η(t),ε(v))H, ∀v∈ V.
In the study of Problem PVwη we have the following result.



Int. J. Anal. Appl. (2023), 21:56 11

Lemma 4.1. The problem PVwη has a unique solution which satisfies uwη ∈ W 1,∞(0,T ; V ) ∩
W 2,∞(0,T ; H)

Proof. By assumptions (3.30), (3.14), (3.17)(3), we see that the operator A is linear, continuous,

and symmetric from V to V
′
and satisfies the condition (2.2) with λ = 0 and ω = mG.

After, we define the set-valued operator ψ : V → V
′
by

ψ = B + ∂2j. (4.7)

From (3.16)(2), we deduce that the operator B defined by (3.31), is monotone. Using (3.31) and

(3.14), we have

‖Bu−Bv‖
V
′ ≤‖Aε(u) −Aε(v)‖H, ∀u,v∈ V,

keeping in mind (3.16)(2), (3.16)(3), (3.16)(4) and Krasnoselski’s theorem (see [15, p.60]), we find

that B : V → V
′
is a continuous operator. Using again (3.31) and (3.16)(2), we find that B is

bounded.

From (3.24) and (3.25), we deduce that j is maximal monotone. Consequently, since B is monotone,

bounded and hemicontinuous from V to V
′
, we conclude (see [4, p.39]) that ψ = B + ∂2j is maximal

monotone.

Moreover, the initial data u0 and u̇0 satisfy (2.3) due to (3.28) and (3.29). Thus, all the requirements

of Theorem 2.1, with A defined by (3.30), M = ψ given in (4.7) and f = F, are satisfied, it follows that

there exists a unique solution uwη to Problem PVwη satisfying the regularity expressed in (4.1). �

Let η ∈ L2(0,T ;H). In the second step, we consider the operator χ : C(0,T ; L2(Γ3)) →
C(0,T ; L2(Γ3)) defined by

χw(t) = kw‖v∗‖
∫ t

0

pν(uwην(s) −w(s))ds ∀t ∈ [0,T ]. (4.8)

Lemma 4.2. The operator χ has a unique fixed point w∗ ∈ C(0,T ; L2(Γ3)).

Proof. Let w1,w2 ∈ C(0,T ; L2(Γ3)) and denote by ui, i = 1, 2 the solutions to the problem PVwη,
for w = wi i.e. ui = uwiη and vi = u̇i = uwiη. From the definition (4.8) of χ, we can write

|χw1(t) −χw2(t)|L2(Γ3) ≤ kw‖v
∗‖
∫ t

0

|(pν(u1ν(t) −w1(t)) −pν(u2ν(t) −w2(t)))|ds

Using (3.21)(2) and (3.15), we get

‖χw1(t) −χw2(t)‖2L2(Γ3) ≤ C
(∫ t

0

‖w1(s) −w2(s)‖2L2(Γ3)ds +
∫ t

0

‖u1(s) −u2(s)‖2V
)
ds (4.9)



12 Int. J. Anal. Appl. (2023), 21:56

Using the relation (4.6), we find

((v̇1(t) − v̇2(t),v1(t) −v2(t)))H + (Au1(t) −Au2(t),v1(t) −v2(t))V

+ (Bv1(t) −Bv2(t),v1(t) −v2(t))V

≤ j(u1(t),v2(t),w1(t)) − j(u2(t),v2(t),w2(t))

+ j(u2(t),v1(t),w2(t)) − j(u1(t),v1,w1(t)).

By virtue of (3.30), (3.31), (3.16)(2), (3.17)(2), (3.21) and (3.14)(2), this inequality becomes

1

2

d

dt
|‖v1(t) −v2(t)|‖2H +

mG
2

d

dt
‖u1(t) −u2(t)‖2V + mA‖v1(t) −v2(t)‖

2
V

≤ Lν
∫

Γ3

(|u1ν −u2ν| + |w1 −w2|)|v2ν|da−Lν
∫

Γ3

(|u1ν −u2ν| + |w1 −w2|)|v1ν|da

+ Lτ

∫
Γ3

(|u1ν −u2ν| + |w1 −w2|)‖v2τ‖da−Lτ
∫

Γ3

(|u1ν −u2ν| + |w1 −w2|)‖v1τ‖da.

Integrating this inequality over the interval time variable (0,t), using (3.15) and the inequality 2ab ≤
a2 + b2 leads to

|‖v1(t) −v2(t)|‖2H +
mG
2
‖u1(t) −u2(t)‖2V + mA

∫ t
0

‖v1(s) −v2(s)‖2V ds

≤ (Lν + Lτ )
C1
2

∫ t
0

(
C1‖u1(s) −u2(s)‖2V + (C1 + 1)‖v1(s) −v2(s)‖

2
V

+ ‖w1(s) −w2(s)‖2L2(Γ3)
)
ds,

and keeping in mind (3.41), we obtain

∫ T
0

‖v1(s) −v2(s)‖2V ds ≤ C
∫ T

0

(
‖u1(s) −u2(s)‖2V + ‖w1(s) −w2(s)‖

2
L2(Γ3)

)
ds. (4.10)

On the other hand, since ui (t) = u0 +
∫ t

0
vi (s) ds, we have

‖u1(t) −u2(t)‖2V ≤
∫ t

0

‖v1(s) −v2(s)‖2V ds, (4.11)

and using this inequality in (4.10) yields

‖u1(t) −u2(t)‖2V ≤ C
∫ T

0

(
‖w1(t) −w2(t)‖2V +

∫ t
0

‖v1(s) −v2(s)‖2V ds
)
.

It follows now from a Gronwall-type argument that

‖u1(t) −v2(t)‖2V ds ≤ C
∫ t

0

‖w1(s) −w2(s)‖2V ds.



Int. J. Anal. Appl. (2023), 21:56 13

which implies for s ≤ t ≤ T∫ t
0

‖u1(s) −u2(s)‖2V ds ≤C
∫ t

0

∫ s
0

‖w1(r) −w2(r)‖2L2(Γ3)drds

≤C
∫ t

0

∫ t
0

‖w1(r) −w2(r)‖2L2(Γ3)drds

≤C
∫ t

0

‖w1(r) −w2(r)‖2L2(Γ3)dr
∫ T

0

ds.

Then ∫ t
0

‖u1(s) −u2(s)‖2V ds ≤ CT
∫ t

0

‖w1(s) −w2(s)‖2L2(Γ3)ds. (4.12)

From (4.8) and (4.12), we deduce that

‖χw1(s) −χw2(s)‖2L2(Γ3) ≤ CT
∫ t

0

‖w1(s) −w2(s)‖2L2(Γ3)ds. (4.13)

Reiterating the last inequality n times, we infer that

‖χnw1 −χnw2‖C(0,T ;L2(Γ3)) ≤
(
CnTn+1

n!

)1
2

‖w1 −w2‖C(0,T ;L2(Γ3)).

Thus, for n sufficiently large, a power χn of χ is a contraction in the Banach space C(0,T ; L2(Γ3)).

Which implies that the operator χ has a unique fixed point w∗ ∈ C(0,T ; L2(Γ3)). �

In the third step, let γ ∈ L2(0,T ; L2(Ω)) be given and consider the following variational problem
for the damage field.

Problem PVγ. Find a damage field ςγ : [0,T ] → H1(Ω) such that

ςγ ∈ K, (ς̇γ,β − ςγ(t))L2(Ω) + b(ςγ(t),β − ςγ(t))

≥ (γ(t),β − ςγ(t))L2(Ω) ∀ β ∈ K,

ςγ(0) = ς0.

(4.14)

Lemma 4.3. The problem PVγ has a unique solution ςγ satisfying

ςγ ∈ H1(0,T ; L2(Ω)) ∩L2(0,T ; H1(Ω)) (4.15)

Proof. Using (3.33) and (3.28), after some algebraic computations and from a classical existence and

uniqueness result of parabolic equations (see for example the reference [22, p.60]), we find that the

problem PVγ has a unique solution ςγ ∈ H1(0,T ; L2(Ω)) ∩L2(0,T ; H1(Ω)). �

In the forth step, let ϕ ∈ L2(0,T ; E
′
), we consider the following variational problem



14 Int. J. Anal. Appl. (2023), 21:56

Problem PVϕ. Find a temperature θϕ : Ω × (0,T ) →R such that

〈θ̇ϕ(t),α〉E′×E + a(θϕ(t),α) = 〈ϕ(t) + q(t),α〉E′×E

∀ α ∈ E a.e t ∈ (0,T ),

θϕ(0) = θ0.

(4.16)

Lemma 4.4. The problem PVϕ has a unique solution θϕ satisfies the regularity (4.3).

Proof. From the Friedrichs-poincaré inequality, we can find that there exists a constant µ2 > 0 such

that ∫
Ω

‖∇(ξ)‖2dx +
µ1
µ0

∫
Γ

‖ξ‖2da ≥ µ2
∫

Ω

‖ξ‖2dx.

Thus, we obtain

a(ξ,ξ) ≥ µ3‖ξ‖2E, (4.17)

where µ3 =
µ0min(1,µ2)

2
, which implies that a is elliptic on E. Consequently, based on a classical

arguments of functional analysis concerning parabolic equations (see [4, p.140]), we conclude that the

problem PVϕ has a unique solution θϕ which satisfies the regularity (4.3). �

Let us now consider the operator Λ : L2(0,T ;H×L2(Ω) ×E) → L2(0,T ;H×L2(Ω) ×E)

Λ(η,γ,ϕ)(t) = (Λ1(η,γ,ϕ)(t), Λ2(η,γ,ϕ)(t), Λ3(η,γ,ϕ)(t)), (4.18)

defined by

Λ1(η,γ,ϕ)(t) =

∫ t
0

B
(
ε(uw∗η)(s),ςγ(s),θϕ(s)

)
ds, (4.19)

Λ2(η,γ,ϕ)(t) = S(ε(uw∗η(t)),ςγ(t),θϕ(t)), (4.20)

Λ3(η,γ,ϕ)(t) = Ψ(ε(uw∗η(t)),ςγ(t),θϕ(t)). (4.21)

Here, w∗ be the fixed point of the operator χ. For every (η,γ,ϕ) ∈ L2(0,T ;H×L2(Ω)×E
′
), uw∗η,ςγ

and θϕ represent the displacement, the damage and the temperature obtained in Lemma 4.1, Lemma

4.3 and Lemma 4.4 respectively.

The last step in the proof of Theorem 4.1 is the next result.

Lemma 4.5. The operator Λ has a unique fixed point (η∗,γ∗,ϕ∗) ∈ L2(0,T ;H×L2(Ω) ×E′).

Proof. Let (η1,γ1,ϕ1), (η2,γ2,ϕ2) ∈ L2(0,T ;H×L2(Ω) ×E
′
).

We use the notation uw∗ηi = ui, u̇w∗ηi = vi,ςγi = ςi and θϕi = θi for i = 1, 2.

Using (4.19)-(4.21) and from (3.18)-(3.20) and (3.14), we get

‖Λ(η1,γ1,ϕ1)(t) − Λ(η2,γ2,ϕ2)‖H×L2(Ω)×E′

≤ LB
∫ t

0

(
‖u1(s) −u2(s)‖V + ‖ς1(s) − ς2(s)‖L2(Ω) + ‖θ1(s) −θ2(s)‖E

)
ds

+ (LS + LΦ)
(
‖u1(t) −u2(t)‖V + ‖ς1(t) − ς2(t)‖L2(Ω) + ‖θ1(t) −θ2(t)‖L2(Ω)

)



Int. J. Anal. Appl. (2023), 21:56 15

Employing Hölder’s and Young’s inequalities, we deduce that

‖Λ(η1,γ1,ϕ1)(t) − Λ(η2,γ2,ϕ2)‖2H×L2(Ω)×E′

≤ C
∫ t

0

(
‖u1(s) −u2(s)‖2V + ‖ς1(s) − ς2(s)‖

2
L2(Ω)

+ ‖θ1(s) −θ2(s)‖2E
)
ds + C

(
‖u1(t) −u2(t)‖2V

+ ‖ς1(t) − ς2(t)‖2L2(Ω) + ‖θ1(t) −θ2(t)‖
2
L2(Ω)

)
(4.22)

Using the relation (4.6), we obtain

(Aε(v1) −Aε(v2),ε(v1 −v2)V + ((v̇1 − v̇2,v1 −v2))H

≤ (Gε(u1) −Gε(u2),ε(v1 −v2))H + j(u1,v2,w) − j(u1,v1,w) + j(u2,v1,w)

− j(u2,v2,w) − (η1 −η2, ,ε(v1) −ε(v2))H

We use similar arguments that those used in the proof of the relation (4.12) to obtain that∫ t
0

‖u1(s) −u2(s)‖2V ds ≤ C
∫ t

0

‖η1(s) −η2(s)‖2Hds. (4.23)

From (4.14), we get

(ς̇1 − ς̇1,ς1 − ς2)L2(Ω) + b(ς1 − ς2,ς1 − ς2) ≤ (ς1 − ς2,ς1 − ς2)L2(Ω) a.e. t ∈ (0,T ).

Integrating the previous inequality with respect to time, using the initial conditions ς1(0) = ς1(0) = ς0

and the inequality b(ς1 − ς2,ς1 − ς2) ≥ 0, we find

1

2
‖ς1(t) − ς2(t)‖2L2(Ω) ≤

∫ t
0

(ς1(s) − ς2(s),ς1(s) − ς2(s))L2(Ω)ds,

which implies that

‖ς1(t) − ς2(t)‖2L2(Ω) ≤
∫ t

0

(
‖γ1(s) −γ2(s)‖2L2(Ω) + ‖ς1(s) − ς2(s)‖

2
L2(Ω)

)
ds,

this inequality combined with Gronwall’s inequality leads to

‖ς1(t) − ς2(t)‖2L2(Ω) ≤ C
∫ t

0

‖γ1(s) −γ2(s)‖2L2(Ω)ds, ∀t ∈ [0,T ]. (4.24)

In order words from (4.16), it follows

(θ̇1 − θ̇1,θ1 −θ2)E′×E + a(θ1 −θ2,θ1 −θ2) = (ϕ1 −ϕ2,θ1 −θ2)E′×E a.e. t ∈ (0,T ).

We integrate the previous equality, using (3.13), the initial conditions θ1(0) = θ1(0) = θ0 and as a is

E-elliptic, we get

‖θ1(t) −θ2(t)‖2L2(Ω) + µ3
∫ t

0

‖θ1(t) −θ2(t)‖2E

≤
∫ t

0

‖ϕ1(s) −ϕ2(s)‖E′‖θ1(s) −θ2(s)‖E a.e. t ∈ (0,T ),



16 Int. J. Anal. Appl. (2023), 21:56

employing Young’s and Hölder’s inequalities, we get

‖θ1(t) −θ2(t)‖2L2(Ω) +
∫ t

0

‖θ1(s) −θ2(s)‖2Eds

≤ C
∫ t

0

‖ϕ1(s) −ϕ2(s)‖2E′ds a.e. t ∈ (0,T ).
(4.25)

Now, we combine (4.23), (4.24) and (4.25), we find that

‖Λ(η1,γ1,ϕ1)(t) − Λ(η2,γ2,ϕ2)‖2H×L2(Ω)×E′ ≤ C
∫ T

0

‖(η1,γ1,ϕ1)(t) − (η2,γ2,ϕ2)‖2H×L2(Ω)×E′.

Reiterating this inequality n times we are led to

‖Λ(η1,γ1,ϕ1)(t) − Λ(η2,γ2,ϕ2)(t)‖2H×L2(Ω)×E′

≤ Cn
∫ t

0

∫ s
0

...

∫ m
0︸ ︷︷ ︸

n integrals

∫ T
0

‖(η1,γ1,ϕ1)(r) − (η2,γ2,ϕ2)(r)‖2H×L2(Ω)×E′ dr...ds,

which implies that

‖Λn(η1,γ1,ϕ1) − Λn(η2,γ2,ϕ2)‖2L2(H×L2(Ω)×E′)

≤
Cn Tn

n!
‖(η1,γ1,ϕ1) − (η2,γ2,ϕ2)‖2L2(H×L2(Ω)×E′).

(4.26)

Since lim
n→∞

Cn Tn

n!
= 0, it follows that there exists a positive integer n such that C

n Tn

n!
< 1 and,

therefore, (4.26) shows that the operator Λn is a contraction on the Banach space L2(H×L2(Ω) ×
E
′
) and, so, there exists a unique fixed point (η∗,γ∗,ϕ∗) ∈ L2(0,T ;H × L2(Ω) × E

′
) such that

Λ(η∗,γ∗,ϕ∗) = (η∗,γ∗,ϕ∗). �

We have now all the ingredient to prove Theorem 4.1 which we complete now.

Existence. Let w∗ be the fixed point of the operator χ given by (4.8) and (η∗,γ∗,ϕ∗) be the fixed

point of the operator Λ given by (4.18)-(4.21) and denote

u∗ = uw∗η∗, ς∗ = ςγ∗, θ∗ = θϕ∗. (4.27)

It follows from (4.19)-(4.21) that

η∗(t) =

∫ t
0

B
(
ε(u∗(s)),ς∗(s),θ∗(s)

)
ds,

γ∗(t) = S(ε(u∗(t)),ς∗(t),θ∗(t)),

ϕ∗(t) = Ψ(ε(u∗(t)),ς∗(t),θ∗(t)),

and, therefore, (4.6), (4.14), (4.16) and (4.8) imply that (u∗,ς∗,θ∗,w∗) is a solution of problem PV .



Int. J. Anal. Appl. (2023), 21:56 17

Uniqueness. The uniqueness of the solution follows from the uniqueness of the fixed point of the

operators χ and Λ defined by (4.8) and (4.18)-(4.21) respectively.

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi-

cation of this paper.

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https://doi.org/10.1016/0020-7683(95)00140-9

	1. Introduction
	2. Notations and Preliminaries
	3. Mechanical and variational formulations
	Problem P
	Problem PV

	4. Existence and uniqueness result
	Problem PVw
	Problem PV
	Problem PV
	Existence
	Uniqueness

	References