Analysis of a history-dependent frictionless contact problem

Abstract

We consider a mathematical model which describes the quasistatic contact between a viscoelastic body and an obstacle, the so-called foundation. The contact is frictionless and is modeled with a new and nonstandard condition which involves both normal compliance, unilateral constraint and memory effects. We derive a variational formulation of the problem which is the form of a history-dependent variational inequality for the displacement field.

Then, using a recent result obtained by Sofonea and Matei, we prove the unique weak solvability of the problem. Next, we study the continuous dependence of the weak solution with respect the data and prove a first convergence result. Finally, we prove that the weak solution of the problem represents the limit of the weak solution of a contact problem with normal compliance and memory term, as the stiffness coefficient of the foundation converges to infinity.

Authors

Mircea Sofonea
Laboratoire de Mathรฉmatiques et Physique, Universitรฉ de Perpignan

Flavius Patrulescu
Tiberiu Popoviciu Institute of Numerical Analysis,
Romanian Academy

Keywords

viscoelastic material; frictionless contact; unilateral constraint; memory term; history-dependent variational inequality; weak solution

Cite this paper as:

M. Sofonea, F. Pฤƒtrulescu, Analysis of a history-dependent frictionless contact problem, Math. Mech. Solids, 18 (2013) no.4, pp. 409-430.

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Mathematics and Mechanics of Solids

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SAGE Publications, Thousand Oaks, CA

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1081-2865

Online ISSN

1741-3028

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3179460

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Analysis of a History-dependent Frictionless Contact Problem

M. Sofonea 1 F. Pฤƒtrulescu 2,
1 Laboratoire de Mathรฉmatiques et Physique
Universitรฉ de Perpignan, 52 Avenue de Paul Alduy, 66860 Perpignan, France
2 Tiberiu Popoviciu Institute of Numerical Analysis
P.O. Box 68-1, 400110 Cluj-Napoca, Romania
Abstract

We consider a mathematical model which describes the quasistatic contact between a viscoelastic body and an obstacle, the so-called foundation. The contact is frictionless and is modelled with a new and nonstandard condition which involves both normal compliance, unilateral constraint and memory effects. We derive a variational of the problem which is the form of a history history-dependent variational inequality for the displacement field. Then, using a recent result obtained in [26], we prove the unique weak solvability of the problem. Next, we study the continuous dependence of the weak solution with respect the data and prove a first convergence result. Finally, we prove that the weak solution of the problem converges to the weak solution of a contact problem with normal compliance and memory term, as the stiffness coefficient of the foundation converges to infinity.

2010 Mathematics Subject Classification : 74M15, 74G25, 74G30, 49J40, 35Q74.
Keywords: viscoelastic material, frictionless contact, unilateral constraint, memory term, history-depdendent variational inequality, weak solution.

1 Introduction

Phenomena of contact between deformable bodies abound in industry and everyday life. Contact of braking pads with wheels, tires with roads, pistons with skirts are just few simple examples. Common industrial processes such as metal forming, metal extrusion, involve contact evolutions. Owing to their inherent complexity, contact
phenomena lead to mathematical models expressed in terms of strongly nonlinear elliptic or evolutionary boundary value problems.

An early attempt to study frictional contact problems within the framework of variational inequalities was made in [3]. An excellent reference on analysis and numerical approximations of contact problems involving elastic materials with or without friction is [7]. The variational analysis of various contact problems, including existence and uniqueness results, can be found in the monographs [4, 6, 15, 22. The state of the art in the field can also be found in the proceedings [10, 18, 29] and in the special issue [21], as well.

To construct a mathematical model which describes a specific contact process we need to precise the materialโ€™s behavior and the contact conditions, among others. In this paper we assume that the material is viscoelastic and we describe its behavior with a constitutive law with long memory of the form

๐ˆ(t)=๐’œ๐œบ(๐’–(t))+โˆซ0tโ„ฌ(tโˆ’s)๐œบ(๐’–(s))๐‘‘s\boldsymbol{\sigma}(t)=\mathcal{A}\boldsymbol{\varepsilon}(\boldsymbol{u}(t))+\int_{0}^{t}\mathcal{B}(t-s)\boldsymbol{\varepsilon}(\boldsymbol{u}(s))ds (1.1)

Here ๐’–\boldsymbol{u} denotes the displacement field, ๐ˆ\boldsymbol{\sigma} represents the stress, ๐œบ(๐’–)\boldsymbol{\varepsilon}(\boldsymbol{u}) is the linearized strain tensor and, finally, ๐’œ\mathcal{A} and โ„ฌ\mathcal{B} are the elasticity operator and the relaxation tensor, respectively. Results and mechanical interpretations in the study of viscoelastic materials of the form (1.1) can be found in [3, 17, 28, for instance. The analysis of various contact problems which such kind of materials was provided in [19, 20, 24. There, the unique solvability of the problems was proved by using existence and uniqueness results for evolutionary variational inequalities involving a Volterra-type integral term; fully discrete schemes for the numerical approximation of the models were considered and error estimates were derived; finally, the schemes were implemented on a computer code and numerical simulations were presented. The analysis of models of antiplane frictional contact problems with viscoelastic materials of the form (1.1), including existence, uniqueness and convergence results, was performed in 25.

We turn now to describe some representative contact conditions used in the literature and, to this end, we denote by uฮฝu_{\nu} and ฯƒฮฝ\sigma_{\nu} the normal displacement and the normal stress on the contact surface, respectively.

The so-called normal compliance contact condition describes a deformable foundation. It assigns a reactive normal pressure that depends on the interpenetration of the asperities on the bodyโ€™s surface and those of the foundation. A general expression for this condition is

โˆ’ฯƒฮฝ=p(uฮฝ)-\sigma_{\nu}=p\left(u_{\nu}\right) (1.2)

where pp is a nonnegative prescribed function which vanishes for negative argument. Indeed, when uฮฝ<0u_{\nu}<0 there is no contact and the normal pressure vanishes. When there is contact then uฮฝu_{\nu} is positive and represents a measure of the interpenetration of the asperities. Then, condition (1.2) shows that the foundation exerts a pressure on the body, which depends on the penetration.

A commonly used example of the normal compliance function pp is

p(r)=cฮฝr+p(r)=c_{\nu}r^{+} (1.3)

Here the constant cฮฝ>0c_{\nu}>0 is the surface stiffness coefficient and r+=maxโก{r,0}r^{+}=\max\{r,0\} denotes the positive part of rr. A second example is provided by the truncated normal compliance function

p(r)={cฮฝr+if rโ‰คฮฑ,cฮฝฮฑ if r>ฮฑ,p(r)=\begin{cases}c_{\nu}r^{+}&\text{if }r\leq\alpha,\\ c_{\nu}\alpha&\text{ if }r>\alpha,\end{cases}

where ฮฑ\alpha is a positive coefficient related to the wear and hardness of the surface. In this case the contact condition (1.2) means that when the penetration is too large, i.e., when it exceeds ฮฑ\alpha, the obstacle offers no additional resistance to penetration.

The normal compliance contact condition was first introduced in [14] and since then used in many publications, see, e.g., [7, 8, 9, 11] and references therein. The term normal compliance was first introduced in [8, 9]. An idealization of the normal compliance, which is used often in engineering literature, and can also be found in mathematical publications, is the Signorini contact condition, in which the foundation is assumed to be perfectly rigid. It is obtained, formally, from the normal compliance condition (1.2), (1.3), in the limit when the surface stiffness coefficient becomes infinite, i.e., cฮฝโ†’โˆžc_{\nu}\rightarrow\infty, and thus interpenetration is not allowed. This leads to the idea of regarding contact with a rigid support as a limiting case of contact with deformable support, whose resistance to compression increases. The Signorini contact condition can be stated in the following complementarity form:

uฮฝโ‰ค0,ฯƒฮฝโ‰ค0,ฯƒฮฝuฮฝ=0u_{\nu}\leq 0,\quad\sigma_{\nu}\leq 0,\quad\sigma_{\nu}u_{\nu}=0 (1.5)

This condition was first introduced in 23 and then used in many papers, see, e.g., [22] for further details and references. Assume now that there is an initial gap g>0g>0 between the body and the foundation. Then the Signorini contact condition in a form with a gap function is given by

uฮฝโ‰คg,ฯƒฮฝโ‰ค0,ฯƒฮฝ(uฮฝโˆ’g)=0u_{\nu}\leq g,\quad\sigma_{\nu}\leq 0,\quad\sigma_{\nu}\left(u_{\nu}-g\right)=0 (1.6)

In various situations the reaction of the foundation at the moment tt depends on the history of the penetration and, therefore, it cannot be determinate as a function of the current value uฮฝ(t)u_{\nu}(t). In this case one can assume that the normal stress satisfies a condition of the form

โˆ’ฯƒฮฝ(t)=โˆซ0tb(tโˆ’s)uฮฝ+(s)๐‘‘s-\sigma_{\nu}(t)=\int_{0}^{t}b(t-s)u_{\nu}^{+}(s)ds (1.7)

in which bb represents a given function, the so-called surface memory function. Contact conditions of the form (1.7) have a simple physical interpretation if there are no cycles of contact and separation during the time interval of interest. For instance, assume in what follows that bb is a positive function. Moreover, assume that in the time interval
[0;t][0;t] there is only penetration (i.e. uฮฝ(s)โ‰ฅ0u_{\nu}(s)\geq 0 for all sโˆˆ[0,t]s\in[0,t] ). Then (1.7) shows that the reaction of the foundation at tt is towards the body (since ฯƒฮฝ(t)โ‰ค0\sigma_{\nu}(t)\leq 0 ). Also, if in the time interval [0;t][0;t] there is separation (i.e. uฮฝ(s)<0u_{\nu}(s)<0 for all sโˆˆ[0,t]s\in[0,t] ) then there is no reaction at the moment tt (since ฯƒฮฝ(t)=0\sigma_{\nu}(t)=0 ). Now, assume a situation in which uฮฝu_{\nu} is positive in time interval [0,t0]\left[0,t_{0}\right] and negative on the time interval [t0,t]\left[t_{0},t\right]. Then, following (1.7) we have

โˆ’ฯƒฮฝ(t)=โˆซ0t0b(tโˆ’s)uฮฝ+(s)๐‘‘s-\sigma_{\nu}(t)=\int_{0}^{t_{0}}b(t-s)u_{\nu}^{+}(s)ds

since the integral on the remaining interval [t0,t]\left[t_{0},t\right] vanishes. Assume, in addition, that the support of the function bb is included in the interval [0,ฮด][0,\delta] with ฮด>0\delta>0. Two possibilities arise. First, if tโˆ’t0>ฮดt-t_{0}>\delta it follows that b(tโˆ’s)=0b(t-s)=0 for all sโˆˆ[0,t0]s\in\left[0,t_{0}\right] and (1.7) shows the normal stress ฯƒฮฝ(t)\sigma_{\nu}(t) vanishes. Second, if tโˆ’t0โ‰คฮดt-t_{0}\leq\delta (1.7) implies that ฯƒฮฝ(t)โ‰ค0\sigma_{\nu}(t)\leq 0 i.e. a residual pression exists at the moment tt on the bodyโ€™s surface. We interpret this as a memory effect in which the foundation prevents the separation, moves towards the body and exerts a pression on a short interval of time of length ฮด\delta. Various other mechanical interpretation of the condition (1.7) could be obtained if bb is assumed to be a negative function or if this condition is associated to the normal compliance condition (1.2), as shown in Section 3 below. Note that conditions of the form (1.7) were considered in [13] in the study of a lumped model with contact and friction.

In the present paper we study a quasistatic frictionless contact problem for viscoelastic materials of the form (1.1). The novelty consists in the fact that the contact condition we use describes a deformable foundation which becomes rigid when the penetration reaches a critical bound and which developes memory effects. This contact condition includes as particular cases both the normal compliance condition (1.2), the Signorini condition (1.6) and the history-dependent condition (1.7). Considering such condition leads to a new and nonstandard mathematical model which, in a variational formulation, is given by a history-dependent variational inequality for the displacement field. We prove the unique weak solvability of the problem then we establish two convergence results.

The rest of the paper is structured as follows. In Section 2 we present the notation we shall use as well as some preliminary material. In Section 3 we describe the model of the contact process. In Section 4 we list the assumptions on the data and derive the variational formulation of the problem. Then we state and prove our main existence and uniqueness result, Theorem 4.1. In Section 5 we state and prove our first converge result, Theorem 5.1. It states the continuous dependence of the solution with respect to the data. Finally, in Section 6 we state and prove our second converge result, Theorem 6.1. It states that the weak solution of the problem with normal compliance, memory term and unilateral constraint can be approached by the weak solution of a problem with normal compliance and memory term, as the stiffness coefficient of the foundation converges to infinity.

2 Notation and preliminaries

Everywhere in this paper we use the notation โ„•โˆ—\mathbb{N}^{*} for the set of positive integers and โ„+\mathbb{R}_{+}will represent the set of non negative real numbers, i.e. โ„+=[0,+โˆž)\mathbb{R}_{+}=[0,+\infty). We denote by ๐•Šd\mathbb{S}^{d} the space of second order symmetric tensors on โ„d\mathbb{R}^{d} or, equivalently, the space of symmetric matrices of order dd. The inner product and norm on โ„d\mathbb{R}^{d} and ๐•Šd\mathbb{S}^{d} are defined by

๐’–โ‹…๐’—=uivi,โ€–๐’—โ€–=(๐’—โ‹…๐’—)12โˆ€๐’–=(ui),๐’—=(vi)โˆˆโ„d๐ˆโ‹…๐‰=ฯƒijฯ„ij,โ€–๐‰โ€–=(๐‰โ‹…๐‰)12โˆ€๐ˆ=(ฯƒij),๐‰=(ฯ„ij)โˆˆ๐•Šd\begin{array}[]{lll}\boldsymbol{u}\cdot\boldsymbol{v}=u_{i}v_{i},&\|\boldsymbol{v}\|=(\boldsymbol{v}\cdot\boldsymbol{v})^{\frac{1}{2}}&\forall\boldsymbol{u}=\left(u_{i}\right),\boldsymbol{v}=\left(v_{i}\right)\in\mathbb{R}^{d}\\ \boldsymbol{\sigma}\cdot\boldsymbol{\tau}=\sigma_{ij}\tau_{ij},&\|\boldsymbol{\tau}\|=(\boldsymbol{\tau}\cdot\boldsymbol{\tau})^{\frac{1}{2}}&\forall\boldsymbol{\sigma}=\left(\sigma_{ij}\right),\boldsymbol{\tau}=\left(\tau_{ij}\right)\in\mathbb{S}^{d}\end{array}

Let ฮฉโŠ‚โ„d(d=1,2,3)\Omega\subset\mathbb{R}^{d}(d=1,2,3) be a bounded domain with Lipschitz continuous boundary ฮ“\Gamma and let ฮ“1,ฮ“2\Gamma_{1},\Gamma_{2}, and ฮ“3\Gamma_{3} be three measurable parts of ฮ“\Gamma such that meas (ฮ“1)>0\left(\Gamma_{1}\right)>0. We use the notation ๐’™=(xi)\boldsymbol{x}=\left(x_{i}\right) for a typical point in ฮฉ\Omega and we denote by ๐‚=(ฮฝi)\boldsymbol{\nu}=\left(\nu_{i}\right) the outward unit normal at ฮ“\Gamma. Here and below the indices i,j,k,li,j,k,l run between 1 and dd and, unless stated otherwise, the summation convention over repeated indices is used. An index that follows a comma represents the partial derivative with respect to the corresponding component of the spatial variable, e.g. ui,j=โˆ‚ui/โˆ‚xju_{i,j}=\partial u_{i}/\partial x_{j}. We use standard notation for the Lebesgue and Sobolev spaces associated to ฮฉ\Omega and ฮ“\Gamma. In particular, we recall that the inner products on the Hilbert spaces L2(ฮฉ)dL^{2}(\Omega)^{d} and L2(ฮ“3)dL^{2}\left(\Gamma_{3}\right)^{d} are given by

(๐’–,๐’—)L2(ฮฉ)d=โˆซฮฉ๐’–โ‹…๐’—๐‘‘x,(๐’–,๐’—)L2(ฮ“2)d=โˆซฮ“2๐’–โ‹…๐’—๐‘‘a(\boldsymbol{u},\boldsymbol{v})_{L^{2}(\Omega)^{d}}=\int_{\Omega}\boldsymbol{u}\cdot\boldsymbol{v}dx,\quad(\boldsymbol{u},\boldsymbol{v})_{L^{2}\left(\Gamma_{2}\right)^{d}}=\int_{\Gamma_{2}}\boldsymbol{u}\cdot\boldsymbol{v}da

and the associated norms will be denoted by โˆฅโ‹…โˆฅL2(ฮฉ)d\|\cdot\|_{L^{2}(\Omega)^{d}} and โˆฅโ‹…โˆฅL2(ฮ“2)d\|\cdot\|_{L^{2}\left(\Gamma_{2}\right)^{d}}, respectively. Moreover, we consider the spaces

V\displaystyle V ={๐’—โˆˆH1(ฮฉ)d:๐’—=๐ŸŽ on ฮ“1}\displaystyle=\left\{\boldsymbol{v}\in H^{1}(\Omega)^{d}:\boldsymbol{v}=\mathbf{0}\text{ on }\Gamma_{1}\right\}
Q\displaystyle Q ={๐‰=(ฯ„ij)โˆˆL2(ฮฉ)d:ฯ„ij=ฯ„ji}\displaystyle=\left\{\boldsymbol{\tau}=\left(\tau_{ij}\right)\in L^{2}(\Omega)^{d}:\tau_{ij}=\tau_{ji}\right\}
Q1\displaystyle Q_{1} ={๐‰=(ฯ„ij)โˆˆQ:ฯ„ij,jโˆˆL2(ฮฉ)}\displaystyle=\left\{\boldsymbol{\tau}=\left(\tau_{ij}\right)\in Q:\tau_{ij,j}\in L^{2}(\Omega)\right\}

These are real Hilbert spaces endowed with the inner products

(๐’–,๐’—)V\displaystyle(\boldsymbol{u},\boldsymbol{v})_{V} =โˆซฮฉ๐œบ(๐’–)โ‹…๐œบ(๐’—)๐‘‘x\displaystyle=\int_{\Omega}\boldsymbol{\varepsilon}(\boldsymbol{u})\cdot\boldsymbol{\varepsilon}(\boldsymbol{v})dx
(๐ˆ,๐‰)Q\displaystyle(\boldsymbol{\sigma},\boldsymbol{\tau})_{Q} =โˆซฮฉ๐ˆโ‹…๐‰๐‘‘x\displaystyle=\int_{\Omega}\boldsymbol{\sigma}\cdot\boldsymbol{\tau}dx
(๐ˆ,๐‰)Q1\displaystyle(\boldsymbol{\sigma},\boldsymbol{\tau})_{Q_{1}} =โˆซฮฉ๐ˆโ‹…๐‰๐‘‘x+โˆซฮฉDivโก๐ˆโ‹…Divโก๐‰dx\displaystyle=\int_{\Omega}\boldsymbol{\sigma}\cdot\boldsymbol{\tau}dx+\int_{\Omega}\operatorname{Div}\boldsymbol{\sigma}\cdot\operatorname{Div}\boldsymbol{\tau}dx

and the associated norms โˆฅโ‹…โˆฅV,โˆฅโ‹…โˆฅQ\|\cdot\|_{V},\|\cdot\|_{Q} and โˆฅโ‹…โˆฅQ1\|\cdot\|_{Q_{1}}, respectively. Here ๐œบ\boldsymbol{\varepsilon} and Div are the deformation and divergence operators given by

๐œบ(๐’—)\displaystyle\boldsymbol{\varepsilon}(\boldsymbol{v}) =(ฮตij(๐’—)),ฮตij(๐’—)=12(vi,j+vj,i)โˆ€๐’—โˆˆH1(ฮฉ)d\displaystyle=\left(\varepsilon_{ij}(\boldsymbol{v})\right),\quad\varepsilon_{ij}(\boldsymbol{v})=\frac{1}{2}\left(v_{i,j}+v_{j,i}\right)\quad\forall\boldsymbol{v}\in H^{1}(\Omega)^{d}
Divโก๐‰\displaystyle\operatorname{Div}\boldsymbol{\tau} =(ฯ„ij,j)โˆ€๐‰โˆˆQ1\displaystyle=\left(\tau_{ij,j}\right)\quad\forall\boldsymbol{\tau}\in Q_{1}

Completeness of the space ( V,โˆฅโ‹…โˆฅVV,\|\cdot\|_{V} ) follows from the assumption meas (ฮ“1)>0\left(\Gamma_{1}\right)>0, which allows the use of Kornโ€™s inequality.

For an element ๐’—โˆˆV\boldsymbol{v}\in V we still write ๐’—\boldsymbol{v} for the trace of VV and we denote by vฮฝv_{\nu} and ๐’—ฯ„\boldsymbol{v}_{\tau} the normal and tangential components of ๐’—\boldsymbol{v} on ฮ“\Gamma given by vฮฝ=๐’—โ‹…๐‚,๐’—ฯ„=๐’—โˆ’vฮฝ๐‚v_{\nu}=\boldsymbol{v}\cdot\boldsymbol{\nu},\boldsymbol{v}_{\tau}=\boldsymbol{v}-v_{\nu}\boldsymbol{\nu}. By the Sobolev trace theorem, there exists a positive constant c0c_{0}, depending on ฮฉ\Omega, ฮ“1\Gamma_{1}, and ฮ“3\Gamma_{3}, such that

โ€–๐’—โ€–L2(ฮ“3)dโ‰คc0โ€–๐’—โ€–Vโˆ€๐’—โˆˆV\|\boldsymbol{v}\|_{L^{2}\left(\Gamma_{3}\right)^{d}}\leq c_{0}\|\boldsymbol{v}\|_{V}\quad\forall\boldsymbol{v}\in V (2.1)

For an regular function ๐ˆ:ฮฉโˆชฮ“โ†’๐•Šd\boldsymbol{\sigma}:\Omega\cup\Gamma\rightarrow\mathbb{S}^{d} we denote by ฯƒฮฝ\sigma_{\nu} and ๐ˆฯ„\boldsymbol{\sigma}_{\tau} the normal and the tangential components of the vector ๐ˆ๐‚\boldsymbol{\sigma}\boldsymbol{\nu} on ฮ“\Gamma, respectively, and we recall that ฯƒฮฝ=๐ˆ๐‚โ‹…๐‚,๐ˆฯ„=๐ˆโˆ’ฯƒฮฝ๐‚\sigma_{\nu}=\boldsymbol{\sigma}\boldsymbol{\nu}\cdot\boldsymbol{\nu},\boldsymbol{\sigma}_{\tau}=\boldsymbol{\sigma}-\sigma_{\nu}\boldsymbol{\nu}. Moreover, the following Greenโ€™s formula holds:

โˆซฮฉ๐ˆโ‹…๐œบ(๐’—)๐‘‘x+โˆซฮฉDivโก๐ˆโ‹…๐’—dx=โˆซฮ“๐ˆ๐‚โ‹…๐’—๐‘‘aโˆ€๐’—โˆˆV\int_{\Omega}\boldsymbol{\sigma}\cdot\boldsymbol{\varepsilon}(\boldsymbol{v})dx+\int_{\Omega}\operatorname{Div}\boldsymbol{\sigma}\cdot\boldsymbol{v}dx=\int_{\Gamma}\boldsymbol{\sigma}\boldsymbol{\nu}\cdot\boldsymbol{v}da\quad\forall\boldsymbol{v}\in V (2.2)

Finally, we denote by ๐โˆž\mathbf{Q}_{\infty} the space of fourth order tensor fields given by

๐โˆž={โ„ฐ=(โ„ฐijkl)โˆฃโ„ฐijkl=โ„ฐjikl=โ„ฐklijโˆˆLโˆž(ฮฉ),1โ‰คi,j,k,lโ‰คd}\mathbf{Q}_{\infty}=\left\{\mathcal{E}=\left(\mathcal{E}_{ijkl}\right)\mid\mathcal{E}_{ijkl}=\mathcal{E}_{jikl}=\mathcal{E}_{klij}\in L^{\infty}(\Omega),\quad 1\leq i,j,k,l\leq d\right\}

We note that ๐โˆž\mathbf{Q}_{\infty} is a real Banach space with the norm

โ€–โ„ฐโ€–๐โˆž=โˆ‘0โ‰คi,j,k,lโ‰คdโ€–โ„ฐijklโ€–Lโˆž(ฮฉ)\|\mathcal{E}\|_{\mathbf{Q}_{\infty}}=\sum_{0\leq i,j,k,l\leq d}\left\|\mathcal{E}_{ijkl}\right\|_{L^{\infty}(\Omega)}

Moreover, a simple calculation shows that

โ€–โ„ฐ๐‰โ€–Qโ‰คdโ€–โ„ฐโ€–๐โˆžโ€–๐‰โ€–Qโˆ€โ„ฐโˆˆ๐โˆž,๐‰โˆˆQ\|\mathcal{E}\boldsymbol{\tau}\|_{Q}\leq d\|\mathcal{E}\|_{\mathbf{Q}_{\infty}}\|\boldsymbol{\tau}\|_{Q}\quad\forall\mathcal{E}\in\mathbf{Q}_{\infty},\boldsymbol{\tau}\in Q (2.3)

For each Banach space XX we use the notation C(โ„+;X)C\left(\mathbb{R}_{+};X\right) for the space of continuously functions defined on โ„+\mathbb{R}_{+}with values on XX. It is well known that C(โ„+;X)C\left(\mathbb{R}_{+};X\right) can be organized in a canonical way as a Frรฉchet space, i.e. as a complete metric space in which the corresponding topology is induced by a countable family of seminorms. Details can be found in 22 and [12], for instance. Here we restrict ourseleves to recall that the convergence of a sequence (xm)m\left(x_{m}\right)_{m} to the element xx, in the space C(โ„+;X)C\left(\mathbb{R}_{+};X\right), can be described as follows:

{xmโ†’x in C(โ„+;X) as mโ†’โˆž if and only if maxrโˆˆ[0,n]โกโ€–xm(r)โˆ’x(r)โ€–Xโ†’0 as mโ†’โˆž, for all nโˆˆโ„•โˆ—\left\{\begin{array}[]{l}x_{m}\rightarrow x\quad\text{ in }C\left(\mathbb{R}_{+};X\right)\text{ as }m\rightarrow\infty\text{ if and only if }\\ \max_{r\in[0,n]}\left\|x_{m}(r)-x(r)\right\|_{X}\rightarrow 0\text{ as }m\rightarrow\infty,\text{ for all }n\in\mathbb{N}^{*}\end{array}\right.

Equivalence (2.4) will be used several times in Section 5 of the paper.
Consider now a real Hilbert space XX with inner product (โ‹…,โ‹…)X(\cdot,\cdot)_{X} and associated norm โˆฅโ‹…โˆฅX\|\cdot\|_{X}. Also, assume given a set KโŠ‚XK\subset X, the operators A:Kโ†’X,๐’ฎA:K\rightarrow X,\mathcal{S} : C(โ„+;X)โ†’C(โ„+;X)C\left(\mathbb{R}_{+};X\right)\rightarrow C\left(\mathbb{R}_{+};X\right) and a function f:โ„+โ†’Xf:\mathbb{R}_{+}\rightarrow X such that:
KK is a closed, convex, nonempty subset of XX.

{ (a) There exists m>0 such that (Au1โˆ’Au2,u1โˆ’u2)Xโ‰ฅmโ€–u1โˆ’u2โ€–X2โˆ€u1,u2โˆˆK. (b) There exists L>0 such that โ€–Au1โˆ’Au2โ€–Xโ‰คLโ€–u1โˆ’u2โ€–Xโˆ€u1,u2โˆˆK.{ For every nโˆˆโ„•โˆ— there exists rn>0 such that โ€–๐’ฎu1(t)โˆ’๐’ฎu2(t)โ€–Yโ‰คrnโˆซ0tโ€–u1(s)โˆ’u2(s)โ€–X๐‘‘sโˆ€u1,u2โˆˆC(โ„+;X),โˆ€tโˆˆ[0,n]fโˆˆC(โ„+;X)\displaystyle\left\{\begin{array}[]{l}\text{ (a) There exists }m>0\text{ such that }\\ \left(Au_{1}-Au_{2},u_{1}-u_{2}\right)_{X}\geq m\left\|u_{1}-u_{2}\right\|_{X}^{2}\quad\forall u_{1},u_{2}\in K.\\ \text{ (b) There exists }L>0\text{ such that }\\ \left\|Au_{1}-Au_{2}\right\|_{X}\leq L\left\|u_{1}-u_{2}\right\|_{X}\quad\forall u_{1},u_{2}\in K.\\ \left\{\begin{array}[]{l}\text{ For every }n\in\mathbb{N}^{*}\text{ there exists }r_{n}>0\text{ such that }\\ \left\|\mathcal{S}u_{1}(t)-\mathcal{S}u_{2}(t)\right\|_{Y}\leq r_{n}\int_{0}^{t}\left\|u_{1}(s)-u_{2}(s)\right\|_{X}ds\\ \forall u_{1},u_{2}\in C\left(\mathbb{R}_{+};X\right),\forall t\in[0,n]\end{array}\right.\\ f\in C\left(\mathbb{R}_{+};X\right)\end{array}\right. (2.6)

We have following result, which represents a particular case of a more general existence and uniqueness result proved in 26 .

Theorem 2.1 Assume that (2.5)-(2.8) hold. Then there exists a unique function uโˆˆC(โ„+;X)u\in C\left(\mathbb{R}_{+};X\right) such that, for all tโˆˆโ„+t\in\mathbb{R}_{+}, the inequality below holds:

u(t)โˆˆK,\displaystyle u(t)\in K, (Au(t),vโˆ’u(t))X+(๐’ฎu(t),vโˆ’u(t))X\displaystyle(Au(t),v-u(t))_{X}+(\mathcal{S}u(t),v-u(t))_{X} (2.9)
โ‰ฅ(f(t),vโˆ’u(t))Xโˆ€vโˆˆK\displaystyle\geq(f(t),v-u(t))_{X}\quad\forall v\in K

Following the terminology introduced in [26] we refer to (2.9) as a history-dependent variational inequality. To avoid any confusion, we note that here and below the notation Au(t)Au(t) and ๐’ฎu(t)\mathcal{S}u(t) are short hand notation for A(u(t))A(u(t)) and (๐’ฎu)(t)(\mathcal{S}u)(t), i.e. Au(t)=A(u(t))Au(t)=A(u(t)) and ๐’ฎu(t)=(๐’ฎu)(t)\mathcal{S}u(t)=(\mathcal{S}u)(t), for all tโˆˆโ„+t\in\mathbb{R}_{+}.

3 The model

The physical setting is as follows. A viscoelastic body occupies a bounded domain ฮฉโŠ‚โ„d(d=1,2,3)\Omega\subset\mathbb{R}^{d}(d=1,2,3) with a Lipschitz continuous boundary ฮ“\Gamma, divided into three measurable parts ฮ“1,ฮ“2\Gamma_{1},\Gamma_{2} and ฮ“3\Gamma_{3} such that meas (ฮ“1)>0\left(\Gamma_{1}\right)>0. The body is subject to the action of body forces of density ๐’‡0\boldsymbol{f}_{0}. We also assume that the body is fixed on ฮ“1\Gamma_{1} and surfaces tractions of density ๐’‡2\boldsymbol{f}_{2} act on ฮ“2\Gamma_{2}. On ฮ“3\Gamma_{3}, the body is in frictionless contact with a obstacle, the so-called foundation. We assume that the foundation is deformable and, therefore, the penetration is allowed. Nevertheless, when the penetration reaches a given bound gg, the foundation becomes rigid. And, finally, there are memory effects during the contact process. The process is quasistatic, and it is studied in the interval of time โ„+=[0,โˆž)\mathbb{R}_{+}=[0,\infty). With these assumption, the classical formulation of the problem is the following.

Problem ๐’ซ\mathcal{P}. Find a displacement field ๐’–:ฮฉร—โ„+โ†’โ„d\boldsymbol{u}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{R}^{d} and a stress field ๐ˆ\boldsymbol{\sigma} : ฮฉร—โ„+โ†’๐•Šd\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{S}^{d} such that, for all tโˆˆโ„+t\in\mathbb{R}_{+},

๐ˆ(t)=๐’œ๐œบ(๐’–(t))+โˆซ0tโ„ฌ(tโˆ’s)๐œบ(๐’–(s)))ds in ฮฉ,Divโก๐ˆ(t)+๐’‡0(t)=๐ŸŽ in ฮฉ,๐’–(t)=๐ŸŽ on ฮ“1,๐ˆ(t)๐‚=๐’‡2(t) on ฮ“2,uฮฝ(t)โ‰คg,ฯƒฮฝ(t)+p(uฮฝ(t))+โˆซ0tb(tโˆ’s)uฮฝ+(s)๐‘‘sโ‰ค0,(uฮฝ(t)โˆ’g)(ฯƒฮฝ(t)+p(uฮฝ(t)+โˆซ0tb(tโˆ’s)uฮฝ+(s)ds)=0} on ฮ“3,\left.\begin{array}[]{rcc}\left.\boldsymbol{\sigma}(t)=\mathcal{A}\boldsymbol{\varepsilon}(\boldsymbol{u}(t))+\int_{0}^{t}\mathcal{B}(t-s)\boldsymbol{\varepsilon}(\boldsymbol{u}(s))\right)ds&\text{ in }\quad\Omega,\\ \operatorname{Div}\boldsymbol{\sigma}(t)+\boldsymbol{f}_{0}(t)=\mathbf{0}&\text{ in }\quad\Omega,\\ \boldsymbol{u}(t)=\mathbf{0}&\text{ on }\quad\Gamma_{1},\\ \boldsymbol{\sigma}(t)\boldsymbol{\nu}=\boldsymbol{f}_{2}(t)&\text{ on }\quad\Gamma_{2},\\ u_{\nu}(t)\leq g,\sigma_{\nu}(t)+p\left(u_{\nu}(t)\right)+\int_{0}^{t}b(t-s)u_{\nu}^{+}(s)ds\leq 0,\\ \left(u_{\nu}(t)-g\right)\left(\sigma_{\nu}(t)+p\left(u_{\nu}(t)+\int_{0}^{t}b(t-s)u_{\nu}^{+}(s)ds\right)=0\right.\end{array}\right\}\quad\text{ on }\quad\Gamma_{3},

Here and below, in order to simplify the notation, we do not indicate explicitly the dependence of various functions on the spatial variable ๐’™โˆˆฮฉโˆชฮ“\boldsymbol{x}\in\Omega\cup\Gamma. Equation (3.1) represents the viscoelastic constitutive law of the material introduced in Section 1 and equation (3.2) is the equilibrium equation. Conditions (3.3) and (3.4) are the displacement and traction boundary conditions, respectively, and condition (3.6) shows that the tangential stress on the contact surface, denoted ๐ˆฯ„\boldsymbol{\sigma}_{\tau}, vanishes. We use it here since we assume that the contact process is frictionless.

We now describe the contact condition (3.5) in which our main interest is. Here ฯƒฮฝ\sigma_{\nu} denotes the normal stress, uฮฝu_{\nu} is the normal displacement and pp is a Lipschitz continuous increasing function which vanishes for a negative argument. Moreover, bb is the surface memory function and g>0g>0 is a given bound for the normal displacement. This condition can be derived in the following way. First, we assume that the penetration is limited by the bound gg and, therefore, at each time moment tโˆˆโ„+t\in\mathbb{R}_{+}, the normal displacement satisfies the inequality

uฮฝ(t)โ‰คg on ฮ“3u_{\nu}(t)\leq g\quad\text{ on }\Gamma_{3} (3.7)

Next, we assume that the normal stress has an additive decomposition of the form

ฯƒฮฝ(t)=ฯƒฮฝD(t)+ฯƒฮฝR(t)+ฯƒฮฝM(t) on ฮ“3\sigma_{\nu}(t)=\sigma_{\nu}^{D}(t)+\sigma_{\nu}^{R}(t)+\sigma_{\nu}^{M}(t)\quad\text{ on }\Gamma_{3} (3.8)

in which the functions ฯƒฮฝD,ฯƒฮฝR\sigma_{\nu}^{D},\sigma_{\nu}^{R} and ฯƒฮฝM\sigma_{\nu}^{M} describe the deformability, the rigidity and the memory properties of the foundation, at each tโˆˆโ„+t\in\mathbb{R}_{+}. We assume that the function ฯƒฮฝD\sigma_{\nu}^{D} satisfies the normal compliance contact condition (1.2), that is

โˆ’ฯƒฮฝD(t)=p(uฮฝ(t)) on ฮ“3-\sigma_{\nu}^{D}(t)=p\left(u_{\nu}(t)\right)\quad\text{ on }\Gamma_{3} (3.9)

The part ฯƒฮฝR\sigma_{\nu}^{R} of the normal stress satisfies the Signorini condition in the form with a gap function (1.6), i.e.

ฯƒฮฝR(t)โ‰ค0,ฯƒฮฝR(t)(uฮฝ(t)โˆ’g)=0 on ฮ“3\sigma_{\nu}^{R}(t)\leq 0,\quad\sigma_{\nu}^{R}(t)\left(u_{\nu}(t)-g\right)=0\quad\text{ on }\Gamma_{3} (3.10)

And, finally, the function ฯƒฮฝM\sigma_{\nu}^{M} satisfies the memory condition (1.7), that is

โˆ’ฯƒฮฝM(t)=โˆซ0tb(tโˆ’s)uฮฝ+(s)๐‘‘s on ฮ“3-\sigma_{\nu}^{M}(t)=\int_{0}^{t}b(t-s)u_{\nu}^{+}(s)ds\quad\text{ on }\Gamma_{3} (3.11)

We combine equalities (3.8), (3.10) and (3.11) to see that

ฯƒฮฝR(t)=ฯƒฮฝ(t)+p(uฮฝ(t))+โˆซ0tb(tโˆ’s)uฮฝ+(s)๐‘‘s on ฮ“3\sigma_{\nu}^{R}(t)=\sigma_{\nu}(t)+p\left(u_{\nu}(t)\right)+\int_{0}^{t}b(t-s)u_{\nu}^{+}(s)ds\quad\text{ on }\Gamma_{3} (3.12)

Then we substitute equality (3.12) in (3.10) and use inequality (3.7) to obtain the contact condition (3.5).

Not that (3.5) describes a condition with unilateral constraint, since inequality (3.7) holds at each time moment. Assume now that at a given moment tt there is penetration which did not reach the bound gg, i.e. 0<uฮฝ(t)<g0<u_{\nu}(t)<g. Then (3.5) yields

โˆ’ฯƒฮฝ(t)=p(uฮฝ(t))+โˆซ0tb(tโˆ’s)uฮฝ+(s)๐‘‘s-\sigma_{\nu}(t)=p\left(u_{\nu}(t)\right)+\int_{0}^{t}b(t-s)u_{\nu}^{+}(s)ds (3.13)

This equality shows that at the moment tt, the reaction of the foundation depend both on the current value of the penetration (represented by the term p(uฮฝ(t)))\left.p\left(u_{\nu}(t)\right)\right) as well as on the history of the penetration (represented by the integral term in (3.13)). When bb is a positive function the reaction of the foundation is larger than that given by the term p(uฮฝ(t))p\left(u_{\nu}(t)\right) and we conclude that equality (3.13) models the hardening phenomenon of the surface. When bb is a negative function the reaction of the foundation is smaller than that given by the term p(uฮฝ(t))p\left(u_{\nu}(t)\right) and we conclude that equality (3.13) models the softening phenomenon of the surface. Hardening and softening of contact surfaces represent an important phenomenon which appear in various industrial applications applications, see for instance [16] and references therein.

In conclusion, condition (3.5) shows that the contact follows a normal compliance condition with memory term of the form (3.13) but up to the limit gg and then, when this limit is reached, the contact follows a Signorini-type unilateral condition with the gap gg. For this reason we refer to this condition as to a normal compliance contact condition with memory term and unilateral constraint. It can be interpreted physically as follows. The foundation is assumed to be made of a hard material covered by a thin layer of a soft material with thickness gg. The soft material has a viscoelastic behaviour, i.e. is deformable, allows penetration and presents memory effects; the contact with this layer is modelled with normal compliance and memory term. The hard material is perfectly rigid and, therefore, it does not allow penetration; the contact with this material is modelled with the Signorini contact condition. To resume, the foundation has a rigid-viscoelastic behavior; its viscoelastic behavior is given by the layer of the soft material while its rigid behavior is given by the hard material.

In the particular case when b=0b=0 the contact condition (3.5) was introduced in [5], in the study of a dynamic frictionless contact problem with elastic-visco-plastic
materials. Then, it was used in [1] and [27] in the study of various quasistatic contact problems. Also, note that when g>0,p=0g>0,p=0 and b=0b=0 condition (3.5) becomes the Signorini contact condition in a form with a gap function, (1.7). And, finally, if b=0b=0 and gโ†’โˆžg\rightarrow\infty, we recover the normal compliance contact condition with a zero gap function, (1.2).

4 Existence and uniqueness results

To derive the variational formulation of the problem ๐’ซ\mathcal{P} we list the assumptions on the problem data. First, we assume that the elasticity operator ๐’œ\mathcal{A} and the relaxation tensor โ„ฌ\mathcal{B} satisfy the following conditions.

{ (a) ๐’œ:ฮฉร—๐•Šdโ†’๐•Šd (b) There exists L๐’œ>0 such that โ€–๐’œ(๐’™,๐œบ1)โˆ’๐’œ(๐’™,๐œบ2)โ€–โ‰คL๐’œโ€–๐œบ1โˆ’๐œบ2โ€–โˆ€๐œบ1,๐œบ2โˆˆ๐•Šd, a.e. ๐’™โˆˆฮฉ (c) There exists m๐’œ>0 such that (๐’œ(๐’™,๐œบ1)โˆ’๐’œ(๐’™,๐œบ2))โ‹…(๐œบ1โˆ’๐œบ2)โ‰ฅm๐’œโ€–๐œบ1โˆ’๐œบ2โ€–2โˆ€๐œบ1,๐œบ2โˆˆ๐•Šd, a.e. ๐’™โˆˆฮฉ (d) The mapping ๐’™โ†ฆ๐’œ(๐’™,๐œบ) is measurable on ฮฉ for any ๐œบโˆˆ๐•Šd (e) The mapping ๐’™โ†ฆ๐’œ(๐’™,๐ŸŽ) belongs to Qโ„ฌโˆˆC(โ„+;๐โˆž)\left\{\begin{array}[]{l}\text{ (a) }\mathcal{A}:\Omega\times\mathbb{S}^{d}\rightarrow\mathbb{S}^{d}\text{. }\\ \text{ (b) There exists }L_{\mathcal{A}}>0\text{ such that }\\ \left\|\mathcal{A}\left(\boldsymbol{x},\boldsymbol{\varepsilon}_{1}\right)-\mathcal{A}\left(\boldsymbol{x},\boldsymbol{\varepsilon}_{2}\right)\right\|\leq L_{\mathcal{A}}\left\|\boldsymbol{\varepsilon}_{1}-\boldsymbol{\varepsilon}_{2}\right\|\\ \quad\forall\boldsymbol{\varepsilon}_{1},\boldsymbol{\varepsilon}_{2}\in\mathbb{S}^{d},\text{ a.e. }\boldsymbol{x}\in\Omega\text{. }\\ \text{ (c) There exists }m_{\mathcal{A}}>0\text{ such that }\\ \quad\left(\mathcal{A}\left(\boldsymbol{x},\boldsymbol{\varepsilon}_{1}\right)-\mathcal{A}\left(\boldsymbol{x},\boldsymbol{\varepsilon}_{2}\right)\right)\cdot\left(\boldsymbol{\varepsilon}_{1}-\boldsymbol{\varepsilon}_{2}\right)\geq m_{\mathcal{A}}\left\|\boldsymbol{\varepsilon}_{1}-\boldsymbol{\varepsilon}_{2}\right\|^{2}\\ \quad\forall\boldsymbol{\varepsilon}_{1},\boldsymbol{\varepsilon}_{2}\in\mathbb{S}^{d},\text{ a.e. }\boldsymbol{x}\in\Omega\text{. }\\ \text{ (d) The mapping }\boldsymbol{x}\mapsto\mathcal{A}(\boldsymbol{x},\boldsymbol{\varepsilon})\text{ is measurable on }\Omega\text{, }\\ \text{ for any }\boldsymbol{\varepsilon}\in\mathbb{S}^{d}\text{. }\\ \text{ (e) The mapping }\boldsymbol{x}\mapsto\mathcal{A}(\boldsymbol{x},\mathbf{0})\text{ belongs to }Q\text{. }\\ \qquad\mathcal{B}\in C\left(\mathbb{R}_{+};\mathbf{Q}_{\infty}\right)\text{. }\end{array}\right.

The normal compliance and the surface memory function satisfy the conditions

{ (a) p:ฮ“3ร—โ„โ†’โ„+โ‹… (b) There exists Lp>0 such that |p(๐’™,r1)โˆ’p(๐’™,r2)|โ‰คLp|r1โˆ’r2|โˆ€r1,r2โˆˆโ„, a.e. ๐’™โˆˆฮ“3 (c) (p(๐’™,r1)โˆ’p(๐’™,r2))(r1โˆ’r2)โ‰ฅ0โˆ€r1,r2โˆˆโ„, a.e. ๐’™โˆˆฮ“3 (d) The mapping ๐’™โ†ฆp(๐’™,r) is measurable on ฮ“3 for any rโˆˆโ„ (e) p(๐’™,r)=0 for all rโ‰ค0, a.e. ๐’™โˆˆฮ“3bโˆˆC(โ„+;Lโˆž(ฮ“3))\left\{\begin{array}[]{l}\text{ (a) }p:\Gamma_{3}\times\mathbb{R}\rightarrow\mathbb{R}_{+}\cdot\\ \text{ (b) There exists }L_{p}>0\text{ such that }\\ \quad\left|p\left(\boldsymbol{x},r_{1}\right)-p\left(\boldsymbol{x},r_{2}\right)\right|\leq L_{p}\left|r_{1}-r_{2}\right|\\ \quad\forall r_{1},r_{2}\in\mathbb{R},\text{ a.e. }\boldsymbol{x}\in\Gamma_{3}\\ \text{ (c) }\left(p\left(\boldsymbol{x},r_{1}\right)-p\left(\boldsymbol{x},r_{2}\right)\right)\left(r_{1}-r_{2}\right)\geq 0\\ \quad\forall r_{1},r_{2}\in\mathbb{R},\text{ a.e. }\boldsymbol{x}\in\Gamma_{3}\\ \text{ (d) The mapping }\boldsymbol{x}\mapsto p(\boldsymbol{x},r)\text{ is measurable on }\Gamma_{3}\\ \quad\text{ for any }r\in\mathbb{R}\text{. }\\ \text{ (e) }p(\boldsymbol{x},r)=0\text{ for all }r\leq 0,\text{ a.e. }\boldsymbol{x}\in\Gamma_{3}\\ \quad b\in C\left(\mathbb{R}_{+};L^{\infty}\left(\Gamma_{3}\right)\right)\end{array}\right.

Finally, we assume that the densities of body forces and surface tractions have the regularity

๐’‡0โˆˆC(โ„+;L2(ฮฉ)d),๐’‡2โˆˆC(โ„+;L2(ฮ“2)d)\boldsymbol{f}_{0}\in C\left(\mathbb{R}_{+};L^{2}(\Omega)^{d}\right),\quad\boldsymbol{f}_{2}\in C\left(\mathbb{R}_{+};L^{2}\left(\Gamma_{2}\right)^{d}\right) (4.5)

and, moreover, we introduce the set of admissible displacements fields defined by

U={๐’—โˆˆV:vฮฝโ‰คg on ฮ“3}.U=\left\{\boldsymbol{v}\in V:v_{\nu}\leq g\text{ on }\Gamma_{3}\right\}. (4.6)

Assume in what follows that ( ๐’–,๐ˆ\boldsymbol{u},\boldsymbol{\sigma} ) are sufficiently regular functions which satisfy (3.1)-(3.6) and let ๐’—โˆˆU\boldsymbol{v}\in U and t>0t>0 be given. We use the Green formula (2.2) and the equilibrium equation (3.2) to obtain

โˆซฮฉ๐ˆ(t)โ‹…(๐œบ(๐’—)โˆ’๐œบ(๐’–(t)))๐‘‘x=โˆซฮฉ๐’‡0(t)โ‹…(๐’—โˆ’๐’–(t))๐‘‘x+โˆซฮ“๐ˆ(t)๐‚โ‹…(๐’—โˆ’๐’–(t))๐‘‘a\int_{\Omega}\boldsymbol{\sigma}(t)\cdot(\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))dx=\int_{\Omega}\boldsymbol{f}_{0}(t)\cdot(\boldsymbol{v}-\boldsymbol{u}(t))dx+\int_{\Gamma}\boldsymbol{\sigma}(t)\boldsymbol{\nu}\cdot(\boldsymbol{v}-\boldsymbol{u}(t))da

We split the boundary integral over ฮ“1,ฮ“2\Gamma_{1},\Gamma_{2} and ฮ“3\Gamma_{3} and, since ๐’—โˆ’๐’–(t)=๐ŸŽ\boldsymbol{v}-\boldsymbol{u}(t)=\mathbf{0} on ฮ“1\Gamma_{1} and ๐ˆ(t)๐‚=๐’‡2(t)\boldsymbol{\sigma}(t)\boldsymbol{\nu}=\boldsymbol{f}_{2}(t) on ฮ“2\Gamma_{2}, we obtain

โˆซฮฉ๐ˆ(t)โ‹…(๐œบ(๐’—)โˆ’๐œบ(๐’–(t)))๐‘‘x=โˆซฮฉ๐’‡0(t)โ‹…(๐’—โˆ’๐’–(t))๐‘‘x\displaystyle\int_{\Omega}\boldsymbol{\sigma}(t)\cdot(\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))dx=\int_{\Omega}\boldsymbol{f}_{0}(t)\cdot(\boldsymbol{v}-\boldsymbol{u}(t))dx (4.7)
+โˆซฮ“2๐’‡2(t)โ‹…(๐’—โˆ’๐’–(t))๐‘‘a+โˆซฮ“3๐ˆ(t)๐‚โ‹…(๐’—โˆ’๐’–(t))๐‘‘a\displaystyle\quad+\int_{\Gamma_{2}}\boldsymbol{f}_{2}(t)\cdot(\boldsymbol{v}-\boldsymbol{u}(t))da+\int_{\Gamma_{3}}\boldsymbol{\sigma}(t)\boldsymbol{\nu}\cdot(\boldsymbol{v}-\boldsymbol{u}(t))da

Moreover, since

๐ˆ(t)๐‚โ‹…(๐’—โˆ’๐’–(t))=ฯƒฮฝ(t)(vฮฝโˆ’uฮฝ(t))+๐ˆฯ„(t)๐‚โ‹…(๐’—ฯ„โˆ’๐’–ฯ„(t)) on ฮ“3,\boldsymbol{\sigma}(t)\boldsymbol{\nu}\cdot(\boldsymbol{v}-\boldsymbol{u}(t))=\sigma_{\nu}(t)\left(v_{\nu}-u_{\nu}(t)\right)+\boldsymbol{\sigma}_{\tau}(t)\boldsymbol{\nu}\cdot\left(\boldsymbol{v}_{\tau}-\boldsymbol{u}_{\tau}(t)\right)\quad\text{ on }\Gamma_{3},

condition (3.6) implies that

โˆซฮ“3๐ˆ(t)๐‚โ‹…(๐’—โˆ’๐’–(t))๐‘‘a=โˆซฮ“3ฯƒฮฝ(t)(vฮฝโˆ’uฮฝ)๐‘‘a\int_{\Gamma_{3}}\boldsymbol{\sigma}(t)\boldsymbol{\nu}\cdot(\boldsymbol{v}-\boldsymbol{u}(t))da=\int_{\Gamma_{3}}\sigma_{\nu}(t)\left(v_{\nu}-u_{\nu}\right)da (4.8)

We write now

ฯƒฮฝ(t)(vฮฝโˆ’uฮฝ(t))=(ฯƒฮฝ(t)+p(uฮฝ(t))+โˆซ0tb(tโˆ’s)uฮฝ+(s)๐‘‘s)(vฮฝโˆ’g)\displaystyle\sigma_{\nu}(t)\left(v_{\nu}-u_{\nu}(t)\right)=\left(\sigma_{\nu}(t)+p\left(u_{\nu}(t)\right)+\int_{0}^{t}b(t-s)u_{\nu}^{+}(s)ds\right)\left(v_{\nu}-g\right)
+(ฯƒฮฝ(t)+p(uฮฝ(t))+โˆซ0tb(tโˆ’s)uฮฝ+(s)๐‘‘s)(gโˆ’uฮฝ(t))\displaystyle\quad+\left(\sigma_{\nu}(t)+p\left(u_{\nu}(t)\right)+\int_{0}^{t}b(t-s)u_{\nu}^{+}(s)ds\right)\left(g-u_{\nu}(t)\right)
โˆ’(p(uฮฝ(t))+โˆซ0tb(tโˆ’s)uฮฝ+(s)๐‘‘s)(vฮฝโˆ’uฮฝ(t)) on ฮ“3\displaystyle\quad-\left(p\left(u_{\nu}(t)\right)+\int_{0}^{t}b(t-s)u_{\nu}^{+}(s)ds\right)\left(v_{\nu}-u_{\nu}(t)\right)\quad\text{ on }\Gamma_{3}

then we use the contact conditions (3.5) and the definition (4.6) of the set UU to see that

ฯƒฮฝ(t)(vฮฝโˆ’uฮฝ(t))โ‰ฅโˆ’(p(uฮฝ(t))+โˆซ0tb(tโˆ’s)uฮฝ+(s)๐‘‘s)(vฮฝโˆ’uฮฝ(t)) on ฮ“3\sigma_{\nu}(t)\left(v_{\nu}-u_{\nu}(t)\right)\geq-\left(p\left(u_{\nu}(t)\right)+\int_{0}^{t}b(t-s)u_{\nu}^{+}(s)ds\right)\left(v_{\nu}-u_{\nu}(t)\right)\quad\text{ on }\Gamma_{3}

and, therefore,

โˆซฮ“3ฯƒฮฝ(t)(vฮฝโˆ’uฮฝ(t))๐‘‘aโ‰ฅโˆ’โˆซฮ“3(p(uฮฝ(t))+โˆซ0tb(tโˆ’s)uฮฝ+(s)๐‘‘s)(vฮฝโˆ’uฮฝ(t))๐‘‘a\int_{\Gamma_{3}}\sigma_{\nu}(t)\left(v_{\nu}-u_{\nu}(t)\right)da\geq-\int_{\Gamma_{3}}\left(p\left(u_{\nu}(t)\right)+\int_{0}^{t}b(t-s)u_{\nu}^{+}(s)ds\right)\left(v_{\nu}-u_{\nu}(t)\right)da (4.9)

We combine now equalities (4.7), (4.8) then we use inequality (4.9) to deduce that

(๐ˆ(t),๐œบ(๐’—)โˆ’๐œบ(๐’–(t)))Q+(p(uฮฝ(t)),vฮฝโˆ’uฮฝ(t))L2(ฮ“3)\displaystyle(\boldsymbol{\sigma}(t),\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))_{Q}+\left(p\left(u_{\nu}(t)\right),v_{\nu}-u_{\nu}(t)\right)_{L^{2}\left(\Gamma_{3}\right)} (4.10)
+(โˆซ0tb(tโˆ’s)uฮฝ+(s)๐‘‘s,vฮฝโˆ’uฮฝ(t))L2(ฮ“3)\displaystyle\quad+\left(\int_{0}^{t}b(t-s)u_{\nu}^{+}(s)ds,v_{\nu}-u_{\nu}(t)\right)_{L^{2}\left(\Gamma_{3}\right)}
โ‰ฅ(๐’‡0(t),๐’—โˆ’๐’–(t))L2(ฮฉ)d+(๐’‡2(t),๐’—โˆ’๐’–(t))L2(ฮ“2)d\displaystyle\quad\geq\left(\boldsymbol{f}_{0}(t),\boldsymbol{v}-\boldsymbol{u}(t)\right)_{L^{2}(\Omega)^{d}}+\left(\boldsymbol{f}_{2}(t),\boldsymbol{v}-\boldsymbol{u}(t)\right)_{L^{2}\left(\Gamma_{2}\right)^{d}}

In addition, we note that the boundary condition (3.3), the first inequality in (3.5) and notation (4.6) imply that ๐’–(t)โˆˆU\boldsymbol{u}(t)\in U. Therefore, using the constitutive law (3.1) and inequality (4.10) we derive the following variational formulation of Problem ๐’ซ\mathcal{P}.

Problem ๐’ซV\mathcal{P}^{V}. Find a displacement field ๐’–:โ„+โ†’V\boldsymbol{u}:\mathbb{R}_{+}\rightarrow V such that, for all tโˆˆโ„+t\in\mathbb{R}_{+}, the inequality below holds:

๐’–(t)\displaystyle\boldsymbol{u}(t) โˆˆU,(๐’œ๐œบ(๐’–(t)),๐œบ(๐’—)โˆ’๐œบ(๐’–(t)))Q\displaystyle\in U,\quad(\mathcal{A}\boldsymbol{\varepsilon}(\boldsymbol{u}(t)),\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))_{Q} (4.11)
+\displaystyle+ (โˆซ0tโ„ฌ(tโˆ’s)๐œบ(๐’–(s)ds,๐œบ(๐’—)โˆ’๐œบ(๐’–(t)))Q\displaystyle\left(\int_{0}^{t}\mathcal{B}(t-s)\boldsymbol{\varepsilon}(\boldsymbol{u}(s)ds,\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))_{Q}\right.
+(p(uฮฝ(t)),vฮฝโˆ’uฮฝ(t))L2(ฮ“3)+(โˆซ0tb(tโˆ’s)uฮฝ+(s)ds,vฮฝโˆ’uฮฝ(t)))L2(ฮ“3)\displaystyle\left.+\left(p\left(u_{\nu}(t)\right),v_{\nu}-u_{\nu}(t)\right)_{L^{2}\left(\Gamma_{3}\right)}+\left(\int_{0}^{t}b(t-s)u_{\nu}^{+}(s)ds,v_{\nu}-u_{\nu}(t)\right)\right)_{L^{2}\left(\Gamma_{3}\right)}
โ‰ฅ(๐’‡0(t),๐’—โˆ’๐’–(t))L2(ฮฉ)d+(๐’‡2(t),๐’—โˆ’๐’–(t))L2(ฮ“2)dโˆ€๐’—โˆˆU\displaystyle\geq\left(\boldsymbol{f}_{0}(t),\boldsymbol{v}-\boldsymbol{u}(t)\right)_{L^{2}(\Omega)^{d}}+\left(\boldsymbol{f}_{2}(t),\boldsymbol{v}-\boldsymbol{u}(t)\right)_{L^{2}\left(\Gamma_{2}\right)^{d}}\quad\forall\boldsymbol{v}\in U

In the study of the problems ๐’ซV\mathcal{P}^{V} we have the following existence and uniqueness result.

Theorem 4.1 Assume that (4.1) -(4.5) hold. Then, Problem ๐’ซV\mathcal{P}^{V} has a unique solution which satisfies ๐’–โˆˆC(โ„+;V)\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right).

Proof. We start with by providing an equivalent form of Problem ๐’ซV\mathcal{P}^{V}. To this end we use the Riesz representation theorem to define the operators P:Vโ†’VP:V\rightarrow V, ๐’ฎ:C(โ„+;V)โ†’C(โ„+;V)\mathcal{S}:C\left(\mathbb{R}_{+};V\right)\rightarrow C\left(\mathbb{R}_{+};V\right) and the function ๐’‡:โ„+โ†’V\boldsymbol{f}:\mathbb{R}_{+}\rightarrow V by equalities

(P๐’–,๐’—)V=โˆซฮ“3p(uฮฝ)vฮฝ๐‘‘aโˆ€๐’–,๐’—โˆˆV\displaystyle(P\boldsymbol{u},\boldsymbol{v})_{V}=\int_{\Gamma_{3}}p\left(u_{\nu}\right)v_{\nu}da\quad\forall\boldsymbol{u},\boldsymbol{v}\in V (4.12)
(๐’ฎ๐’–(t),๐’—)V=(โˆซ0tโ„ฌ(tโˆ’s)๐œบ(๐’–(s))๐‘‘s,๐œบ(๐’—))Q\displaystyle(\mathcal{S}\boldsymbol{u}(t),\boldsymbol{v})_{V}=\left(\int_{0}^{t}\mathcal{B}(t-s)\boldsymbol{\varepsilon}(\boldsymbol{u}(s))ds,\boldsymbol{\varepsilon}(\boldsymbol{v})\right)_{Q} (4.13)
+(โˆซ0tb(tโˆ’s)uฮฝ+(s)๐‘‘s,vฮฝ)L2(ฮ“3)โˆ€๐’–โˆˆC(โ„+;V),๐’—โˆˆV\displaystyle\quad+\left(\int_{0}^{t}b(t-s)u_{\nu}^{+}(s)ds,v_{\nu}\right)_{L^{2}\left(\Gamma_{3}\right)}\quad\forall\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right),\boldsymbol{v}\in V
(๐’‡(t),๐’—)V=โˆซฮฉ๐’‡0(t)โ‹…๐’—๐‘‘x+โˆซฮ“2๐’‡2(t)โ‹…๐’—๐‘‘aโˆ€๐’–,๐’—โˆˆV,tโˆˆโ„+\displaystyle(\boldsymbol{f}(t),\boldsymbol{v})_{V}=\int_{\Omega}\boldsymbol{f}_{0}(t)\cdot\boldsymbol{v}dx+\int_{\Gamma_{2}}\boldsymbol{f}_{2}(t)\cdot\boldsymbol{v}da\quad\forall\boldsymbol{u},\boldsymbol{v}\in V,t\in\mathbb{R}_{+} (4.14)

Then, it is easy to see that Problem ๐’ซV\mathcal{P}^{V} is equivalent to the problem of finding a function ๐’–:โ„+โ†’V\boldsymbol{u}:\mathbb{R}_{+}\rightarrow V such that the inequality below holds, for all tโˆˆโ„+t\in\mathbb{R}_{+}:

๐’–(t)\displaystyle\boldsymbol{u}(t) โˆˆU,(๐’œ๐œบ(๐’–(t)),๐œบ(๐’—)โˆ’๐œบ(๐’–(t)))Q+(P๐’–(t),๐’—โˆ’๐’–(t))V\displaystyle\in U,\quad(\mathcal{A}\boldsymbol{\varepsilon}(\boldsymbol{u}(t)),\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))_{Q}+(P\boldsymbol{u}(t),\boldsymbol{v}-\boldsymbol{u}(t))_{V} (4.15)
+(๐’ฎ๐’–(t),๐’—โˆ’๐’–(t))Vโ‰ฅ(๐’‡(t),๐’—โˆ’๐’–(t))Vโˆ€๐’—โˆˆU\displaystyle+(\mathcal{S}\boldsymbol{u}(t),\boldsymbol{v}-\boldsymbol{u}(t))_{V}\geq(\boldsymbol{f}(t),\boldsymbol{v}-\boldsymbol{u}(t))_{V}\quad\forall\boldsymbol{v}\in U

To solve the variational inequality (4.15) we use Theorem 2.1 with X=VX=V and K=UK=U. To this end we consider the operator A:Vโ†’VA:V\rightarrow V defined by

(A๐’–,๐’—)V=(๐’œฮต(๐’–),ฮต(๐’—))Q+(P๐’–,๐’—)Vโˆ€๐’–,๐’—โˆˆV(A\boldsymbol{u},\boldsymbol{v})_{V}=(\mathcal{A}\varepsilon(\boldsymbol{u}),\varepsilon(\boldsymbol{v}))_{Q}+(P\boldsymbol{u},\boldsymbol{v})_{V}\quad\forall\boldsymbol{u},\boldsymbol{v}\in V (4.16)

It is easy to see that condition (2.5) holds. Next, we use (4.1), (4.3) and (2.1) to see that the operator AA verifies condition (2.6). Let nโˆˆโ„•โˆ—n\in\mathbb{N}^{*}. Then, a simple calculation based on assumptions (4.2), (4.4) and inequalities (2.1), (2.3) shows that

{โ€–๐’ฎ๐’–1(t)โˆ’๐’ฎ๐’–2(t)โ€–Vโ‰คdmaxrโˆˆ[0,n]โกโ€–โ„ฌ(r)โ€–๐โˆžโˆซ0tโ€–๐’–1(s)โˆ’๐’–2(s)โ€–V๐‘‘s+c02maxrโˆˆ[0,n]โกโ€–b(r)โ€–Lโˆž(ฮ“3)โˆซ0tโ€–๐’–1(s)โˆ’๐’–2(s)โ€–V๐‘‘sโˆ€๐’–1,๐’–2โˆˆC(โ„+;V),โˆ€tโˆˆ[0,n]\left\{\begin{array}[]{c}\left\|\mathcal{S}\boldsymbol{u}_{1}(t)-\mathcal{S}\boldsymbol{u}_{2}(t)\right\|_{V}\leq d\max_{r\in[0,n]}\|\mathcal{B}(r)\|_{\mathbf{Q}_{\infty}}\int_{0}^{t}\left\|\boldsymbol{u}_{1}(s)-\boldsymbol{u}_{2}(s)\right\|_{V}ds\\ +c_{0}^{2}\max_{r\in[0,n]}\|b(r)\|_{L^{\infty}\left(\Gamma_{3}\right)}\int_{0}^{t}\left\|\boldsymbol{u}_{1}(s)-\boldsymbol{u}_{2}(s)\right\|_{V}ds\\ \forall\boldsymbol{u}_{1},\boldsymbol{u}_{2}\in C\left(\mathbb{R}_{+};V\right),\forall t\in[0,n]\end{array}\right.

This inequality implies that the operator (4.13) satisfies condition (2.7) with

rn=dmaxrโˆˆ[0,n]โกโ€–โ„ฌ(r)โ€–๐โˆž+c02maxrโˆˆ[0,n]โกโ€–b(r)โ€–Lโˆž(ฮ“3)r_{n}=d\max_{r\in[0,n]}\|\mathcal{B}(r)\|_{\mathbf{Q}_{\infty}}+c_{0}^{2}\max_{r\in[0,n]}\|b(r)\|_{L^{\infty}\left(\Gamma_{3}\right)} (4.18)

Finally, using (4.5) and (4.14) we deduce that ๐’‡โˆˆC(โ„+;V)\boldsymbol{f}\in C\left(\mathbb{R}_{+};V\right) and, therefore, (2.8) holds. It follows now from Theorem 2.1 that there exists a unique function ๐’–โˆˆC(โ„+;V)\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right) which satisfies the inequality

๐’–(t)โˆˆU,(A๐’–(t),๐’—โˆ’๐’–(t))V+(๐’ฎ๐’–(t),๐’—โˆ’๐’–(t))V\displaystyle\boldsymbol{u}(t)\in U,\quad(A\boldsymbol{u}(t),\boldsymbol{v}-\boldsymbol{u}(t))_{V}+(\mathcal{S}\boldsymbol{u}(t),\boldsymbol{v}-\boldsymbol{u}(t))_{V} (4.19)
โ‰ฅ(๐’‡(t),๐’—โˆ’๐’–(t))Vโˆ€๐’—โˆˆU\displaystyle\geq(\boldsymbol{f}(t),\boldsymbol{v}-\boldsymbol{u}(t))_{V}\quad\forall\boldsymbol{v}\in U

for all tโˆˆโ„+t\in\mathbb{R}_{+}. And, using (4.16) we deduce that that there exists a unique function ๐’–โˆˆC(โ„+;V)\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right) such that (4.15) holds for all tโˆˆโ„+t\in\mathbb{R}_{+}, which concludes the proof.

Let ฯƒ\sigma be the function defined by (3.1). Then, it follows (4.1) and (4.2) that ๐ˆโˆˆC(โ„+;Q)\boldsymbol{\sigma}\in C\left(\mathbb{R}_{+};Q\right). Moreover, it is easy to see that (4.10) holds for all tโˆˆโ„+t\in\mathbb{R}_{+}and, using standard arguments, it result from here that

Divโก๐ˆ(t)+๐’‡0(t)=๐ŸŽโˆ€tโˆˆโ„+\operatorname{Div}\boldsymbol{\sigma}(t)+\boldsymbol{f}_{0}(t)=\mathbf{0}\quad\forall t\in\mathbb{R}_{+} (4.20)

Therefore, using the regularity ๐’‡0โˆˆC(โ„+;L2(ฮฉ)d)\boldsymbol{f}_{0}\in C\left(\mathbb{R}_{+};L^{2}(\Omega)^{d}\right) in (4.5) we deduce that Divโก๐ˆโˆˆC(โ„+;L2(ฮฉ)d)\operatorname{Div}\boldsymbol{\sigma}\in C\left(\mathbb{R}_{+};L^{2}(\Omega)^{d}\right) which implies that ๐ˆโˆˆC(โ„+;Q1)\boldsymbol{\sigma}\in C\left(\mathbb{R}_{+};Q_{1}\right). A couple of functions ( ๐’–,๐ˆ\boldsymbol{u},\boldsymbol{\sigma} ) which satisfies (3.1), (4.11) for all tโˆˆโ„+t\in\mathbb{R}_{+}is called a weak solution to the contact problem ๐’ซ\mathcal{P}. We conclude that Theorem 4.1 provides the unique weak solvability of Problem ๐’ซ\mathcal{P}. Moreover, the regularity of the weak solution is ๐’–โˆˆC(โ„+;V),๐ˆโˆˆC(โ„+;Q1)\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right),\boldsymbol{\sigma}\in C\left(\mathbb{R}_{+};Q_{1}\right).

5 A first convergence result

We study now the dependence of the solution of Problem ๐’ซV\mathcal{P}^{V} with respect to perturbations of the data. To this end, we assume in what follows that (4.1)-(4.5) hold and we denote by ๐’–\boldsymbol{u} the solution of Problem ๐’ซV\mathcal{P}^{V} obtained in Theorem 4.1. For each ฯ>0\rho>0 let โ„ฌฯ,pฯ,bฯ,๐’‡0ฯ\mathcal{B}_{\rho},p_{\rho},b_{\rho},\boldsymbol{f}_{0\rho} and ๐’‡2ฯ\boldsymbol{f}_{2\rho} be perturbations of โ„ฌ,p,b,๐’‡0\mathcal{B},p,b,\boldsymbol{f}_{0} and ๐’‡2\boldsymbol{f}_{2} which satisfy conditions (4.2), (4.3), (4.4) and (4.5), respectively. We consider the following variational problem.

Problem ๐’ซฯV\mathcal{P}_{\rho}^{V}. Find a displacement field ๐’–ฯ:โ„+โ†’V\boldsymbol{u}_{\rho}:\mathbb{R}_{+}\rightarrow V such that, for all tโˆˆโ„+t\in\mathbb{R}_{+}, the inequality below holds :

๐’–ฯ(t)\displaystyle\boldsymbol{u}_{\rho}(t) โˆˆU,(๐’œ๐œบ(๐’–ฯ(t)),๐œบ(๐’—)โˆ’๐œบ(๐’–ฯ(t)))Q\displaystyle\in U,\quad\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\rho}(t)\right),\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\rho}(t)\right)\right)_{Q} (5.1)
+\displaystyle+ (โˆซ0tโ„ฌฯ(tโˆ’s)๐œบ(๐’–ฯ(s))๐‘‘s,๐œบ(๐’—)โˆ’๐œบ(๐’–ฯ(t)))Q\displaystyle\left(\int_{0}^{t}\mathcal{B}_{\rho}(t-s)\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\rho}(s)\right)ds,\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\rho}(t)\right)\right)_{Q}
+\displaystyle+ (pฯ(uฯฮฝ(t)),vฮฝโˆ’uฯฮฝ(t))L2(ฮ“3)+(โˆซ0tbฯ(tโˆ’s)uฯฮฝ+(s)๐‘‘s,vฮฝโˆ’uฯฮฝ(t))L2(ฮ“3)\displaystyle\left(p_{\rho}\left(u_{\rho\nu}(t)\right),v_{\nu}-u_{\rho\nu}(t)\right)_{L^{2}\left(\Gamma_{3}\right)}+\left(\int_{0}^{t}b_{\rho}(t-s)u_{\rho\nu}^{+}(s)ds,v_{\nu}-u_{\rho\nu}(t)\right)_{L^{2}\left(\Gamma_{3}\right)}
โ‰ฅ(๐’‡0ฯ(t),๐’—โˆ’๐’–ฯ(t))L2(ฮฉ)d+(๐’‡2ฯ(t),๐’—โˆ’๐’–ฯ(t))L2(ฮ“2)dโˆ€๐’—โˆˆU\displaystyle\geq\left(\boldsymbol{f}_{0\rho}(t),\boldsymbol{v}-\boldsymbol{u}_{\rho}(t)\right)_{L^{2}(\Omega)^{d}}+\left(\boldsymbol{f}_{2\rho}(t),\boldsymbol{v}-\boldsymbol{u}_{\rho}(t)\right)_{L^{2}\left(\Gamma_{2}\right)^{d}}\quad\forall\boldsymbol{v}\in U

Note that, here and below, uฯฮฝu_{\rho\nu} represents the normal component of the function ๐’–ฯ\boldsymbol{u}_{\rho}.
It follows from Theorem 4.1 that, for each ฯ>0\rho>0 Problem ๐’ซฯV\mathcal{P}_{\rho}^{V} has a unique solution ๐’–ฯโˆˆC(โ„+;V)\boldsymbol{u}_{\rho}\in C\left(\mathbb{R}_{+};V\right). Consider now the following assumptions:

โ„ฌฯโ†’โ„ฌbฯโ†’b๐’‡0ฯโ†’๐’‡0 in C(โ„+;Qโˆž) in C(โ„+;Lโˆž(ฮ“3)) as ฯโ†’0.๐’‡2ฯโ†’๐’‡2 in C(โ„+;L2(ฮ“2)d) as ฯโ†’0.\displaystyle\quad\begin{array}[]{l}\mathcal{B}_{\rho}\rightarrow\mathcal{B}\\ b_{\rho}\rightarrow b\\ \quad\boldsymbol{f}_{0\rho}\rightarrow\boldsymbol{f}_{0}\quad\text{ in }C\left(\mathbb{R}_{+};\mathrm{Q}_{\infty}\right)\quad\text{ in }C\left(\mathbb{R}_{+};L^{\infty}\left(\Gamma_{3}\right)\right)\quad\text{ as }\quad\rho\rightarrow 0.\\ \quad\boldsymbol{f}_{2\rho}\rightarrow\boldsymbol{f}_{2}\quad\text{ in }C\left(\mathbb{R}_{+};L^{2}\left(\Gamma_{2}\right)^{d}\right)\quad\text{ as }\quad\rho\rightarrow 0.\end{array} (5.2)
{ There exists G:โ„+โ†’โ„+and ฮฒโˆˆโ„+such that  (a) |pฯ(๐’™,r)โˆ’p(๐’™,r)|โ‰คG(ฯ)(|r|+ฮฒ)โˆ€rโˆˆโ„, a.e. ๐’™โˆˆฮ“3, for each ฯ>0. (b) G(ฯ)โ†’0 as ฯโ†’0.\displaystyle\left\{\begin{array}[]{l}\text{ There exists }G:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}\text{and }\beta\in\mathbb{R}_{+}\text{such that }\\ \text{ (a) }\left|p_{\rho}(\boldsymbol{x},r)-p(\boldsymbol{x},r)\right|\leq G(\rho)(|r|+\beta)\\ \forall r\in\mathbb{R},\text{ a.e. }\boldsymbol{x}\in\Gamma_{3},\text{ for each }\rho>0.\\ \text{ (b) }G(\rho)\rightarrow 0\quad\text{ as }\quad\rho\rightarrow 0.\end{array}\right. (5.3)

We have the following convergence result.
Theorem 5.1 Under assumptions (5.2)-(5.6), the solution ๐’–ฯ\boldsymbol{u}_{\rho} of Problem ๐’ซฯV\mathcal{P}_{\rho}^{V} converges to the solution ๐’–\boldsymbol{u} of Problem ๐’ซV\mathcal{P}^{V}, i.e.,

๐’–ฯโ†’๐’– in C(โ„+;V) as ฯโ†’0.\boldsymbol{u}_{\rho}\rightarrow\boldsymbol{u}\quad\text{ in }\quad C\left(\mathbb{R}_{+};V\right)\quad\text{ as }\quad\rho\rightarrow 0. (5.7)

Proof. Let ฯ>0\rho>0. We use the Riesz representation theorem to define the operators Pฯ:Vโ†’V,๐’ฎฯ:C(โ„+;V)โ†’C(โ„+;V)P_{\rho}:V\rightarrow V,\mathcal{S}_{\rho}:C\left(\mathbb{R}_{+};V\right)\rightarrow C\left(\mathbb{R}_{+};V\right) and the function ๐’‡ฯ:โ„+โ†’V\boldsymbol{f}_{\rho}:\mathbb{R}_{+}\rightarrow V by equalities

(Pฯ๐’–,๐’—)V=โˆซฮ“3pฯ(uฮฝ)vฮฝ๐‘‘aโˆ€๐’–,๐’—โˆˆV\displaystyle\left(P_{\rho}\boldsymbol{u},\boldsymbol{v}\right)_{V}=\int_{\Gamma_{3}}p_{\rho}\left(u_{\nu}\right)v_{\nu}da\quad\forall\boldsymbol{u},\boldsymbol{v}\in V (5.8)
(๐’ฎฯ๐’–(t),๐’—)V=(โˆซ0tโ„ฌฯ(tโˆ’s)๐œบ(๐’–(s))๐‘‘s,๐œบ(๐’—))Q\displaystyle\left(\mathcal{S}_{\rho}\boldsymbol{u}(t),\boldsymbol{v}\right)_{V}=\left(\int_{0}^{t}\mathcal{B}_{\rho}(t-s)\boldsymbol{\varepsilon}(\boldsymbol{u}(s))ds,\boldsymbol{\varepsilon}(\boldsymbol{v})\right)_{Q} (5.9)
+(โˆซ0tbฯ(tโˆ’s)uฮฝ+(s)๐‘‘s,vฮฝ)L2(ฮ“3)โˆ€๐’–โˆˆC(โ„+;V),๐’—โˆˆV\displaystyle\quad+\left(\int_{0}^{t}b_{\rho}(t-s)u_{\nu}^{+}(s)ds,v_{\nu}\right)_{L^{2}\left(\Gamma_{3}\right)}\quad\forall\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right),\boldsymbol{v}\in V
(๐’‡ฯ(t),๐’—)V=โˆซฮฉ๐’‡0ฯ(t)โ‹…๐’—๐‘‘x+โˆซฮ“2๐’‡2ฯ(t)โ‹…๐’—๐‘‘aโˆ€๐’–,๐’—โˆˆV,tโˆˆโ„+\displaystyle\left(\boldsymbol{f}_{\rho}(t),\boldsymbol{v}\right)_{V}=\int_{\Omega}\boldsymbol{f}_{0\rho}(t)\cdot\boldsymbol{v}dx+\int_{\Gamma_{2}}\boldsymbol{f}_{2\rho}(t)\cdot\boldsymbol{v}da\quad\forall\boldsymbol{u},\boldsymbol{v}\in V,t\in\mathbb{R}_{+} (5.10)

It follows from the proof of Theorem 4.1 that ๐’–\boldsymbol{u} is a solution of Problem ๐’ซV\mathcal{P}^{V} iff ๐’–\boldsymbol{u} solves inequality (4.15), for all tโˆˆโ„+t\in\mathbb{R}_{+}. In a similar way, ๐’–ฯ\boldsymbol{u}_{\rho} is a solution of Problem ๐’ซฯV\mathcal{P}_{\rho}^{V} iff, for all tโˆˆโ„+t\in\mathbb{R}_{+}, the inequality below holds:

๐’–ฯ(t)\displaystyle\boldsymbol{u}_{\rho}(t) โˆˆU,(๐’œ๐œบ(๐’–ฯ(t)),๐œบ(๐’—)โˆ’๐œบ(๐’–ฯ(t)))Q+(Pฯ๐’–ฯ(t),๐’—โˆ’๐’–ฯ(t))V\displaystyle\in U,\quad\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\rho}(t)\right),\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\rho}(t)\right)\right)_{Q}+\left(P_{\rho}\boldsymbol{u}_{\rho}(t),\boldsymbol{v}-\boldsymbol{u}_{\rho}(t)\right)_{V} (5.11)
+(๐’ฎฯ๐’–ฯ(t),๐’—โˆ’๐’–ฯ(t)Vโ‰ฅ(๐’‡ฯ(t),๐’—โˆ’๐’–ฯ(t))Vโˆ€๐’—โˆˆU\displaystyle+\left(\mathcal{S}_{\rho}\boldsymbol{u}_{\rho}(t),\boldsymbol{v}-\boldsymbol{u}_{\rho}(t)_{V}\geq\left(\boldsymbol{f}_{\rho}(t),\boldsymbol{v}-\boldsymbol{u}_{\rho}(t)\right)_{V}\quad\forall\boldsymbol{v}\in U\right.

Let nโˆˆโ„•โˆ—n\in\mathbb{N}^{*} and let tโˆˆ[0,n]t\in[0,n]. We take ๐’—=๐’–(t)\boldsymbol{v}=\boldsymbol{u}(t) in (5.11) and ๐’—=๐’–ฯ(t)\boldsymbol{v}=\boldsymbol{u}_{\rho}(t) in (4.15) and add the resulting inequalities to obtain

(๐’œฮต\displaystyle(\mathcal{A}\varepsilon (๐’–ฯ(t))โˆ’๐’œฮต(๐’–(t)),๐œบ(๐’–ฯ(t))โˆ’๐œบ(๐’–(t)))Q\displaystyle\left.\left(\boldsymbol{u}_{\rho}(t)\right)-\mathcal{A}\varepsilon(\boldsymbol{u}(t)),\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\rho}(t)\right)-\boldsymbol{\varepsilon}(\boldsymbol{u}(t))\right)_{Q} (5.12)
โ‰ค\displaystyle\leq (Pฯ๐’–ฯ(t)โˆ’P๐’–(t),๐’–(t)โˆ’๐’–ฯ(t))V\displaystyle\left(P_{\rho}\boldsymbol{u}_{\rho}(t)-P\boldsymbol{u}(t),\boldsymbol{u}(t)-\boldsymbol{u}_{\rho}(t)\right)_{V}
+(๐’ฎฯ๐’–ฯ(t)โˆ’๐’ฎ๐’–(t),๐’–(t)โˆ’๐’–ฯ(t))V+(๐’‡ฯ(t)โˆ’๐’‡(t),๐’–ฯ(t)โˆ’๐’–(t))V\displaystyle+\left(\mathcal{S}_{\rho}\boldsymbol{u}_{\rho}(t)-\mathcal{S}\boldsymbol{u}(t),\boldsymbol{u}(t)-\boldsymbol{u}_{\rho}(t)\right)_{V}+\left(\boldsymbol{f}_{\rho}(t)-\boldsymbol{f}(t),\boldsymbol{u}_{\rho}(t)-\boldsymbol{u}(t)\right)_{V}

Next, we use the definitions (5.8) and (4.12), the monotonicity of the function pฯp_{\rho} and assumption (5.6) to see that

(Pฯ๐’–ฯ(t)โˆ’P๐’–(t),๐’–(t)โˆ’๐’–ฯ(t))V=โˆซฮ“3(pฯ(uฯฮฝ)โˆ’p(uฮฝ))(uฮฝ(t)โˆ’uฯฮฝ(t))๐‘‘a\displaystyle\left(P_{\rho}\boldsymbol{u}_{\rho}(t)-P\boldsymbol{u}(t),\boldsymbol{u}(t)-\boldsymbol{u}_{\rho}(t)\right)_{V}=\int_{\Gamma_{3}}\left(p_{\rho}\left(u_{\rho\nu}\right)-p\left(u_{\nu}\right)\right)\left(u_{\nu}(t)-u_{\rho\nu}(t)\right)da
โ‰คโˆซฮ“3(pฯ(uฮฝ(t))โˆ’p(uฮฝ(t)))(uฮฝ(t)โˆ’uฯฮฝ(t))๐‘‘a\displaystyle\quad\leq\int_{\Gamma_{3}}\left(p_{\rho}\left(u_{\nu}(t)\right)-p\left(u_{\nu}(t)\right)\right)\left(u_{\nu}(t)-u_{\rho\nu}(t)\right)da
โ‰คโˆซฮ“3|pฯ(uฮฝ(t))โˆ’p(uฮฝ(t))||uฮฝ(t)โˆ’uฯฮฝ(t)|๐‘‘a\displaystyle\quad\leq\int_{\Gamma_{3}}\left|p_{\rho}\left(u_{\nu}(t)\right)-p\left(u_{\nu}(t)\right)\right|\left|u_{\nu}(t)-u_{\rho\nu}(t)\right|da
โ‰คโˆซฮ“3G(ฯ)(|uฮฝ(t)|+ฮฒ)|uฮฝ(t)โˆ’uฯฮฝ(t)|๐‘‘a\displaystyle\quad\leq\int_{\Gamma_{3}}G(\rho)\left(\left|u_{\nu}(t)\right|+\beta\right)\left|u_{\nu}(t)-u_{\rho\nu}(t)\right|da

Therefore, using the trace inequality (2.1), after some elementary calculus we find
that

(Pฯ๐’–ฯ(t)โˆ’P๐’–(t),๐’–(t)โˆ’๐’–ฯ(t))V\displaystyle\left(P_{\rho}\boldsymbol{u}_{\rho}(t)-P\boldsymbol{u}(t),\boldsymbol{u}(t)-\boldsymbol{u}_{\rho}(t)\right)_{V} (5.13)
โ‰คG(ฯ)(c02โ€–๐’–(t)โ€–V+c0ฮฒ meas (ฮ“3)12)โ€–๐’–ฯ(t)โˆ’๐’–(t)โ€–V\displaystyle\quad\leq G(\rho)\left(c_{0}^{2}\|\boldsymbol{u}(t)\|_{V}+c_{0}\beta\text{ meas }\left(\Gamma_{3}\right)^{\frac{1}{2}}\right)\left\|\boldsymbol{u}_{\rho}(t)-\boldsymbol{u}(t)\right\|_{V}

On the other hand, using assumptions (4.2), (4.4) and arguments similar to those used in the proof of (4.17) we find that

โ€–๐’ฎฯ๐’–ฯ(t)โˆ’๐’ฎ๐’–(t)โ€–Vโ‰คโ€–๐’ฎฯ๐’–ฯ(t)โˆ’๐’ฎฯ๐’–(t)โ€–V+โ€–๐’ฎฯ๐’–(t)โˆ’๐’ฎ๐’–(t)โ€–V\displaystyle\left\|\mathcal{S}_{\rho}\boldsymbol{u}_{\rho}(t)-\mathcal{S}\boldsymbol{u}(t)\right\|_{V}\leq\left\|\mathcal{S}_{\rho}\boldsymbol{u}_{\rho}(t)-\mathcal{S}_{\rho}\boldsymbol{u}(t)\right\|_{V}+\left\|\mathcal{S}_{\rho}\boldsymbol{u}(t)-\mathcal{S}\boldsymbol{u}(t)\right\|_{V}
โ‰คdmaxrโˆˆ[0,n]โกโ€–โ„ฌฯ(r)โ€–๐โˆžโˆซ0tโ€–๐’–ฯ(s)โˆ’๐’–(s)โ€–V๐‘‘s\displaystyle\quad\leq d\max_{r\in[0,n]}\left\|\mathcal{B}_{\rho}(r)\right\|_{\mathbf{Q}_{\infty}}\int_{0}^{t}\left\|\boldsymbol{u}_{\rho}(s)-\boldsymbol{u}(s)\right\|_{V}ds
+c02maxrโˆˆ[0,n]โกโ€–bฯ(r)โ€–Lโˆž(ฮ“3)โˆซ0tโ€–๐’–ฯ(s)โˆ’๐’–(s)โ€–V๐‘‘s\displaystyle\quad+c_{0}^{2}\max_{r\in[0,n]}\left\|b_{\rho}(r)\right\|_{L^{\infty}\left(\Gamma_{3}\right)}\int_{0}^{t}\left\|\boldsymbol{u}_{\rho}(s)-\boldsymbol{u}(s)\right\|_{V}ds
+dmaxrโˆˆ[0,n]โกโ€–โ„ฌฯ(r)โˆ’โ„ฌ(r)โ€–๐โˆžโˆซ0tโ€–๐’–(s)โ€–V๐‘‘s\displaystyle\quad+d\max_{r\in[0,n]}\left\|\mathcal{B}_{\rho}(r)-\mathcal{B}(r)\right\|_{\mathbf{Q}_{\infty}}\int_{0}^{t}\|\boldsymbol{u}(s)\|_{V}ds
+c02maxrโˆˆ[0,n]โกโ€–bฯ(r)โˆ’b(r)โ€–Lโˆž(ฮ“3)โˆซ0tโ€–๐’–(s)โ€–V๐‘‘s\displaystyle\quad+c_{0}^{2}\max_{r\in[0,n]}\left\|b_{\rho}(r)-b(r)\right\|_{L^{\infty}\left(\Gamma_{3}\right)}\int_{0}^{t}\|\boldsymbol{u}(s)\|_{V}ds

Therefore,

(๐’ฎฯ๐’–ฯ(t)โˆ’๐’ฎ๐’–(t),๐’–(t)โˆ’๐’–ฯ(t))V\displaystyle\left(\mathcal{S}_{\rho}\boldsymbol{u}_{\rho}(t)-\mathcal{S}\boldsymbol{u}(t),\boldsymbol{u}(t)-\boldsymbol{u}_{\rho}(t)\right)_{V} (5.14)
โ‰คโ€–๐’ฎฯ๐’–ฯ(t)โˆ’๐’ฎ๐’–(t)โ€–Vโ€–๐’–ฯ(t)โˆ’๐’–(t)โ€–V\displaystyle\quad\leq\left\|\mathcal{S}_{\rho}\boldsymbol{u}_{\rho}(t)-\mathcal{S}\boldsymbol{u}(t)\right\|_{V}\left\|\boldsymbol{u}_{\rho}(t)-\boldsymbol{u}(t)\right\|_{V}
โ‰ค(ฮธฯnโˆซ0tโ€–๐’–ฯ(s)โˆ’๐’–(s)โ€–V๐‘‘s+ฯ‰ฯnโˆซ0tโ€–๐’–(s)โ€–V๐‘‘s)โ€–๐’–ฯ(t)โˆ’๐’–(t)โ€–V\displaystyle\quad\leq\left(\theta_{\rho n}\int_{0}^{t}\left\|\boldsymbol{u}_{\rho}(s)-\boldsymbol{u}(s)\right\|_{V}ds+\omega_{\rho n}\int_{0}^{t}\|\boldsymbol{u}(s)\|_{V}ds\right)\left\|\boldsymbol{u}_{\rho}(t)-\boldsymbol{u}(t)\right\|_{V}

where

ฮธฯn=dmaxrโˆˆ[0,n]โกโ€–โ„ฌฯ(r)โ€–๐โˆž+c02maxrโˆˆ[0,n]โกโ€–bฯ(r)โ€–Lโˆž(ฮ“3)\displaystyle\theta_{\rho n}=d\max_{r\in[0,n]}\left\|\mathcal{B}_{\rho}(r)\right\|_{\mathbf{Q}_{\infty}}+c_{0}^{2}\max_{r\in[0,n]}\left\|b_{\rho}(r)\right\|_{L^{\infty}\left(\Gamma_{3}\right)} (5.15)
ฯ‰ฯn=dmaxrโˆˆ[0,n]โกโ€–โ„ฌฯ(r)โˆ’โ„ฌ(r)โ€–๐โˆž+c02maxrโˆˆ[0,n]โกโ€–bฯ(r)โˆ’b(r)โ€–Lโˆž(ฮ“3)\displaystyle\omega_{\rho n}=d\max_{r\in[0,n]}\left\|\mathcal{B}_{\rho}(r)-\mathcal{B}(r)\right\|_{\mathbf{Q}_{\infty}}+c_{0}^{2}\max_{r\in[0,n]}\left\|b_{\rho}(r)-b(r)\right\|_{L^{\infty}\left(\Gamma_{3}\right)} (5.16)

Finally, we note that

(๐’‡ฯ(t)โˆ’๐’‡(t),๐’–ฯ(t)โˆ’๐’–(t))Vโ‰คฮดฯnโ€–๐’–ฯ(t)โˆ’๐’–(t)โ€–V\left(\boldsymbol{f}_{\rho}(t)-\boldsymbol{f}(t),\boldsymbol{u}_{\rho}(t)-\boldsymbol{u}(t)\right)_{V}\leq\delta_{\rho n}\left\|\boldsymbol{u}_{\rho}(t)-\boldsymbol{u}(t)\right\|_{V} (5.17)

where

ฮดฯn=maxrโˆˆ[0,n]โกโ€–๐’‡ฯ(r)โˆ’๐’‡(r)โ€–V\delta_{\rho n}=\max_{r\in[0,n]}\left\|\boldsymbol{f}_{\rho}(r)-\boldsymbol{f}(r)\right\|_{V} (5.18)

and, using assumption (4.1) it follows that

(๐’œ๐œบ(๐’–ฯ(t))โˆ’๐’œ๐œบ(๐’–(t)),๐œบ(๐’–ฯ(t))โˆ’๐œบ(๐’–(t)))Qโ‰ฅm๐’œโ€–๐’–ฯ(t)โˆ’๐’–(t)โ€–V2.\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\rho}(t)\right)-\mathcal{A}\boldsymbol{\varepsilon}(\boldsymbol{u}(t)),\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\rho}(t)\right)-\boldsymbol{\varepsilon}(\boldsymbol{u}(t))\right)_{Q}\geq m_{\mathcal{A}}\left\|\boldsymbol{u}_{\rho}(t)-\boldsymbol{u}(t)\right\|_{V}^{2}. (5.19)

We combine now inequalities (5.12)-(5.14), (5.17) and (5.19) to deduce that

โ€–๐’–ฯ(t)โˆ’๐’–(t)โ€–Vโ‰คG(ฯ)m๐’œ(c02โ€–๐’–(t)โ€–V+c0ฮฒ meas (ฮ“3)12)\displaystyle\left\|\boldsymbol{u}_{\rho}(t)-\boldsymbol{u}(t)\right\|_{V}\leq\frac{G(\rho)}{m_{\mathcal{A}}}\left(c_{0}^{2}\|\boldsymbol{u}(t)\|_{V}+c_{0}\beta\text{ meas }\left(\Gamma_{3}\right)^{\frac{1}{2}}\right) (5.20)
+ฮธฯnm๐’œโˆซ0tโ€–๐’–ฯ(s)โˆ’๐’–(s)โ€–V๐‘‘s+ฯ‰ฯnm๐’œโˆซ0tโ€–๐’–(s)โ€–V๐‘‘s+ฮดฯnm๐’œ\displaystyle\quad+\frac{\theta_{\rho n}}{m_{\mathcal{A}}}\int_{0}^{t}\left\|\boldsymbol{u}_{\rho}(s)-\boldsymbol{u}(s)\right\|_{V}ds+\frac{\omega_{\rho n}}{m_{\mathcal{A}}}\int_{0}^{t}\|\boldsymbol{u}(s)\|_{V}ds+\frac{\delta_{\rho n}}{m_{\mathcal{A}}}

Denote

ฮพn,u=max{1m๐’œ(c02maxrโˆˆ[0,n]โˆฅ๐’–(r)โˆฅV+c0ฮฒmeas(ฮ“3)12),1m๐’œโˆซ0nโˆฅ๐’–(s)โˆฅVds,1m๐’œ}\xi_{n,u}=\max\left\{\frac{1}{m_{\mathcal{A}}}\left(c_{0}^{2}\max_{r\in[0,n]}\|\boldsymbol{u}(r)\|_{V}+c_{0}\beta\operatorname{meas}\left(\Gamma_{3}\right)^{\frac{1}{2}}\right),\frac{1}{m_{\mathcal{A}}}\int_{0}^{n}\|\boldsymbol{u}(s)\|_{V}ds,\frac{1}{m_{\mathcal{A}}}\right\}

Then, (5.20) yields

โ€–๐’–ฯ(t)โˆ’๐’–(t)โ€–Vโ‰ค(G(ฯ)+ฯ‰ฯn+ฮดฯn)ฮพn,u+ฮธฯnm๐’œโˆซ0tโ€–๐’–ฯ(s)โˆ’๐’–(s)โ€–V๐‘‘s\left\|\boldsymbol{u}_{\rho}(t)-\boldsymbol{u}(t)\right\|_{V}\leq\left(G(\rho)+\omega_{\rho n}+\delta_{\rho n}\right)\xi_{n,u}+\frac{\theta_{\rho n}}{m_{\mathcal{A}}}\int_{0}^{t}\left\|\boldsymbol{u}_{\rho}(s)-\boldsymbol{u}(s)\right\|_{V}ds

and, using the Gronwall inequality we obtain

โ€–๐’–ฯ(t)โˆ’๐’–(t)โ€–Vโ‰ค(G(ฯ)+ฯ‰ฯn+ฮดฯn)ฮพn,ueฮธฯnm๐’œt\left\|\boldsymbol{u}_{\rho}(t)-\boldsymbol{u}(t)\right\|_{V}\leq\left(G(\rho)+\omega_{\rho n}+\delta_{\rho n}\right)\xi_{n,u}e^{\frac{\theta_{\rho n}}{m_{\mathcal{A}}}t} (5.21)

We use the assumptions (5.2), (5.3) and the equivalence (2.4) to see that the sequence (ฮธฯn)ฯ\left(\theta_{\rho n}\right)_{\rho} defined by (5.15) is bounded. Therefore, there exists ฮถn>0\zeta_{n}>0 which depends on nn and is independent of ฯ\rho such that

0โ‰คฮธฯnโ‰คฮถn for all ฯ>00\leq\theta_{\rho n}\leq\zeta_{n}\quad\text{ for all }\quad\rho>0 (5.22)

We pas to the upper bound as tโˆˆ[0,n]t\in[0,n] in (5.21) and use (5.22) to obtain

maxtโˆˆ[0,n]โกโ€–๐’–ฯ(t)โˆ’๐’–(t)โ€–Vโ‰ค(G(ฯ)+ฯ‰ฯn+ฮดฯn)ฮพn,uenฮถnm๐’œ for all ฯ>0\max_{t\in[0,n]}\left\|\boldsymbol{u}_{\rho}(t)-\boldsymbol{u}(t)\right\|_{V}\leq\left(G(\rho)+\omega_{\rho n}+\delta_{\rho n}\right)\xi_{n,u}e^{\frac{n\zeta_{n}}{m_{\mathcal{A}}}}\quad\text{ for all }\quad\rho>0 (5.23)

We use now assumption (5.2)-(5.5) and definitions (5.16), (5.18) to see that

ฯ‰ฯnโ†’0 and ฮดฯnโ†’0 as ฯโ†’0\omega_{\rho n}\rightarrow 0\quad\text{ and }\quad\delta_{\rho n}\rightarrow 0\quad\text{ as }\rho\rightarrow 0 (5.24)

We combine now the convergences (5.24) and (5.6) (b) with inequality (5.23) to obtain that

maxtโˆˆ[0,n]โกโ€–๐’–ฯ(t)โˆ’๐’–(t)โ€–Vโ†’0 as ฯโ†’0\max_{t\in[0,n]}\left\|\boldsymbol{u}_{\rho}(t)-\boldsymbol{u}(t)\right\|_{V}\rightarrow 0\quad\text{ as }\rho\rightarrow 0 (5.25)

Since the convergence (5.25) holds for each nโˆˆโ„•โˆ—n\in\mathbb{N}^{*}, we deduce from (2.4) that (5.7) holds, which concludes the proof.

Note that the convergence result in Theorem 5.1 can be easily extended to the corresponding stress functions. Indeed, let ๐ˆ\boldsymbol{\sigma} be the function defined by (3.1) and, for all ฯ>0\rho>0, denote by ๐ˆฯ\boldsymbol{\sigma}_{\rho} the function given by

๐ˆฯ(t)=๐’œ๐œบ(๐’–ฯ(t))+โˆซ0tโ„ฌฯ(tโˆ’s)๐œบ(๐’–ฯ(s))๐‘‘s\boldsymbol{\sigma}_{\rho}(t)=\mathcal{A}\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\rho}(t)\right)+\int_{0}^{t}\mathcal{B}_{\rho}(t-s)\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\rho}(s)\right)ds (5.26)

for all tโˆˆโ„+t\in\mathbb{R}_{+}. Then, it follows that ๐ˆฯโˆˆC(โ„+;Q1)\boldsymbol{\sigma}_{\rho}\in C\left(\mathbb{R}_{+};Q_{1}\right) and, moreover, (5.1) yields

Divโก๐ˆฯ(t)+๐’‡ฯ0(t)=๐ŸŽโˆ€tโˆˆโ„+.\operatorname{Div}\boldsymbol{\sigma}_{\rho}(t)+\boldsymbol{f}_{\rho 0}(t)=\mathbf{0}\quad\forall t\in\mathbb{R}_{+}. (5.27)

We combine now equalities (3.1), (4.20), (5.26) and (5.27), then we use the convergences (5.2), (5.4) and (5.7) to see that

๐ˆฯโ†’๐ˆ in C(โ„+;Q1) as ฯโ†’0.\boldsymbol{\sigma}_{\rho}\rightarrow\boldsymbol{\sigma}\quad\text{ in }\quad C\left(\mathbb{R}_{+};Q_{1}\right)\quad\text{ as }\rho\rightarrow 0. (5.28)

In addition to the mathematical interest in the convergence result (5.7), (5.28), it is of importance from mechanical point of view, since it states that the weak solution of problem (3.1)-(3.5) depends continuously on the relaxation operator, the normal compliance function, the surface memory function and the densities of body forces and surface tractions.

6 A second convergence result

In this section we provide a second convergence result in the study of Problem ๐’ซ\mathcal{P}, based on the penalization of the unilateral constraint. For simplicity we assume that the function pp does not depend on ๐’™โˆˆฮ“3\boldsymbol{x}\in\Gamma_{3}, i.e. we consider the homogeneous case. Note that in this case assumption (4.3) can be written as follows:

{ (a) p:โ„โ†’โ„+.(b) There exists Lp>0 such that |p(r1)โˆ’p(r2)|โ‰คLp|r1โˆ’r2|โˆ€r1,r2โˆˆโ„. (c) (p(r1)โˆ’p(r2))(r1โˆ’r2)โ‰ฅ0โˆ€r1,r2โˆˆโ„. (d) p(r)=0 for all rโ‰ค0.\left\{\begin{array}[]{l}\text{ (a) }p:\mathbb{R}\rightarrow\mathbb{R}_{+}.\\ \text{(b) There exists }L_{p}>0\text{ such that }\\ \quad\left|p\left(r_{1}\right)-p\left(r_{2}\right)\right|\leq L_{p}\left|r_{1}-r_{2}\right|\quad\forall r_{1},r_{2}\in\mathbb{R}.\\ \text{ (c) }\left(p\left(r_{1}\right)-p\left(r_{2}\right)\right)\left(r_{1}-r_{2}\right)\geq 0\quad\forall r_{1},r_{2}\in\mathbb{R}.\\ \text{ (d) }p(r)=0\text{ for all }r\leq 0.\end{array}\right.

Let qq be a function which satisfies

{ (a) q:[g,+โˆž]โ†’โ„+.(b) There exists Lq>0 such that |q(r1)โˆ’q(r2)|โ‰คLq|r1โˆ’r2|โˆ€r1,r2โ‰ฅg. (c) (q(r1)โˆ’q(r2))(r1โˆ’r2)>0โˆ€r1,r2โ‰ฅg,r1โ‰ r2. (d) q(g)=0.\left\{\begin{array}[]{l}\text{ (a) }q:[g,+\infty]\rightarrow\mathbb{R}_{+}.\\ \text{(b) There exists }L_{q}>0\text{ such that }\\ \quad\left|q\left(r_{1}\right)-q\left(r_{2}\right)\right|\leq L_{q}\left|r_{1}-r_{2}\right|\quad\forall r_{1},r_{2}\geq g.\\ \text{ (c) }\left(q\left(r_{1}\right)-q\left(r_{2}\right)\right)\left(r_{1}-r_{2}\right)>0\quad\forall r_{1},r_{2}\geq g,r_{1}\neq r_{2}.\\ \text{ (d) }q(g)=0.\end{array}\right.

Also, let ฮผ>0\mu>0 and consider the function pฮผ:โ„โ†’โ„p_{\mu}:\mathbb{R}\rightarrow\mathbb{R} defined by

pฮผ(r)={p(r) if rโ‰คg,1ฮผq(r)+p(g) if r>g.p_{\mu}(r)=\left\{\begin{array}[]{cc}p(r)&\text{ if }\quad r\leq g,\\ \frac{1}{\mu}q(r)+p(g)&\text{ if }\quad r>g.\end{array}\right.

Using assumptions (6.1) and (6.2) it follows that the function pฮผp_{\mu} satisfies condition (6.1), i.e.

{ (a) pฮผ:โ„โ†’โ„+ (b) There exists Lฮผ>0 such that |pฮผ(r1)โˆ’pฮผ(r2)|โ‰คLฮผ|r1โˆ’r2|โˆ€r1,r2โˆˆโ„ (c) (pฮผ(r1)โˆ’pฮผ(r2))(r1โˆ’r2)โ‰ฅ0โˆ€r1,r2โˆˆโ„ (d) pฮผ(r)=0 for all r<0\left\{\begin{array}[]{l}\text{ (a) }p_{\mu}:\mathbb{R}\rightarrow\mathbb{R}_{+}\text{. }\\ \text{ (b) There exists }L_{\mu}>0\text{ such that }\\ \quad\left|p_{\mu}\left(r_{1}\right)-p_{\mu}\left(r_{2}\right)\right|\leq L_{\mu}\left|r_{1}-r_{2}\right|\quad\forall r_{1},r_{2}\in\mathbb{R}\text{. }\\ \text{ (c) }\left(p_{\mu}\left(r_{1}\right)-p_{\mu}\left(r_{2}\right)\right)\left(r_{1}-r_{2}\right)\geq 0\quad\forall r_{1},r_{2}\in\mathbb{R}\text{. }\\ \text{ (d) }p_{\mu}(r)=0\text{ for all }r<0\text{. }\end{array}\right.

With these preliminaries, we consider the following contact problem.
Problem ๐’ซฮผ\mathcal{P}_{\mu}. Find a displacement field ๐’–ฮผ:ฮฉร—โ„+โ†’โ„d\boldsymbol{u}_{\mu}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{R}^{d} and a stress field ฯƒฮผ:ฮฉร—โ„+โ†’๐•Šd\sigma_{\mu}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{S}^{d} such that, for all tโˆˆโ„+t\in\mathbb{R}_{+},

๐ˆฮผ(t)=๐’œ๐œบ(๐’–ฮผ(t))+โˆซ0tโ„ฌ(tโˆ’s)๐œบ(๐’–ฮผ(s)))ds in ฮฉDivโก๐ˆฮผ(t)+๐’‡0(t)=๐ŸŽ in ฮฉ๐’–ฮผ(t)=๐ŸŽ on ฮ“1๐ˆฮผ(t)๐‚=๐’‡2(t) on ฮ“2โˆ’ฯƒฮผฮฝ(t)=pฮผ(uฮผฮฝ(t))+โˆซ0tb(tโˆ’s)uฮผฮฝ+(s)๐‘‘s on ฮ“3๐ˆฮผฯ„(t)=๐ŸŽ on ฮ“3\begin{array}[]{rcc}\left.\boldsymbol{\sigma}_{\mu}(t)=\mathcal{A}\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(t)\right)+\int_{0}^{t}\mathcal{B}(t-s)\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(s)\right)\right)ds&\text{ in }\quad\Omega\\ \operatorname{Div}\boldsymbol{\sigma}_{\mu}(t)+\boldsymbol{f}_{0}(t)=\mathbf{0}&\text{ in }\quad\Omega\\ \boldsymbol{u}_{\mu}(t)=\mathbf{0}&\text{ on }\quad\Gamma_{1}\\ \boldsymbol{\sigma}_{\mu}(t)\boldsymbol{\nu}=\boldsymbol{f}_{2}(t)&\text{ on }\quad\Gamma_{2}\\ -\sigma_{\mu\nu}(t)=p_{\mu}\left(u_{\mu\nu}(t)\right)+\int_{0}^{t}b(t-s)u_{\mu\nu}^{+}(s)ds&\text{ on }\quad\Gamma_{3}\\ \boldsymbol{\sigma}_{\mu\tau}(t)=\mathbf{0}&\text{ on }\quad\Gamma_{3}\end{array}

Note that here and below uฮผฮฝu_{\mu\nu} is the normal component of the displacement field ๐’–ฮผ\boldsymbol{u}_{\mu} and ฯƒฮผฮฝ,๐ˆฮผฯ„\sigma_{\mu\nu},\boldsymbol{\sigma}_{\mu\tau} represent the normal and tangential components of the stress tensor ๐ˆฮผ\boldsymbol{\sigma}_{\mu}, respectively. The equations and boundary conditions in problem (6.5)-(6.10) have a similar interpretations as those in problem (3.1)-(3.6). The difference arises in the fact that here we replace the contact condition with normal compliance, memory term and unilateral constraint (3.5) with the contact condition with normal compliance and memory term (6.9). In this condition ฮผ\mu represents a penalization parameter which may be interpreted as a deformability coefficient of the foundation, and then 1ฮผ\frac{1}{\mu} is the surface stiffness coefficient. Indeed, when ฮผ\mu is smaller the reaction force of the foundation to penetration is larger and so the same force will result in a smaller penetration, which means that the foundation is less deformable. When ฮผ\mu is larger the reaction force of the foundation to penetration is smaller, and so the foundation is less stiff and more deformable.

Assume now that (4.1), (4.2), (4.4), (4.5), (6.1) and (6.2) hold. Then using arguments similar to those used in the study of Problem ๐’ซ\mathcal{P} we obtain the following variational formulation of Problem ๐’ซฮผ\mathcal{P}_{\mu}.

Problem ๐’ซฮผV\mathcal{P}_{\mu}^{V}. Find a displacement field ๐’–ฮผ:โ„+โ†’V\boldsymbol{u}_{\mu}:\mathbb{R}_{+}\rightarrow V such that the equality below holds, for all tโˆˆโ„+t\in\mathbb{R}_{+}:

(๐’œ๐œบ(๐’–ฮผ(t)),๐œบ(๐’—))Q+(โˆซ0tโ„ฌ(tโˆ’s)๐œบ(๐’–ฮผ(s))๐‘‘s,๐œบ(๐’—))Q\displaystyle\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(t)\right),\boldsymbol{\varepsilon}(\boldsymbol{v})\right)_{Q}+\left(\int_{0}^{t}\mathcal{B}(t-s)\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(s)\right)ds,\boldsymbol{\varepsilon}(\boldsymbol{v})\right)_{Q} (6.11)
+(pฮผ(uฮผฮฝ(t)),vฮฝ)L2(ฮ“3)+(โˆซ0tb(tโˆ’s)uฮผฮฝ+(s)ds,vฮฝ))L2(ฮ“3)\displaystyle\left.+\left(p_{\mu}\left(u_{\mu\nu}(t)\right),v_{\nu}\right)_{L^{2}\left(\Gamma_{3}\right)}+\left(\int_{0}^{t}b(t-s)u_{\mu\nu}^{+}(s)ds,v_{\nu}\right)\right)_{L^{2}\left(\Gamma_{3}\right)}
=(๐’‡0(t),๐’—)L2(ฮฉ)d+(๐’‡2(t),๐’—)L2(ฮ“2)dโˆ€๐’—โˆˆV\displaystyle=\left(\boldsymbol{f}_{0}(t),\boldsymbol{v}\right)_{L^{2}(\Omega)^{d}}+\left(\boldsymbol{f}_{2}(t),\boldsymbol{v}\right)_{L^{2}\left(\Gamma_{2}\right)^{d}}\quad\forall\boldsymbol{v}\in V

Our main result in this section, which states unique solvability of Problem ๐’ซฮผV\mathcal{P}_{\mu}^{V} and describes the behavior of its solution as ฮผโ†’0\mu\rightarrow 0, is the following.

Theorem 6.1 Assume that (4.1), (4.2), (4.4), (4.5), (6.1) and (6.2) hold. Then:

  1. 1.

    For each ฮผ>0\mu>0 Problem ๐’ซฮผV\mathcal{P}_{\mu}^{V} has a unique solution which satisfies ๐’–ฮผโˆˆC(โ„+;V)\boldsymbol{u}_{\mu}\in C\left(\mathbb{R}_{+};V\right).

  2. 2.

    The solution ๐’–ฮผ\boldsymbol{u}_{\mu} of the Problem ๐’ซฮผV\mathcal{P}_{\mu}^{V} converges to the solution ๐’–\boldsymbol{u} of the Problem ๐’ซV\mathcal{P}^{V}, that is

โ€–๐’–ฮผ(t)โˆ’๐’–(t)โ€–Vโ†’0\left\|\boldsymbol{u}_{\mu}(t)-\boldsymbol{u}(t)\right\|_{V}\rightarrow 0 (6.12)

as ฮผโ†’0\mu\rightarrow 0, for all tโˆˆโ„+t\in\mathbb{R}_{+}.

The proof of Theorem 6.1 is carried out in several steps that we present in what follows. In the rest of this section we suppose that the assumption of Theorem 6.1 hold and we denote by cc a positive generic constant that may depend on time but does not depend on ฮผ\mu, and whose value may change from line to line. We use notation (4.12), (4.13) and (4.14). Moreover, condition (6.4) allows us to consider the operator Pฮผ:Vโ†’VP_{\mu}:V\rightarrow V defined by

(Pฮผ๐’–,๐’—)V=โˆซฮ“3pฮผ(uฮฝ)vฮฝ๐‘‘aโˆ€๐’–,๐’—โˆˆV\left(P_{\mu}\boldsymbol{u},\boldsymbol{v}\right)_{V}=\int_{\Gamma_{3}}p_{\mu}\left(u_{\nu}\right)v_{\nu}da\quad\forall\boldsymbol{u},\boldsymbol{v}\in V (6.13)

Then, it is easy to see that Problem ๐’ซฮผV\mathcal{P}_{\mu}^{V} is equivalent to the problem of finding a function ๐’–ฮผ:โ„+โ†’V\boldsymbol{u}_{\mu}:\mathbb{R}_{+}\rightarrow V such that, for all tโˆˆโ„+t\in\mathbb{R}_{+},

(๐’œฮต(๐’–ฮผ(t)),ฮต(๐’—))Q+(Pฮผ๐’–ฮผ(t),๐’—)V+(๐’ฎ๐’–ฮผ(t),๐’—)V\displaystyle\left(\mathcal{A}\varepsilon\left(\boldsymbol{u}_{\mu}(t)\right),\varepsilon(\boldsymbol{v})\right)_{Q}+\left(P_{\mu}\boldsymbol{u}_{\mu}(t),\boldsymbol{v}\right)_{V}+\left(\mathcal{S}\boldsymbol{u}_{\mu}(t),\boldsymbol{v}\right)_{V} (6.14)
=(๐’‡(t),๐’—)Vโˆ€๐’—โˆˆV.\displaystyle=(\boldsymbol{f}(t),\boldsymbol{v})_{V}\quad\forall\boldsymbol{v}\in V.

For this reason, we start by proving the unique solvability of this variational equation.

Lemma 6.2 There exists a unique solution ๐’–ฮผโˆˆC(โ„+;V)\boldsymbol{u}_{\mu}\in C\left(\mathbb{R}_{+};V\right) which satisfies (6.14), for all tโˆˆโ„+t\in\mathbb{R}_{+}.

Proof. We use Theorem 2.1 with K=X=VK=X=V. Let Aฮผ:Vโ†’VA_{\mu}:V\rightarrow V be the operator defined by

(Aฮผ๐’–,๐’—)V=(๐’œฮต(๐’–),ฮต(๐’—))Q+(Pฮผ๐’–,๐’—)Vโˆ€๐’–,๐’—โˆˆV,\left(A_{\mu}\boldsymbol{u},\boldsymbol{v}\right)_{V}=(\mathcal{A}\varepsilon(\boldsymbol{u}),\varepsilon(\boldsymbol{v}))_{Q}+\left(P_{\mu}\boldsymbol{u},\boldsymbol{v}\right)_{V}\quad\forall\boldsymbol{u},\boldsymbol{v}\in V, (6.15)

We use (4.1), (6.4) and (6.13) to see that AฮผA_{\mu} is a strongly monotone and Lipschitz continuous operator i.e. it verifies condition (2.6). Therefore, it follows from Theorem 2.1 that there exists a unique function ๐’–ฮผโˆˆC(โ„+;V)\boldsymbol{u}_{\mu}\in C\left(\mathbb{R}_{+};V\right) which satisfies the inequality

(Aฮผ๐’–ฮผ(t),๐’—โˆ’๐’–ฮผ(t))V+(๐’ฎ๐’–ฮผ(t),๐’—โˆ’๐’–ฮผ(t))Vโ‰ฅ(๐’‡(t),๐’—โˆ’๐’–ฮผ(t))Vโˆ€๐’—โˆˆV,\begin{gathered}\left(A_{\mu}\boldsymbol{u}_{\mu}(t),\boldsymbol{v}-\boldsymbol{u}_{\mu}(t)\right)_{V}+\left(\mathcal{S}\boldsymbol{u}_{\mu}(t),\boldsymbol{v}-\boldsymbol{u}_{\mu}(t)\right)_{V}\\ \geq\left(\boldsymbol{f}(t),\boldsymbol{v}-\boldsymbol{u}_{\mu}(t)\right)_{V}\quad\forall\boldsymbol{v}\in V,\end{gathered}

for all tโˆˆโ„+t\in\mathbb{R}_{+}. We replace ๐’—\boldsymbol{v} with ๐’–ฮผ(t)ยฑ๐’—\boldsymbol{u}_{\mu}(t)\pm\boldsymbol{v} to see that the previous inequality is equivalent to the variational equation

(A๐’–ฮผ(t),๐’—)V+(๐’ฎ๐’–ฮผ(t),๐’—)V=(๐’‡(t),๐’—)Vโˆ€๐’—โˆˆV,\left(A\boldsymbol{u}_{\mu}(t),\boldsymbol{v}\right)_{V}+\left(\mathcal{S}\boldsymbol{u}_{\mu}(t),\boldsymbol{v}\right)_{V}=(\boldsymbol{f}(t),\boldsymbol{v})_{V}\quad\forall\boldsymbol{v}\in V,

for all tโˆˆโ„+t\in\mathbb{R}_{+}. Therefore, using (6.15) we deduce that that there exists a unique function ๐’–ฮผโˆˆC(โ„+;V)\boldsymbol{u}_{\mu}\in C\left(\mathbb{R}_{+};V\right) which satisfies the inequality (6.14) for all tโˆˆโ„+t\in\mathbb{R}_{+}, which concludes the proof.

In the second step we consider the auxiliary problem of finding a displacement field ๐’–~ฮผ:โ„+โ†’V\tilde{\boldsymbol{u}}_{\mu}:\mathbb{R}_{+}\rightarrow V such that, for all tโˆˆโ„+t\in\mathbb{R}_{+},

(๐’œฮต(๐’–~ฮผ(t)),ฮต(๐’—))Q+(Pฮผ๐’–~ฮผ(t),๐’—)V+(๐’ฎ๐’–(t),๐’—)V\displaystyle\left(\mathcal{A}\varepsilon\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right),\varepsilon(\boldsymbol{v})\right)_{Q}+\left(P_{\mu}\tilde{\boldsymbol{u}}_{\mu}(t),\boldsymbol{v}\right)_{V}+(\mathcal{S}\boldsymbol{u}(t),\boldsymbol{v})_{V} (6.16)
=(๐’‡(t),๐’—)Vโˆ€๐’—โˆˆV.\displaystyle=(\boldsymbol{f}(t),\boldsymbol{v})_{V}\quad\forall\boldsymbol{v}\in V.

Note that the difference between problems (6.14) and (6.16) arises in the fact that in (6.16) the operator ๐’ฎ\mathcal{S} is applied to a known function. We have the following existence and uniqueness result.

Lemma 6.3 There exists a unique solution ๐’–~ฮผโˆˆC(โ„+;V)\tilde{\boldsymbol{u}}_{\mu}\in C\left(\mathbb{R}_{+};V\right) which satisfies (6.16), for all tโˆˆโ„+t\in\mathbb{R}_{+}.

Proof. Besides the operator Aฮผ:Vโ†’VA_{\mu}:V\rightarrow V defined by (6.15) we define the function ๐’‡~:โ„+โ†’V\tilde{\boldsymbol{f}}:\mathbb{R}_{+}\rightarrow V by equality

(๐’‡~(t),๐’—)V=(๐’‡(t),๐’—)Vโˆ’(๐’ฎ๐’–(t),๐’—)Vโˆ€๐’—โˆˆV,tโˆˆโ„+(\tilde{\boldsymbol{f}}(t),\boldsymbol{v})_{V}=(\boldsymbol{f}(t),\boldsymbol{v})_{V}-(\mathcal{S}\boldsymbol{u}(t),\boldsymbol{v})_{V}\quad\forall\boldsymbol{v}\in V,t\in\mathbb{R}_{+} (6.17)

and we note that assumptions on ๐’‡0,๐’‡2,โ„ฌ\boldsymbol{f}_{0},\boldsymbol{f}_{2},\mathcal{B} and bb yield

f~โˆˆC(โ„+;V).\tilde{f}\in C\left(\mathbb{R}_{+};V\right). (6.18)

Let tโˆˆโ„+t\in\mathbb{R}_{+}. Based on (6.15) and (6.17), it is easy to see that (6.16) is equivalent to equality

Aฮผ๐’–~ฮผ(t)=๐’‡~(t).A_{\mu}\tilde{\boldsymbol{u}}_{\mu}(t)=\tilde{\boldsymbol{f}}(t). (6.19)

Recall that AฮผA_{\mu} is a strongly monotone and Lipschitz continuous operator. Therefore, by standard arguments we deduce the existence of a unique function ๐’–~ฮผโˆˆC(โ„+;V)\tilde{\boldsymbol{u}}_{\mu}\in C\left(\mathbb{R}_{+};V\right) such that (6.19) holds for all tโˆˆโ„+t\in\mathbb{R}_{+}, which concludes the proof.

We proceed with the following convergence result.
Lemma 6.4 As ฮผโ†’0\mu\rightarrow 0,

๐’–~ฮผ(t)โ‡€๐’–(t) in V,\tilde{\boldsymbol{u}}_{\mu}(t)\rightharpoonup\boldsymbol{u}(t)\quad\text{ in }V,

for all tโˆˆโ„+t\in\mathbb{R}_{+}.
Proof. Let tโˆˆโ„+t\in\mathbb{R}_{+}. We take ๐’—=๐’–~ฮผ(t)\boldsymbol{v}=\tilde{\boldsymbol{u}}_{\mu}(t) in (6.16) to obtain

(๐’œ๐œบ\displaystyle(\mathcal{A}\boldsymbol{\varepsilon} (๐’–~ฮผ(t)),๐œบ(๐’–~ฮผ(t)))Q+(Pฮผ๐’–~ฮผ(t),๐’–~ฮผ(t))V\displaystyle\left.\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right),\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)\right)_{Q}+\left(P_{\mu}\tilde{\boldsymbol{u}}_{\mu}(t),\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V} (6.20)
+(๐’ฎ๐’–(t),๐’–~ฮผ(t))V=(๐’‡(t),๐’–~ฮผ(t))V\displaystyle+\left(\mathcal{S}\boldsymbol{u}(t),\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}=\left(\boldsymbol{f}(t),\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}

On the other hand, the properties (6.4) of the function pฮผp_{\mu} yield

(Pฮผ๐’–~ฮผ(t),๐’–~ฮผ(t))Vโ‰ฅ0\left(P_{\mu}\tilde{\boldsymbol{u}}_{\mu}(t),\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}\geq 0 (6.21)

We combine (6.20), (6.21) and use (4.1) to obtain that

โ€–๐’–~ฮผ(t)โ€–Vโ‰คc(โ€–๐’‡(t)โ€–V+โ€–๐’ฎ๐’–(t)โ€–V+โ€–๐’œ๐ŸŽโ€–Q)\left\|\tilde{\boldsymbol{u}}_{\mu}(t)\right\|_{V}\leq c\left(\|\boldsymbol{f}(t)\|_{V}+\|\mathcal{S}\boldsymbol{u}(t)\|_{V}+\|\mathcal{A}\mathbf{0}\|_{Q}\right) (6.22)

This inequality shows that the sequence {๐’–~ฮผ(t)}ฮผโŠ‚V\left\{\tilde{\boldsymbol{u}}_{\mu}(t)\right\}_{\mu}\subset V is bounded. Hence, there exists a subsequence of the sequence {๐’–~ฮผ(t)}ฮผ\left\{\tilde{\boldsymbol{u}}_{\mu}(t)\right\}_{\mu}, still denoted {๐’–~ฮผ(t)}ฮผ\left\{\tilde{\boldsymbol{u}}_{\mu}(t)\right\}_{\mu}, and an element ๐’–~(t)โˆˆV\tilde{\boldsymbol{u}}(t)\in V such that

๐’–~ฮผ(t)โ‡€๐’–~(t) in V.\tilde{\boldsymbol{u}}_{\mu}(t)\rightharpoonup\tilde{\boldsymbol{u}}(t)\quad\text{ in }V. (6.23)

It follows from (6.20) that

(Pฮผ๐’–~ฮผ(t),๐’–~ฮผ(t))V=(๐’‡(t),๐’–~ฮผ(t))Vโˆ’(๐’œ๐œบ(๐’–~ฮผ(t)),๐œบ(๐’–~ฮผ(t)))Qโˆ’(๐’ฎ๐’–(t),๐’–~ฮผ(t))V\left(P_{\mu}\tilde{\boldsymbol{u}}_{\mu}(t),\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}=\left(\boldsymbol{f}(t),\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}-\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right),\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)\right)_{Q}-\left(\mathcal{S}\boldsymbol{u}(t),\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}

and, since {๐’–~ฮผ(t)}ฮผ\left\{\tilde{\boldsymbol{u}}_{\mu}(t)\right\}_{\mu} is a bounded sequence in VV, using (4.1) we deduce that

(Pฮผ๐’–~ฮผ(t),๐’–~ฮผ(t))Vโ‰คc\left(P_{\mu}\tilde{\boldsymbol{u}}_{\mu}(t),\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}\leq c

This implies that

โˆซฮ“3pฮผ(u~ฮผฮฝ(t))u~ฮผฮฝ(t)๐‘‘aโ‰คc\int_{\Gamma_{3}}p_{\mu}\left(\tilde{u}_{\mu\nu}(t)\right)\tilde{u}_{\mu\nu}(t)da\leq c

and, since pฮผp_{\mu} and gg are positive, it follows that

โˆซฮ“3pฮผ(u~ฮผฮฝ(t))(u~ฮผฮฝ(t)โˆ’g)๐‘‘aโ‰คc\int_{\Gamma_{3}}p_{\mu}\left(\tilde{u}_{\mu\nu}(t)\right)\left(\tilde{u}_{\mu\nu}(t)-g\right)da\leq c (6.24)

We consider now the measurable subsets of ฮ“3\Gamma_{3} defined by

ฮ“31={๐’™โˆˆฮ“3:u~ฮผฮฝ(t)(๐’™)โ‰คg},ฮ“32={๐’™โˆˆฮ“3:u~ฮผฮฝ(t)(๐’™)>g}\Gamma_{31}=\left\{\boldsymbol{x}\in\Gamma_{3}:\tilde{u}_{\mu\nu}(t)(\boldsymbol{x})\leq g\right\},\quad\Gamma_{32}=\left\{\boldsymbol{x}\in\Gamma_{3}:\tilde{u}_{\mu\nu}(t)(\boldsymbol{x})>g\right\} (6.25)

Clearly, both ฮ“31\Gamma_{31} and ฮ“32\Gamma_{32} depend on tt and ฮผ\mu but, for simplicity, we do not indicate explicitly this dependence. We use (6.24) to write

โˆซฮ“31pฮผ(u~ฮผฮฝ(t))(u~ฮผฮฝ(t)โˆ’g)๐‘‘a+โˆซฮ“32pฮผ(u~ฮผฮฝ(t))(u~ฮผฮฝ(t)โˆ’g)๐‘‘aโ‰คc\int_{\Gamma_{31}}p_{\mu}\left(\tilde{u}_{\mu\nu}(t)\right)\left(\tilde{u}_{\mu\nu}(t)-g\right)da+\int_{\Gamma_{32}}p_{\mu}\left(\tilde{u}_{\mu\nu}(t)\right)\left(\tilde{u}_{\mu\nu}(t)-g\right)da\leq c

and, since

โˆซฮ“31pฮผ(u~ฮผฮฝ(t))u~ฮผฮฝ(t)๐‘‘aโ‰ฅ0\int_{\Gamma_{31}}p_{\mu}\left(\tilde{u}_{\mu\nu}(t)\right)\tilde{u}_{\mu\nu}(t)da\geq 0

we obtain

โˆซฮ“32pฮผ(u~ฮผฮฝ(t))(u~ฮผฮฝ(t)โˆ’g)๐‘‘aโ‰คโˆซฮ“31pฮผ(u~ฮผฮฝ(t))g๐‘‘a+c\int_{\Gamma_{32}}p_{\mu}\left(\tilde{u}_{\mu\nu}(t)\right)\left(\tilde{u}_{\mu\nu}(t)-g\right)da\leq\int_{\Gamma_{31}}p_{\mu}\left(\tilde{u}_{\mu\nu}(t)\right)gda+c

Thus, taking into account that pฮผ(r)=p(r)p_{\mu}(r)=p(r) for rโ‰คgr\leq g, by the monotonicity of the function pp we can write

โˆซฮ“32pฮผ(u~ฮผฮฝ(t))(u~ฮผฮฝ(t)โˆ’g)๐‘‘aโ‰คโˆซฮ“31p(u~ฮผฮฝ)g๐‘‘a+cโ‰คโˆซฮ“3p(g)g๐‘‘a+c\int_{\Gamma_{32}}p_{\mu}\left(\tilde{u}_{\mu\nu}(t)\right)\left(\tilde{u}_{\mu\nu}(t)-g\right)da\leq\int_{\Gamma_{31}}p\left(\tilde{u}_{\mu\nu}\right)gda+c\leq\int_{\Gamma_{3}}p(g)gda+c

Therefore, we deduce that

โˆซฮ“32pฮผ(u~ฮผฮฝ(t))(u~ฮผฮฝ(t)โˆ’g)๐‘‘aโ‰คc\int_{\Gamma_{32}}p_{\mu}\left(\tilde{u}_{\mu\nu}(t)\right)\left(\tilde{u}_{\mu\nu}(t)-g\right)da\leq c (6.26)

We use now the definitions (6.3) and (6.25) to see that

pฮผ(u~ฮผฮฝ(t))=1ฮผq(u~ฮผฮฝ(t))+p(g),p(g)(u~ฮผฮฝ(t)โˆ’g)โ‰ฅ0 a.e. on ฮ“32p_{\mu}\left(\tilde{u}_{\mu\nu}(t)\right)=\frac{1}{\mu}q\left(\tilde{u}_{\mu\nu}(t)\right)+p(g),\quad p(g)\left(\tilde{u}_{\mu\nu}(t)-g\right)\geq 0\quad\text{ a.e. on }\Gamma_{32}\text{. }

Consequently, the inequality (6.26) yields

โˆซฮ“32q(u~ฮผฮฝ(t))(u~ฮผฮฝ(t)โˆ’g)๐‘‘aโ‰คcฮผ\int_{\Gamma_{32}}q\left(\tilde{u}_{\mu\nu}(t)\right)\left(\tilde{u}_{\mu\nu}(t)-g\right)da\leq c\mu (6.27)

Next, we consider the function defined by

p~:โ„โ†’โ„+p~(r)={0 if rโ‰คgq(r) if r>g\tilde{p}:\mathbb{R}\rightarrow\mathbb{R}_{+}\quad\tilde{p}(r)=\left\{\begin{array}[]{cl}0&\text{ if }\quad r\leq g\\ q(r)&\text{ if }\quad r>g\end{array}\right.

and we note that by (6.2) it follows that p~\tilde{p} is a continuous increasing function and, moreover,

p~(r)=0 iff rโ‰คg.\tilde{p}(r)=0\quad\text{ iff }\quad r\leq g. (6.28)

We use (6.27), equality q(u~ฮผฮฝ(t))=p~(u~ฮผฮฝ(t))q\left(\tilde{u}_{\mu\nu}(t)\right)=\tilde{p}\left(\tilde{u}_{\mu\nu}(t)\right) a.e. on ฮ“32\Gamma_{32} and (6.25) to deduce that

โˆซฮ“3p~(u~ฮผฮฝ(t))(u~ฮผฮฝ(t)โˆ’g)+โ‰คcฮผ\int_{\Gamma_{3}}\tilde{p}\left(\tilde{u}_{\mu\nu}(t)\right)\left(\tilde{u}_{\mu\nu}(t)-g\right)^{+}\leq c\mu

where, recall, (u~ฮผฮฝ(t)โˆ’g)+\left(\tilde{u}_{\mu\nu}(t)-g\right)^{+}denotes the positive part of u~ฮผฮฝ(t)โˆ’g\tilde{u}_{\mu\nu}(t)-g. Therefore, passing to the limit as ฮผโ†’0\mu\rightarrow 0, using (6.23) as well as compactness of the trace operator we find that

โˆซฮ“3p~(u~ฮฝ(t))(u~ฮฝ(t)โˆ’g)+๐‘‘aโ‰ค0\int_{\Gamma_{3}}\tilde{p}\left(\tilde{u}_{\nu}(t)\right)\left(\tilde{u}_{\nu}(t)-g\right)^{+}da\leq 0

Since the integrand p~(u~ฮฝ(t))(u~ฮฝ(t)โˆ’g)+\tilde{p}\left(\tilde{u}_{\nu}(t)\right)\left(\tilde{u}_{\nu}(t)-g\right)^{+}is positive a.e. on ฮ“3\Gamma_{3}, the last inequality yields

p~(u~ฮฝ(t))(u~ฮฝ(t)โˆ’g)+=0 a.e. on ฮ“3\tilde{p}\left(\tilde{u}_{\nu}(t)\right)\left(\tilde{u}_{\nu}(t)-g\right)^{+}=0\text{ a.e. on }\Gamma_{3}

and, using (6.28) and definition (4.6) we conclude that

๐’–~(t)โˆˆU\tilde{\boldsymbol{u}}(t)\in U (6.29)

Next, we test in (6.16) with ๐’—โˆ’๐’–~ฮผ(t)\boldsymbol{v}-\tilde{\boldsymbol{u}}_{\mu}(t), where ๐’—โˆˆU\boldsymbol{v}\in U, to obtain

(๐’œ๐œบ(๐’–~ฮผ(t)),๐œบ(๐’—)โˆ’๐œบ(๐’–~ฮผ(t)))V+(Pฮผ๐’–~ฮผ(t),๐’—โˆ’๐’–~ฮผ(t))V\displaystyle\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right),\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)\right)_{V}+\left(P_{\mu}\tilde{\boldsymbol{u}}_{\mu}(t),\boldsymbol{v}-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V} (6.30)
+(๐’ฎ๐’–(t),๐’—โˆ’๐’–~ฮผ(t))V=(๐’‡(t),๐’—โˆ’๐’–~ฮผ(t))V\displaystyle\quad+\left(\mathcal{S}\boldsymbol{u}(t),\boldsymbol{v}-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}=\left(\boldsymbol{f}(t),\boldsymbol{v}-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}

Since ๐’—โˆˆU\boldsymbol{v}\in U we have pฮผ(vฮฝ)=p(vฮฝ)p_{\mu}\left(v_{\nu}\right)=p\left(v_{\nu}\right) a.e. on ฮ“3\Gamma_{3}. Thus, taking into account the monotonicity of the function pฮผp_{\mu} yields

p(vฮฝ)(vฮฝโˆ’u~ฮผฮฝ(t))โ‰ฅpฮผ(u~ฮผฮฝ(t))(vฮฝโˆ’u~ฮผฮฝ(t)) a.e. on ฮ“3p\left(v_{\nu}\right)\left(v_{\nu}-\tilde{u}_{\mu\nu}(t)\right)\geq p_{\mu}\left(\tilde{u}_{\mu\nu}(t)\right)\left(v_{\nu}-\tilde{u}_{\mu\nu}(t)\right)\text{ a.e. on }\Gamma_{3}

and, therefore, we obtain

(P๐’—,๐’—โˆ’๐’–~ฮผ(t))Vโ‰ฅ(Pฮผ๐’–~ฮผ(t),๐’—โˆ’๐’–~ฮผ(t))V\left(P\boldsymbol{v},\boldsymbol{v}-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}\geq\left(P_{\mu}\tilde{\boldsymbol{u}}_{\mu}(t),\boldsymbol{v}-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V} (6.31)

Then, using (6.31) and (6.30) we find that

(๐’œฮต(๐’–~ฮผ(t)),ฮต(๐’—)โˆ’๐œบ(๐’–~ฮผ(t)))Q+(P๐’—,๐’—โˆ’๐’–~ฮผ(t))V\displaystyle\left(\mathcal{A}\varepsilon\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right),\varepsilon(\boldsymbol{v})-\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)\right)_{Q}+\left(P\boldsymbol{v},\boldsymbol{v}-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V} (6.32)
+(๐’ฎ๐’–(t),๐’—โˆ’๐’–~ฮผ(t))Vโ‰ฅ(๐’‡(t),๐’—โˆ’๐’–~ฮผ(t))Vโˆ€๐’—โˆˆU\displaystyle\quad+\left(\mathcal{S}\boldsymbol{u}(t),\boldsymbol{v}-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}\geq\left(\boldsymbol{f}(t),\boldsymbol{v}-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}\quad\forall\boldsymbol{v}\in U

We take ๐’—=๐’–~(t)\boldsymbol{v}=\tilde{\boldsymbol{u}}(t) in (6.32) to obtain

(๐’œ๐œบ(๐’–~ฮผ(t)),๐œบ(๐’–~ฮผ(t))โˆ’๐œบ(๐’–~(t)))Qโ‰ค(P๐’–~(t),๐’–~(t)โˆ’๐’–~ฮผ(t))V\displaystyle\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right),\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)-\boldsymbol{\varepsilon}(\tilde{\boldsymbol{u}}(t))\right)_{Q}\leq\left(P\tilde{\boldsymbol{u}}(t),\tilde{\boldsymbol{u}}(t)-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V} (6.33)
+(๐’ฎ๐’–(t),๐’–~(t)โˆ’๐’–~ฮผ(t))V+(๐’‡(t),๐’–~ฮผ(t)โˆ’๐’–~(t))V\displaystyle+\left(\mathcal{S}\boldsymbol{u}(t),\tilde{\boldsymbol{u}}(t)-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}+\left(\boldsymbol{f}(t),\tilde{\boldsymbol{u}}_{\mu}(t)-\tilde{\boldsymbol{u}}(t)\right)_{V}

and, passing to the upper limit as ฮผโ†’0\mu\rightarrow 0, by (6.23) we find that

lim supฮผโ†’0(๐’œ๐œบ(๐’–~ฮผ(t)),๐œบ(๐’–~ฮผ(t))โˆ’๐œบ(๐’–~(t)))Qโ‰ค0\limsup_{\mu\rightarrow 0}\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right),\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)-\boldsymbol{\varepsilon}(\tilde{\boldsymbol{u}}(t))\right)_{Q}\leq 0

Therefore, by a pseudomonotonicity argument is follows that

lim infฮผโ†’0\displaystyle\liminf_{\mu\rightarrow 0} (๐’œ๐œบ(๐’–~ฮผ(t)),๐œบ(๐’–~ฮผ(t))โˆ’๐œบ(๐’—))Q\displaystyle\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right),\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)-\boldsymbol{\varepsilon}(\boldsymbol{v})\right)_{Q} (6.34)
โ‰ฅ(๐’œ๐œบ(๐’–~(t)),๐œบ(๐’–~(t))โˆ’๐œบ(๐’—))Qโˆ€๐’—โˆˆV\displaystyle\geq(\mathcal{A}\boldsymbol{\varepsilon}(\tilde{\boldsymbol{u}}(t)),\boldsymbol{\varepsilon}(\tilde{\boldsymbol{u}}(t))-\boldsymbol{\varepsilon}(\boldsymbol{v}))_{Q}\quad\forall\boldsymbol{v}\in V

We use now (6.32) to see that

(P๐’—,๐’—โˆ’๐’–~ฮผ(t))V+(๐’ฎ๐’–(t),๐’—โˆ’๐’–~ฮผ(t))V+(๐’‡(t),๐’–~ฮผ(t)โˆ’๐’—)V\displaystyle\left(P\boldsymbol{v},\boldsymbol{v}-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}+\left(\mathcal{S}\boldsymbol{u}(t),\boldsymbol{v}-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}+\left(\boldsymbol{f}(t),\tilde{\boldsymbol{u}}_{\mu}(t)-\boldsymbol{v}\right)_{V}
โ‰ฅ(๐’œ๐œบ(๐’–~ฮผ(t)),๐œบ(๐’–~ฮผ(t))โˆ’๐œบ(๐’—))Qโˆ€๐’—โˆˆU\displaystyle\quad\geq\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right),\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)-\boldsymbol{\varepsilon}(\boldsymbol{v})\right)_{Q}\quad\forall\boldsymbol{v}\in U

then we pass to the lower limit in this inequality and use (6.34) and (6.23) to find that

(๐’œ๐œบ(๐’–~(t)),๐œบ(๐’—)โˆ’๐œบ(๐’–~(t)))Q+(P๐’—,๐’—โˆ’๐’–~(t))V\displaystyle(\mathcal{A}\boldsymbol{\varepsilon}(\tilde{\boldsymbol{u}}(t)),\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\tilde{\boldsymbol{u}}(t)))_{Q}+(P\boldsymbol{v},\boldsymbol{v}-\tilde{\boldsymbol{u}}(t))_{V} (6.35)
+(๐’ฎ๐’–(t),๐’—โˆ’๐’–~(t))Vโ‰ฅ(๐’‡(t),๐’—โˆ’๐’–~(t))Vโˆ€๐’—โˆˆU.\displaystyle\quad+(\mathcal{S}\boldsymbol{u}(t),\boldsymbol{v}-\tilde{\boldsymbol{u}}(t))_{V}\geq(\boldsymbol{f}(t),\boldsymbol{v}-\tilde{\boldsymbol{u}}(t))_{V}\quad\forall\boldsymbol{v}\in U.

Next, we take ๐’—=๐’–(t)\boldsymbol{v}=\boldsymbol{u}(t) in (6.35) and ๐’—=๐’–~(t)\boldsymbol{v}=\tilde{\boldsymbol{u}}(t) in (4.15). Then, adding the resulting inequalities we obtain

(๐’œ๐œบ(๐’–~(t))โˆ’๐’œ๐œบ(๐’–(t)),๐œบ(๐’–~(t)โˆ’๐œบ(๐’–(t)))Qโ‰ค0\left(\mathcal{A}\boldsymbol{\varepsilon}(\tilde{\boldsymbol{u}}(t))-\mathcal{A}\boldsymbol{\varepsilon}(\boldsymbol{u}(t)),\boldsymbol{\varepsilon}(\tilde{\boldsymbol{u}}(t)-\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))_{Q}\leq 0\right.

This inequality combined with (4.1) implies that

๐’–~(t)=๐’–(t)\tilde{\boldsymbol{u}}(t)=\boldsymbol{u}(t)

It follows from here that the whole sequence {๐’–~ฮผ(t)}ฮผ\left\{\tilde{\boldsymbol{u}}_{\mu}(t)\right\}_{\mu} is weakly convergent to the element ๐’–(t)โˆˆV\boldsymbol{u}(t)\in V, which concludes the proof.

We proceed with the following strong convergence result.
Lemma 6.5 As ฮผโ†’0\mu\rightarrow 0,

โ€–๐’–~ฮผ(t)โˆ’๐’–(t)โ€–Vโ†’0\left\|\tilde{\boldsymbol{u}}_{\mu}(t)-\boldsymbol{u}(t)\right\|_{V}\rightarrow 0

for all tโˆˆโ„+t\in\mathbb{R}_{+}.
Proof. Let tโˆˆโ„+t\in\mathbb{R}_{+}. Using (4.1) we write

m๐’œโ€–๐’–~ฮผ(t)โˆ’๐’–(t)โ€–V2โ‰ค(๐’œ๐œบ(๐’–~ฮผ(t))โˆ’๐’œ๐œบ(๐’–(t)),๐œบ(๐’–~ฮผ(t))โˆ’๐œบ(๐’–(t)))Q\displaystyle m_{\mathcal{A}}\left\|\tilde{\boldsymbol{u}}_{\mu}(t)-\boldsymbol{u}(t)\right\|_{V}^{2}\leq\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)-\mathcal{A}\boldsymbol{\varepsilon}(\boldsymbol{u}(t)),\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)-\boldsymbol{\varepsilon}(\boldsymbol{u}(t))\right)_{Q}
=(๐’œ๐œบ(๐’–(t)),๐œบ(๐’–(t))โˆ’๐œบ(๐’–~ฮผ(t)))Qโˆ’(๐’œ๐œบ(๐’–~ฮผ(t)),๐œบ(๐’–(t))โˆ’๐œบ(๐’–~ฮผ(t)))Q\displaystyle\quad=\left(\mathcal{A}\boldsymbol{\varepsilon}(\boldsymbol{u}(t)),\boldsymbol{\varepsilon}(\boldsymbol{u}(t))-\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)\right)_{Q}-\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right),\boldsymbol{\varepsilon}(\boldsymbol{u}(t))-\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)\right)_{Q}

Next, we take ๐’—=๐’–(t)\boldsymbol{v}=\boldsymbol{u}(t) in (6.32) to obtain

โˆ’(๐’œ๐œบ(๐’–~ฮผ(t)),๐œบ(๐’–(t))โˆ’๐œบ(๐’–~ฮผ(t)))Qโ‰ค(P๐’–(t),๐’–(t)โˆ’๐’–~ฮผ(t))V+(๐’ฎ๐’–(t),๐’–(t)โˆ’๐’–~ฮผ(t)))Vโˆ’(๐’‡(t),๐’–(t)โˆ’๐’–~ฮผ(t))V\begin{gathered}-\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right),\boldsymbol{\varepsilon}(\boldsymbol{u}(t))-\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)\right)_{Q}\leq\left(P\boldsymbol{u}(t),\boldsymbol{u}(t)-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}\\ \left.+\left(\mathcal{S}\boldsymbol{u}(t),\boldsymbol{u}(t)-\tilde{\boldsymbol{u}}_{\mu}(t)\right)\right)_{V}-\left(\boldsymbol{f}(t),\boldsymbol{u}(t)-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}\end{gathered}

and, combining the previous two inequalities, we find that

m๐’œโ€–๐’–~ฮผ(t)โˆ’๐’–(t)โ€–V2โ‰ค(๐’œ๐œบ(๐’–(t)),๐œบ(๐’–(t))โˆ’๐œบ(๐’–~ฮผ(t)))Q+(P๐’–(t),๐’–(t)โˆ’๐’–~ฮผ(t))V\displaystyle m_{\mathcal{A}}\left\|\tilde{\boldsymbol{u}}_{\mu}(t)-\boldsymbol{u}(t)\right\|_{V}^{2}\leq\left(\mathcal{A}\boldsymbol{\varepsilon}(\boldsymbol{u}(t)),\boldsymbol{\varepsilon}(\boldsymbol{u}(t))-\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)\right)_{Q}+\left(P\boldsymbol{u}(t),\boldsymbol{u}(t)-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}
+(๐’ฎ๐’–(t),๐’–(t)โˆ’๐’–~(t))Vโˆ’(๐’‡(t),๐’–(t)โˆ’๐’–~ฮผ(t))V\displaystyle\quad+(\mathcal{S}\boldsymbol{u}(t),\boldsymbol{u}(t)-\tilde{\boldsymbol{u}}(t))_{V}-\left(\boldsymbol{f}(t),\boldsymbol{u}(t)-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}

We pass to the upper limit in this inequality and use Lemma 6.4 to conclude that ๐’–~ฮผ(t)โ†’๐’–(t)\tilde{\boldsymbol{u}}_{\mu}(t)\rightarrow\boldsymbol{u}(t) in VV, as ฮผโ†’0\mu\rightarrow 0.

We are now in position to provide the proof of Theorem 6.1.
Proof. 1) Is easy to see that Problem ๐’ซฮผV\mathcal{P}_{\mu}^{V} is equivalent to the problem of finding a function ๐’–ฮผ:โ„+โ†’V\boldsymbol{u}_{\mu}:\mathbb{R}_{+}\rightarrow V such that, for all tโˆˆโ„+t\in\mathbb{R}_{+}, (6.14) holds. Therefore, the existence of a unique solution ๐’–ฮผโˆˆC(โ„+;V)\boldsymbol{u}_{\mu}\in C\left(\mathbb{R}_{+};V\right) to Problem ๐’ซฮผV\mathcal{P}_{\mu}^{V} is a direct consequence of Lemma 6.2
2) Let tโˆˆโ„+t\in\mathbb{R}_{+}and let nโˆˆโ„•โˆ—n\in\mathbb{N}^{*} be such that tโˆˆ[0,n]t\in[0,n]. Let also ฮผ>0\mu>0. Then, testing with ๐’—=๐’–ฮผ(t)โˆ’๐’–~ฮผ(t)\boldsymbol{v}=\boldsymbol{u}_{\mu}(t)-\tilde{\boldsymbol{u}}_{\mu}(t) in (6.16) and (6.14) we have

(๐’œ๐œบ(๐’–~ฮผ(t)),๐œบ(๐’–ฮผ(t))โˆ’๐œบ(๐’–~ฮผ(t)))Q+(Pฮผ๐’–~ฮผ(t),๐’–ฮผ(t)โˆ’๐’–~ฮผ(t))V+(๐’ฎ๐’–(t),๐’–ฮผ(t))โˆ’๐’–~ฮผ(t))V=(๐’‡(t),๐’–ฮผ(t)โˆ’๐’–~ฮผ(t))V(๐’œ๐œบ(๐’–ฮผ(t)),๐œบ(๐’–ฮผ(t))โˆ’๐œบ(๐’–~ฮผ(t)))V+(Pฮผ๐’–ฮผ(t),๐’–ฮผ(t)โˆ’๐’–~ฮผ(t))V+(๐’ฎ๐’–ฮผ(t),๐’–ฮผ(t))โˆ’๐’–~ฮผ(t))V=(๐’‡(t),๐’–ฮผ(t)โˆ’๐’–~ฮผ(t))V\begin{gathered}\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right),\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(t)\right)-\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)\right)_{Q}+\left(P_{\mu}\tilde{\boldsymbol{u}}_{\mu}(t),\boldsymbol{u}_{\mu}(t)-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}\\ \left.+\left(\mathcal{S}\boldsymbol{u}(t),\boldsymbol{u}_{\mu}(t)\right)-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}=\left(\boldsymbol{f}(t),\boldsymbol{u}_{\mu}(t)-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}\\ \left(\mathcal{A}\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(t)\right),\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(t)\right)-\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)\right)_{V}+\left(P_{\mu}\boldsymbol{u}_{\mu}(t),\boldsymbol{u}_{\mu}(t)-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}\\ \left.+\left(\mathcal{S}\boldsymbol{u}_{\mu}(t),\boldsymbol{u}_{\mu}(t)\right)-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}=\left(\boldsymbol{f}(t),\boldsymbol{u}_{\mu}(t)-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}\end{gathered}

We subtract the previous inequalities and use the monotonicity of the operator PฮผP_{\mu} to deduce that

(๐’œ๐œบ(๐’–ฮผ(t))โˆ’๐’œ๐œบ(๐’–~ฮผ(t)),๐œบ(๐’–ฮผ(t))โˆ’๐œบ(๐’–~ฮผ(t)))Qโ‰ค(๐’ฎ๐’–(t)โˆ’๐’ฎ๐’–ฮผ(t),๐’–ฮผ(t))โˆ’๐’–~ฮผ(t))V\begin{gathered}\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(t)\right)-\mathcal{A}\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right),\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(t)\right)-\boldsymbol{\varepsilon}\left(\tilde{\boldsymbol{u}}_{\mu}(t)\right)\right)_{Q}\\ \left.\leq\left(\mathcal{S}\boldsymbol{u}(t)-\mathcal{S}\boldsymbol{u}_{\mu}(t),\boldsymbol{u}_{\mu}(t)\right)-\tilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}\end{gathered}

and, therefore,

โ€–๐’–ฮผ(t)โˆ’๐’–~ฮผ(t)โ€–Vโ‰ค1m๐’œโ€–๐’ฎ๐’–(t)โˆ’๐’ฎ๐’–ฮผ(t)โ€–V.\left\|\boldsymbol{u}_{\mu}(t)-\tilde{\boldsymbol{u}}_{\mu}(t)\right\|_{V}\leq\frac{1}{m_{\mathcal{A}}}\left\|\mathcal{S}\boldsymbol{u}(t)-\mathcal{S}\boldsymbol{u}_{\mu}(t)\right\|_{V}. (6.36)

We combine now (6.36) and with (4.17), (4.18) to find that

โ€–๐’–ฮผ(t)โˆ’๐’–~ฮผ(t)โ€–Vโ‰คrnm๐’œโˆซ0tโ€–๐’–(s)โˆ’๐’–ฮผ(s)โ€–V๐‘‘s\left\|\boldsymbol{u}_{\mu}(t)-\tilde{\boldsymbol{u}}_{\mu}(t)\right\|_{V}\leq\frac{r_{n}}{m_{\mathcal{A}}}\int_{0}^{t}\left\|\boldsymbol{u}(s)-\boldsymbol{u}_{\mu}(s)\right\|_{V}ds

It follows from here that

โ€–๐’–ฮผ(t)โˆ’๐’–(t)โ€–Vโ‰คโ€–๐’–~ฮผ(t)โˆ’๐’–(t)โ€–V+rnm๐’œโˆซ0tโ€–๐’–ฮผ(s)โˆ’๐’–(s)โ€–V๐‘‘s\left\|\boldsymbol{u}_{\mu}(t)-\boldsymbol{u}(t)\right\|_{V}\leq\left\|\tilde{\boldsymbol{u}}_{\mu}(t)-\boldsymbol{u}(t)\right\|_{V}+\frac{r_{n}}{m_{\mathcal{A}}}\int_{0}^{t}\left\|\boldsymbol{u}_{\mu}(s)-\boldsymbol{u}(s)\right\|_{V}ds

and, using a Gronwallโ€™s argument, we obtain

โ€–๐’–ฮผ(t)โˆ’๐’–(t)โ€–Vโ‰คโ€–๐’–~ฮผ(t)โˆ’๐’–(t)โ€–V+rnm๐’œโˆซ0ternm๐’œ(tโˆ’s)โ€–๐’–~ฮผ(s)โˆ’๐’–(s)โ€–V๐‘‘s\left\|\boldsymbol{u}_{\mu}(t)-\boldsymbol{u}(t)\right\|_{V}\leq\left\|\tilde{\boldsymbol{u}}_{\mu}(t)-\boldsymbol{u}(t)\right\|_{V}+\frac{r_{n}}{m_{\mathcal{A}}}\int_{0}^{t}e^{\frac{r_{n}}{m_{\mathcal{A}}}(t-s)}\left\|\tilde{\boldsymbol{u}}_{\mu}(s)-\boldsymbol{u}(s)\right\|_{V}ds (6.37)

Note that ernm๐’œ(tโˆ’s)โ‰คernm๐’œtโ‰คenrnm๐’œe^{\frac{r_{n}}{m_{\mathcal{A}}}(t-s)}\leq e^{\frac{r_{n}}{m_{\mathcal{A}}}t}\leq e^{\frac{nr_{n}}{m_{\mathcal{A}}}} for all sโˆˆ[0,t]s\in[0,t] and, therefore, (6.37) yields

โ€–๐’–ฮผ(t)โˆ’๐’–(t)โ€–Vโ‰คโ€–๐’–~ฮผ(t)โˆ’๐’–(t)โ€–V+rnm๐’œenrnm๐’œโˆซ0tโ€–๐’–~ฮผ(s)โˆ’๐’–(s)โ€–V๐‘‘s\left\|\boldsymbol{u}_{\mu}(t)-\boldsymbol{u}(t)\right\|_{V}\leq\left\|\tilde{\boldsymbol{u}}_{\mu}(t)-\boldsymbol{u}(t)\right\|_{V}+\frac{r_{n}}{m_{\mathcal{A}}}e^{\frac{nr_{n}}{m_{\mathcal{A}}}}\int_{0}^{t}\left\|\tilde{\boldsymbol{u}}_{\mu}(s)-\boldsymbol{u}(s)\right\|_{V}ds (6.38)

On the other hand, by estimate (6.22), Lemma 6.5 and Lebesgueโ€™s convergence theorem it follows that

โˆซ0tโ€–๐’–~ฮผ(s)โˆ’๐’–(s)โ€–V๐‘‘sโ†’0 as ฮผโ†’0\int_{0}^{t}\left\|\tilde{\boldsymbol{u}}_{\mu}(s)-\boldsymbol{u}(s)\right\|_{V}ds\rightarrow 0\quad\text{ as }\quad\mu\rightarrow 0 (6.39)

We use now (6.38), (6.39) and Lemma 6.5 to obtain the convergence (6.12), which concludes the proof.

We extend now the convergence result in Theorem 6.1 to the weak solution of the corresponding contact problems ๐’ซ\mathcal{P} and ๐’ซฮผ\mathcal{P}{}_{\mu}. Let nโˆˆโ„•โˆ—n\in\mathbb{N}^{*} be such that tโˆˆ[0,n]t\in[0,n]. Then, using (6.5) and (3.1) we obtain

โˆฅ๐ˆฮผ(t)โˆ’๐ˆ(t)\displaystyle\|\boldsymbol{\sigma}_{\mu}(t)-\boldsymbol{\sigma}(t) โˆฅโ‰คQโˆฅ๐’œ๐œบ(๐’–ฮผ(t))โˆ’๐’œ๐œบ(๐’–(t))โˆฅQ\displaystyle\left\|{}_{Q}\leq\right\|\mathcal{A}\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(t)\right)-\mathcal{A}\boldsymbol{\varepsilon}(\boldsymbol{u}(t))\|_{Q}
+โˆซ0tโ€–โ„ฌ(tโˆ’s)(๐œบ(๐’–ฮผ(t))โˆ’๐œบ(๐’–(t)))โ€–Q\displaystyle+\int_{0}^{t}\left\|\mathcal{B}(t-s)\left(\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(t)\right)-\boldsymbol{\varepsilon}(\boldsymbol{u}(t))\right)\right\|_{Q}

and, using (4.1) and arguments similar to those used to obtain (4.17) it follows that

โˆฅ๐ˆฮผ(t)โˆ’๐ˆ(t)โˆฅQโ‰คcโˆฅ๐’–ฮผ(t)โˆ’๐’–(t))โˆฅV\displaystyle\left.\left\|\boldsymbol{\sigma}_{\mu}(t)-\boldsymbol{\sigma}(t)\right\|_{Q}\leq c\|\boldsymbol{u}_{\mu}(t)-\boldsymbol{u}(t)\right)\|_{V} (6.40)
+dmaxrโˆˆ[0,n]โกโ€–โ„ฌ(r)โ€–๐โˆžโˆซ0tโ€–๐’–ฮผ(s)โˆ’๐’–(s)โ€–V๐‘‘s\displaystyle\quad+d\max_{r\in[0,n]}\|\mathcal{B}(r)\|_{\mathbf{Q}_{\infty}}\int_{0}^{t}\left\|\boldsymbol{u}_{\mu}(s)-\boldsymbol{u}(s)\right\|_{V}ds

Moreover, taking ๐’—=๐’–ฮผ(t)\boldsymbol{v}=\boldsymbol{u}_{\mu}(t) in (6.14) and using the monotonicity of PฮผP_{\mu} and ๐’œ\mathcal{A} we find that

โˆฅ๐’–ฮผ(t)โˆฅVโ‰คc(โˆฅ๐’‡(t)โˆฅV+โˆฅ๐‘บ๐’–ฮผ(t)โˆฅQ\left\|\boldsymbol{u}_{\mu}(t)\right\|_{V}\leq c\left(\|\boldsymbol{f}(t)\|_{V}+\left\|\boldsymbol{S}\boldsymbol{u}_{\mu}(t)\right\|_{Q}\right.

We use now the property (4.17) of the operator ๐’ฎ\mathcal{S} and the Gronwall argument to see that

โ€–๐’–ฮผ(t)โ€–Vโ‰คcn\left\|\boldsymbol{u}_{\mu}(t)\right\|_{V}\leq c_{n} (6.41)

where cnc_{n} is a positive constant which depends on n,โ„ฌn,\mathcal{B} and bb. Then, we use the convergence (6.12), estimate (6.41) and Lebesqueโ€™s theorem, again, and pass to the limit in (6.40). As a result we find that

โ€–๐ˆฮผ(t)โˆ’๐ˆ(t)โ€–Qโ†’0 as ฮผโ†’0\left\|\boldsymbol{\sigma}_{\mu}(t)-\boldsymbol{\sigma}(t)\right\|_{Q}\rightarrow 0\quad\text{ as }\quad\mu\rightarrow 0 (6.42)

Finally, since (4.20) implies that Divโก๐ˆฮผ(t)=Divโก๐ˆ(t)=โˆ’๐’‡0(t)\operatorname{Div}\boldsymbol{\sigma}_{\mu}(t)=\operatorname{Div}\boldsymbol{\sigma}(t)=-\boldsymbol{f}_{0}(t), we conclude that

โ€–๐ˆฮผ(t)โˆ’๐ˆ(t)โ€–Q1=โ€–๐ˆฮผ(t)โˆ’๐ˆ(t)โ€–Q\left\|\boldsymbol{\sigma}_{\mu}(t)-\boldsymbol{\sigma}(t)\right\|_{Q_{1}}=\left\|\boldsymbol{\sigma}_{\mu}(t)-\boldsymbol{\sigma}(t)\right\|_{Q}

and, therefore, (6.42) yields

โ€–๐ˆฮผ(t)โˆ’๐ˆ(t)โ€–Q1โ†’0 as ฮผโ†’0\left\|\boldsymbol{\sigma}_{\mu}(t)-\boldsymbol{\sigma}(t)\right\|_{Q_{1}}\rightarrow 0\quad\text{ as }\quad\mu\rightarrow 0 (6.43)

In addition to the mathematical interest in the convergence result (6.12), (6.43), it is important from the mechanical point of view, since it shows that the weak solution of the viscoelastic contact problem with normal compliance memory term and unilateral constraint may be approached as closely as one wishes by the solution of the viscoplastic contact problem with normal compliance and memory term, with a sufficiently small deformability coefficient.

A brief comparison between the convergence results (5.7), (5.28) on one hand, and the convergence results (6.12), (6.43) on the other hand, show that the convergences (5.7), (5.28) hold in the Frรฉchet spaces C(โ„+;V)C\left(\mathbb{R}_{+};V\right) and C(โ„+;Q1)C\left(\mathbb{R}_{+};Q_{1}\right), respectively, and, in contrast, the convergences (6.12), (6.43) hold in the spaces VV and Q1Q_{1}, respectively, at each tโˆˆโ„+t\in\mathbb{R}_{+}. This feature arises from the mathematical tools we use on the proof of Theorem 6.1. The extension of (6.12), (6.43) to convergence results on the spaces C(โ„+;V)C\left(\mathbb{R}_{+};V\right) and C(โ„+;Q1)C\left(\mathbb{R}_{+};Q_{1}\right) remain an open problem which deserves to be investigated in the future.

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