A variational analysis of a class of dynamic problems with slip dependent friction

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with slip dependent friction

Marius Cocou

To cite this version:

Marius Cocou. A variational analysis of a class of dynamic problems with slip dependent friction.

Annals of the University of Bucharest. Mathematical series, București : Editura Universității din

București, 2014, 5 (LXIII) (2), pp.279-297. �hal-01098604�

problems with slip dependent friction

Marius Cocou

Dedicated to Professor Nicolae Cristescu on the occasion of his eighty fifth birthday

Abstract - This work is concerned with the extension of some recent ex- istence results proved for a class of nonsmooth dynamic frictional contact problem, to the case of a coefficient of friction depending on the slip veloc- ity. Based on existence and approximation results for some general implicit variational inequalities, which are established by using Ky Fan’s fixed point theorem, several estimates and compactness arguments, relaxed unilateral conditions with slip dependent friction between two viscoelastic bodies of Kelvin-Voigt type are analyzed.

Key words and phrases : Implicit inequalities, set-valued mapping, dynamic problems, pointwise conditions, Coulomb friction.

Mathematics Subject Classification (2010) : 35Q74, 49J40, 74A55, 74D05, 74H20

1 Introduction

This paper deals with the analysis of a dynamic contact problem with some relaxed unilateral contact conditions, adhesion, and slip dependent pointwise friction, between two Kelvin-Voigt viscoelastic bodies.

The quasistatic unilateral contact problems with local Coulomb friction have been studied in [1, 29, 30] and adhesion laws based on the evolution of intensity of adhesion were investigated in [28, 10]. Also, the normal compliance model, which can be seen as a particular regularization of the Signorini’s conditions, has been considered by several authors, see e.g. [18, 16, 31] and references therein.

A recent unified approach including unilateral and bilateral contact with nonlocal friction, or normal compliance conditions, in the quasistatic case and for a nonlinear elastic behavior, has been proposed in [2].

In the dynamic case, viscoelastic contact problems with nonlocal friction laws were considered in [17, 20, 21, 14, 6, 11] and the corresponding prob- lems with normal compliance laws have been analyzed in [23, 18, 19, 5, 24].

1

Dynamic frictionless problems with adhesion have been studied by several authors, see, e.g [33] and references therein, and dynamic viscoelastic prob- lems coupling unilateral contact, recoverable adhesion and nonlocal friction have been investigated in [12, 8]. Using the Clarke subdifferential, vari- ous variational contact problems can be analyzed by using the theory of hemivariational inequalities, which represent a broad generalization of the variational inequalities to locally Lipschitz functions, see [24, 25, 26] and references therein.

Based on new abstract formulations and on Ky Fan’s fixed point theo- rem, a static contact problem with relaxed unilateral conditions and point- wise Coulomb friction was studied in [27]. The extension of this interesting approach to an elastic quasistatic contact problem was considered in [7] and to a dynamic viscoelastic contact problem with slip independent coefficient of friction was investigated in [9].

This paper extends the results presented in [9] to the case of a coeffi- cient of friction depending on the slip velocity, which enables to treat more realistic situations.

The paper is organized as follows. In Section 2 the classical formulation of the dynamic contact problem is presented. In Section 3 two variational formulations are given as a two-field problem. In Section 4 a more general evolution implicit variational inequality is considered and some auxiliary results are proved. Section 5 is devoted to the study of a fixed point problem, which is equivalent to the previous variational inequality. Using the Ky Fan’s theorem, the existence of a fixed point is proved. In Section 6 this abstract result is used to prove the existence of a variational solution of the dynamic contact problem with slip dependent friction.

The applications presented in this paper concern the contact between two linear viscoelastic bodies but these results can be extended to more general constitutive laws, as, for example, the ones characterizing some elastovis- coplastic materials investigated in [13].

2 Classical formulation

Let Ω

be the reference domains of R

, d = 2 or 3, occupied by two vis- coelastic bodies, characterized by a Kelvin-Voigt constitutive law. Suppose that the bodies have Lipschitz boundaries Γ

= ∂Ω

, α = 1, 2.

Let Γ

, Γ

and Γ

be three open disjoint sufficiently smooth parts of Γ

such that Γ

= Γ

∪ Γ

∪ Γ

and, to simplify the estimates, meas(Γ

) >

0, α = 1, 2. We shall assume the small deformation hypothesis and we shall use Cartesian coordinate representations.

Let y

(x

, t) denote the position at time t ∈ [0, T ], where 0 < T < +∞,

of the material point represented by x

in the reference configuration, and

u

(x

, t) := y

(x

, t) − x

denote the displacement vector of x

at time

t, with the Cartesian coordinates u

= (u

, ..., u

) = (¯ u

, u

). Let ε

, with the Cartesian coordinates ε

= (ε

(u

)), and σ

, with the Cartesian coordinates σ

= (σ

), be the infinitesimal strain tensor and the stress tensor, respectively, corresponding to Ω

, α = 1, 2. The usual summation convention will be used for i, j, k, l = 1, . . . , d.

Assume that the displacement U

= 0 on Γ

× (0, T ), α = 1, 2, and that the densities of both bodies are equal to 1. Let f

= (f

, f

) denote the given body forces in Ω

∪ Ω

and f

= (f

, f

) denote the tractions on Γ

∪ Γ

. The initial displacements and velocities of the bodies are denoted by u

= (u

, u

), u

= (u

, u

).

Suppose that the solids can be in contact between the potential contact surfaces Γ

and Γ

which are parametrized by two C

functions, ϕ

, ϕ

, defined on an open and bounded subset Ξ of R

, such that ϕ

(ξ)−ϕ

(ξ) ≥ 0 ∀ ξ ∈ Ξ and each Γ

is the graph of ϕ

on Ξ that is Γ

= { (ξ, ϕ

(ξ)) ∈ R

; ξ ∈ Ξ}, α = 1, 2. Define the initial normalized gap between the two contact surfaces by

g

(ξ) = ϕ

(ξ) − ϕ

(ξ)

p 1 + |∇ϕ

(ξ)|

∀ ξ ∈ Ξ

and suppose that this initial gap is sufficiently small. Let n

denote the unit outward normal vector to Γ

, α = 1, 2. We shall use the following notations for the normal and tangential components of a displacement field v

, α = 1, 2, of the relative displacement corresponding to v := (v

, v

) and of the stress vector σ

n

on Γ

:

v

(ξ, t) := v

(ξ, ϕ

(ξ), t), v

(ξ, t) := v

(ξ, t) · n

(ξ), v

(ξ, t) := v

(ξ, t) + v

(ξ, t), [v

](ξ, t) := v

(ξ, t) − g

(ξ), v

(ξ, t) := v

(ξ, t) − v

(ξ, t)n

(ξ), v

(ξ, t) := v

(ξ, t) − v

(ξ, t), σ

(ξ, t) := (σ

(ξ, t)n

(ξ)) · n

(ξ),

σ

(ξ, t) = σ

(ξ, t)n

(ξ) − σ

(ξ, t)n

(ξ),

for all ξ ∈ Ξ and for all t ∈ [0, T ]. Let g := −[u

] = g

− u

− u

be the gap corresponding to the solution u := (u

, u

). Using a similar method as the one presented in [3] (see also [11], [8]) we obtain the following unilateral contact condition at time t in the set Ξ: [u

] (ξ, t) = −g(ξ, t) ≤ 0 ∀ ξ ∈ Ξ.

Let A

, B

denote two fourth-order tensors, the elasticity tensor and the viscosity tensor corresponding to Ω

, with the components A

= (A

) and B

= (B

), respectively. Assume that these components satisfy the following classical symmetry and ellipticity conditions: C

= C

= C

∈ L

(Ω

), ∀ i, j, k, l = 1, . . . , d, ∃ α

> 0 such that C

τ

τ

≥ α

τ

τ

∀ τ = (τ

) verifying τ

= τ

, ∀ i, j = 1, . . . , d, where C

= A

, C

=

A

or C

= B

, C

= B

∀ i, j, k, l = 1, . . . , d, α = 1, 2.

Let µ = µ(ξ, u ˙

) be the slip rate dependent coefficient of friction and assume that µ : Ξ × R

→ R

is a bounded function such that for a.e.

ξ ∈ Ξ µ(ξ, ·) is Lipschitz continuous with the Lipschitz constant, denoted by C

, independent of ξ, and for every v ∈ R

µ(·, v) is measurable.

Let κ, κ : R → R be two mappings with κ lower semicontinuous and κ upper semicontinuous, satisfying the following conditions:

κ(s) ≤ κ(s) and 0 ∈ / (κ(s), κ(s)) ∀ s ∈ R , (2.1)

∃ r

≥ 0 such that max(|κ(s)|, |κ(s)|) ≤ r

∀ s ∈ R . (2.2) We consider the following dynamic viscoelastic contact problem.

Problem P

: Find u = (u

, u

) such that u(0) = u

, ˙ u(0) = u

and, for all t ∈ (0, T ),

¨

u

− div σ

(u

, u ˙

) = f

in Ω

, (2.3) σ

(u

, u ˙

) = A

ε(u

) + B

ε( ˙ u

) in Ω

, (2.4) u

= 0 on Γ

, σ

n

= f

on Γ

, α = 1, 2, (2.5)

σ

n

+ σ

n

= 0 in Ξ, (2.6)

κ([u

]) ≤ σ

≤ κ([u

]) in Ξ, (2.7)

|σ

| ≤ µ( ˙ u

) |σ

| in Ξ and (2.8)

|σ

| < µ( ˙ u

) |σ

| ⇒ u ˙

= 0,

|σ

| = µ( ˙ u

) |σ

| ⇒ ∃ ϑ ≥ 0, u ˙

= −ϑσ

, where σ

= σ

(u

, u ˙

), α = 1, 2, σ

:= σ

and σ

:= σ

.

Some contact and friction conditions, corresponding to particular κ and κ, with a general coefficient of friction, are presented in the following exam- ples.

Example 1. (Adhesion and friction conditions) Let s

≥ 0, M ≥ 0 be constants, k : R → R be a continuous function such that k ≥ 0 with k(0) = 0 and define

κ(s) =







0 if s ≤ −s

,

k(s) if − s

< s < 0,

−M if s ≥ 0,

κ(s) =







0 if s < −s

,

k(s) if − s

≤ s ≤ 0,

−M if s > 0.

Example 2. (Friction condition)

In Example 1 we set k = s

= 0 and define κ

(s) =

0 if s < 0,

−M if s ≥ 0, κ

(s) =

0 if s ≤ 0,

−M if s > 0.

The classical Signorini’s conditions correspond formally to M = +∞.

Example 3. (General normal compliance conditions)

Various normal compliance conditions, friction and adhesion laws can be obtained from the previous general formulation if one considers κ = κ = κ, where κ : R → R is some bounded Lipschitz continuous function with κ(0) = 0, so that σ

is given by the relation σ

= κ([u

]), see e.g. [8], where the intensity of adhesion was also considered.

3 Variational formulations

We shall consider two different variational formulations of problem P

. We adopt the following notations:

H

(Ω

) := H

(Ω

; R

), α = 1, 2, H

:= H

(Ω

) × H

(Ω

), hv, wi

= hv

, w

i

+ hv

, w

i

∀ v = (v

, v

) ∈ H

, ∀ w = (w

, w

) ∈ H

, ∀ s ∈ R .

Define the Hilbert spaces (H , |.|) with the associated inner product denoted by (. , .), (V , k.k) with the associated inner product (of H

) denoted by h. , .i, and the closed convex cones L

(Ξ), L

(Ξ × (0, T )) as follows:

H := H

= L

(Ω

; R

) × L

(Ω

; R

), V := V

× V

, where V

= {v

∈ H

(Ω

); v

= 0 a.e. on Γ

}, α = 1, 2,

L

(Ξ) := {δ ∈ L

(Ξ); δ ≥ 0 a.e. in Ξ},

L

(Ξ × (0, T )) := {η ∈ L

(0, T ; L

(Ξ)); η ≥ 0 a.e. in Ξ × (0, T )}.

Let a, b be two bilinear, continuous and symmetric mappings defined on H

× H

→ R by

a(v, w) = a

(v

, w

) + a

(v

, w

), b(v, w) = b

(v

, w

) + b

(v

, w

)

∀ v = (v

, v

), w = (w

, w

) ∈ H

, where, for α = 1, 2, a

(v

, w

) =

Z

Ω^α

A

ε(v

)·ε(w

) dx, b

(v

, w

) = Z

Ω^α

B

ε(v

)·ε(w

) dx.

Assume f

∈ W

(0, T ; L

(Ω

; R

)), f

∈ W

(0, T ; L

(Γ

; R

)), α = 1, 2, u

, u

∈ V , g

∈ L

(Ξ), and define the following mappings:

J : L

(Ξ) × (H

)

→ R , J (δ, v, w) = Z

Ξ

µ(v

) |δ| |w

| dξ

∀ δ ∈ L

(Ξ), ∀ v = (v

, v

), w = (w

, w

) ∈ H

, f ∈ W

(0, T ; H

), hf , vi = P

α=1,2

Z

Ω^α

f

· v

dx + X

α=1,2

Z

Γ^α_F

f

· v

ds

∀ v = (v

, v

) ∈ H

, ∀ t ∈ [0, T ].

Assume the following compatibility conditions: [u

] ≤ 0, κ([u

]) = 0 a.e.

in Ξ and ∃ p

∈ H such that

(p

, w) + a(u

, w) + b(u

, w) = hf (0), wi ∀ w ∈ V . (3.1) For every ζ ∈ L

(0, T ; L

(Ξ)) = L

(Ξ × (0, T )), define the following sets:

Λ(ζ) = {η ∈ L

(0, T ; L

(Ξ)); κ ◦ ζ ≤ η ≤ κ ◦ ζ a.e. in Ξ × (0, T ) }, Λ

(ζ ) = {η ∈ L

(Ξ × (0, T )); κ

◦ ζ ≤ η ≤ κ

◦ ζ a.e. in Ξ × (0, T ) }, Λ

(ζ ) = {η ∈ L

(Ξ × (0, T )); κ

◦ ζ ≤ η ≤ κ

◦ ζ a.e. in Ξ × (0, T ) }, where, for each r ∈ R , r

:= max(0, r) and r

:= max(0, −r) denote the positive and negative parts, respectively.

For each ζ ∈ L

(0, T ; L

(Ξ)) the sets Λ(ζ), Λ

(ζ) and Λ

(ζ) are clearly closed, convex and nonempty, because the bounding functions belong to the respective set. Since meas(Ξ) < ∞ and κ, κ satisfy (2.2), it follows that for all ζ ∈ L

(0, T ; L

(Ξ)) these three sets are also bounded in norm in L

(Ξ × (0, T )) by r

, and in L

(0, T ; L

(Ξ)) by r

= r

T

meas(Ξ)

.

A first variational formulation of the problem P

is the following.

Problem P

: Find u ∈ C

([0, T ]; H

)∩W

(0, T ; V ), λ ∈ L

(0, T ; L

(Ξ)) such that u(0) = u

, ˙ u(0) = u

, λ ∈ Λ([u

]) and

h u(T ˙ ), v(T) − u(T )i

− (u

, v(0) − u

) − Z

0

( ˙ u, v ˙ − u) ˙ dt +

Z

a(u, v − u) + b( ˙ u, v − u) − (λ, v

− u

)

dt (3.2) +

Z

0

{J (λ, u, ˙ v + k u ˙ − u) − J (λ, u, k ˙ u)} ˙ dt ≥ Z

0

hf , v − ui dt

∀ v ∈ L

(0, T ; V ) ∩ W

(0, T ; H ), where 1 > ι > 1

2 , k > 0.

The formal equivalence between the variational problem P

and the classical problem (2.3)–(2.8) can be easily proved by using Green’s formula and an integration by parts, where the Lagrange multiplier λ satisfies the relation λ = σ

.

Let φ : (L

(Ξ))

× (V )

→ R be defined by φ(δ

, δ

, v, w) = −(δ

− δ

, w

)

+

Z

Ξ

µ(v

) (δ

+ δ

) |w

| dξ

∀ (δ

, δ

) ∈ (L

(Ξ))

, ∀ v = (v

, v

), w = (w

, w

) ∈ V .

(3.3)

Since η ∈ Λ(ζ) if and only if (η

, η

) ∈ Λ

(ζ ) × Λ

(ζ), it follows that the

variational problem P

is equivalent with the following problem.

Problem P

: Find u ∈ C

([0, T ]; H

)∩W

(0, T ; V ), λ ∈ L

(0, T ; L

(Ξ)) such that u(0) = u

, ˙ u(0) = u

, (λ

, λ

) ∈ Λ

([u

]) × Λ

([u

]) and

h u(T ˙ ), v(T ) − u(T)i

− (u

, v(0) − u

) +

Z

{−( ˙ u, v ˙ − u) + ˙ a(u, v − u) + b( ˙ u, v − u)} dt

+ Z

0

{φ(λ

, λ

, u, ˙ v + k u ˙ − u) − φ(λ

, λ

, u, k ˙ u)} ˙ dt

≥ Z

0

hf , v − ui dt ∀ v ∈ L

(0, T ; V ) ∩ W

(0, T ; H).

(3.4)

The existence of variational solutions of the problem P

will follow from some general existence results that will be proved in the next sections.

4 Existence results for some variational inequali- ties

Let U

, (V

, k.k, h. , .i), (U, k.k

) and (H

, |.|, (. , .)) be four Hilbert spaces such that U

is a closed linear subspace of V

dense in H

, V

⊂ U ⊆ H

with continuous embeddings and the embedding from V

into U is compact.

Let B

(Ξ), B

(Ξ

) denote the closed balls with center 0 and radius r in L

(Ξ), L

(Ξ

), respectively, where Ξ

:= Ξ × (0, T ) and r > 0.

Let a

, b

: V

× V

→ R be two bilinear and symmetric forms such that

∃ M

, M

> 0 a

(u, v) ≤ M

kuk kvk, b

(u, v) ≤ M

kuk kvk, (4.1)

∃ m

, m

> 0 a

(v, v) ≥ m

kvk

, b

(v, v) ≥ m

kvk

∀ u, v ∈ V

. (4.2) Let l : V

→ L

(Ξ) and φ

: [0, T ] ×(L

(Ξ))

×(V

)

→ R be two mappings satisfying the following conditions:

∃ k

> 0 such that ∀ v

, v

∈ V

,

kl(v

) − l(v

)k

≤ k

kv

− v

k

, (4.3)

∀ t ∈ [0, T ], ∀ γ

, γ

∈ L

(Ξ), ∀ v, v

, v

∈ V

,

φ

(t, γ

, γ

, v, v

+ v

) ≤ φ

(t, γ

, γ

, v, v

) + φ

(t, γ

, γ

, v, v

), (4.4) φ

(t, γ

, γ

, v, θv

) = θ φ

(t, γ

, γ

, v, v

), ∀ θ ≥ 0, (4.5) φ

(t, γ

, γ

, v, w) = 0, ∀ w ∈ U

, (4.6)

φ

(0, 0, 0, 0, v) = 0, (4.7)

∀ r > 0, ∃ k

(r) > 0 such that ∀ t

, t

∈ [0, T ],

∀ γ

, γ

, δ

, δ

∈ L

(Ξ) ∩ B

(Ξ), ∀ v

, v

, w

, w

∈ V

,

|φ

(t

, γ

, γ

, v

, w

) − φ

(t

, γ

, γ

, v

, w

) +φ

(t

, δ

, δ

, v

, w

) − φ

(t

, δ

, δ

, v

, w

)|

≤ k

(r)(|t

− t

| + kγ

− δ

k

+ kγ

− δ

k

+kv

− v

k

)kw

− w

k

,

(4.8)

if (γ

, γ

) ∈ (L

(Ξ

))

for all n ∈ N

and (γ

, γ

) ⇀ (γ

, γ

) in (L

(0, T ; L

(Ξ)))

, then Z

0

φ

(s, γ

, γ

, v, w) ds → Z

0

φ

(s, γ

, γ

, v, w) ds ∀ v, w ∈ L

(0, T ; V

).

(4.9)

Remark 4.1 i) Since by (4.5) φ

(·, ·, ·, ·, 0) = 0, from (4.8), for w

= w, w

= 0, we have

∀ t

, t

∈ [0, T ], ∀ γ

, γ

, δ

, δ

∈ L

(Ξ) ∩ B

(Ξ), ∀ v

, v

, w ∈ V

,

|φ

(t

, γ

, γ

, v

, w) − φ

(t

, δ

, δ

, v

, w)|

≤ k

(r)(|t

− t

| + kγ

− δ

k

+ kγ

− δ

k

+ kv

− v

k

)kwk

. (4.10) ii) From (4.7) and (4.8), for t

= t, t

= 0, δ

= δ

= 0 and v

= v, v

= 0 we derive

∀ t ∈ [0, T ], ∀ γ

, γ

∈ L

(Ξ) ∩ B

(Ξ), ∀ v, w

, w

∈ V

,

|φ

(t, γ

, γ

, v, w

) − φ

(t, γ

, γ

, v, w

)|

≤ k

(r)(t + kγ

k

+ kγ

k

+ kvk

)kw

− w

k

.

(4.11) iii) If v

→ v, w

→ w in L

(0, T ; U ), (γ

, γ

) ∈ (L

(Ξ

) ∩ B

(Ξ

))

, for all n ∈ N , and (γ

, γ

) ⇀ (γ

, γ

) in (L

(0, T ; L

(Ξ)))

, then

n,m

lim

Z

φ

(s, γ

, γ

, v

, w

) ds → Z

0

φ

(s, γ

, γ

, v, w) ds, (4.12) which can be proved by taking into account (4.11) in the following relations:

| Z

0

{φ

(s, γ

, γ

, v

, w

) − φ

(s, γ

, γ

, v, w)} ds|

≤ Z

0

|φ

(s, γ

, γ

, v

, w

) − φ

(s, γ

, γ

, v

, w)| ds +

Z

|{φ

(s, γ

, γ

, v

, w) − φ

(s, γ

, γ

, v, w)}| ds +|

Z

{φ

(s, γ

, γ

, v, w) − φ

(s, γ

, γ

, v, w)} ds|

≤ Z

0

k

(r)(kγ

k

+ kγ

k

+ kv

k

)kw

− wk

ds +

Z

k

(r)(kγ

k

+ kγ

k

+ kv

− vk

)kwk

ds +|

Z

{φ

(s, γ

, γ

, v, w) − φ

(s, γ

, γ

, v, w)} ds|,

and passing to limits by using that (γ

)

are bounded and (4.9).

Assume that f

∈ W

(0, T ; V

), u

, u

∈ V

are given, and that the following compatibility condition holds: κ(l(u

)) = 0 and ∃ p

∈ H

such that

(p

, w) + a

(u

, w) + b

(u

, w) = hf

(0), wi ∀ w ∈ V

. (4.13) We consider the following problem.

Problem Q

: Find u ∈ W

, λ ∈ L

(0, T ; L

(Ξ)) such that u(0) = u

,

˙

u(0) = u

, (λ

, λ

) ∈ Λ

(l(u)) × Λ

(l(u)) and h u(T ˙ ), v(T ) − u(T )i

− (u

, v(0) − u

) +

Z

{−( ˙ u, v ˙ − u) + ˙ a

(u, v − u) + b

( ˙ u, v − u)} dt

+ Z

0

{φ

(t, λ

, λ

, u, v ˙ + k u ˙ − u) − φ

(t, λ

, λ

, u, k ˙ u)} ˙ dt

≥ Z

0

hf

, v − ui dt ∀ v ∈ L

(0, T ; V

) ∩ W

(0, T ; H

), where W

:= C

([0, T ]; U

) ∩ W

(0, T ; V

).

The sets Λ

(ζ), Λ

(ζ ) and Λ(ζ ) have the following useful properties, proved in [8], see also [27].

Lemma 4.1 Let ζ ∈ L

(0, T ; L

(Ξ)) and (η

, η

) ∈ Λ

(ζ) × Λ

(ζ). Then η

η

= 0 a.e. in Ξ

and there exists η ∈ Λ(ζ) such that η

= η

, η

= η

a.e. in Ξ

.

Based on the previous lemma, consider the following problem, which has the same solution u as the problem Q

, and the solutions λ

, λ

satisfy the relation λ = λ

− λ

, where λ is a solution of Q

.

Problem Q

: Find u ∈ W

, λ

, λ

∈ L

(0, T ; L

(Ξ)) such that u(0) = u

,

˙

u(0) = u

, (λ

, λ

) ∈ Λ

(l(u)) × Λ

(l(u)) and h u(T ˙ ), v(T) − u(T )i

− (u

, v(0) − u

) +

Z

{−( ˙ u, v ˙ − u) + ˙ a

(u, v − u) + b

( ˙ u, v − u)} dt

+ Z

0

{φ

(t, λ

, λ

, u, v ˙ + k u ˙ − u) − φ

(t, λ

, λ

, u, k ˙ u)} ˙ dt

≥ Z

0

hf

, v − ui dt ∀ v ∈ L

(0, T ; V

) ∩ W

(0, T ; H

).

(4.14)

For the convenience of the reader, an existence and uniqueness result proved in [11] will be restated here under an adapted and more general form that will enable to study problem Q

.

Let β : V

→ R and φ

: [0, T ] × V

→ R be two sequentially weakly continuous mappings such that

β(0) = 0 and φ

(t, z, v, w

+ w

) ≤ φ

(t, z, v, w

) + φ

(t, z, v, w

), (4.15)

φ

(t, z, v, θw) = θ φ

(t, z, v, w), (4.16)

φ

(0, 0, 0, w) = 0 ∀ t ∈ [0, T ], ∀ z, v, w, w

∈ V

, ∀ θ ≥ 0, (4.17)

∃ k

> 0 such that ∀ t

∈ [0, T ], ∀ u

, v

, w ∈ V

,

|φ

(t

, u

, v

, w) − φ

(t

, u

, v

, w)|

≤ k

(|t

− t

| + |β (u

− u

)| + kv

− v

k

) kwk,

(4.18)

∃ k

> 0 such that ∀ t

∈ [0, T ], ∀ u

, v

, w

∈ V

,

|φ

(t

, u

, v

, w

) − φ

(t

, u

, v

, w

) + φ

(t

, u

, v

, w

)

−φ

(t

, u

, v

, w

)| ≤ k

( |t

− t

| + ku

− u

k + kv

− v

k

) kw

− w

k.

(4.19) Let L ∈ W

(0, T ; V

) and assume the following compatibility condition on the initial data: ∃ p

∈ H

such that

(p

, w)+a

(u

, w)+b

(u

, w)+φ

(0, u

, u

, w) = hL(0), wi ∀ w ∈ V

. (4.20) Consider the following problem.

Problem Q

: Find u ∈ W

(0, T ; H

)∩W

(0, T ; V

) such that u(0) = u

,

˙

u(0) = u

, and for almost all t ∈ (0, T ) (¨ u, v − u) + ˙ a

(u, v − u) + ˙ b

( ˙ u, v − u) ˙

+φ

(t, u, u, v) ˙ − φ

(t, u, u, ˙ u) ˙ ≥ hL, v − ui ∀ ˙ v ∈ V

. (4.21) We have the following existence and uniqueness result.

Theorem 4.1 Under the assumptions (4.1), (4.2), (4.15)-(4.20), there ex- ists a unique solution to the problem Q

.

The proof, which will be presented in a forthcoming paper, is based on a similar method to that used to prove the Theorem 3.2 established in [11]

and on a useful estimate, see [22] or [32], which, when applied to the spaces V

⊂ U ⊆ H

, implies the following result: for every ǫ > 0 there exists C

> 0 such that

kuk

≤ ǫ kuk + C

|u| ∀ u ∈ V

. (4.22)

Lemma 4.2 Assume that (4.1), (4.2), (4.4), (4.5), (4.7), (4.8) and (4.13)

hold. If r > 0 then, for each (γ

, γ

) ∈ (W

(0, T ; L

(Ξ)))

∩ (L

(Ξ

))

∩

(B

(Ξ

))

with γ

(0) = γ

(0) = 0, there exists a unique solution u = u

of the following evolution variational inequality: find u ∈ W

(0, T ; H

) ∩ W

(0, T ; V

) such that u(0) = u

, u(0) = ˙ u

, and for almost all t ∈ (0, T )

(¨ u, v − u) + ˙ a

(u, v − u) + ˙ b

( ˙ u, v − u) ˙

+φ

(t, γ

, γ

, u, v) ˙ − φ

(t, γ

, γ

, u, ˙ u) ˙ ≥ hf

, v − ui ˙ ∀ v ∈ V

. (4.23) Proof. We apply Theorem 4.1 to β = 0, L = f

and

φ

(t, z, v, w) = φ

(t, γ

(t), γ

(t), v, w) ∀ t ∈ [0, T ], ∀ z, v, w ∈ V

. Since φ

satisfies (4.4), (4.5) and (4.7), one can easily verify the proper- ties (4.15)-(4.17). Also, (4.13) and (4.17) imply the condition (4.20).

Using (4.8), for some k

= k

(r) > 0 we have

∀ t

∈ [0, T ], ∀ u

, v

, w

∈ V

,

|φ

(t

, u

, v

, w

) − φ

(t

, u

, v

, w

) + φ

(t

, u

, v

, w

) − φ

(t

, u

, v

, w

)|

= |φ

(t

, γ

(t

), γ

(t

), v

, w

) − φ

(t

, γ

(t

), γ

(t

), v

, w

)|

+φ

(t

, γ

(t

), γ

(t

), v

, w

) − φ

(t

, γ

(t

), γ

(t

), v

, w

)|

≤ k

(|t

− t

| + kγ

(t

) − γ

(t

)k

+ kγ

(t

) − γ

(t

)k

+kv

− v

k

)kw

− w

k

≤ k

((1 + C

+ C

)|t

− t

| + kv

− v

k

)kw

− w

k

,

≤ k

(|t

− t

| + kv

− v

k

)kw

− w

k

,

where C

, C

denote the Lipschitz constants of γ

, γ

, respectively, and k

= k

(1 + C

+ C

).

Thus,

|φ

(t

, u

, v

, w

) − φ

(t

, u

, v

, w

) + φ

(t

, u

, v

, w

)

−φ

(t

, u

, v

, w

)| ≤ k

(|t

− t

| + kv

− v

k

)kw

− w

k

∀ t

∈ [0, T ], ∀ u

, v

, w

∈ V

,

(4.24)

and, since by the continuous embedding V

⊂ U there exists C

> 0 such that kwk

≤ C

kwk ∀ w ∈ V

, it follows that φ

satisfies (4.19) with k

= k

C

.

Taking in (4.24) w

= w, w

= 0, by (4.16) with θ = 0, we obtain

|φ

(t

, u

, v

, w) − φ

(t

, u

, v

, w)| ≤ k

(|t

− t

| + kv

− v

k

)kwk

∀ t

∈ [0, T ], ∀ u

, v

, w ∈ V

, (4.25)

and using the continuous embedding V

⊂ U , it follows that φ

satisfies

(4.18) with k

= k

C

.