# Quasistatic Elastic Contact with Adhesion

Authors: Boudjemaa Teniou; Sabrina Benferdi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

The aim of this paper is the variational study of the contact with adhesion
between an elastic material and a rigid foundation in the quasistatic
process where the deformations are supposed to be small. The behavior of
this material is modelled by a nonlinear elastic law and the contact is
modelled with Signorini's conditions and adhesion. The evolution of bonding
field is described by a nonlinear differential equation. We derive a
variational formulation of the mechanical problem, and we prove the existence
and uniqueness of the weak solution using a theorem on variational
inequalities, the theorem of Cauchy-Lipschitz, a lemma of Gronwall, as well
as the fixed point of Banach.


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## Body

## 1. Introduction

The phenomena of contact with or without friction between deformable bodies or between deformable and rigid bodies abound in industry and everyday life. The contact of the tires with the roads, the shoe with disc of break, pistons with skirts are current examples. Because of the importance of contact process in structural and mechanical systems, a considerable effort has been made in its mathematical modeling, mathematical analysis, and numerical simulations.Process of adhesion is important in many industrial settings where parts, nonmetallic, are glued together. Recently, composite materials reached prominence, since they are very strong and light, and therefore, of considerable importance in aviation, space exploration, and in the automotive industry. Composite materials may undergo delamination under stress, a process in which different layers debond and move relatively to each other. To model the process when bonding is not permanent and debonding may take place, there is a need to add adhesion to the description of the contact process. For these reasons, adhesive contact between bodies when a glue is added to prevent the surfaces from relative motion, has recently received increased attention in the mathematical literature. In this paper we introduce an internal variable of surface, known as bonding field and denoted in this paper byβ, which describes the fractional density of active bonds on the contact surface. When β=1 at a point of contact surface, the adhesion is complete and all the bonds are active; when β=0 all the bonds are inactive, severed, and there is no adhesion; when 0<β<1 the adhesion is partial and only a fraction of the bonds is active. The problems of contact with adhesion were studied by several authors. Significant results on these problems can be found in [1–7] and references therein.Here, the novelty consists in the introduction of the bonding field into the contact between a material elastic and a rigid foundation where the process is quasistatic and the material is modelled by a nonlinear elastic law. The main contribution of this study lies in the proof of existence and uniqueness of the weak solution of the mechanical problem.This work is organized as follows. In Section2 we present some notations and preliminaries. In Section 3 we state the mechanical models of elastic contact with adhesion, list the assumptions on the data of the mechanical problem and deduce its variational formulation. In Section 4 we state and prove the existence of a unique weak solution to the mechanical problem; the proof is based on arguments of evolutionary equations and Banach fixed point.


## 2. Notations and Preliminaries

In this section, we specify the standard notations used and we remind of some definitions and necessary results for the study of this mechanical problem.We denote bySN the space of second order symmetric tensors on ℝN (N=2,3) while “·” and |·| represent the inner product and the Euclidean norm on ℝN and SN, respectively. Thus, for every u,v∈ℝN, and σ,τ∈SN we have(2.1)u⋅v=uivi,|u|=(u⋅u)1/2,σ⋅τ=σijτij,|σ|=(σ⋅σ)1/2.Here and below, the indicesi, j run between 1, N and the summation convention over repeated indices is adopted.LetΩ⊂ℝN be a bounded domain with a Lipschitz boundary Γ and let ν denote the unit outer normal on Γ. Moreover, we use also the spaces(2.2)H={u=(ui)/ui∈L2(Ω)},H={σ=(σij)/σij=σji∈L2(Ω)},H1={u∈H/ɛ(u)∈H},H1={σ∈H/Divσ∈H},
where ɛ:H1→ℋ, Div:ℋ→H are the deformation and the divergence operators, respectively, defined by(2.3)ɛ(u)=(ɛij(u)),ɛij(u)=12(∂jui+∂iuj),Divσ=(∂jσij).The spacesH, ℋ, H1, and ℋ1 are real Hilbert spaces endowed with the canonical inner products given by(2.4)〈u,v〉H=∫Ωuividx,∀u,v∈H,〈σ,τ〉H=∫Ωσijτijdx,∀σ,τ∈H,〈u,v〉H1=〈u,v〉H+〈ɛ(u),ɛ(v)〉H,∀u,v∈H,〈σ,τ〉H1=〈σ,τ〉H+〈Divσ,Divτ〉H,∀σ,τ∈H.The associated norms are denoted by|·|H, |·|ℋ, |·|H1, and |·|ℋ1, respectively.Since the boundaryΓ is Lipschitz continuous, the unit outward normal vector ν on the boundary is defined almost everywhere for every vector field u∈H1, we also use the notation u for the trace of u on Γ and we denote by uν and uτ the normal and tangential components of u on the boundary Γ given by(2.5)uν=u⋅ν,uτ=u-uνν.For a regular (sayC1) stress field σ, the application of its trace on the boundary to ν is the Cauchy stress vector σν. We define, similarly, the normal and tangential components of the stress on the boundary Γ by(2.6)σν=(σν)⋅ν,στ=σν-σνν.And we recall that the following Green's formula holds(2.7)〈σ,ɛ(u)〉H+〈Divσ,u〉H=∫Γσνuds,∀u∈H1.LetΓ1 be a measurable part of Γ such that meas Γ1>0 and let V be the closed subset of H1 defined by(2.8)V={v∈H1/v=0on⁡Γ1}.
Since measΓ1>0, then Korn's inequality holds, and thus there exists a constant c>0, depending only on Ω and Γ1 such that(2.9)|ɛ(u)|H≥c|u|H1∀u∈V.A proof of Korn's inequality may be found in [8, page 79].We define the inner product over the spaceV by(2.10)〈u,v〉V=〈ɛ(u),ɛ(v)〉H,∀u,v∈V.It follows from Korn's inequality that|·|V and |·|H1are equivalent norms on V. Therefore (V,|·|V) is a real Hilbert space. Moreover, by the Sobolev trace theorem, there exists a positive constant c>0, depending only on Ω, Γ1 and Γ3 such that(2.11)|v|L2(Γ3)N≤c|v|V,∀v∈V.Finally, we shall use the notation𝒬 for the set defined by(2.12)Q={β∈C(0,T;L2(Γ3)):0≤β(t)≤1∀t∈[0,T]a.e.on⁡Γ3,β(0)=β0}.We end this preliminary with the following version of the classical theorem of Cauchy-Lipschitz which can be found in [9, page 60].Theorem 2.1.
Assume that(X,|·|X) is a real Banach space. Let F(t,·):X→X be an operator defined almost everywhere on ]0,T[, satisfying the following conditions:(1)
|F(t,u)-F(t,v)|X≤LF|u-v|X∀u,v∈X  a.e. t∈]0,T[, for some LF,(2)
t↦F(t,u)∈Lp([0,T];X),∀u∈X, and some p≥1.Then, for every u0∈X, there exists a unique function u∈W1,p([0,T];X) such that
(2.13)u̇(t)=F(t,u(t))a.e.t∈]0,T[,u(0)=u0.This theorem will be used in Section4 to prove the theorem of existence and uniqueness of weak solution of the mechanical problem.


## 3. Problem Statement and Variational Formulation

We consider an elastic material which occupies a bounded domainΩ⊂ℝN (N=2,3) and assume that its boundary Γ is regular and partitioned into three disjoint measurable parts Γ1, Γ2, and  Γ3 such that meas Γ1>0. Let [0,T] be the time interval of interest, where T>0. The material is clamped on Γ1×]0,T[ and therefore the displacement field vanishes there. We also assume that a volume force of density f0 acts in Ω×]0,T[ and that a surface traction of density f2 acts on Γ2×]0,T[. On Γ3×]0,T[ the material may come in contact with arigid foundation. Moreover, the process is quasistatic and the evolution of the bonding field is described by a nonlinear differential equation.Under these conditions, the formulation of the mechanical problem is the following.Problem P.
Find a displacement fieldu:Ω×[0,T]→ℝN, a stress field σ:Ω×[0,T]→SN and a bonding field  β:Γ3×[0,T]→[0,1] such that
(3.1)σ(t)=F(ε(u(t)))inΩ×]0,T[,(3.2)Divσ(t)+f0(t)=0inΩ×]0,T[,(3.3)u=0,on⁡Γ1×]0,T[,(3.4)σν=f2,on⁡Γ2×]0,T[,(3.5)uν≤0,σν+γνR(uν)β2≤0,(σν+R(uν)β2)uν=0,on⁡Γ3×]0,T[,(3.6)στ=0,on⁡Γ3×]0,T[,(3.7)β̇=-(γνβ[(-R(uν))+]2-ϵa)+,on⁡Γ3×]0,T[,(3.8)β(0)=0,on⁡Γ3.Here (3.1) is the nonlinear elastic constitutive law. Equation (3.2) represents the equilibrium (3.3) and (3.4) are the displacement-traction boundary conditions, respectively. Conditions (3.5) represent the Signorini conditions with adhesion where γν is a given adhesion coefficient and R is the truncation operator defined by(3.9)R(s)={L,ifs≥L,s,if|s|<L,-L,ifs≤-L.
Here L>0 being the characteristic length of the bond, beyond which it does not offer additional traction. The introduction of the operator R, defined below, is motivated by the mathematical argument but is not restrictive in terms of application, since no restriction on the size of the parameter L is made in the sequel. Thus, by choosing L very large, we can assume that R(uν)=uν and, therefore, from (3.5) we recover the contact conditions:(3.10)uν≤0,σν+γνuνβ2≤0,(σν+uνβ2)uν=0,on⁡Γ3×]0,T[,it follows from (3.5) that there is no penetration between the material and the foundation, since uν≤0 during the process. Also, note that when the bonding vanishes, then the contact conditions (3.5) become the classical Signironi contact conditions with zero gap function, that is,(3.11)uν≤0,σν≤0,στ=0,σνuν=0,on⁡Γ3×]0,T[.Condition (3.6) represents the frictionless contact and shows that the tangential stress vanishes on the contact surface during the process. Equation (3.7) describes the evolution of the bonding field with given material parameters γν and ϵa. Also, the data β0 in (3.8) is the given initial bonding field.Assumptions.
For the variational study of the mechanical problem, we assume that the operatorF satisfies the following conditions:(3.12a)F:Ω×SN⟶SNsuchthat∃m>0such that(F(x,ε1)-F(x,ε2))⋅(ε1-ε2)≥m|ε1-ε2|2a.e.x∈Ω∀ε1,ε2∈SN,(3.12b)∃L>0such that|F(x,ε1)-F(x,ε2)|≤L|ε1-ε2|a.e.x∈Ω∀ε1,ε2∈SN,(3.12c)Themappingx⟼F(x,ε)isLebesguemeasurablea.e.x∈Ω,∀ε∈SN,(3.12d)Themappingx⟼F(x,0)∈H.We suppose that the adhesion coefficients satisfy(3.13)γν∈L∞(Γ3),ϵa∈L2(Γ3),γν,ϵa≥0a.e.on⁡Γ3.
And the body forces and surface traction have the regularity:(3.14)f0∈W1,∞(0,T;H),f2∈W1,∞(0,T;L2(Γ2)N).
The initial data satisfy(3.15)β0∈Q.
We use the convex subset of admissible displacements defined by(3.16)U={u∈Vsuch thatuν≤0on⁡Γ3}.
It follows from (3.14) and Riesz-Frechet representation theorem that there exists a unique function f:[0,T]→V such that:(3.17)〈f(t),v〉V=〈f0(t),v〉H+〈f2(t),v〉L2(Γ2)N∀v∈V,t∈]0,T[.
Moreover, we note that the conditions (3.14) imply(3.18)f∈W1,∞(0,T;V).
Finally, we define the adhesion functional j:L∞(Γ3)×V×V→ℝ by(3.19)j(β,u,v)=-∫Γ3γνβ2(-R(uν))+vνds.


### 3.1. Variational Formulation

By  applying Green's formula, and using the equilibrium equation and the boundary conditions, we easily deduce the following variational formulation of the mechanical Problem P.Problem PV.
Find a displacementu:[0,T]→V, and a bonding field β:[0,T]→L∞(Γ3) such that:
(3.20)u(t)∈U,〈F(ε(u(t))),ε(v-u(t))〉H+j(β(t),u(t),v-u(t))≥〈f(t),v-u(t)〉V∀v∈U,t∈]0,T[,(3.21)β̇(t)=-(γνβ(t)[(-R(uν(t)))+]2-ϵa)+a.e.t∈]0,T[,(3.22)β(0)=β0.Note that the variational Problem PV is formulated in terms of displacement and bonding field, since the stress field was eliminated. However, if the solution(u,β) of these variational problems is known, the corresponding stress field σ can be easily obtained by using the nonlinear elastic law (3.1).Below in this subsectionβ, β1, and β2 denote various elements of 𝒬, and u, u1, u2, and v represent elements of V, and c>0 represent constants which may depend on Ω, Γ1, Γ3, γν, and L. Note that the adhesion functional j is linear with respect to the last argument and therefore(3.23)j(β,u,-v)=-j(β,u,v).
Next using (3.19), as well as the properties (3.9) of the truncation operator R, we find(3.24)j(β1,u1,u2-u1)+j(β2,u2,u1-u2)≤c∫Γ3|β1-β2||u1-u2|ds.And by the Sobolev theorem trace we obtain(3.25)j(β1,u1,u2-u1)+j(β2,u2,u1-u2)≤c|β1-β2|L2(Γ3)|u1-u2|V.
We now take β=β1=β2 in (3.25) to deduce(3.26)j(β,u1,u2-u1)+j(β,u2,u1-u2)≤0.
Similar computations, based on the Lipschitz continuity of R, show that the following inequality also holds(3.27)|j(β,u1,v)-j(β,u2,v)|≤c|u1-u2|V|v|V.
We take u1=v and u2=0 in (3.26) then we use equality R(0)=0 and (3.23) to obtain(3.28)j(β,v,v)≥0.
The inequalities (3.25)–(3.28) will be used in various places to prove the theorem of existence and uniqueness of the weak solution.


## 3.1. Variational Formulation

By  applying Green's formula, and using the equilibrium equation and the boundary conditions, we easily deduce the following variational formulation of the mechanical Problem P.Problem PV.
Find a displacementu:[0,T]→V, and a bonding field β:[0,T]→L∞(Γ3) such that:
(3.20)u(t)∈U,〈F(ε(u(t))),ε(v-u(t))〉H+j(β(t),u(t),v-u(t))≥〈f(t),v-u(t)〉V∀v∈U,t∈]0,T[,(3.21)β̇(t)=-(γνβ(t)[(-R(uν(t)))+]2-ϵa)+a.e.t∈]0,T[,(3.22)β(0)=β0.Note that the variational Problem PV is formulated in terms of displacement and bonding field, since the stress field was eliminated. However, if the solution(u,β) of these variational problems is known, the corresponding stress field σ can be easily obtained by using the nonlinear elastic law (3.1).Below in this subsectionβ, β1, and β2 denote various elements of 𝒬, and u, u1, u2, and v represent elements of V, and c>0 represent constants which may depend on Ω, Γ1, Γ3, γν, and L. Note that the adhesion functional j is linear with respect to the last argument and therefore(3.23)j(β,u,-v)=-j(β,u,v).
Next using (3.19), as well as the properties (3.9) of the truncation operator R, we find(3.24)j(β1,u1,u2-u1)+j(β2,u2,u1-u2)≤c∫Γ3|β1-β2||u1-u2|ds.And by the Sobolev theorem trace we obtain(3.25)j(β1,u1,u2-u1)+j(β2,u2,u1-u2)≤c|β1-β2|L2(Γ3)|u1-u2|V.
We now take β=β1=β2 in (3.25) to deduce(3.26)j(β,u1,u2-u1)+j(β,u2,u1-u2)≤0.
Similar computations, based on the Lipschitz continuity of R, show that the following inequality also holds(3.27)|j(β,u1,v)-j(β,u2,v)|≤c|u1-u2|V|v|V.
We take u1=v and u2=0 in (3.26) then we use equality R(0)=0 and (3.23) to obtain(3.28)j(β,v,v)≥0.
The inequalities (3.25)–(3.28) will be used in various places to prove the theorem of existence and uniqueness of the weak solution.


## 4. Existence and Uniqueness of Weak Solution

Theorem 4.1.
Assume that  (3.12a)–(3.15) hold. Then there exists a unique solution (u,β) to Problem PV and it satisfies
(4.1)u∈W1,∞(0,T;V),β∈W1,∞(0,T;L∞(Γ3)).A triple(u,σ,β) which satisfies (3.1) and (3.20)–(3.22) is called weak solution of the mechanical Problem P. We conclude that under the stated assumptions, the mechanical problem has a unique weak solution. The regularity of the weak solution in terms of stress is given by(4.2)σ∈W1,∞(0,T;H1).
Indeed, taking v=φ∈D∞(Ω) in (3.20) and using (3.1), (3.17) we find: (4.3)Divσ(t)+f0(t)=0.
Now (3.14) and (4.3) imply that Divσ∈W1,∞(0,T;H), which in its turn implies σ∈W1,∞(0,T;ℋ1).The proof of Theorem4.1 will be carried out in several steps. In the first step we consider the following problem in which β∈𝒬, is given.Problem 1.
Find a displacementuβ:[0,T]→V such that
(4.4)uβ(t)∈U,F〈(ε(uβ(t))),ε(v-uβ(t))〉H+j(β(t),uβ(t),v-uβ(t))≥〈f(t),v-uβ(t)〉V∀v∈U,t∈]0,T[.
We have the following result.Lemma 4.2.
There exists a unique solution to Problem1 which satisfies
(4.5)uβ∈C(0,T;V).Proof.
Lett∈[0,T]. We consider the operator At:V→V defined by
(4.6)〈Atu,v〉V=〈F(ɛ(u(t))),ɛ(v)〉H+j(β(t),u(t),v)∀u,v∈V.
Letu1,u2∈V. We have
(4.7)〈Atu1-Atu2,u1-u2〉V=〈F(ɛ(u1))-F(ɛ(u2)),ɛ(u1-u2)〉H-j(β,u1,u2-u1)-j(β,u2,u1-u2).
We use (3.26), (3.12a), Korn's inequality and the equivalence of the two norms |·|H1 and |·|V to show that
(4.8)〈Atu1-Atu2,u1-u2〉V≥c|u1-u2|V2.
The operatorAt is, therefore, strongly monotone.
Letu1,u2,v∈V. We have
(4.9)|〈Atu1-Atu2,v〉V|≤|〈F(ɛ(u1))-F(ɛ(u2)),ɛ(v)〉H|+|j(β,u1,v)-j(β,u2,v)|.
Using Cauchy-Schwartz's inequality, (3.12b) and (3.27), one obtains
(4.10)|Atu1-Atu2|V≤c|u1-u2|V.
Therefore,At is a continuous Lipschitz operator. Since U is a nonempty convex closed subset of V, it follows from the standard results on elliptic variational inequalities that there exists a unique element uβ, such that
(4.11)uβ∈U,〈Atuβ,v-uβ〉≥〈f,v-uβ〉V∀v∈U.
Using (4.6) we get
(4.12)uβ(t)∈U,〈F(ɛ(uβ(t))),ɛ(v-uβ(t))〉H+j(β(t),uβ(t),v-uβ(t))≥〈f(t),v-uβ(t)〉V∀v∈U,t∈]0,T[.
We  now show thatuβ∈C(0,T;V).
Lett1, t2∈[0,T]. Denote uβ(ti), β(ti), and f(ti) by ui, βi and fi (for i=1,2), respectively. Then we have
(4.13)〈F(ɛ(u1))-F(ɛ(u2)),ɛ(u1-u2)〉H≤〈f1-f2,u1-u2〉V+j(β1,u1,u2-u1)+j(β2,u2,u1-u2).
Using (3.12a) we obtain
(4.14)m|ɛ(u1-u2)|H2≤〈f1-f2,u1-u2〉V+j(β1,u1,u2-u1)+j(β2,u2,u1-u2).
Using Korn's inequality and the fact that|·|V, |·|H1 are equivalent norms on V, Cauchy-Schwartz's inequality and (3.25) we get
(4.15)|u1-u2|V≤c(|f1-f2|V+|β1-β2|L2(Γ3)).
From the previous inequality, the fact thatβ∈𝒬 and the regularity of the function f given by (3.18), it follows that uβ∈C(0,T;V).In the second step we use the displacement fielduβ, obtained in Lemma 4.2 and we consider the following auxiliary problem.Problem 2.
Find a bonding fieldθβ:[0,T]→L∞(Γ3) such that
(4.16)θ̇β(t)=-(γνθβ(t)[(-R(uβν(t)))+]2-ϵa)+a.e.t∈]0,T[,(4.17)θβ(0)=β0.
We have the following result.Lemma 4.3.
There exists a unique solution to Problem2 which satisfies
(4.18)θβ∈W1,∞(0,T;L∞(Γ3))∩Q.Proof.
Consider the mappingFβ:[0,T]×L∞(Γ3)→L∞(Γ3) defined by
(4.19)Fβ(t,θβ)=-(γνθβ(t)[(-R(uβν(t)))+]2-ϵa)+.
It follows from the properties of the truncation operatorR, that Fβ is Lipschitz continuous with respect to the second argument, uniformly in time. Moreover, for any θβ∈L∞(Γ3), the mapping
t↦Fβ(t,θβ) belongs to L∞(0,T;L∞(Γ3)). Then from Theorem 2.1, we deduce the existence of a unique function θβ∈W1,∞(0,T;L∞(Γ3)), which satisfies (4.16)-(4.17). The regularity θβ∈𝒬, follows from (4.16)-(4.17) and assumption 0≤β0≤1 a.e. on Γ3. Indeed, (4.16) implies that for a.e. x∈Γ3, the function t→θβ(x,t) is decreasing and its derivative vanishes when γνθβ(t)[(-R(uβν(t)))+]2≤ϵa. Combining these properties with the inequality 0≤β0≤1 we deduce that 0≤θβ(t)≤1, for all t∈[0,t], a.e. on Γ3, which shows that θβ∈𝒬.In the third step we denote byuβ the solution of Problem 1 and θβ the solution of Problem 2, for every β∈𝒬. Moreover, we define the operator  Λ:𝒬→𝒬  by(4.20)Λβ=θβ.Lemma 4.4.
The operatorΛ has a unique fixed point β*.Proof.
We show that for a positive integerp the mapping Λp is a contraction on 𝒬. For this, suppose that βi are two functions of 𝒬 and we denote by ui, θi the functions obtained in Lemmas 4.2 and 4.3, respectively, for β=βi, (i=1,2).
Lett∈[0,T]. We use (4.4) and (3.25) to deduce that
(4.21)〈F(ɛ(u1(t))-F(ɛ(u2(t)))),ɛ(u1(t)-u2(t))〉H≤c|β1(t)-β2(t)|L2(Γ3)|u1(t)-u2(t)|V.
Using the fact thatF is a strongly monotone, Korn's inequality and |·|V, |·|H1are equivalent norms on V, we get
(4.22)|u1(t)-u2(t)|V≤c|β1(t)-β2(t)|L2(Γ3),
which implies
(4.23)∫0t|u1(s)-u2(s)|Vds≤c∫0t|β1(s)-β2(s)|L2(Γ3)ds.
On the other hand, it follows from (4.16) and (4.17) that
(4.24)θi(t)=β0-∫0t(γνθi(s)[(-R(uiν(s)))+]2-ϵa)+dsi=1,2.
And then(4.25)|θ1(t)-θ2(t)|L2(Γ3)≤c∫0t|θ1(s)[(-R(u1ν(s)))+]2-θ2(s)[(-R(u2ν(s)))+]2|L2(Γ3)ds.
Using the definition (3.9) and writing θ1=θ1-θ2+θ2, we get
(4.26)|θ1(t)-θ2(t)|L2(Γ3)≤c∫0t|θ1(s)-θ2(s)|L2(Γ3)ds+c∫0t|u1ν(s)-u2ν(s)|L2(Γ3)ds.
By Gronwall's inequality and the Sobolev trace theorem, it follows that:(4.27)|θ1(t)-θ2(t)|L2(Γ3)≤c∫0t|u1(s)-u2(s)|Vds.
Using (4.20) and (4.27), we obtain
(4.28)|Λβ1(t)-Λβ2(t)|L2(Γ3)≤c∫0t|u1(s)-u2(s)|Vds.
We now combine (4.23) and (4.28) to see that
(4.29)|Λβ1(t)-Λβ2(t)|L2(Γ3)≤c∫0t|β1(s)-β2(s)|L2(Γ3)ds.
And reiterating this inequalityp times, yields
(4.30)|Λpβ1-Λpβ2|C(0,T;L2(Γ3))≤cpTpp!|β1-β2|C(0,T;L2(Γ3))∀p∈N.
Recall that𝒬 is a nonempty closed set in the Banach space C(0,T;L2(Γ3)) and note that inequality (4.30) shows that for p sufficiently large Λp:𝒬→𝒬, then Λ has a unique fixed point β*∈𝒬.Now, the proof of the Theorem4.1, is a consequence of the previous lemmas.Proof.
Existence. Let β*∈𝒬 be the fixed point of Λ and u* be the solution of Problem 1 for β=β*, that is, u*=uβ*. Arguments similar to those used in proof of (4.22) lead to
(4.31)|u*(t1)-u*(t2)|V≤c|β*(t1)-β*(t2)|L2(Γ3)∀t1,t2∈[0,T].
Sinceβ*=θβ* it follows from Lemma 4.3 that β*∈W1,∞(0,T;L∞(Γ3)) and therefore (4.31) implies that u*∈W1,∞(0,T;V). From (4.4), (4.16) and (4.17) we conclude that (u*,β*) is a solution of the Problem PV which satisfies (4.1).
Uniqueness. The uniqueness of the solution is a consequence of the fixed point of the operator Λ defined by (4.20). Indeed, let (u,β) be a solution of Problem PV which satisfies (4.1). Since β∈𝒬, it follows from (3.20) that u is solution to Problem 1, but the Lemma 4.2 implies that this problem has a unique solution denoted uβ, we get
(4.32)u=uβ.
We putu=uβ in (3.21) and using the initial condition (3.22) then we can see that β is a solution to Problem 2, but the Lemma 4.3 implies that this last problem has a unique solution denoted θβ, we get
(4.33)β=θβ.
We use now (4.20) and (4.33) to see that Λβ=β, that is, β is a fixed point of the operator Λ. It follows now from Lemma 4.4 that
(4.34)β=β*.
The uniqueness of the solution is now a consequence of (4.32) and (4.34).



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*Source: 686139-2011-10-31.xml*