A Quasistatic Contact Problem for Viscoelastic Materials with Slip-Dependent Friction and Time Delay

Abstract

A mathematical model which describes an explicit time-dependent quasistatic frictional contact problem between a deformable body and a foundation is introduced and studied, in which the contact is bilateral, the friction is modeled with Tresca's friction law with the friction bound depending on the total slip, and the behavior of the material is described with a viscoelastic constitutive law with time delay. The variational formulation of the mathematical model is given as a quasistatic integro-differential variational inequality system. Based on arguments of the time-dependent variational inequality and Banach's fixed point theorem, an existence and uniqueness of the solution for the quasistatic integro-differential variational inequality system is proved under some suitable conditions. Furthermore, the behavior of the solution with respect to perturbations of time-delay term is considered and a convergence result is also given.

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Si-sheng Yao 1, 2 and Nan-jing Huang 1

Recommended by Marco Paggi

1, Department of Mathematics, Sichuan University, Chengdu 610064, China
2, Department of Mathematics, Kunming University, Kunming 650221, China

Received 27 June 2012; Revised 26 September 2012; Accepted 16 November 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The phenomena of contact between deformable bodies or between deformable and rigid bodies are abound in industry and daily life. Contact of braking pads with wheels and that of tires with roads are just a few simple examples [ 1]. Because of the importance of contact processes in structural and mechanical systems, a considerable effort has been made in their modeling and numerical simulations (see [ 1- 4] and the references therein). What is worth to be taken particularly is some engineering papers that discussed the developed mathematical modeling to a practically interesting problem [ 5, 6]. Owing to their inherent complexity, contact phenomena are modeled by nonlinear evolutionary problems that are difficult to analyze (see [ 1]). The first work concerned with the study of frictional contact problems within the framework of variational inequalities was made in [ 7]. Comprehensive references for the analysis and numerical approximation of contact problems include [ 1, 8]. Mathematical, mechanical, and numerical state-of-the-art on contact mechanics can be found in the proceedings [ 9, 10] and in the special issue [ 11].

When a viscoelastic material undergoes a small deformation gradient with a relatively slow force applied, the process of the relative contact problem can be modeled by a quasistatic system. At a relatively short duration, the effect of temperature changes caused by energy dissipation on the deformation of the material is usually negligible. Rigorous mathematical treatment of quasistatic problems is a test of recent years. The reason lies in the considerable difficulties that the process of frictional contact presents in the modeling and analysis because of the complicated surface phenomena involved (see [ 12]). By employing Banach's fixed point theorem, Chau et al. [ 13] got some existence and uniqueness results for two quasistatic problems which describe the frictional contact between a deformable body and an obstacle. They also proved that the solution of the viscoelastic problem converges to the solution of the corresponding elastic problem. By using arguments for time-dependent elliptic variational inequalities and Banach's fixed point theorem, Rodriguez-Aros et al. [ 10] dealt with the existence of a unique solution to an evolutionary variational inequality with Volterra-type integral term. Delost and Fabre [ 14] presented a valid approximation method for a quasistatic abstract variational inequality with time-independent constraint and applied its results to the approximation of the quasistatic evolution of an elastic body in bilateral contact with a rigid foundation. Very recently, Vollebregt and Schuttelaars [ 15] studied the quasistatic analysis of a contact problem with slip-velocity-dependent friction. For more works concerned with the quasistatic contact problems, we refer to [ 1, 16] and the references therein.

Quasistatic contact problems for viscoelastic or other materials with explicit time-dependent operators were investigated in a large number of papers. Applying the theory of evolutionary hemivariational inequality, Migórski et al. [ 17] proved the existence and the regularity properties of the unique weak solution to a nonlinear explicit time-dependent elastic-viscoplastic frictional contact problem with multivalued subdifferential boundary conditions. In [ 18], Migórski et al. considered a class of quasistatic contact problems for explicit time-dependent viscoelastic materials with subdifferential frictional contact conditions. Based on the fixed point theorem for multivalued mappings and variational-hemivariational inequality theory, Costea and Matei [ 19] proved the existence of weak solution for the general and unified framework contact models. They also discussed the uniqueness, the boundedness, and the stability of the weak solution under some suitable conditions.

It is well known that time-delay phenomena are frequently encountered in various technical systems, such as electric, pneumatic and hydraulic networks, and chemical processes. For example, regarding polymer under the action of alternative stress, the stress will lag behind the strain, which is just a time delay phenomenon. It gives us a mechanism of the time-delay phenomena appeared in contact problems. Comincioli [ 20] proved the existence and uniqueness for a kind of variational inequality with time-delay. For general results of variational inequalities with time delay, we refer to [ 21- 23]. However, to the best of our knowledge, there is no papers to study contact problems for viscoelastic materials with time-delay.

Motivated and inspired by the work mentioned above, in this paper, we introduce and study a mathematical model which describes an explicit time-dependent quasistatic frictional contact problem between a deformable body and a foundation, in which the contact is bilateral, the friction is modeled with Tresca's friction law with the friction bound depending on the total slip, and the behavior of the material is described with a viscoelastic constitutive law with time delay. We give the variational formulation of the mathematical model as a quasistatic integro-differential variational inequality system. By using the arguments of time-dependent variational inequality and Banach's fixed point theorem, we prove an existence and uniqueness of the solution for the quasistatic integro-differential variational inequality system under some suitable conditions. Furthermore, we consider the behavior of the solution with respect to perturbations of time-delay term and show a convergence result. The results presented in this paper generalize and improve some known results of [ 1, 24].

The paper is structured as follows. In Section 2, we list the necessary assumptions on the data and derive the variational formulation for the problem. In this part, an example which is assumed to the Kelvin-Voigt viscoelastic constitutive law with long memory is given, which represents a constitutive equation of the form ( 2.19). In Section 3, we prove the existence and uniqueness of the solution to the quasistatic integro-differential variational inequality system. In Section 4, we study the behavior of the solution with respect to perturbations of time-delay term and derive the convergence result.

2. Preliminaries

Let ^{... d} be a d -dimensional Euclidean space and ^{...AE; d} the space of second order symmetric tensors on ^{... d} . Let Ω ⊂ ^{... d} be open, connected, and bounded with a Lipschitz boundary Γ that is divided into three disjoint measurable parts _{Γ 1} , _{Γ 2} , and _{Γ 3} such that meas ( _{Γ 1} ) >0 . Let ^{L 2} ( Ω ) be the Lebesgue space of 2 -integrable functions and ^{W k ,p} ( Ω ) the Sobolev space of functions whose weak derivatives of orders less than or equal to k are p -integrable on Ω . Let ^{H k} ( Ω ) = ^{W k ,2} ( Ω ) .

Since the boundary is Lipschitz continuous, the outward unit normal which is denoted by ν exists a.e. on Γ . For T >0 , and let I ¯ = [0 ,T ] be the bounded time interval of interest. Let [real] (u ) be the range of displacement u . Since the body is clamped on _{Γ 1} , the displacement field vanishes there. Surface traction of density _{f 2} acts on _{Γ 2} and a body force of density _{f 0} is applied in Ω . The contact is bilateral, that is, the normal displacement _{u ν} vanishes on _{Γ 3} at any time.

The canonical inner products and corresponding norms on ^{... d} and ^{...AE; d} are defined as follows: [figure omitted; refer to PDF] Everywhere in the sequel the index i and j run between 1 and d and the summation convention over repeated indices is implied.

In the following we denote [figure omitted; refer to PDF] where H and Q are Hilbert spaces with the canonical inner products. The associated norms on the spaces will be denoted by || · _{|| H} and || · _{|| Q} , respectively.

Define [figure omitted; refer to PDF] It is easy to verify that ( _{H 1} , || · _{|| H 1} ) is a real Hilbert space. Since V is a closed subspace of the space _{H 1} and meas ( _{Γ 1} ) >0 , the following Korn's inequality holds: [figure omitted; refer to PDF] where ι denotes a positive constant depending only on Ω and _{Γ 1} . We define the inner product ( · , · _{) V} and the norm || · _{|| V} on V by [figure omitted; refer to PDF] It follows that || · _{|| H 1} and || · _{|| V} are equivalent norms on V . Thus, (V , || · _{|| V} ) is a real Hilbert space and _{V 1} is also a real Hilbert space under the inner product of the space V given by ( 2.5).

For every element v ∈ _{H 1} , we also use the notation v for the trace of v on Γ and we denote by _{v ν} and _{v τ} the normal and the tangential components of v on Γ given by [figure omitted; refer to PDF] We also denote by _{σ ν} and _{σ τ} the normal and the tangential traces of a function σ ∈Q , and we recall that when σ is a regular function, that is, σ ∈ ^{C 1} ( Ω ¯ ^{) s d ×d} , then [figure omitted; refer to PDF] and the following Green's formula holds: [figure omitted; refer to PDF]

We model the friction with Tresca's friction law, where the friction bound g is assumed to depend on the accumulated slip of the surface. In this model we try to incorporate changes in the contact surface structure resulted from sliding. Therefore, g =g (t , _{S t} ( u ) ) on _{Γ 3} ×I with _{S t} ( u ) (x ) being the accumulated slip at the point x on _{Γ 3} over the time period I ¯ as [figure omitted; refer to PDF] It follows that || _{σ τ} || ...4;g ( _{S t} ( u ) ) on _{Γ 3} . When the strict inequality holds, the material point is in the stick zone: _{u τ} =0 , while when the equality holds, || _{σ τ} || =g ( _{S t} ( u ) ) , the material point is in the slip zone: _{σ τ} = - λ _{u τ} for some λ >0 .

Let r be a constant satisfying 0 <r <T and set _{Q -r} = Ω × ( -r ,0 ) . Let ... be the Borel σ -algebra of the interval [ -r ,0 ] and μ ( · ) be a given finite signed measure defined on ( [ -r ,0 ] , ... ) . Zhu [ 22] defined the time-delay operator G as follows: for any h ∈ ^{L 2} ( Ω × ( -r , ∞ ) ^{) ...AE; d ×d} , [figure omitted; refer to PDF] In order to make the above integral coherent, we always take the integrand to be a Borel correction of h (by which we mean a Borel measurable function that is equal to h almost everywhere).

Some special cases of the operator G are as follows:

(i) Let _{Ω 1} = { _{ω 1} , _{ω 2} , ... , _{ω n} , ... } , _{[Lagrangian (script capital L)] 1} = ^{2 _{Ω 1}} , and [figure omitted; refer to PDF]

where _{μ 1} ( _{ω i} ) = _{p i} , i =1,2 , ... ,n , ... , _{p i} ∈ ^{... +} and _{μ 1} ( ∅ ) =0 . Then it is easy to see that [figure omitted; refer to PDF]

which can be used to describe the countably many discrete delays.

(ii) Let _{Ω 2} = ... , _{[Lagrangian (script capital L)] 2} be a σ -algebra of _{Ω 2} , m a Lebesgue measure, f a Lebesgue measurable function, and [figure omitted; refer to PDF]

Then [figure omitted; refer to PDF]

Moreover, letting f ...1;1 , then [figure omitted; refer to PDF]

Remark 2.1.

It is easy to see that, for any h ∈ ^{L 2} ( Ω × ( -r , ∞ ) ^{) ...AE; d ×d} , Gh as an element in ^{L 2} ( Ω × (0 , ∞ ) ^{) ...AE; d ×d} is independent of the choices of Borel corrections for h .

Remark 2.2.

Since μ is a very general regular measure, ( 2.10) can be used in many cases such as finitely many and countably many discrete delays. At this stage, we note that ( 2.10) contains a very wide class of time-delay operators.

The following lemma is a fundamental result for operator G .

Lemma 2.3 (see [ 22]).

For h ∈ ^{L 2} ( ( -r , ∞ ) × Ω ; ^{... d} ) , we have Gh ∈ ^{L 2} ( (0 , ∞ ) × Ω ; ^{... d} ) . Furthermore, for any g ∈ ^{L 2} ( (0 , ∞ ) × Ω ; ^{... d} ) ,0 ...4;s ...4; + ∞ and 0 ...4; _{s 0} ...4;r , the following inequality holds: [figure omitted; refer to PDF]

Now we consider the contact problem. For any u ~ ∈ [real] (u ) , based on ( 2.10), we derive the time-delay operator ...A2; of the form [figure omitted; refer to PDF]

Remark 2.4.

Replacing x with [straight epsilon] ( u ~ ) in ( 2.10) and letting _{s 0} =0 and g =Gh in Lemma 2.3, it is easy to know that [figure omitted; refer to PDF]

Under the previous assumptions, the classical formulation of the frictional contact problem with total slip dependent friction bound and the time-delay is as follows. For any u ~ ∈ [real] (u ) , find a displacement field u : Ω × I ¯ [arrow right] ^{... d} and a stress field σ : Ω × I ¯ [arrow right] ^{...AE; d} such that [figure omitted; refer to PDF]

We present a short description of the equations and conditions in Problems ( 2.19)-( 2.24). For more details and mechanical interpretation, we refer to [ 1, 16]. Here ( 2.19) represents the viscoelastic constitutive law in which ...9C; , [Bernoulli] , and ...A2; are given nonlinear operators, called the viscosity operator, elasticity operator, and time-delay operator, respectively. The prime represents the derivative with respect to the time variable, and therefore u represents the velocity field. Note that the explicit dependence of the viscosity, elasticity, and time-delay operators ...9C; , [Bernoulli] , and ...A2; with respect to the time variable means that the model involve the situations when the properties of the material depend on the temperature, that is, its evolution in time is prescribed. Equality ( 2.20) represents the equilibrium equation where Div σ = ( _{σ ij ,j} ) represents the divergence of stress. Conditions ( 2.21) and ( 2.22) are the displacement and traction boundary conditions, respectively. Equation ( 2.23) represents the frictional contact conditions and ( 2.24) is the initial condition in which the function _{u 0} denotes the initial displacement field.

In the study of mechanical problems ( 2.19)-( 2.24), we assume that ...9C; , [Bernoulli] , g , and h satisfy the following conditions.

H( ...9C; ): ...9C; : _{Q 1} × ^{...AE; d} [arrow right] ^{...AE; d} is an operator such that

(i) || ...9C; (x , _{t 1} , _{[straight epsilon] 1} ) - ...9C; (x , _{t 2} , _{[straight epsilon] 2} ) _{|| Q} ...4;L ( | _{t 1} - _{t 2} | + || _{[straight epsilon] 1} - _{[straight epsilon] 2 || Q} ) , for all _{[straight epsilon] 1} , _{[straight epsilon] 2} ∈ ^{...AE; d} , _{t 1} , _{t 2} ∈ I ¯ , a.e. x ∈ Ω with L >0 ;

(ii) ( ( ...9C; (x ,t , _{[straight epsilon] 1} ) - ...9C; (x ,t , _{[straight epsilon] 2} ) ) , ( _{[straight epsilon] 1} - _{[straight epsilon] 2} ) _{) Q} ...5;M || _{[straight epsilon] 1} - _{[straight epsilon] 2} ^{|| Q 2} , for all _{[straight epsilon] 1} , _{[straight epsilon] 2} ∈ ^{...AE; d} , a.e. (x ,t ) ∈ _{Q 1} with M >0 ;

(iii): for any [straight epsilon] ∈ ^{...AE; d} , (x ,t ) ... ...9C; (x ,t , [straight epsilon] ) is measurable on _{Q 1} ;

(iv) the mapping (x ,t ) ... ...9C; (x ,t ,0 ) ∈ ^{L 2} ( _{Q 1} ^{) d ×d} .

H( [Bernoulli] ): [Bernoulli] : _{Q 1} × ^{...AE; d} [arrow right] ^{...AE; d} is an operator such that

(i) || [Bernoulli] (x ,t , _{[straight epsilon] 1} ) - [Bernoulli] (x ,t , _{[straight epsilon] 2} ) _{|| Q} ...4; _{L 1} || _{[straight epsilon] 1} - _{[straight epsilon] 2 || Q} , for all _{[straight epsilon] 1} , _{[straight epsilon] 2} ∈ ^{...AE; d} , a.e. (x ,t ) ∈ _{Q 1} with _{L 1} >0 ;

(ii) for any [straight epsilon] ∈ ^{...AE; d} , (x ,t ) ... [Bernoulli] (x ,t , [straight epsilon] ) is measurable on Q ;

(iii): the mapping (x ,t ) ... [Bernoulli] (x ,t ,0 ) ∈ ^{L 2} ( _{Q 1} ^{) d ×d} .

H( g ): g : _{Γ 3} ×I × ... [arrow right] ^{... +} is an operator such that

(i) there exists _{L 2} >0 such that for all _{t 1} , _{t 2} ∈ I ¯ , _{u 1} , _{u 2} ∈ ... , x ∈ Ω , |g (x , _{t 1} , _{u 1} ) -g (x , _{t 2} , _{u 2} ) | ...4; _{L 2} ( | _{t 1} - _{t 2} | + | _{u 1} - _{u 2} | ) ;

(ii) for any u ∈ ... , (x ,t ) ...g (x ,t ,u ) is measurable;

(iii): the mapping (x ,t ) ...g (x ,t ,0 ) ∈ ^{L 2} ( _{Γ 3} ×I ) ;

(iv) || _{σ τ} || <g (t , _{S t} ( u (t ) ) ) [implies] _{u τ} =0 , || _{σ τ} || =g (t , _{S t} ( u (t ) ) ) [implies] exists λ ...5;0 such that _{σ τ} = - λ _{u τ} .

H( h ): h : _{Q 1} × ^{...AE; d} [arrow right] ^{...AE; d} is an operator such that

(i) ||h (x ,t , _{[straight epsilon] 1} ) -h (x ,t , _{[straight epsilon] 2} ) _{|| Q} ...4; _{L 3} || _{[straight epsilon] 1} - _{[straight epsilon] 2 || Q} , for all _{[straight epsilon] 1} , _{[straight epsilon] 2} ∈ ^{...AE; d} , a.e. (x ,t ) ∈ _{Q 1} with _{L 3} >0 ;

(ii) for any [straight epsilon] ∈ ^{...AE; d} , (x ,t ) ...h (x ,t , [straight epsilon] ) is measurable on _{Q 1} ;

(iii): (h (x ,t , _{[straight epsilon] 1} ) -h (x ,t , _{[straight epsilon] 2} ) , ( _{[straight epsilon] 1} - _{[straight epsilon] 2} ) _{) Q} ...5;M ' || _{[straight epsilon] 1} - _{[straight epsilon] 2} ^{|| Q 2} , for all _{[straight epsilon] 1} , _{[straight epsilon] 2} ∈ ^{...AE; d} , a.e. (x ,t ) ∈ _{Q 1} with M ' >0 ;

(iv) the mapping (x ,t ) ...h (x ,t ,0 ) ∈ ^{L 2} ( _{Q 1} ^{) d ×d} .

In the following, we provide an elementary example of the mechanical problem which hold the constitutive law equation ( 2.19).

Example 2.5.

Let ...9C; and [Bernoulli] be nonlinear operators which describe the viscous and the elastic properties of the material and satisfy the conditions H( ...9C; ) and H( [Bernoulli] ), respectively, while ...9E; is the linear relaxation operator. The following example is assumed to be the Kelvin-Voigt viscoelastic constitutive law with long memory of the form [figure omitted; refer to PDF] which represents a constitutive equation of the form ( 2.19).

Contact problems involving viscoelastic materials with long memory have been studied in [ 25, 26]. For more detail on the long memory models, we refer to [ 27, 28].

The famous time-temperature superposition principle tells us that when materials are applied with the alternating stress, the reaction time is an inverse proportion to the effect of the frequency. Hence, the influence of increasing the time (or reducing the frequency) and elevating temperature to materials is equivalent.

The sinusoidally driven indentation test was shown to be effective for viability characterization of articular cartilage. Based on the viscoelastic correspondence principle, Argatov [ 5] described the mechanical response of the articular cartilage layer in the framework of viscoelastic model. Using the asymptotic modeling approach, Argatov analyzed and interpreted the results of the indentation test. Now, deriving from the (30 ) and (115 ) in [ 5], and noting the relationship between time and frequency, we write the viscoelastic constitutive law in the following form: [figure omitted; refer to PDF] where _{a 1} , _{a 2} and _{b 1} , _{b 2} , _{b 3} are some parameters which rely on the characteristic relaxation time of strain under an applied step in stress, the equilibrium elastic modulus, and the glass elastic modulus.

It is easy to verify that ...9C; (t , [straight epsilon] ( u (t ) ) ) = ( _{a 1} ^{t 2} / ( _{a 2} + ^{t 2} ) ) [straight epsilon] ( u (t ) ) and [Bernoulli] (t , [straight epsilon] (u (t ) ) ) = ( ( _{b 1} + _{b 2} ^{t 2} ) / ( _{b 3} + ^{t 2} ) ) [straight epsilon] (u (t ) ) satisfy the assumption H( ...9C; ) and H( [Bernoulli] ), respectively.

Next, we denote by f (t ) the element of _{V 1} given by [figure omitted; refer to PDF]

When we assume that the body force and surface traction satisfy _{f 0} ∈C ( I ¯ ;H ) and _{f 2} ∈C ( I ¯ ; ^{L 2} ( _{Γ 2} ) ) , we can get [figure omitted; refer to PDF]

Let j :I × ^{L 2} ( _{Γ 3} ) × _{V 1} [arrow right] ... be the functional defined as follows: [figure omitted; refer to PDF]

We notice that, by the assumption H( g ), the integral in ( 2.29) is well defined.

Lemma 2.6 (Gronwall's inequality).

Assume that f ,g ∈C [a ,b ] satisfy [figure omitted; refer to PDF] where c >0 is a constant. Then [figure omitted; refer to PDF] Moreover, if g is nondecreasing, then [figure omitted; refer to PDF]

Proceeding in a standard way with these notations, we combine ( 2.8)-( 2.24) to obtain the following variational formulation.

Problem 1.

Find a displacement u : I ¯ [arrow right] _{V 1} such that ( 2.24) holds and [figure omitted; refer to PDF] We first introduce the following problem.

Problem 2.

Find a displacement u : I ¯ [arrow right] _{V 1} such that ( 2.24) holds and [figure omitted; refer to PDF]

For solving the above problems, we derive some results for an elliptic variational inequality of the second kind: Given f ∈X , find u ∈V such that [figure omitted; refer to PDF]

Lemma 2.7 (see [ 1]).

Let j :V [arrow right] ... ¯ be a proper, convex, and lower semicontinuous functional. Then for any f ∈V , there exists a unique element u : =Pro _{x j} (f ) such that [figure omitted; refer to PDF]

Lemma 2.8.

Let V be a Hilbert space. Assume that H( ...9C; ) holds and j :V [arrow right] ... ¯ is a proper, convex, lower semicontinuous functional. Then for any f ∈V , variational inequality ( 2.35) has a unique solution.

Proof.

For any f ∈V , let ρ >0 be a parameter to be chosen later. Since ρj :V [arrow right] ... ¯ is again a proper, convex, and lower semicontinuous functional, we can define an operator T : I ¯ ×V [arrow right]V by [figure omitted; refer to PDF] where (A (t ,u ) ,v _{) V} = ( ...9C; (x ,t , [straight epsilon] (u ) ) , [straight epsilon] (v ) _{) Q} (see ( 2.5)). We will show that with a suitable choice of ρ the operator T is a contractive mapping on I ¯ ×V . To this end, let u ,v ∈V . Since Prox is a nonexpansive mapping, it follows from ( 2.37) that [figure omitted; refer to PDF] Using the assumption H( ...9C; ) and ( 2.5), we obtain [figure omitted; refer to PDF] If 0 ...4; ρ ...4; 2M / ^{L 2} , then [figure omitted; refer to PDF] Taking [figure omitted; refer to PDF] we deduce that α ∈ (0,1 ) and [figure omitted; refer to PDF] which shows that T : I ¯ ×V [arrow right]V is a contractive mapping. Therefore, T has a fixed point u , that is, [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] It follows that [figure omitted; refer to PDF] Since ρ >0 , we deduce from the above inequality that u is a solution of variational inequality ( 2.35).

To show the uniqueness, we assume that there exist two solutions _{u 1} , _{u 2} ∈V of variational inequality ( 2.35). Then for any v ∈V and a.e . t ∈ I ¯ , we have [figure omitted; refer to PDF] Since j is proper, we know that j ( _{u 1} ) < ∞ and j ( _{u 2} ) < ∞ . Taking v = _{u 2} in the first inequality and v = _{u 1} in the second one and adding the corresponding inequalities, we get [figure omitted; refer to PDF] Using ( 2.5) and H( ...9C; ), we obtain that _{u 1} = _{u 2} , which completes the proof of Lemma 2.8.

Remark 2.9.

Lemma 2.8is a generalization of Theorem 4.1 of [ 1].

3. Main Results

In this section, we present an existence and uniqueness result concerned with the solution of Problem 1. Throughout this section, we assume that H( ...9C; ), H( [Bernoulli] ), H( g ), H( h ), and ( 2.28) hold.

Theorem 3.1.

Problem 2has a unique solution u ∈ ^{C 1} ( I ¯ ; _{V 1} ) .

The proof of Theorem 3.1is based on fixed point arguments and is established in several steps. Let η ∈C ( I ¯ ;Q ) and ξ ∈C ( I ¯ ; _{V 1} ) be arbitrarily given. We consider the following auxiliary variational problem.

Problem 3.

Find _{w η ξ} : I ¯ [arrow right] _{V 1} such that for any v ∈ _{V 1} [figure omitted; refer to PDF]

Lemma 3.2.

There exists a unique solution _{w η ξ} ∈C ( I ¯ ; _{V 1} ) to Problem 3.

Proof.

For each fixed t ∈ I ¯ , in terms of hypotheses ( 2.9), H( ...9C; ), H( g ), and ( 2.29), Problem 3is an elliptic variational inequality on Q . It follows from Lemma 2.8that Problem 3is uniquely solvable. Let _{w η ξ} (t ) ∈ _{V 1} be the unique solution of Problem 3. Now we show that _{w η ξ} (t ) ∈C ( I ¯ , _{V 1} ) .

Suppose that _{t 1} , _{t 2} ∈ I ¯ . For simplicity we write _{w η ξ} ( _{t i} ) = _{w i} , η ( _{t i} ) = _{η i} and f ( _{t i} ) = _{f i} with i =1,2 . Using ( 3.1) for t = _{t 1} , _{t 2} , we have [figure omitted; refer to PDF] By adding two inequalities with v = _{w 2} in ( 3.2) and v = _{w 1} in ( 3.3), we get [figure omitted; refer to PDF] where [figure omitted; refer to PDF] It implies that [figure omitted; refer to PDF] By H( ...9C; ), we get [figure omitted; refer to PDF] Constituting a trace operator γ :V [arrow right] ^{L 2} ( _{Γ 3} ) that γv =v _{|" Γ 3} , since γ is a linear continuous operator, it implies that there exists a constant c >0 such that [figure omitted; refer to PDF]

It follows from ( 2.9), ( 2.29), ( 3.8), and H( g ) that [figure omitted; refer to PDF] By ( 3.6)-( 3.7) and ( 3.9), we have [figure omitted; refer to PDF] which implies that _{w η ξ} ∈C ( I ¯ ; _{V 1} ) . This completes the proof of Lemma 3.2.

In order to get the unique solution of Problem 2, we derive the following operator _{Λ η} :C ( I ¯ ; _{V 1} ) [arrow right]C ( I ¯ ; _{V 1} ) defined by [figure omitted; refer to PDF]

Lemma 3.3.

For any η ∈C ( I ¯ ;Q ) , the operator _{Λ η} has a unique fixed point _{ξ η} ∈C ( I ¯ ; _{V 1} ) .

Proof.

Let η ∈C ( I ¯ ;Q ) and _{ξ 1} , _{ξ 2} ∈C ( I ¯ ; _{V 1} ) . We denote by _{w i} the solution of Problem 3with ξ = _{ξ i} for i =1,2 . By an argument similar to that used in obtaining ( 3.6), we get [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Using ( 2.29), ( 3.8), ( 2.9), and H( g ), we deduce that, for any t ∈ I ¯ , [figure omitted; refer to PDF] where _{c 1} =c _{L 2} . By using the similar method in obtaining ( 3.10), we have [figure omitted; refer to PDF] Since _{w i} = _{w η ξ i} = _{Λ η ξ i} , we rewrite the above inequality as [figure omitted; refer to PDF] For w ∈C ( I ¯ ; _{V 1} ) , let [figure omitted; refer to PDF] where β >0 is a constant which will be chosen later. Clearly, || · _{|| β} defines a norm on the space C ( I ¯ ; _{V 1} ) and [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] and so the operator _{Λ η} is a contraction on the space C ( I ¯ ; _{V 1} ) endowed with the equivalent norm || · _{|| β} if we choose β such that M β > _{c 1} . Therefore, the operator _{Λ η} has a unique fixed point _{ξ η} ∈C ( I ¯ ; _{V 1} ) , which completes the proof of Lemma 3.3.

In what follows, for any η ∈C ( I ¯ ;Q ) , we write [figure omitted; refer to PDF] By _{Λ η ξ η} = _{ξ η} , ( 3.11), and ( 3.20), we have [figure omitted; refer to PDF] Taking ξ = _{ξ η} in ( 3.1) and using ( 3.20) and ( 3.21), we deduce that, for any v ∈ _{V 1} , [figure omitted; refer to PDF] Let _{w η} : I ¯ [arrow right] _{V 1} be the function given by [figure omitted; refer to PDF] In addition, we define the operator Λ :C ( I ¯ ;Q ) [arrow right]C ( I ¯ ;Q ) by [figure omitted; refer to PDF]

Lemma 3.4.

The operator Λ has a unique fixed point ^{η *} ∈C ( I ¯ ;Q ) .

Proof.

For any _{η 1} , _{η 2} ∈C ( I ¯ ;Q ) , let _{u i} = _{u η i} , _{u ~ i} = _{u ~ η i} and _{w i} = _{w η i} with i =1,2 . Using ( 3.22) and arguments similar to those used in the proof of Lemma 3.3, we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] An application of the Gronwall inequality yields [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] where _{c 4} = _{c 3} (1 +T ) . For the operator Λ defined in ( 3.24), by u ~ ∈ [real] (u ) , ( 2.5), Remark 2.4, and H( h ), we obtain [figure omitted; refer to PDF] where _{c 5} = _{L 3} | μ | ( [ -r ,0 ] ) _{c 4} . By using ( 3.30) and the similar proof of Lemma 3.3, we get the result of Lemma 3.4. This completes the proof.

Now we prove Theorem 3.1.

Proof of Theorem 3.1.

Let ^{η *} ∈C ( I ¯ ;Q ) be the fixed point of Λ and let _{u ^{η *}} ∈ ^{C 1} ( I ¯ ;Q ) be the function defined by ( 3.24) for η = ^{η *} . For any v ∈ _{V 1} and a.e. t ∈ I ¯ , it follows from _{u ^{η *}} = _{w ^{η *}} and ( 3.22) that [figure omitted; refer to PDF] Now inequality ( 2.34) follows from ( 3.24) and ( 3.30). Moreover, since ( 3.23) implies _{u ^{η *}} (0 ) = _{u 0} , we conclude that _{u ^{η *}} is a solution of Problem 2.

Let _{u 1} , _{u 2} ∈C ( I ¯ ; _{V 1} ) be two solutions to Problem 2and let _{w i} = _{u i} for i =1,2 . Then we have [figure omitted; refer to PDF] For a.e. t ∈ I ¯ , by the similar argument used in obtaining ( 3.6), we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Using ( 2.29), ( 3.8), ( 2.9), and H( g ), we deduce that [figure omitted; refer to PDF] From the assumption H( ...9C; ) and relations ( 3.30)-( 3.35), we know that, for a.e. t ∈ I ¯ , [figure omitted; refer to PDF] where [figure omitted; refer to PDF] An application of the Gronwall inequality yields [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Recalling the definition ( 3.32) of _{u 1} and _{u 2} , and letting _{c 8} = _{c 7} (1 +T ) , we obtain [figure omitted; refer to PDF] and so the Gronwall inequality implies that _{w 1} = _{w 2} . By definition ( 3.32), we see that _{u 1} = _{u 2} , which completes the proof of Theorem 3.1.

Theorem 3.5.

Problem 1has a unique solution u ∈ ^{C 1} ( I ¯ ; _{V 1} ) .

Proof.

Let ζ ∈C ( I ¯ ; _{V 1} ) and denote by _{u ζ} ∈C ( I ¯ ; _{V 1} ) the solution of the following problem: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] From Theorem 3.1, we know that there exists a unique solution _{u ζ} for problem ( 3.41).

Consider the operator Υ :C ( I ¯ ; ^{V *} ) [arrow right]C ( I ¯ ; ^{V *} ) defined by [figure omitted; refer to PDF] Now we show that the operator Υ has a unique fixed point. In fact, for any _{ζ 1} , _{ζ 2} ∈C ( I ¯ ; _{V 1} ) , let _{u 1} = _{u ζ 1} and _{u 2} = _{u ζ 2} be the corresponding solutions to ( 3.41). Then it is easy to see that _{u 1} , _{u 2} ∈C ( I ¯ ; _{V 1} ) . For any _{u ~ 1} ∈ [real] ( _{u 1} ) and _{u ~ 2} ∈ [real] ( _{u 2} ) , by the similar argument used in obtaining ( 3.33), we have [figure omitted; refer to PDF] By H( ...9C; ), H( h ), ( 2.5), ( 3.30), ( 3.32), and ( 3.35), we get [figure omitted; refer to PDF] where [figure omitted; refer to PDF] An application of the Gronwall inequality yields [figure omitted; refer to PDF] where _{c 10} = (1 +T )max { _{c 9} , ^{c 9 2 e _{c 9} T} } . From H( [Bernoulli] ), ( 3.43), and ( 3.47), we have [figure omitted; refer to PDF] where _{c 11} =T ^{c 10 2 L 1 2} . Iterating the last inequality p times, we obtain [figure omitted; refer to PDF] which leads to [figure omitted; refer to PDF] Since _{lim p [arrow right] ∞} ( ^{c 11 p T p} / (p -1 ) ! ^{) 1 / 2} =0 , the previous inequality implies that, for p large enough, a power ^{Υ p} of Υ is a contraction. It follows that there exists a unique element ^{ζ *} ∈ _{V 1} such that ^{Υ p ζ *} = ^{ζ *} . Moreover, since [figure omitted; refer to PDF] we deduce that Υ ^{ζ *} is also a fixed point of the operator ^{Υ p} . By the uniqueness of the fixed point that Υ ^{ζ *} = ^{ζ *} , we know that ^{ζ *} is a fixed point of Υ . The uniqueness of the fixed point of Υ results straightforward from the uniqueness of the fixed point of ^{Υ p} . This implies that _{u ^{ζ *}} is the unique solution of Problem 1, which completes the proof of Theorem 3.5.

Remark 3.6.

When ...A2; =0 and all the viscosity and elasticity operators ...9C; and [Bernoulli] are explicitly time dependent, Theorem 3.5reduces to Theorem 10.2 of [ 1]. Furthermore, Theorem 3.5is also a generalization of Theorem 2.1 of [ 24].

4. A Convergence Result

In this section, we study the dependence of the solution to Problem 1with respect to perturbations of the operator h . We assume that H( ...9C; ), H( [Bernoulli] ), H( g ), and H( h ) hold and, for any β >0 , let _{h β} be a perturbation of the operator h .

We consider the following problem.

Problem 4.

Find _{u β} : I ¯ [arrow right] _{V 1} such that [figure omitted; refer to PDF] It follows from Theorem 3.5that, for each β >0 , Problem 4has a unique solution denoted by _{u β} ∈C ( I ¯ ; _{V 1} ) .

In order to get the convergence result, we need the following assumption: [figure omitted; refer to PDF]

Now we give the convergence result.

Theorem 4.1.

Assume that H ( ...9C; ), H ( [Bernoulli] ), H (g ), H (h ), and ( 2.28) hold. Then the solution _{u β} of Problem 4converges to the solution u of Problem 1, that is, [figure omitted; refer to PDF]

Proof.

For any β >0 and a.e. t ∈ I ¯ , let [figure omitted; refer to PDF] Keeping in mind ( 4.1) and ( 2.33), and using H( ...9C; ) and H( [Bernoulli] ), by the similar argument used in obtaining ( 3.36), we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Hence, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] For a.e. t ∈ I ¯ , it follows from ( 4.5), ( 2.5), and Remark 2.4that [figure omitted; refer to PDF] Noting _{u ~ β} ∈ [real] ( _{u β} ) , from the last inequality, ( 4.3), ( 4.8), and H( h ), we have [figure omitted; refer to PDF] It follows from the Gronwall inequality that _{w β} =w and so the convergence result ( 4.4) is a consequence of ( 3.32). This completes the proof.

Acknowledgments

The authors are grateful to the editor and the referees for their valuable comments and suggestions. This work was supported by the Key Program of NSFC (Grant no. 70831005), the National Natural Science Foundation of China (11171237, 11101069), and the Scientific Research Foundation of Yunnan Provincial Education Office (2010Z024).

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