\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 75, pp. 1--20.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/75\hfil Dynamic frictional contact] {Dynamic frictional contact for elastic viscoplastic material} \author[K. L. Kuttler\hfil EJDE-2007/75\hfilneg] {Kenneth L. Kuttler} \address{Kenneth L. Kuttler \newline Department of Mathematics Brigham Young University Provo, UT 84602, USA} \email{klkuttle@math.byu.edu} \thanks{Submitted May 10, 2007. Published May 22, 2007.} \subjclass[2000]{74H20, 74H25, 74M15, 74M10, 74D10} \keywords{Dynamic frictional contact; set-valued inclusions; existence; \hfill\break\indent uniqueness; discontinuous friction coefficient; normal compliance; elastic visco-plastic material} \begin{abstract} Using a general theory for evolution inclusions, existence and uniqueness theorems are obtained for weak solutions to a frictional dynamic contact problem for elastic visco-plastic material. An existence theorem in the case where the friction coefficient is discontinuous is also presented. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} The purpose of this paper is to consider a model involving frictional contact between an elastic visco-plastic material and a foundation. The balance of momentum and initial conditions are of the form \begin{gather} \ddot{\mathbf{u}}=\mathop{\rm div}(\mathbf{\sigma })+\mathbf{f}\quad \text{for }\;(t,\mathbf{x})\in (0,T)\times \Omega , \label{a1.1} \\ \mathbf{u}(0,\mathbf{x})=\mathbf{u}_{0}(\mathbf{x}), \label{a1.2}\\ \dot{\mathbf{u}}(0,\mathbf{x})=\mathbf{v}_{0}(\mathbf{x}), \label{a1.3} \end{gather} where for convenience, in the top balance of momentum equation, the density has been taken to equal 1. The domain $\Omega$ is a bounded open subset of $\mathbb{R}^{d}$ for $d=2$ or $3$ having Lipschitz boundary consisting of the union of three disjoint sets, $\Gamma _{C},\Gamma _{0}$, and $\Gamma _{N}$, any of which could be empty. Dirichlet conditions for $\mathbf{u}$ will be given on $\Gamma _{0}$, and on $\Gamma _{N}$, the traction density will be specified, while on $\Gamma _{C}$ are the complicated contact conditions involving friction. The following will be needed to describe these. Let $\mathbf{n}$ be the unit outward normal to $\partial \Omega$. Then $u_{n},\mathbf{u}_{T},\mathbf{\sigma }_{T}$, and $\sigma _{n}$ are defined by the following. \begin{gather*} u_{n}=\mathbf{u\cdot n}, \\ \mathbf{u}_{T}=\mathbf{u-}(\mathbf{u}\cdot \mathbf{n})\mathbf{n} ,\\ \sigma _{n}=\sigma _{ij}n_{j}n_{i}, \\ \sigma _{Ti}=\sigma _{ij}n_{j}-\sigma _{n}n_{i}. \end{gather*} Then on $\Gamma _{C}$ the boundary conditions are of the form \begin{gather} \sigma _{n}=-p((u_{n}-g)_{+})C_{n}, \label{a1.4} \\ |\mathbf{\sigma }_{T}|\leq F((u_{n}-g)_{+})\mu (|\dot{\mathbf{u}} _{T}-\dot{\mathbf{U}}_{T}|), \label{a1.5} \\ |\mathbf{\sigma }_{T}|0$such that $$|\mathcal{A}(\mathbf{x,\varepsilon }_{1})-\mathcal{A} (\mathbf{x,\varepsilon }_{2})|_{\mathfrak{S}^{d}}\leq L_{ \mathcal{A}}|\mathbf{\varepsilon }_{1}-\mathbf{\varepsilon } _{2}|_{\mathfrak{S}^{d}} \label{1maye2}$$ for all$\mathbf{\varepsilon }_{1},\mathbf{\varepsilon }_{2}\in \mathfrak{S} ^{d}$, a.e.$\mathbf{x}\in \Omega $. There exists$m_{\mathcal{A}}>0$such that $$(\mathcal{A}(\mathbf{x,\varepsilon }_{1})-\mathcal{A} (\mathbf{x,\varepsilon }_{2}))\cdot (\mathbf{ \varepsilon }_{1}-\mathbf{\varepsilon }_{2})\geq m_{\mathcal{A} }|\mathbf{\varepsilon }_{1}-\mathbf{\varepsilon }_{2}|_{\mathfrak{S} ^{d}}^{2} \label{1maye3}$$ for all$\mathbf{\varepsilon }_{1},\mathbf{\varepsilon }_{2}\in \mathfrak{S} ^{d}$, a.e.$\mathbf{x}\in \Omega $. For any$\mathbf{\varepsilon }\in \mathfrak{S}^{d},\mathbf{x\to } \mathcal{A}(\mathbf{x,\varepsilon })$is measurable on$\Omega $and the mapping$\mathbf{x\to }\mathcal{A}(\mathbf{x,0})$is in$\mathcal{H}$. $$\mathcal{E}:\Omega \times \mathfrak{S}^{d}\to \mathfrak{S}^{d}. \label{1maye4}$$ For any$\mathbf{\varepsilon }\in \mathfrak{S}^{d},\mathbf{x\to } \mathcal{A}(\mathbf{x,\varepsilon })$is measurable on$\Omega $and the mapping$\mathbf{x\to }\mathcal{A}(\mathbf{x,0}) $is in$\mathcal{H}$. There exists$L_{\mathcal{E}}>0$such that $$\|\mathcal{E}(\mathbf{x,\varepsilon }_{1})-\mathcal{ E}(\mathbf{x,\varepsilon }_{2})\|\leq L_{\mathcal{ E}}(\|\mathbf{\varepsilon }_{1}-\mathbf{\varepsilon } _{2}\|)\label{1maye5}$$ for all$\mathbf{\varepsilon }_{1},\mathbf{\varepsilon }_{2}\in \mathfrak{S} ^{d},a.e$.$\mathbf{x}\in \Omega $. For any$\mathbf{\varepsilon}\in \mathfrak{S}^{d},\mathbf{x\to }\mathcal{ E}(\mathbf{x,\varepsilon })$is measurable on$\Omega $and the mapping$\mathbf{x\to }\mathcal{E}(\mathbf{x,0})$is in$\mathcal{H}$. $$\mathcal{G}:\Omega \times \mathfrak{S}^{d}\times \mathfrak{S}^{d}\times \mathfrak{S} ^{d}\to \mathfrak{S}^{d}. \label{1maye6}$$ There exists$L_{\mathcal{G}}>0$such that $$\|\mathcal{G}(\mathbf{x,\varepsilon }_{1}', \mathbf{\varepsilon }_{1},\mathbf{\sigma }_{1})-\mathcal{G}( \mathbf{x,\varepsilon }_{2}',\mathbf{\varepsilon }_{2}, \mathbf{\sigma }_{2})\| \leq L_{\mathcal{G}}(\|\mathbf{\varepsilon }_{1}'- \mathbf{\varepsilon }_{2}'\|+\|\mathbf{ \varepsilon }_{1}-\mathbf{\varepsilon }_{2}\|+\| \mathbf{\sigma }_{1}-\mathbf{\sigma }_{2}\|) \label{1maye7}$$ for all$\mathbf{\varepsilon }_{1}',\mathbf{\varepsilon } _{2}',\mathbf{\varepsilon }_{1},\mathbf{\varepsilon }_{2},\mathbf{ \sigma }_{1},\mathbf{\sigma }_{2}\in \mathfrak{S}^{d}$, a.e.$\mathbf{x}\in \Omega $. For any$\mathbf{\varepsilon }',\mathbf{\varepsilon ,\sigma }\in \mathfrak{S}^{d},\mathbf{x\to }\mathcal{G}(\mathbf{x,\varepsilon } ',\mathbf{\varepsilon ,\sigma })$is measurable on$\Omega $and the mapping \begin{equation*} \mathbf{x\to }\mathcal{G}(\mathbf{x,0,0,0}) \end{equation*} is in$\mathcal{H}$. Also assume the following on$p$and$F$. The functions$p$and$F$are increasing and \begin{gather} \delta ^{2}r-K\leq p(r)\leq K(1+r),\quad r\geq 0, \label{a1.8}\\ p(r)=0,\;r<0, \notag\\ F(r)\leq K(1+r)\quad r\geq 0, \label{a1.9} \\ F(r)=0\quad \text{if } r<0, \notag \\ |\mu (r_{1})-\mu (r_{2})|\leq \mathop{\rm Lip}(\mu )|r_{1}-r_{2}|,\quad \|\mu \|_{\infty }\leq C, \label{a1.9a} \end{gather} and for$a=F,p$, and$r_{1},r_{2}\geq 0$, $$|a(r_{1})-a(r_{2})|\leq K|r_{1}-r_{2}|. \label{a1.10}$$ To allow for dependence on$\mathbf{x}$of the functions,$p$and$F$, \begin{gather} \mathbf{x} \to p(\mathbf{x},r)\quad\text{is measurable on }\Gamma _{C} \notag \\ \mathbf{x} \to F(\mathbf{x},r)\quad\text{is measurable on }\Gamma _{C}. \label{4maye4} \end{gather} However, this dependence on$\mathbf{x}$will be usually ignored for the sake of simpler notation. With the above conventions and definitions, the following existence theorem will be obtained. \begin{theorem}\label{4mayt1} Let$\Omega $be a bounded open set in$\mathbb{R}^{d}$having Lipschitz boundary. Then there exists a weak solution to the partial differential equation given by \eqref{a1.1} - \eqref{a1.3}, the boundary conditions given by \eqref{a1.4} - \eqref{4maye2} with the constitutive equation for$\mathbf{\sigma }$given in \eqref{1maye10} under the conditions given in \eqref{1maye1} - \eqref{4maye4}. \end{theorem} The plan is to show the conditions of a fundamental existence theorem, presented in the next section are satisfied. First here are some function spaces and definitions of the same sort used in \cite{sof2007}.$\mathfrak{S}^{d}$denotes the space of second order symmetric tensors on$ \mathbb{R}^{d}$with the usual Frobenius inner product, \begin{equation*} A\cdot B\equiv A_{ij}B_{ij}=\mathop{\rm trace}(AB^{T}). \end{equation*} In which the repeated index summation convention is used as will be the case whenever convenient. It is always assumed$\Omega $is a bounded open set having Lipschitz boundary. Define the following spaces. \begin{equation*} \mathcal{H\equiv }\{ \mathbf{\sigma }=(\sigma _{ij}) :\sigma _{ij}=\sigma _{ji}\in L^{2}(\Omega )\} , \end{equation*} with the norm and inner product given by \begin{equation*} \|\mathbf{\sigma }\|_{\mathcal{H}}^{2}\equiv \int_{\Omega }\sigma _{ij}\sigma _{ij}dx,\quad (\mathbf{\sigma ,\tau })_{\mathcal{H}}\equiv \int_{\Omega }\sigma _{ij}\tau _{ij}dx. \end{equation*} Also define \begin{equation*} H_{1}\equiv \{ \mathbf{u}=(u_{i}):\mathbf{\varepsilon } (\mathbf{u})\in \mathcal{H}\} , \end{equation*} with an inner product given by \begin{equation*} (\mathbf{u,v})_{H_{1}}\equiv (\mathbf{u,v}) _{L^{2}(\Omega )^{d}}+(\mathbf{\varepsilon }( \mathbf{u}),\mathbf{\varepsilon }(\mathbf{v}))_{ \mathcal{H}} \end{equation*} \section{A fundamental existence theorem} The monograph, \cite{nan95} describes the theory and application of set valued pseudomonotone maps. The definition given there is as follows. \begin{definition}\label{d0.1}$A:V\to \mathcal{P}(V')$, for$V$a reflexive real Banach space and$V'$the space of linear functionals, is pseudomonotone if the following conditions hold. \begin{enumerate} \item \label{a1} The set$Au$is non empty, bounded, closed, and convex for all$u\in V$. \item \label{a2} If$F$is a finite dimensional subspace of$V$,$u\in F$, and if$U$is a weakly open set in$V'$such that$U\supseteq Au$, then there exists$\delta >0$such that if$v\in B(u,\delta ) \cap F$, then$Av\subseteq U$. \item \label{a3} If$u_{i}\rightharpoonup u$in$V$and if$u_{i}^{\ast }\in Au_{i}$is such that $$\lim \sup_{i\to \infty }\langle u_{i}^{\ast },u_{i}-u\rangle _{V}\leq 0, \label{0.1}$$ then for each$v\in V$there exists$u^{\ast }(v)\in Au$such that $$\lim \inf_{i\to \infty }\langle u_{i}^{\ast },u_{i}-v\rangle _{V}\geq \langle u^{\ast }(v),u-v\rangle _{V}. \label{0.2}$$ \end{enumerate} \end{definition} As a special case, the above is implied if the following simpler conditions hold. \begin{definition} \label{def2.2}\rm$A:V\to \mathcal{P}(V')$, for$V$a reflexive real Banach space and$V'$the space of bounded linear functionals, is pseudomonotone if the following conditions hold. \begin{enumerate} \item The set$Au$is non empty, bounded, closed, and convex for all$u\in V $and the set, \begin{equation*} \{ u^{\ast }:u^{\ast }\in Au\text{ for }u\in B\} \end{equation*} for$B$a bounded set is bounded. Simply stated,$A$is bounded. \item If$u_{i}\rightharpoonup u$in$V$and if$u_{i}^{\ast }\in Au_{i}$is such that $$\lim \sup_{i\to \infty }\langle u_{i}^{\ast },u_{i}-u\rangle _{V}\leq 0, \label{1maye8}$$ then for each$v\in V$there exists$u^{\ast }(v)\in Au$such that $$\lim \inf_{i\to \infty }\langle u_{i}^{\ast },u_{i}-v\rangle _{V}\geq \langle u^{\ast }(v),u-v\rangle _{V}. \label{1maye9}$$ \end{enumerate} \end{definition} The existence theorems in this paper are obtained from reducing to a situation in which the following theorem can be applied. \cite{KSpsm} \begin{theorem}\label{3mayt1} Let$V$be a real Banach space and let$H$be a real Hilbert space containing$V$such that$V$is dense in$H$. Identify$H$and$ H'$. Suppose$p\geq 2$, and define the space of solutions as follows: \begin{gather*} X \equiv \big\{ u\in L^{p}(0,T;V):u'\in L^{p'}(0,T;V')\big\} \\ \|u\|_{X} \equiv \|u\|_{L^{p}(0,T;V)}+\|u'\| _{L^{p'}(0,T;V')} \end{gather*} where the derivative is taken in the sense of$V'$valued distributions, \begin{equation*} u'(\phi )\equiv -\int_{0}^{T}\phi '(t)u(t)dt \end{equation*} for all$\phi \in C_{c}^{\infty }(0,T)$, the space of test functions having compact support in$(0,T)$. Then suppose \begin{equation*} A:X\to \mathcal{P}(X') \end{equation*} is pseudomonotone and for$\mathcal{V}\equiv L^{p}(0,T;V)$, \begin{equation*} A:\mathcal{V}\to \mathcal{P}(\mathcal{V}') \end{equation*} is bounded and coercive in the sense that \begin{equation*} \lim_{\|u\|_{\mathcal{V}}\to \infty ,u\in X}\frac{\inf \{ \langle u^{\ast },u\rangle :u^{\ast }\in Au\} }{\|u\|_{\mathcal{V}}}=\infty \end{equation*} Also let$f\in \mathcal{V}'$. Then there exists a solution to the initial value problem, \begin{equation*} u'+Au\ni f,\quad u(0)=u_{0}\in H. \end{equation*} \end{theorem} In the problem considered in this paper,$\mathcal{V}_{t}$will equal$ L^{2}(0,t;V)$where$V$is a closed subspace of$H_{1}$described above which also contains the functions,$C_{c}^{\infty }( \Omega )^{d}$. Specifically, \begin{equation*} V\equiv \{ \mathbf{u}\in H_{1}:\mathbf{u=0}\text{ on }\Gamma _{0}\} . \end{equation*} If no subscript is placed on$\mathcal{V}$it will mean$t=T$. The Hilbert space mentioned in the above will be$L^2(\Omega)^d$\section{The abstract formulation} In the formula for$\mathbf{\sigma }(t)$given in \eqref{1maye10} , denote by$\mathbf{v}$the function,$\dot{\mathbf{u}.}$Then in terms of these functions, \begin{equation*} \mathbf{\sigma }(t)\equiv \mathcal{A}\mathbf{\varepsilon } (\mathbf{v}(t))+\mathcal{E}\mathbf{\varepsilon } (\mathbf{u}(t))+\int_{0}^{t}\mathcal{G}( \mathbf{\sigma }(s),\mathbf{\varepsilon }(\mathbf{v} (s)),\mathbf{\varepsilon }(\mathbf{u}( s)))ds \end{equation*} where $$\mathbf{u}(t)=\mathbf{u}_{0}+\int_{0}^{t}\mathbf{v}( s)ds \label{1maye11}$$ and it will always be assumed that$\mathbf{u}_{0}\in V$. I will also denote by$K$a constant which is larger than all the Lipschitz constants which could occur. Thus, for each$\mathbf{v}\in \mathcal{V}$,$ \mathbf{\sigma }$is a fixed point of the operator, $$\Psi (\mathbf{v})\mathbf{\sigma }(t) \equiv \mathcal{A}\mathbf{\varepsilon }(\mathbf{v}(t))+ \mathcal{E}\mathbf{\varepsilon }(\mathbf{u}(t)) +\int_{0}^{t}\mathcal{G}(\mathbf{\sigma }(s),\mathbf{ \varepsilon }(\mathbf{v}(s)),\mathbf{\varepsilon } (\mathbf{u}(s)))ds \label{1maye13}$$ \begin{lemma}\label{3mayl1} For each$\mathbf{v}\in \mathcal{V}$, there exists a unique fixed point$\mathbf{\sigma }(t)\in L^{2}(0,T;\mathcal{H} )$for$\Psi (\mathbf{v})$. Also, letting$\mathbf{\sigma }_{i}$be the fixed point corresponding to$\mathbf{v}_{i}$, $$\label{2maye5} |\mathbf{\sigma }_{1}(t)-\mathbf{\sigma }_{2}(t)|_{\mathcal{H}} \leq K|\mathbf{\varepsilon }(\mathbf{v}_{1})( t)-\mathbf{\varepsilon }(\mathbf{v}_{2}(t))|_{\mathcal{H}} +K\int_{0}^{t}|\mathbf{\varepsilon }(\mathbf{v}_{1}( s))-\mathbf{\varepsilon }(\mathbf{v}_{2}(s) )|_{\mathcal{H}}ds$$ It also follows there exist constants$\delta $,$C$and$K$such that $$(\mathbf{\sigma ,\varepsilon }(\mathbf{v}))_{ \mathcal{H}}\geq \delta ^{2}|\mathbf{\varepsilon }(\mathbf{v} (t))|_{\mathcal{H}}^{2}-C-K\int_{0}^{t}| \mathbf{\varepsilon }(\mathbf{v}(s))|_{ \mathcal{H}}^{2}ds. \label{3maye1}$$ Letting$\mathbf{\sigma }_{i}$be the fixed point corresponding to$\mathbf{v}_{i}$, there exist constants,$\delta ,Ksuch that \begin{aligned} &(\mathbf{\sigma }_{1}(t)-\mathbf{\sigma }_{2}( t)\mathbf{,\varepsilon }(\mathbf{v}_{1}(t))- \mathbf{\varepsilon }(\mathbf{v}_{2}(t)))_{ \mathcal{H}}\\ &\geq \delta ^{2}|\mathbf{\varepsilon }(\mathbf{v} _{1}(t))-\mathbf{\varepsilon }(\mathbf{v} _{2}(t))|_{\mathcal{H}}^{2} -K\int_{0}^{t}|\mathbf{\varepsilon }(\mathbf{v}_{1}( s))-\mathbf{\varepsilon }(\mathbf{v}_{2}(s) )|_{\mathcal{H}}^{2}ds. \end{aligned} \label{3maye11} \end{lemma} \begin{proof} Consider the equivalent norm onL^{2}(0,T;\mathcal{H})$, \begin{equation*} \|\mathbf{\sigma }\|_{\lambda }^{2}\equiv \int_{0}^{T}e^{-\lambda t}\|\mathbf{\sigma }(t)\|^{2}dt \end{equation*} I will show if$\lambda $is large enough,$\Psi (\mathbf{v})$is a contraction map. Let$\mathbf{\sigma }_{i}\in L^{2}(0,T;\mathcal{H})$,$i=1,2. \begin{align*} &\|\Psi (\mathbf{v})\mathbf{\sigma }_{1}-\Psi ( \mathbf{v})\mathbf{\sigma }_{2}\|_{\lambda }^{2} \\ &\equiv \int_{0}^{T}e^{-\lambda t}\Big\|\int_{0}^{t}\mathcal{G}( \mathbf{\sigma }_{1}(s),\mathbf{\varepsilon }(\mathbf{v} (s)),\mathbf{\varepsilon }(\mathbf{u}( s)))-\mathcal{G}(\mathbf{\sigma }_{2}( s),\mathbf{\varepsilon }(\mathbf{v}(s)), \mathbf{\varepsilon }(\mathbf{u}(s))) ds\Big\|^{2}dt \\ &\leq K\int_{0}^{T}e^{-\lambda t}t\int_{0}^{t}\|\mathbf{\sigma } _{1}(s)-\mathbf{\sigma }_{2}(s)\| ^{2}dsdt \\ &=K\int_{0}^{T}\|\mathbf{\sigma }_{1}(s)-\mathbf{ \sigma }_{2}(s)\|^{2}\int_{s}^{T}te^{-\lambda t}dtds \\ &=\int_{0}^{T}\|\mathbf{\sigma }_{1}(s)-\mathbf{ \sigma }_{2}(s)\|^{2}e^{-\lambda s}\int_{s}^{T}te^{\lambda (s-t)}dtds \\ &\leq T(\frac{K}{\lambda })\int_{0}^{T}\|\mathbf{ \sigma }_{1}(s)-\mathbf{\sigma }_{2}(s)| |^{2}e^{-\lambda s}ds \\ &=\frac{TK}{\lambda }\|\mathbf{\sigma }_{1}(s)- \mathbf{\sigma }_{2}(s)\|_{\lambda }^{2} \end{align*} Thus there exists a unique fixed point for\Psi (\mathbf{v})$as claimed. Now consider \eqref{2maye5}. From the description of$\mathbf{\sigma }$in \eqref{2maye2}, it follows there is a suitable constant,$Ksuch that \begin{align*} |\mathbf{\sigma }_{1}(t)-\mathbf{\sigma }_{2}(t)|_{\mathcal{H}} &\leq K(|\mathbf{\varepsilon } (\mathbf{v}_{1}(t))-\mathbf{\varepsilon }( \mathbf{v}_{2}(t))|_{\mathcal{H} }+\int_{0}^{t}|\mathbf{\varepsilon }(\mathbf{v}_{1}( s))-\mathbf{\varepsilon }(\mathbf{v}_{2}(s))|_{\mathcal{H}}ds)\\ &\quad +K\int_{0}^{t}|\mathbf{\sigma }_{1}(s)-\mathbf{\sigma } _{2}(s)|_{\mathcal{H}}ds \end{align*} and now the desired result follows from Gronwall's inequality and adjusting constants. Consider \eqref{3maye1}. First, it follows from the description of\mathbf{\sigma }$in \eqref{2maye2} and the assumption that$\mathbf{x\to }\mathcal{G}(\mathbf{x,0,0,0})$is in$\mathcal{H}$that for some constants,$C$and$K$depending on the Lipschitz constants for$ \mathcal{G}and the initial data, \begin{align*} |\mathbf{\sigma }(t)|_{\mathcal{H}}^{2} &\leq K\Big(|\mathbf{\varepsilon }(\mathbf{v}(t) )|_{\mathcal{H}}^{2}+C+\int_{0}^{t}|\mathbf{\varepsilon } (\mathbf{v}(s))|_{\mathcal{H}}^{2}ds\Big) \\ &\quad +C+K\Big(\int_{0}^{t}|\mathbf{\sigma }(s)|_{ \mathcal{H}}^{2}+|\mathbf{\varepsilon }(\mathbf{v}( s))|_{\mathcal{H}}^{2}ds\Big) \end{align*} and after adjusting the constants and using Gronwall's inequality, $$|\mathbf{\sigma }(t)|_{\mathcal{H}}^{2}\leq C+K|\mathbf{\varepsilon }(\mathbf{v}(t)) |_{\mathcal{H}}^{2}+K\int_{0}^{t}|\mathbf{\varepsilon }( \mathbf{v}(s))|_{\mathcal{H}}^{2}ds. \label{3maye2}$$ Now from \eqref{2maye2} and the assumptions on\mathcal{A},\mathcal{E}$,$\mathcal{G}$, there exist constants$\delta ,C,Ksuch that \begin{align*} (\mathbf{\sigma ,\varepsilon }(\mathbf{v}))_{ \mathcal{H}} &\geq \delta ^{2}|\mathbf{\varepsilon }(\mathbf{v} (t))|_{\mathcal{H}}^{2}-K|\mathbf{ \varepsilon }(\mathbf{u}(t))|_{\mathcal{H} }^{2}-K\int_{0}^{t}|\mathbf{\sigma }(s)|_{\mathcal{ H}}^{2}ds \\ &\quad -K\int_{0}^{t}|\mathbf{\varepsilon }(\mathbf{v}(s) )|_{\mathcal{H}}^{2}ds-K\int_{0}^{t}|\mathbf{\varepsilon }(\mathbf{u}(s))|_{\mathcal{H}}^{2}ds-C \end{align*} and so, adjusting these constants, yields \begin{equation*} (\mathbf{\sigma ,\varepsilon }(\mathbf{v}))_{ \mathcal{H}}\geq \delta ^{2}|\mathbf{\varepsilon }(\mathbf{v} (t))|_{\mathcal{H}}^{2}-K\int_{0}^{t}| \mathbf{\varepsilon }(\mathbf{v}(s))|_{ \mathcal{H}}^{2}ds-K\int_{0}^{t}|\mathbf{\sigma }(s) |_{\mathcal{H}}^{2}ds-C. \end{equation*} Now from \eqref{3maye2}, a further adjusting of constants yields \begin{equation*} (\mathbf{\sigma }(t)\mathbf{,\varepsilon }(\mathbf{v }(t)))_{\mathcal{H}}\geq \delta ^{2}| \mathbf{\varepsilon }(\mathbf{v}(t))|_{ \mathcal{H}}^{2}-K\int_{0}^{t}|\mathbf{\varepsilon }(\mathbf{v} (s))|_{\mathcal{H}}^{2}ds-C \end{equation*} Finally, let\mathbf{\sigma }_{i}$correspond to$\mathbf{v}_{i}$. Then from the properties of$\mathcal{A},\mathcal{E},\mathcal{G}$, it follows there exists a constant,$Ksuch that \begin{align*} |\mathbf{\sigma }_{1}(t)-\mathbf{\sigma }_{2}( t)|_{\mathcal{H}} &\leq K\Big(|\mathbf{\varepsilon } (\mathbf{v}_{1}(t))-\mathbf{\varepsilon }( \mathbf{v}_{2}(t))|_{\mathcal{H} }+\int_{0}^{t}|\mathbf{\varepsilon }(\mathbf{v}_{1}( s))-\mathbf{\varepsilon }(\mathbf{v}_{2}(s))|_{\mathcal{H}}ds\Big)\\ &\quad +K\int_{0}^{t}|\mathbf{\sigma }_{1}(s)-\mathbf{\sigma } _{2}(s)|_{\mathcal{H}}ds \end{align*} and so by Gronwall's inequality, it follows that after adjusting the constant, \begin{equation*} |\mathbf{\sigma }_{1}(t)-\mathbf{\sigma }_{2}( t)|_{\mathcal{H}}\leq K\Big(|\mathbf{\varepsilon } (\mathbf{v}_{1}(t))-\mathbf{\varepsilon }( \mathbf{v}_{2}(t))|_{\mathcal{H} }+\int_{0}^{t}|\mathbf{\varepsilon }(\mathbf{v}_{1}( s))-\mathbf{\varepsilon }(\mathbf{v}_{2}(s) )|_{\mathcal{H}}ds\Big) \end{equation*} which implies that on adjusting the constant again, $$|\mathbf{\sigma }_{1}(t)-\mathbf{\sigma }_{2}( t)|_{\mathcal{H}}^{2}\leq K\Big(|\mathbf{\varepsilon } (\mathbf{v}_{1}(t))-\mathbf{\varepsilon }( \mathbf{v}_{2}(t))|_{\mathcal{H} }^{2}+\int_{0}^{t}|\mathbf{\varepsilon }(\mathbf{v}_{1}( s))-\mathbf{\varepsilon }(\mathbf{v}_{2}(s) )|_{\mathcal{H}}^{2}ds\Big). \label{3maye9}$$ Now \begin{align*} &(\mathbf{\sigma }_{1}(t)-\mathbf{\sigma }_{2}( t)\mathbf{,\varepsilon }(\mathbf{v}_{1}(t))- \mathbf{\varepsilon }(\mathbf{v}_{2}(t)))_{ \mathcal{H}} \\ &\geq \delta ^{2}|\mathbf{\varepsilon }(\mathbf{v}_{1}( t))-\mathbf{\varepsilon }(\mathbf{v}_{2}(t) )|_{\mathcal{H}}^{2}-K\int_{0}^{t}|\mathbf{\varepsilon } (\mathbf{v}_{1}(s))-\mathbf{\varepsilon }( \mathbf{v}_{2}(s))|_{\mathcal{H}}^{2}ds\\ &\quad -K\int_{0}^{t}|\mathbf{\sigma }_{1}(s)-\mathbf{\sigma } _{2}(s)|_{\mathcal{H}}^{2}ds \end{align*} which, from \eqref{3maye9}, is greater than or equal to \begin{align*} &\delta ^{2}|\mathbf{\varepsilon }(\mathbf{v}_{1}( t))-\mathbf{\varepsilon }(\mathbf{v}_{2}(t) )|_{\mathcal{H}}^{2}-K\int_{0}^{t}|\mathbf{\varepsilon } (\mathbf{v}_{1}(s))-\mathbf{\varepsilon }( \mathbf{v}_{2}(s))|_{\mathcal{H}}^{2}ds \\ &-K\int_{0}^{t}\int_{0}^{s}|\mathbf{\varepsilon }(\mathbf{v} _{1}(r))-\mathbf{\varepsilon }(\mathbf{v} _{2}(r))|_{\mathcal{H}}^{2}drds \end{align*} and adjusting the constants, is greater than or equal to \begin{equation*} \delta ^{2}|\mathbf{\varepsilon }(\mathbf{v}_{1}( t))-\mathbf{\varepsilon }(\mathbf{v}_{2}(t) )|_{\mathcal{H}}^{2}-K\int_{0}^{t}|\mathbf{\varepsilon } (\mathbf{v}_{1}(s))-\mathbf{\varepsilon }( \mathbf{v}_{2}(s))|_{\mathcal{H}}^{2}ds. \end{equation*} This proves the lemma. \end{proof} For the rest of this article,\mathbf{\sigma }$will be this unique fixed point satisfying $$\mathbf{\sigma }(t)\equiv \mathcal{A}\mathbf{\varepsilon } (\mathbf{v}(t))+\mathcal{E}\mathbf{\varepsilon } (\mathbf{u}(t))+\int_{0}^{t}\mathcal{G}( \mathbf{\sigma }(s),\mathbf{\varepsilon }(\mathbf{v} (s)),\mathbf{\varepsilon }(\mathbf{u}( s)))ds. \label{8maye2}$$ Recall \begin{equation*} V\equiv \{ \mathbf{u}\in H_{1}:\mathbf{u=0}\text{ on }\Gamma_{0}\} \end{equation*} and$\mathbf{f}\in \mathcal{V}'$is defined by $$\langle \mathbf{f,w}\rangle \equiv \int_{0}^{T}(\mathbf{f} _{b},\mathbf{w})_{L^{2}(\Omega )}dt+\int_{0}^{T}( \mathbf{f}_{n},\mathbf{w})_{L^{2}(\Gamma _{N})}dt \label{2maye3}$$ where$\mathbf{f}_{b}\in L^{2}(0,T;L^{2}(\Omega ) ^{d})$and$\mathbf{f}_{n}\in L^{2}(0,T;L^{2}(\Omega )^{d})$. Also, I will continue to use the convention that$ \mathbf{v=\dot{u}}$as described above. Now let$\mathbf{w}\in V$and consider the term,$\mathop{\rm div}(\mathbf{\sigma } ). Then from the boundary conditions, \begin{align*} \int_{\Omega }\mathop{\rm div}(\mathbf{\sigma })\cdot \mathbf{w}dx &= -\int_{\Omega }\mathbf{\sigma \cdot \varepsilon }(\mathbf{w}) dx+\int_{\Gamma _{N}}\mathbf{f}_{n}\cdot \mathbf{w}d\alpha +\int_{\Gamma _{C}}\mathbf{\sigma n\cdot w}d\alpha \\ &= -\int_{\Omega }\mathbf{\sigma \cdot \varepsilon }(\mathbf{w}) dx+\int_{\Gamma _{N}}\mathbf{f}_{n}\cdot \mathbf{w}d\alpha +\int_{\Gamma _{C}}\sigma _{n}\mathbf{n\cdot w}d\alpha \\ &\quad +\int_{\Gamma _{C}}\mathbf{\sigma }_{T}\cdot \mathbf{w}d\alpha \\ &=-\int_{\Omega }\mathbf{\sigma \cdot \varepsilon }(\mathbf{w}) dx+\int_{\Gamma _{N}}\mathbf{f}_{n}\cdot \mathbf{w}d\alpha \\ &\quad -\int_{\Gamma _{C}}p((u_{n}-g)_{+})C_{n}\mathbf{n\cdot w}d\alpha +\int_{\Gamma _{C}}\mathbf{\sigma }_{T}\cdot \mathbf{w}_{T}d\alpha \end{align*} Let\gamma _{T}$denote the operator \begin{equation*} \gamma _{T}\mathbf{w}\equiv (\gamma \mathbf{w})_{T}=\gamma \mathbf{w}-\gamma (\mathbf{w\cdot n})\mathbf{n} \end{equation*} where$\gamma $is the trace map on the boundary. Thus$\gamma _{T}$gives the tangential value of$\mathbf{w}$on$\partial \Omega $. Then from Lemma \eqref{8mayel1}, the boundary condition for the friction on$\Gamma _{C}$is of the form \begin{equation*} \mathbf{\sigma }_{T}\in -\gamma _{T}^{\ast }F((u_{n}-g)_{+})\mu ( |\mathbf{v}_{T}-\dot{\mathbf{U}}_{T}|)\partial \eta (\mathbf{v}_{T}-\dot{\mathbf{U}}_{T}) \end{equation*} where$\eta (\mathbf{x})\equiv |\mathbf{x}|$for$\mathbf{x}\in \mathbb{R}^{d}$. Now define an operator,$\Sigma :\mathcal{V\to V}'$as \begin{equation*} \langle \Sigma \mathbf{v,w}\rangle _{\mathcal{V}}\equiv \int_{0}^{T}\int_{\Omega }\mathbf{\sigma \cdot \varepsilon }(\mathbf{w} )dxdt \end{equation*} where$\mathbf{\sigma }satisfies \eqref{8maye2}. Thus from Lemma \eqref{3mayl1} \begin{aligned} \langle \Sigma \mathbf{v,v}\rangle _{\mathcal{V}} &\geq \int_{0}^{T}(\delta ^{2}|\mathbf{\varepsilon }(\mathbf{v} (t))|_{\mathcal{H}}^{2}-C-K\int_{0}^{t}| \mathbf{\varepsilon }(\mathbf{v}(s))|_{ \mathcal{H}}^{2})ds \\ &=\delta ^{2}\int_{0}^{T}|\mathbf{\varepsilon }(\mathbf{v} (t))|_{\mathcal{H}}^{2}dt-K\int_{0}^{T} \int_{0}^{t}|\mathbf{\varepsilon }(\mathbf{v}(s) )|_{\mathcal{H}}^{2}ds\,dt-C \end{aligned} \label{3maye5} after adjusting the constants. Also letQ:\mathcal{V\to P}(\mathcal{V})$be defined by saying that$\mathbf{v}^{\ast }\in Q\mathbf{v}$means there exists$\mathbf{z}\in L^{\infty }(0,T;L^{\infty }(\Gamma _{C})^{d})$such that for all$\mathbf{w}\in \mathcal{V}$, $$\int_{0}^{T}\int_{\Gamma _{C}}\mathbf{z\cdot w}_{T}d\alpha\,dt\leq \int_{0}^{T}\int_{\Gamma _{C}}|\mathbf{v}_{T}-\dot{\mathbf{U}}_{T}+ \mathbf{w}_{T}|-|\mathbf{v}_{T}-\dot{\mathbf{U}}_{T}| d\alpha \,dt. \label{4maye7}$$ and $$\langle \mathbf{v}^{\ast }\mathbf{,w}\rangle =\int_{0}^{T}\int_{\Gamma _{C}}F((u_{n}-g)_{+})\mu (|\mathbf{v} _{T}-\dot{\mathbf{U}}_{T}|)\mathbf{z\cdot w}_{T}d\alpha \,dt \label{4maye3}$$ The following lemma will be useful later. \begin{lemma}\label{4mayl1} In case the function$F$is bounded, there exists a constant,$K$such that if$\mathbf{v}_{i}^{\ast }\in Q\mathbf{v}_{i}$for$i=1,2$, \begin{equation*} \langle \mathbf{v}_{1}^{\ast }-\mathbf{v}_{2}^{\ast },\mathbf{v}_{1}- \mathbf{v}_{2}\rangle _{\mathcal{V}_{t}}\geq -K\int_{0}^{t}| |\mathbf{v}_{1}(s)-\mathbf{v}_{2}(s)| |_{U}^{2}ds \end{equation*} where$U$is a Sobolev space with the property that$V$embeds compactly into$U$and the trace map from$U$to$L^{2}(\Gamma _{C})^{d}$is continuous. \end{lemma} \begin{proof} Let$(\mathbf{v}_{i},\mathbf{z}_{i})$for$i=1,2$be such that$\mathbf{v}_{i}^{\ast }\in Q\mathbf{v}_{i}is given by \eqref{4maye3}. Then \label{4maye9} %\label{4maye5} \begin{aligned} &\langle \mathbf{v}_{1}^{\ast }-\mathbf{v}_{2}^{\ast },\mathbf{v}_{1}- \mathbf{v}_{2}\rangle _{\mathcal{V}_{t}}\\ &=\int_{0}^{t}\int_{\Gamma _{C}}\Big[ F((u_{1n}-g)_{+})\mu (| \mathbf{v}_{1T}-\dot{\mathbf{U}}_{T}|)\mathbf{z}_{1} \\ & \quad - F((u_{2n}-g)_{+})\mu (|\mathbf{v}_{2T}- \dot{\mathbf{U}}_{T}|)\mathbf{z}_{2}\Big] \cdot (\mathbf{v}_{1}-\mathbf{v}_{2})_{T}d\alpha\,ds\\ &=\int_{0}^{t}\int_{\Gamma _{C}}F((u_{1n}-g)_{+})\mu (|\mathbf{v} _{1T}-\dot{\mathbf{U}}_{T}|)(\mathbf{z}_{1}-\mathbf{z} _{2})\cdot (\mathbf{v}_{1}-\mathbf{v}_{2})_{T}d\alpha\,ds \\ &\quad +\int_{0}^{t}\int_{\Gamma _{C}}\Big(F((u_{1n}-g)_{+})\mu (| \mathbf{v}_{1T}-\dot{\mathbf{U}}_{T}|) \\ &\quad -F((u_{2n}-g)_{+})\mu (|\mathbf{v}_{2T}-\dot{\mathbf{U}} _{T}|)\Big)\mathbf{z}_{2}\cdot (\mathbf{v}_{1}- \mathbf{v}_{2})_{T}d\alpha\,ds \end{aligned} Now the expression in the first term of the right hand side is nonnegative because of \eqref{4maye7}. Therefore,\langle \mathbf{v}_{1}^{\ast }-\mathbf{v}_{2}^{\ast }, \mathbf{v}_{1}-\mathbf{v}_{2}\rangle _{\mathcal{V}}is bounded below by the expression in \eqref{4maye9}. The absolute value of this is bounded above by an expression of the form \begin{align*} &K\int_{0}^{t}\|\mathbf{u}_{1}(s)-\mathbf{u} _{2}(s)\|_{U}\|\mathbf{v}_{1}( s)-\mathbf{v}_{2}(s)\|_{U}ds+ \\ &K\int_{0}^{t}\|\mathbf{v}_{1}(s)-\mathbf{v} _{2}(s)\|_{U}^{2}ds. \end{align*} Now adjusting the constants, this is dominated by an expression of the form \begin{equation*} K\int_{0}^{t}\|\mathbf{v}_{1}(s)-\mathbf{v}_{2}(s)\|_{U}^{2}ds \end{equation*} which proves the lemma. \end{proof} Finally, defineP:\mathcal{V}_{t}\mathcal{\to V}_{t}'$by \begin{equation*} \langle P\mathbf{u,w}\rangle _{\mathcal{V}}\equiv \int_{0}^{t}\int_{\Gamma _{C}}p((u_{n}-g)_{+})C_{n}\mathbf{n\cdot w}d\alpha . \end{equation*} Thus, letting$J'(r)\equiv C_{n}p(r_{+}), \begin{align*} \langle P\mathbf{u,v}\rangle _{\mathcal{V}} &=\int_{0}^{t}\int_{\Gamma _{C}}p((u_{n}-g)_{+})C_{n}v_{n}d\alpha \\ &=\int_{\Gamma _{C}}(J(u_{n}(t)-g)-J( u_{0n}-g))d\alpha \geq C \end{align*} whereC$depends on$u_{0n}$. Also from the assumptions on$p$, for each$\varepsilon >0$there exists a constant,$K_{\varepsilon }$such that $$|\langle P\mathbf{u}_{1}-P\mathbf{u}_{2},\mathbf{v}_{1}-\mathbf{v} _{2}\rangle _{\mathcal{V}_{t}}|\leq \varepsilon \int_{0}^{t}\|\mathbf{v}_{1}-\mathbf{v}_{2}\| _{V}^{2}ds+K_{\varepsilon }\int_{0}^{t}\int_{0}^{s}\|\mathbf{v} _{1}-\mathbf{v}_{2}\|_{V}^{2}drds \label{4maye13}$$ \section{Existence and uniqueness for an abstract formulation} It follows an abstract formulation of the above initial boundary value problem, \eqref{a1.1} - \eqref{4maye2} where the stress is given by \eqref{8maye2} is $$\mathbf{v}'+\Sigma \mathbf{v}+Q\mathbf{v}+P\mathbf{u\ni f}\text{ in }\mathcal{V}',\quad \mathbf{v}(0)=\mathbf{v}_{0}\in L^{2}(\Omega )^{d}. \label{2maye4}$$ Solutions to this abstract inclusion are the weak solutions of Theorem \ref {4mayt1}. From now on a prime will denote the weak time derivative in the sense of$V^{\prime}$valued distributions. \begin{theorem}\label{4mayt2} Let$\mathbf{u}_{0}\in V$and let$\mathbf{v}_{0}\in L^{2}(\Omega )^{d}$. Then there exists a solution to \eqref{2maye4}. This solution satisfies \begin{equation*} \mathbf{v}\in \mathcal{V}, \quad \mathbf{v}\in C(0,T;L^{2}(\Omega )^{d}). \end{equation*} \end{theorem} \begin{proof} I will not consider \eqref{2maye4} directly. Instead, it will be reformulated in terms of a new dependent variable,$\mathbf{v}_{\lambda }$defined by \begin{equation*} e^{\lambda t}\mathbf{v}_{\lambda }(t)\equiv \mathbf{v}(t) \end{equation*} because the problem in terms of this new dependent variable will satisfy the hypotheses of the Theorem \ref{3mayt1} stated above. Then with this definition,$\mathbf{v}$is a solution of \eqref{2maye4} if and only if$\mathbf{v}_{\lambda }$is a solution of $$\begin{gathered} \mathbf{v}_{\lambda }'+\lambda \mathbf{v}_{\lambda }+e^{-\lambda (\cdot )}\Sigma (e^{\lambda (\cdot )}\mathbf{ v}_{\lambda })+e^{-\lambda (\cdot )}Q(e^{\lambda (\cdot )}\mathbf{v}_{\lambda }) +e^{-\lambda (\cdot )}P(\mathbf{u})\ni e^{-\lambda (\cdot )}\mathbf{f}\quad \text{in }\mathcal{V}',\\ \mathbf{v}_{\lambda }(0)=\mathbf{v}_{0}. \end{gathered} \label{3maye7}$$ From \eqref{3maye1} and the definition of$\Sigma , \begin{align*} &\langle e^{-\lambda (\cdot )}\Sigma (e^{\lambda (\cdot )}\mathbf{v}_{\lambda }),\mathbf{v}_{\lambda }\rangle _{\mathcal{V}}\\ &=\langle e^{-2\lambda (\cdot )}\Sigma (e^{\lambda (\cdot )}\mathbf{v}_{\lambda }) ,e^{\lambda (\cdot )}\mathbf{v}_{\lambda }\rangle _{ \mathcal{V}} \\ &\geq \int_{0}^{T}e^{-2\lambda t}\Big(e^{2\lambda t}\delta ^{2}| \mathbf{\varepsilon }(\mathbf{v}_{\lambda }(t)) |_{\mathcal{H}}^{2}-C-K\int_{0}^{t}e^{2\lambda s}|\mathbf{ \varepsilon }(\mathbf{v}_{\lambda }(s))|_{ \mathcal{H}}^{2}ds\Big)dt \\ &\geq \delta ^{2}\|\mathbf{\varepsilon }(\mathbf{v} _{\lambda })\|_{L^{2}(0,T;\mathcal{H}) }^{2}-C-K\int_{0}^{T}\int_{0}^{t}e^{-2\lambda (t-s)}| \mathbf{\varepsilon }(\mathbf{v}_{\lambda }(s)) |_{\mathcal{H}}^{2}dsdt \\ &\geq \delta ^{2}\|\mathbf{\varepsilon }(\mathbf{v} _{\lambda })\|_{L^{2}(0,T;\mathcal{H}) }^{2}-C-K\int_{0}^{T}\int_{s}^{T}e^{-2\lambda (t-s)}| \mathbf{\varepsilon }(\mathbf{v}_{\lambda }(s)) |_{\mathcal{H}}^{2}dtds \\ &\geq \delta ^{2}\|\mathbf{\varepsilon }(\mathbf{v} _{\lambda })\|_{L^{2}(0,T;\mathcal{H}) }^{2}-C-K\frac{1}{2\lambda }\int_{0}^{T}|\mathbf{\varepsilon }( \mathbf{v}_{\lambda }(s))|_{\mathcal{H}}^{2}ds \\ &\geq \delta ^{2}\|\mathbf{v}_{\lambda }\|_{ \mathcal{V}}^{2}-\delta ^{2}|\mathbf{v}_{\lambda }| _{L^{2}(0,T;L^{2}(\Omega )^{d})}^{2}-C-K\frac{1}{ 2\lambda }\|\mathbf{v}_{\lambda }\|_{\mathcal{V}}^{2} \end{align*} Thus, if\lambda $is large enough, the above expression is greater than or equal to \begin{equation*} (\delta ^{2}/2)\|\mathbf{v}_{\lambda }| |_{\mathcal{V}}^{2}-C-\delta ^{2}|\mathbf{v}_{\lambda }| _{L^{2}(0,T;L^{2}(\Omega )^{d})}^{2}. \end{equation*} From the assumptions on$F,\mu , \begin{align*} &|\langle e^{-\lambda (\cdot )}Q(e^{\lambda (\cdot )}\mathbf{v}_{\lambda }),\mathbf{v}_{\lambda }\rangle _{\mathcal{V}}| \\ &\leq C\int_{0}^{T}e^{-\lambda t}\|\mathbf{u}(t)\|_{V}\| \mathbf{v}_{\lambda }(t)\|_{V}dt \\ &\leq C+\int_{0}^{T}e^{-\lambda t}\int_{0}^{t}\|\mathbf{v} _{\lambda }(s)\|ds\|\mathbf{v} _{\lambda }(t)\|_{V}dt \\ &\leq C+(\delta ^{2}/16)\|\mathbf{v}_{\lambda }\|_{\mathcal{V}}^{2}+C_{\delta }\int_{0}^{T}\Big( e^{-\lambda t}\int_{0}^{t}\|\mathbf{v}_{\lambda }(s) \|ds\Big)^{2}dt \\ &\leq C+(\delta ^{2}/16)\|\mathbf{v}_{\lambda }\|_{\mathcal{V}}^{2}+C_{\delta }\int_{0}^{T}e^{-\lambda t}t\int_{0}^{t}\|\mathbf{v}_{\lambda }(s)| |^{2}dsdt \\ &= C+(\delta ^{2}/16)\|\mathbf{v}_{\lambda }| |_{\mathcal{V}}^{2}+C_{\delta }\int_{0}^{T}\int_{s}^{T}e^{-\lambda t}tdt\|\mathbf{v}_{\lambda }(s)\| ^{2}ds \\ &\leq C+(\delta ^{2}/16)\|\mathbf{v}_{\lambda }\|_{\mathcal{V}}^{2}+C_{\delta }T(\frac{1}{\lambda } )\int_{0}^{T}\|\mathbf{v}_{\lambda }(s) \|^{2}ds \\ &\leq C+(\delta ^{2}/8)\|\mathbf{v}_{\lambda }\|_{\mathcal{V}}^{2} \end{align*} provided\lambda $is large enough. Next using the growth condition for$pand adjusting the constants as the computation proceeds, \begin{align*} |\langle e^{-\lambda (\cdot )}P(\mathbf{u} ),\mathbf{v}_{\lambda }\rangle _{\mathcal{V}}| &=|\int_{0}^{T}e^{-\lambda t}\int_{\Gamma _{C}}p((u_{n}-g)_{+})C_{n}v_{\lambda n}d\alpha\,dt|\\ &\leq C_{n}K\int_{0}^{T}e^{-\lambda t}\int_{\Gamma _{C}}(1+| u_{n}(t)|)|v_{\lambda n}(t) |d\alpha\,dt \\ &\leq (\delta ^{2}/16)\|\mathbf{v}_{\lambda }\|_{\mathcal{V}}^{2}+C_{\delta }\int_{0}^{T}( e^{-\lambda t}\int_{\Gamma _{C}}(1+|u_{n}(t)| )d\alpha )^{2}dt \\ &\leq (\delta ^{2}/16)\|\mathbf{v}_{\lambda }\|_{\mathcal{V}}^{2}+C_{\delta }\int_{0}^{T}e^{-2\lambda t}\int_{\Gamma _{C}}(1+|u_{n}(t)|^{2}) d\alpha\,dt \\ &\leq (\delta ^{2}/16)\|\mathbf{v}_{\lambda }\|_{\mathcal{V}}^{2}+C_{\delta }\int_{0}^{T}e^{-2\lambda t}(1+\|\mathbf{u}(t)\| _{V}^{2})dt \\ &\leq (\delta ^{2}/16)\|\mathbf{v}_{\lambda }\|_{\mathcal{V}}^{2}+C_{\delta }\int_{0}^{T}e^{-2\lambda t}\int_{0}^{t}\|\mathbf{v}(s)\| ^{2}dsdt+C_{\delta }/\lambda \\ &\leq (\delta ^{2}/16)\|\mathbf{v}_{\lambda }\|_{\mathcal{V}}^{2}+C_{\delta }/\lambda \int_{0}^{T}| |\mathbf{v}(s)\|^{2}ds+C_{\delta }/\lambda \\ &\leq (\delta ^{2}/8)\|\mathbf{v}_{\lambda }\|_{\mathcal{V}}^{2}+1 \end{align*} whenever\lambda $is large enough. Letting \begin{equation*} A\mathbf{v}_{\lambda }\equiv \lambda \mathbf{v}_{\lambda }+e^{-\lambda (\cdot )}\Sigma (e^{\lambda (\cdot )}\mathbf{ v}_{\lambda })+e^{-\lambda (\cdot )}Q(e^{\lambda (\cdot )}\mathbf{v}_{\lambda })+e^{-\lambda (\cdot )}P(\mathbf{u}) \end{equation*} It follows \eqref{3maye7} is of the form $$\mathbf{v}_{\lambda }'+A\mathbf{v}_{\lambda }\ni e^{-\lambda ( \cdot )}\mathbf{f} \label{5maye2}$$ and$A:\mathcal{V\to P}(\mathcal{V})$is coercive as described in Theorem \ref{3mayt1} whenever$\lambda $is sufficiently large. It is clear from the definition that$A$is bounded. I\ need to verify$A$is pseudomonotone and then the existence of a solution will follow from Theorem \ref{3mayt1}. Letting$\mathbf{u}_{i}$correspond to$\mathbf{v}_{i}$as described above and then$\mathbf{v}_{\lambda i}also be given as above, it follows from \eqref{3maye11} \begin{align*} &\Big\langle e^{-\lambda (\cdot )}\Sigma (e^{\lambda (\cdot )}\mathbf{v}_{\lambda 1})+e^{-\lambda ( \cdot )}P(\mathbf{u}_{1})-(e^{-\lambda ( \cdot )}\Sigma (e^{\lambda (\cdot )}\mathbf{v} _{\lambda 1})+e^{-\lambda (\cdot )}P(\mathbf{u} _{1})),\mathbf{v}_{\lambda 1}-\mathbf{v}_{\lambda 2}\Big\rangle _{\mathcal{V}} \\ &\geq \int_{0}^{T}\delta ^{2}|\mathbf{\varepsilon }(\mathbf{v} _{\lambda 1}(t))-\mathbf{\varepsilon }(\mathbf{v} _{\lambda 2}(t))|_{\mathcal{H}}^{2}-Ke^{-2\lambda t}\int_{0}^{t}e^{2\lambda s}|\mathbf{\varepsilon }(\mathbf{v} _{\lambda 1}(s))-\mathbf{\varepsilon }(\mathbf{v} _{\lambda 2}(s))|_{\mathcal{H}}^{2}ds\,dt \\ &-K\int_{0}^{T}e^{-\lambda t}\int_{\Gamma _{C}}|u_{1n}-u_{2n}| |v_{\lambda 1n}-v_{\lambda 2n}|d\alpha \,dt \end{align*} Using the continuity of the trace maps, this dominates \begin{align*} &\int_{0}^{T}\delta ^{2}|\mathbf{\varepsilon }(\mathbf{v} _{\lambda 1}(t))-\mathbf{\varepsilon }(\mathbf{v} _{\lambda 2}(t))|_{\mathcal{H}}^{2}-Ke^{-2\lambda t}\int_{0}^{t}e^{2\lambda s}|\mathbf{\varepsilon }(\mathbf{v} _{\lambda 1}(s))-\mathbf{\varepsilon }(\mathbf{v} _{\lambda 2}(s))|_{\mathcal{H}}^{2}ds\,dt \\ &-C_{\delta }\int_{0}^{T}e^{-2\lambda t}\|\mathbf{u}_{1}-\mathbf{u }_{2}\|_{V}^{2}dt-(\delta ^{2}/2) \int_{0}^{T}\|\mathbf{v}_{\lambda 1}\mathbf{-v}_{\lambda 2}\|_{V}^{2}dt \\ &\geq \delta ^{2}/2\int_{0}^{T}\|\mathbf{v}_{\lambda 1}\mathbf{ -v}_{\lambda 2}\|_{V}^{2}-Ke^{-2\lambda t}\int_{0}^{t}e^{2\lambda s}|\mathbf{\varepsilon }(\mathbf{v} _{\lambda 1}(s))-\mathbf{\varepsilon }(\mathbf{v} _{\lambda 2}(s))|_{\mathcal{H}}^{2}dsdt \\ &\quad -\delta ^{2}/2\int_{0}^{T}|\mathbf{v}_{\lambda 1}\mathbf{-v} _{\lambda 2}|_{L^{2}(\Omega )^{d}}^{2}dt-C_{\delta }\int_{0}^{T}e^{-2\lambda t}\int_{0}^{t}e^{2\lambda s}\|\mathbf{v }_{\lambda 1}\mathbf{-v}_{\lambda 2}\|^{2}dsdt \\ &\geq \delta ^{2}/2\int_{0}^{T}\|\mathbf{v}_{\lambda 1}\mathbf{-v} _{\lambda 2}\|_{V}^{2}dt-\frac{K+C_{\delta }}{\lambda } \int_{0}^{T}\|\mathbf{v}_{\lambda 1}\mathbf{-v}_{\lambda 2}\|^{2}dt\\ &\quad -\frac{\delta ^{2}}{2}\int_{0}^{T}|\mathbf{v} _{\lambda 1}\mathbf{-v}_{\lambda 2}|_{L^{2}(\Omega )^{d}}^{2}dt \end{align*} It follows that for all\lambda $large enough,$B\mathbf{v}_{\lambda }$given by $$B\mathbf{v}_{\lambda }\equiv \lambda \mathbf{v}_{\lambda }+e^{-\lambda (\cdot )}\Sigma (e^{\lambda (\cdot )}\mathbf{ v}_{\lambda })+e^{-\lambda (\cdot )}P(\mathbf{u} )\label{5maye1}$$ is monotone and bounded as a map from$\mathcal{V}$to$\mathcal{V}'$. It is also clearly hemicontinuous, meaning it is continuous on line segments. Therefore, this operator is pseudomonotone. Since$A=B+e^{-\lambda(\cdot )}Q$, it only remains to verify$e^{-\lambda (\cdot )}Q$is also pseudomonotone. This operator is clearly bounded. It only remains to verify the pseudomonotone condition as a map from the space of solutions,$X$described above to$\mathcal{P}(X)$. To do this, it is helpful to use the following two interesting Theorems found in Lions \cite{lio69} and Seidman \cite{sei89}. \begin{theorem} \label{t2.3a} If$p\geq 1$,$q>1$, and$W\subseteq U\subseteq Y$where the inclusion map of$W$into$U$is compact and the inclusion map of$U$into$Y $is continuous, let \[ S=\{\mathbf{u}\in L^{p}(0,T;W):\mathbf{u}'\in L^{q}(0,T;Y) \text{ and } \|\mathbf{u}\|_{L^{p}(0,T;W)}+\|\mathbf{u}'\|_{L^{q}(0,T;Y)}1$. Then $S$ is pre compact in $C(0,T;U)$. \end{theorem} It suffices to verify that if $\mathbf{v}_{k}$ converges to $\mathbf{v}$ weakly in $X$ and if $\mathbf{v}_{k}^{\ast }\in$ $Q\mathbf{v}_{k}$ then for any $\mathbf{w}\in X$, there exists $\mathbf{v}^{\ast }(\mathbf{w} )\in Q\mathbf{v}$ such that \begin{equation*} \lim \inf_{k\to \infty }\langle \mathbf{v}_{k}^{\ast },\mathbf{v }_{k}-\mathbf{w}\rangle \geq \langle \mathbf{v}^{\ast }( \mathbf{w})\mathbf{,v}-\mathbf{w}\rangle . \end{equation*} This will imply the same is true of $e^{-\lambda (\cdot )}Qe^{\lambda (\cdot )}$ which will show $e^{-\lambda (\cdot )}Qe^{\lambda (\cdot )}$ is pseudomonotone on $X$. Suppose then that $\mathbf{v}_{k}$ converges weakly to $\mathbf{v}$ in $X$ and that $\mathbf{v}_{k}^{\ast }\in Q\mathbf{v}_{k}$ and let $\mathbf{z}_{k}$ be the element of $\partial \eta (\mathbf{v}_{T}-\dot{\mathbf{U}} _{T})$ for $\eta (\mathbf{x})\equiv |\mathbf{x}|$ which satisfies \begin{equation*} \langle \mathbf{v}_{k}^{\ast }\mathbf{,w}\rangle =\int_{0}^{T}\int_{\Gamma _{C}}F((u_{kn}-g)_{+})\mu (|\mathbf{v} _{kT}-\dot{\mathbf{U}}_{T}|)\mathbf{z}_{k}\cdot\mathbf{w} _{T}d\alpha\,dt. \end{equation*} I need to verify that for all $\mathbf{w}\in X$ there exists $\mathbf{v} ^{\ast }(\mathbf{w})\in Q\mathbf{v}$, such that \begin{equation*} \liminf_{k\to \infty }\langle \mathbf{v}_{k}^{\ast }\mathbf{,v }_{k}-\mathbf{w}\rangle \geq \langle \mathbf{v}^{\ast }( \mathbf{w})\mathbf{,v}-\mathbf{w}\rangle \end{equation*} Suppose this does not happen. Then there exists a sequence as described above and $\mathbf{w}\in X$ such that for every $\mathbf{v}^{\ast }\in Q \mathbf{v}$, $$\liminf_{k\to \infty }\langle \mathbf{v}_{k}^{\ast }\mathbf{,v }_{k}-\mathbf{w}\rangle <\langle \mathbf{v}^{\ast }\mathbf{,v}- \mathbf{w}\rangle \label{3maye13}$$ Since $\{ \mathbf{z}_{k}\}$ is bounded in $L^{\infty }(0,T;L^{\infty }(\Gamma _{C})^{d})$ it has a subsequence which converges weak $\ast$ to $\mathbf{z}\in L^{\infty }(0,T;L^{\infty }(\Gamma _{C})^{d})$. Let $U$ be a Sobolev space such that $V$ embeds compactly into $U$ and the trace map from $U$ to $L^{2}(\Gamma _{C})^{d}$ is continuous. By Theorem \ref{t2.4} there is a further subsequence such that $\mathbf{u}_{k}$ converges strongly to $\mathbf{u}$ in $C(0,T;U)$ and by Theorem \ref{t2.3a} there is a further subsequence such that $\mathbf{v}_{k}$ converges strongly to $\mathbf{v}$ in $L^{2}(0,T;U)$. Also $\mathbf{v}_{k}^{\ast }$ is bounded in $\mathcal{V}'$ so a further subsequence converges weak $\ast$ to $\mathbf{v}^{\ast }$. Now using this final subsequence, \begin{equation*} \int_{0}^{T}\int_{\Gamma _{C}}\mathbf{z}_{k}\cdot\mathbf{ w}_{T}d\alpha\,dt \leq \int_{0}^{T}\int_{\Gamma _{C}}|\mathbf{v}_{kT}-\dot{\mathbf{U}} _{T}+\mathbf{w}_{T}|-|\mathbf{v}_{kT}-\dot{\mathbf{U}} _{T}|d\alpha\,dt. \end{equation*} and so, passing to the limit yields \begin{equation*} \int_{0}^{T}\int_{\Gamma _{C}}\mathbf{z\cdot w}_{T}d\alpha\,dt\leq \int_{0}^{T}\int_{\Gamma _{C}}|\mathbf{v}_{T}-\dot{\mathbf{U}}_{T}+ \mathbf{w}_{T}|-|\mathbf{v}_{T}-\dot{\mathbf{U}}_{T}| d\alpha\,dt. \end{equation*} and \begin{equation*} \langle \mathbf{v}_{k}^{\ast }\mathbf{,v}_{k}-\mathbf{w}\rangle =\int_{0}^{T}\int_{\Gamma _{C}}F((u_{kn}-g)_{+})\mu (|\mathbf{v} _{kT}-\dot{\mathbf{U}}_{T}|)\mathbf{z}_{k}\cdot\mathbf{ } (\mathbf{v}_{k}-\mathbf{w})_{T}d\alpha\,dt \end{equation*} so passing to a limit in this expression yields \begin{equation*} \langle \mathbf{v}^{\ast }\mathbf{,v}-\mathbf{w}\rangle =\int_{0}^{T}\int_{\Gamma _{C}}F((u_{n}-g)_{+})\mu (|\mathbf{v} _{T}-\dot{\mathbf{U}}_{T}|)\mathbf{z\cdot }(\mathbf{v}- \mathbf{w})_{T}d\alpha\,dt \end{equation*} showing that \begin{equation*} \lim_{k\to \infty }\langle \mathbf{v}_{k}^{\ast }\mathbf{,v} _{k}-\mathbf{w}\rangle =\langle \mathbf{v}^{\ast }\mathbf{,v}- \mathbf{w}\rangle \end{equation*} and that $\mathbf{v}^{\ast }\in Q\mathbf{v}$ contradicting \eqref{3maye13}. This shows $A$ satisfies all the conditions of Theorem \ref{3mayt1} and this proves Theorem \ref{4mayt2}. \end{proof} Next consider the question of uniqueness. \begin{theorem}\label{5mayt1} In the case that the function $F$ is bounded, the solution to Theorem \ref{4mayt2} is unique. \end{theorem} \begin{proof} Recall the abstract equation of Theorem \ref{4mayt2} is \begin{equation*} \mathbf{v}'+\Sigma \mathbf{v}+Q\mathbf{v}+P\mathbf{u\ni f}\text{ in }\mathcal{V}',\quad \mathbf{v}(0)=\mathbf{v}_{0}\in L^{2}(\Omega )^{d} \end{equation*} where the operators are defined above. Suppose $\mathbf{v}_{i}$ each are solutions for $i=1,2$ and denote by $\mathbf{v}_{i}^{\ast }\in Q\mathbf{v} _{i}$ that which makes the above inclusion an equality. Then it follows from Lemma \eqref{4mayl1} there exists a constant, $K$ such that \begin{equation*} \langle \mathbf{v}_{1}^{\ast }-\mathbf{v}_{2}^{\ast },\mathbf{v}_{1}- \mathbf{v}_{2}\rangle _{\mathcal{V}_{t}}\geq -K\int_{0}^{t}| |\mathbf{v}_{1}(s)-\mathbf{v}_{2}(s)| |_{U}^{2}ds \end{equation*} where $U$ is a Sobolev space such that the embedding of $V$ into $U$ is compact. From \eqref{4maye13}, \begin{equation*} \langle P\mathbf{u}_{1}-P\mathbf{u}_{2},\mathbf{v}_{1}-\mathbf{v} _{2}\rangle _{\mathcal{V}_{t}}\geq -\varepsilon \int_{0}^{t}| |\mathbf{v}_{1}-\mathbf{v}_{2}\| _{V}^{2}ds-K_{\varepsilon }\int_{0}^{t}\int_{0}^{s}\|\mathbf{v} _{1}-\mathbf{v}_{2}\|_{V}^{2}dr\,ds \end{equation*} Also from \eqref{3maye11} it follows \begin{equation*} \langle \Sigma \mathbf{v}_{1}-\Sigma \mathbf{v}_{2},\mathbf{v}_{1}- \mathbf{v}_{2}\rangle _{\mathcal{V}_{t}}\geq \delta ^{2}\int_{0}^{t}\|\mathbf{v}_{1}(t)-\mathbf{v} _{2}(t)\| _{V}^{2}ds-K\int_{0}^{t}\int_{0}^{s}\|\mathbf{v}_{1}-\mathbf{v} _{2}\|_{V}^{2}drds. \end{equation*} Letting $\varepsilon <\delta ^{2}/2$ and adjusting the constants, it follows \begin{aligned} &\frac{1}{2}|\mathbf{v}_{1}(t)-\mathbf{v}_{2}( t)|_{L^{2}(\Omega )^{d}}^{2}+\frac{\delta ^{2}}{2} \int_{0}^{t}\|\mathbf{v}_{1}(t)-\mathbf{v} _{2}(t)\|_{V}^{2}ds \\ &\leq K_{\varepsilon }\int_{0}^{t}\int_{0}^{s}\|\mathbf{v}_{1}- \mathbf{v}_{2}\|_{V}^{2}drds+K\int_{0}^{t}\| \mathbf{v}_{1}(s)-\mathbf{v}_{2}(s)\| _{U}^{2}ds \end{aligned} \label{4maye17} Now the compactness of the embedding of $V$ into $U$ implies for every $\varepsilon >0$ there exists a constant $C_{\varepsilon }$ such that \begin{equation*} \|\mathbf{w}\|_{U}^{2}\leq \varepsilon | |\mathbf{w}\|_{V}^{2}+C_{\varepsilon }|\mathbf{w} |_{L^{2}(\Omega )^{d}}^{2}. \end{equation*} Choosing $\varepsilon$ small enough, \eqref{4maye17} implies \begin{align*} &\frac{1}{2}|\mathbf{v}_{1}(t)-\mathbf{v}_{2}( t)|_{L^{2}(\Omega )^{d}}^{2}+\frac{\delta ^{2}}{4} \int_{0}^{t}\|\mathbf{v}_{1}(t)-\mathbf{v} _{2}(t)\|_{V}^{2}ds \\ &\leq K_{\varepsilon }\int_{0}^{t}\int_{0}^{s}\|\mathbf{v}_{1}- \mathbf{v}_{2}\|_{V}^{2}drds+K_{\varepsilon }\int_{0}^{t}|\mathbf{v}_{1}(s)-\mathbf{v}_{2}( s)|_{L^{2}(\Omega )^{d}}^{2}ds \end{align*} and now the conclusion that $\mathbf{v}_{1}=\mathbf{v}_{2}$ follows from Gronwall's inequality. This proves the theorem. \end{proof} \section{The case of discontinuous $\mu$} Assuming $F$ is bounded, it is possible, as in \cite{KSpsm,kut2002} to extend the existence part of the above results to the case where the coefficient of friction, $\mu$ is discontinuous. This is the situation discussed in every elementary physics book where static friction is greater than sliding friction. Specifically, assume the function $\mu$, has a jump discontinuity at 0, becoming smaller when the sliding speed is positive. Because of the discontinuity of $\mu$ the definition of the friction operator, $Q$ will be modified slightly as follows: $\mathbf{v}^{\ast }\in Q\mathbf{v}$ will mean \begin{equation*} \langle \mathbf{v}^{\ast },\mathbf{w}\rangle _{\mathcal{V}}\leq \int_{0}^{T}\int_{\Gamma _{C}}F((u_{n}-g)_{+})\psi \cdot \big[ |\mathbf{v}_{T}-\dot{\mathbf{U}}_{T}+\mathbf{w} _{T}|-|\mathbf{v}_{T}-\dot{\mathbf{U}}_{T}|\big] d\alpha\,dt \end{equation*} where $\psi$ is in the graph of $\mu$ a.e. $(t,\mathbf{x})$ where $\mu$ is the coefficient of friction, assumed to be decreasing and with a jump at 0. The question of uniqueness is open but the existence theorem is the following. \begin{theorem} Let $\mathbf{u}_{0}\in V$, $\mathbf{v}_{0}\in L^{2}(\Omega ) ^{d}$. Also let $\mu (0+)<\mu (0)$ and $\mu$ is decreasing and Lipschitz continuous on $(0,\infty )$. Then there exists a solution, $\mathbf{v}$, to the following problem. \begin{gather*} \mathbf{v}\in \mathcal{V},\quad \mathbf{v}'\in \mathcal{V}',\quad (u_{n}-g)_{+}\in L^{\infty }(0,T;L^{2}(\Gamma _{C})), \\ \mathbf{v}'+\Sigma \mathbf{v}+P(\mathbf{u})+Q(\mathbf{v})\ni \mathbf{f} \quad \text{in }\mathcal{V}', \\ \mathbf{v}(0)=\mathbf{v}_{0},\quad \mathbf{u}(t)=\mathbf{u}_{0}+\int_{0}^{t} \mathbf{v}(s)ds, \end{gather*} where $\mathbf{v}^{\ast }\in Q(\mathbf{v})$ means \begin{equation*} \langle \mathbf{v}^{\ast },\mathbf{w}\rangle _{\mathcal{V}}\leq \int_{0}^{T}\int_{\Gamma _{C}}F((u_{n}-g)_{+})\psi \big[ |\mathbf{v}_{T}-\dot{\mathbf{U}}_{T}+\mathbf{w}_{T}| -|\mathbf{v}_{T}-\dot{\mathbf{U}}_{T}|\big] d\alpha\,dt \end{equation*} where for a.e. $(t,\mathbf{x})$, \begin{equation*} \psi (t,\mathbf{x})\in [ \mu (0+),\mu (0)] \end{equation*} whenever $(\mathbf{v}_{T}-\dot{\mathbf{U}}_{T})(t,\mathbf{x})=0$ and if $(\mathbf{v}_{T}-\dot{\mathbf{U}}_{T})(t,\mathbf{x})\neq 0$, then for a.e. $(t,\mathbf{x})$, \begin{equation*} \psi (t,\mathbf{x})=\mu (|\mathbf{v}_{T}-\mathbf{ \dot{U}}_{T}|(t,\mathbf{x})). \end{equation*} This solution is the weak limit in $X$ of solutions to the friction contact problem in which the coefficient of friction is Lipschitz continuous. \end{theorem} \begin{proof} In the following argument, it is assumed that whenever necessary, the functions involved are product measurable representatives. Let $\mu _{\varepsilon }(r)=\mu (r)$ for all $r>\varepsilon$, $\mu _{\varepsilon }$ a decreasing function, and $\mu _{\varepsilon }$ is Lipschitz continuous. Thus for $r>0,$ \begin{equation*} \lim_{\varepsilon \to 0}\mu _{\varepsilon }(r)=\mu (r). \end{equation*} See the following picture which describes the situation. \begin{center} \begin{picture}(200,85)(-20,-12) \put(0,-20){\line(0,1){90}} \put(-20,0){\line(1,0){200}} \put(0,30){\qbezier(0,0)(85,-22)(170,-25)} \put(0,30){\circle{5}} \put(5,60){$\mu(0)$} \put(0,60){\circle*{4}} \put(100,20){$\mu(r)$} \put(0,60){\line(2,-5){13.2}} \put(13.2,0){\qbezier(0,-2)(0,0)(0,2)} \put(12,-9){$\varepsilon$} \end{picture} \end{center} Then let $Q_{\varepsilon }$ be defined as before but with $\mu _{\varepsilon}$ in place of $\mu$. Thus by Theorems \ref{4mayt2} and \ref{5mayt1}, there exists a unique solution, $\mathbf{v}_{\varepsilon }$ to the abstract problem \begin{equation*} \mathbf{v}_{\varepsilon }'+\Sigma \mathbf{v}_{\varepsilon }+Q_{\varepsilon }\mathbf{v}_{\varepsilon }+P\mathbf{u}_{\varepsilon } \mathbf{\ni f}\text{ in }\mathcal{V}',\quad \mathbf{v}_{\varepsilon}(0)=\mathbf{v}_{0}\in L^{2}(\Omega )^{d}. \end{equation*} As before, it is convenient to consider an equivalent problem in which the dependent variable, $\mathbf{v}_{\varepsilon \lambda }$ is defined by \begin{equation*} \mathbf{v}_{\varepsilon \lambda }(t)e^{\lambda t} =\mathbf{v}_{\varepsilon }(t). \end{equation*} for $\lambda$ large and positive. As before, this yields $$\begin{gathered} \mathbf{v}_{\varepsilon \lambda }'+\lambda \mathbf{v}_{\varepsilon \lambda }+e^{-\lambda (\cdot )}\Sigma (e^{\lambda ( \cdot )}\mathbf{v}_{\varepsilon \lambda })+e^{-\lambda ( \cdot )}Q_{\varepsilon }(e^{\lambda (\cdot )} \mathbf{v}_{\varepsilon \lambda })+ e^{-\lambda (\cdot )}P(\mathbf{u}_{\varepsilon }) \ni e^{-\lambda (\cdot )}\mathbf{f}\text{ in }\mathcal{V}',\\ \mathbf{v}_{\varepsilon \lambda }(0)=\mathbf{v}_{0} \end{gathered} \label{5maye3}$$ and the operator $B:\mathcal{V\to V}'$ given in \eqref{5maye1} is pseudomonotone, in fact monotone bounded and hemicontinuous while the operator $A_{\varepsilon }$ of \eqref{5maye2} defined by \begin{equation*} \langle A_{\varepsilon }\mathbf{v,v}\rangle _{\mathcal{V}} =\langle \lambda \mathbf{v}+e^{-\lambda (\cdot )}\Sigma (e^{\lambda (\cdot )}\mathbf{v})+e^{-\lambda (\cdot )}Q_{\varepsilon }(e^{\lambda (\cdot ) }\mathbf{v})+e^{-\lambda (\cdot )}P(\mathbf{u} ),\mathbf{v}\rangle _{\mathcal{V}} \end{equation*} is coercive. Furthermore, the coercivity is independent of $\varepsilon$ in the sense that \begin{equation*} \lim_{\|\mathbf{v}\|_{\mathcal{V}}\to \infty }\frac{\langle A_{\varepsilon }\mathbf{v},\mathbf{v} \rangle _{\mathcal{V}}}{\|\mathbf{v}\|_{\mathcal{V}}}=\infty \end{equation*} independent of $\varepsilon >0$. Therefore, there exists a constant $C$ independent of $\varepsilon$ such that for $\mathbf{v}_{\varepsilon \lambda }$ the solution to \eqref{5maye3}, \begin{equation*} \|\mathbf{v}_{\varepsilon \lambda }\|_{\mathcal{V}}\leq C. \end{equation*} Since the various operators in \eqref{5maye3} are bounded, it follows $\{ \mathbf{v}_{\varepsilon \lambda }\}$ is also bounded in $X$, the space of solutions defined above. It follows there exists a subsequence, $\varepsilon \to 0$, still denoted by $\{ \mathbf{v}_{\varepsilon \lambda }\}$ converging weakly to $\mathbf{v}_{\lambda}\in X$. Thus \eqref{5maye3} is of the form $$\mathbf{v}_{\varepsilon \lambda }'+B\mathbf{v}_{\varepsilon \lambda }+e^{-\lambda (\cdot )}Q_{\varepsilon }(e^{\lambda ( \cdot )}\mathbf{v}_{\varepsilon \lambda })\ni e^{-\lambda (\cdot )}\mathbf{f}\text{ in }\mathcal{V}',\;\mathbf{v} _{\varepsilon \lambda }(0)=\mathbf{v}_{0} \label{5maye11}$$ where $B$ is pseudomonotone on $\mathcal{V}$. Let $\mathbf{v}_{\varepsilon \lambda }^{\ast }\in e^{-\lambda (\cdot )}Q_{\varepsilon }(e^{\lambda (\cdot )}\mathbf{v}_{\varepsilon \lambda })$ be such that equality holds in the above inclusion. Letting $\mathbf{v}_{\varepsilon }\equiv e^{\lambda (\cdot )}\mathbf{v} _{\varepsilon \lambda }$ and $\mathbf{v}_{\varepsilon }^{\ast }\equiv e^{\lambda (\cdot )}\mathbf{v}_{\varepsilon \lambda }^{\ast }$, it follows \begin{equation*} \mathbf{v}_{\varepsilon }^{\ast }\in Q_{\varepsilon }(\mathbf{v} _{\varepsilon }). \end{equation*} Then taking a further subsequence, it can be assumed $\mathbf{v} _{\varepsilon }^{\ast }\to \mathbf{v}^{\ast }\in X'$. Let $\mathbf{z}_{\varepsilon }$ be the element of $L^{\infty }( 0,T;L^{\infty }(\Omega )^{d})$ such that $$\langle \mathbf{v}_{\varepsilon }^{\ast },\mathbf{w}\rangle _{ \mathcal{V}}=\int_{0}^{T}\int_{\Gamma _{C}}F((u_{\varepsilon n}-g)_{+})\mu _{\varepsilon }(|\mathbf{v} _{\varepsilon T}-\dot{\mathbf{U}}_{T}|)\mathbf{z}_{\varepsilon }\cdot \mathbf{w}_{T}d\alpha\,dt \label{5maye7}$$ and for all $\mathbf{w}\in \mathcal{V}$, $$\int_{0}^{T}\int_{\Gamma _{C}}\mathbf{z}_{\varepsilon }\cdot\mathbf{w} _{T}d\alpha\,dt \leq \int_{0}^{T}\int_{\Gamma _{C}}|\mathbf{v}_{\varepsilon T} -\dot{\mathbf{U}}_{T}+\mathbf{w}_{T}|-|\mathbf{v}_{\varepsilon T} -\dot{\mathbf{U}}_{T}|d\alpha\,dt. \label{5maye9}$$ By Theorems \ref{t2.3a} and \ref{t2.4}, a subsequence satisfies \begin{equation*} u_{\varepsilon n}(t)\to u_{n}(t) \end{equation*} uniformly in $L^{2}(\Gamma _{C})$ and so \begin{equation*} F((u_{\varepsilon n}-g)_{+})\to F( (u_{n}-g)_{+})\text{ in }L^{2}(\Gamma _{C}) \end{equation*} uniformly which implies a subsequence converges pointwise $a.e$. Taking a further subsequence, \begin{equation*} \mu _{\varepsilon }(|\mathbf{v}_{\varepsilon T}-\dot{\mathbf{U}} _{T}|)\to \psi \text{ weak }\ast \text{ in }L^{\infty }(\left[ 0,T\right] \times \Omega ). \end{equation*} Also, $\mathbf{v}_{\varepsilon }\to \mathbf{v}$ in $L^{2}(0,T;L^{2}(\Gamma _{C})^{d})$ and so a subsequence has the property that the convergence is also pointwise $a.e$. If $(\mathbf{v}_{T}-\dot{\mathbf{U}}_{T})(t,\mathbf{x})>0$, then for all $\varepsilon$ small enough, $(\mathbf{v}_{\varepsilon T}-\dot{\mathbf{U}}_{T})(t,\mathbf{x})$ is bounded away from 0 also and so for such $(t,\mathbf{x})$, \begin{equation*} \lim_{\varepsilon \to 0}\mu _{\varepsilon }(|\mathbf{v} _{\varepsilon T}-\dot{\mathbf{U}}_{T}|)=\mu (| \mathbf{v}_{T}-\dot{\mathbf{U}}_{T}|). \end{equation*} On the other hand, if $(\mathbf{v}_{T}-\dot{\mathbf{U}}_{T}) (t,\mathbf{x})=0$ then for $\eta$ any small positive number it follows that for all $\varepsilon$ small enough, \begin{equation*} \mu _{\varepsilon }(|\mathbf{v}_{\varepsilon T}-\dot{\mathbf{U}} _{T}|(t,\mathbf{x}))\in \left[ \mu ( 0+)-\eta ,\mu (0)\right] \end{equation*} Thus if $E=(\mathbf{v}_{T}-\dot{\mathbf{U}}_{T})^{-1}( 0),$ \begin{equation*} (\alpha \times m)(E)(\mu (0+) -\eta )\leq \int_{0}^{T}\int_{\Gamma _{C}}\mathcal{X}_{E}\psi d\alpha\,dt \leq (\alpha \times m)(E)\mu (0) \end{equation*} which requires $\psi (t,\mathbf{x})\in [ (\mu ( 0+)-\eta ),\mu (0)]$ a.e. Since $\eta$ is arbitrary, it follows that for these values of $(t,\mathbf{x}) ,\psi (t,\mathbf{x})\in [ \mu (0+),\mu ( 0)]$ a.e. Thus for a.e. $(t,\mathbf{x}),\psi (t,\mathbf{x})$ is in the graph of $(t,\mathbf{x}) \to \mu (|\mathbf{v}_{T}-\dot{\mathbf{U}}_{T}|)$. Now consider \eqref{5maye7}. It follows \begin{aligned} \langle \mathbf{v}_{\varepsilon }^{\ast },\mathbf{w}\rangle _{ \mathcal{V}} &=\int_{0}^{T}\int_{\Gamma _{C}}F((u_{\varepsilon n}-g)_{+})\mu _{\varepsilon }(|\mathbf{v} _{\varepsilon T}-\dot{\mathbf{U}}_{T}|)\mathbf{z}_{\varepsilon }\cdot \mathbf{w}_{T}d\alpha\,dt\\ &\leq \int_{0}^{T}\int_{\Gamma _{C}}F((u_{\varepsilon n}-g)_{+})\mu _{\varepsilon }(|\mathbf{v} _{\varepsilon T}-\dot{\mathbf{U}}_{T}|) \\ &\quad\cdot \big[ |\mathbf{v}_{\varepsilon T}-\dot{\mathbf{U}}_{T}+\mathbf{w} _{T}|-|\mathbf{v}_{\varepsilon T}-\dot{\mathbf{U}}_{T}| \big] d\alpha\,dt \end{aligned}\label{5maye17} Therefore, passing to the limit as $\varepsilon \to 0$ in the above, it follows $$\langle \mathbf{v}^{\ast },\mathbf{w}\rangle _{\mathcal{V}} \leq \int_{0}^{T}\int_{\Gamma _{C}}F((u_{n}-g)_{+})\psi (|\mathbf{v}_{T}-\dot{\mathbf{U}}_{T}+\mathbf{w}_{T}| -|\mathbf{v}_{T}-\dot{\mathbf{U}}_{T}|)d\alpha\,dt \label{5maye19}$$ Note the strong convergence properties of $\{ \mathbf{v}_{\varepsilon T}\}$ also imply \begin{equation*} \lim \sup_{\varepsilon \to 0}\langle \mathbf{v}_{\varepsilon }^{\ast },\mathbf{v}_{\varepsilon }-\mathbf{v}\rangle _{\mathcal{V} }\leq 0. \end{equation*} To see this consider \eqref{5maye17} with $\mathbf{w}$ replaced with $\mathbf{v}_{\varepsilon }-\mathbf{v}$. However, you can also replace $\mathbf{w}$ with $\mathbf{v-v}_{\varepsilon }$ and conclude \begin{equation*} 0\geq \limsup_{\varepsilon \to 0}\langle \mathbf{v} _{\varepsilon }^{\ast },\mathbf{v}-\mathbf{v}_{\varepsilon }\rangle _{ \mathcal{V}}=-\liminf_{\varepsilon \to 0}\langle \mathbf{v} _{\varepsilon }^{\ast },\mathbf{v}_{\varepsilon }-\mathbf{v}\rangle _{ \mathcal{V}} \end{equation*} so that $$0\leq \liminf_{\varepsilon \to 0}\langle \mathbf{v} _{\varepsilon }^{\ast },\mathbf{v}_{\varepsilon }-\mathbf{v}\rangle _{ \mathcal{V}}\leq \limsup_{\varepsilon \to 0}\langle \mathbf{v} _{\varepsilon }^{\ast },\mathbf{v}_{\varepsilon }-\mathbf{v}\rangle _{ \mathcal{V}}\leq 0. \label{5maye15}$$ Now recall $$\mathbf{v}_{\varepsilon \lambda }'+B\mathbf{v}_{\varepsilon \lambda }+\mathbf{v}_{\varepsilon \lambda }^{\ast }=e^{-\lambda (\cdot ) }\mathbf{f}\text{ in }\mathcal{V}',\quad \mathbf{v}_{\varepsilon \lambda }(0)=\mathbf{v}_{0} \label{5maye13}$$ where \begin{equation*} \mathbf{v}_{\varepsilon \lambda }^{\ast }\in e^{-\lambda (\cdot )}Q_{\varepsilon }(e^{\lambda (\cdot )}\mathbf{v} _{\varepsilon \lambda }) \end{equation*} Hence letting $\mathbf{v}_{\lambda }^{\ast }\equiv$ $e^{-\lambda ( \cdot )}\mathbf{v}^{\ast }$ where $\mathbf{v}^{\ast }$ is from \eqref {5maye19} and $e^{-\lambda (\cdot )}\mathbf{v}\equiv \mathbf{v} _{\lambda }$, it follows $\mathbf{v}_{\lambda }^{\ast }\in e^{-\lambda (\cdot )}Q(e^{\lambda (\cdot )}\mathbf{v} _{\lambda })$, where the following limits hold. \begin{gather*} \mathbf{v}_{\varepsilon \lambda }^{\ast } \to \mathbf{v}_{\lambda }^{\ast }\quad \text{weak }\ast \text{ in }\mathcal{V}' \\ \mathbf{v}_{\varepsilon \lambda } \to \mathbf{v}_{\lambda }\quad \text{ weakly in }\mathcal{V} \\ \mathbf{v}_{\varepsilon \lambda }' \to \mathbf{v} _{\lambda }'\quad \text{weak }\ast \text{ in }\mathcal{V}' \end{gather*} Taking another subsequence if necessary, it can also be assumed \begin{equation*} B\mathbf{v}_{\varepsilon \lambda }\to \mathbf{g}\quad \text{ weak }\ast \text{ in }\mathcal{V}'. \end{equation*} Passing to a limit in \eqref{5maye13}, \begin{equation*} \mathbf{v}_{\lambda }'+\mathbf{g}+\mathbf{v}_{\lambda }^{\ast }=e^{-\lambda (\cdot )}\mathbf{f}\text{ in }\mathcal{V}',\quad \mathbf{v}_{\lambda }(0)=\mathbf{v}_{0} \end{equation*} and it only remains to identify $\mathbf{g}$ with $B\mathbf{v}_{\lambda }$. From \eqref{5maye13}, \begin{align*} &\Big(\langle \mathbf{v}_{\varepsilon \lambda }'-\mathbf{v} _{\lambda }',\mathbf{v}_{\varepsilon \lambda }-\mathbf{v}_{\lambda }\rangle +\langle \mathbf{v}_{\lambda }',\mathbf{v} _{\varepsilon \lambda }-\mathbf{v}_{\lambda }\rangle +\langle B\mathbf{v}_{\varepsilon \lambda },\mathbf{v} _{\varepsilon \lambda }-\mathbf{v}_{\lambda }\rangle +\langle \mathbf{v}_{\varepsilon \lambda }^{\ast },\mathbf{v}_{\varepsilon \lambda }- \mathbf{v}_{\lambda }\rangle \Big)\\ &=\langle e^{-\lambda (\cdot )}\mathbf{f},\mathbf{v}_{\varepsilon \lambda } -\mathbf{v}_{\lambda }\rangle \end{align*} and so \begin{equation*} \langle \mathbf{v}_{\lambda }',\mathbf{v}_{\varepsilon \lambda }-\mathbf{v}_{\lambda }\rangle +\langle B\mathbf{v}_{\varepsilon \lambda },\mathbf{v}_{\varepsilon \lambda }-\mathbf{v}_{\lambda }\rangle +\langle \mathbf{v}_{\varepsilon \lambda }^{\ast }, \mathbf{v}_{\varepsilon \lambda }-\mathbf{v}_{\lambda }\rangle \leq \langle e^{-\lambda (\cdot )}\mathbf{f},\mathbf{v} _{\varepsilon \lambda }-\mathbf{v}_{\lambda }\rangle . \end{equation*} Hence from \eqref{5maye15}, \begin{align*} &\lim \sup_{\varepsilon \to 0}(\langle \mathbf{v} _{\lambda }',\mathbf{v}_{\varepsilon \lambda }-\mathbf{v}_{\lambda }\rangle +\langle B\mathbf{v}_{\varepsilon \lambda },\mathbf{v} _{\varepsilon \lambda }-\mathbf{v}_{\lambda }\rangle +\langle \mathbf{v}_{\varepsilon \lambda }^{\ast },\mathbf{v}_{\varepsilon \lambda }- \mathbf{v}_{\lambda }\rangle )\\ &=\lim \sup_{\varepsilon \to 0}\langle B\mathbf{v} _{\varepsilon \lambda },\mathbf{v}_{\varepsilon \lambda }-\mathbf{v} _{\lambda }\rangle \leq 0. \end{align*} which implies \begin{equation*} \lim \inf_{\varepsilon \to 0}\langle B\mathbf{v}_{\varepsilon \lambda },\mathbf{v}_{\varepsilon \lambda }-\mathbf{v}_{\lambda }\rangle \geq \langle B\mathbf{v}_{\lambda },\mathbf{v}_{\lambda }-\mathbf{v}_{\lambda }\rangle =0 \end{equation*} and so $\lim_{\varepsilon \to 0}\langle B\mathbf{v} _{\varepsilon \lambda },\mathbf{v}_{\varepsilon \lambda }-\mathbf{v} _{\lambda }\rangle =0$. Since $B$ is pseudomonotone, it follows that for all $\mathbf{w}\in \mathcal{V}$, \begin{align*} \lim \inf_{\varepsilon \to 0}\langle B\mathbf{v}_{\varepsilon \lambda },\mathbf{v}_{\varepsilon \lambda }-\mathbf{w}\rangle &= \lim \inf_{\varepsilon \to 0}(\langle B\mathbf{v}_{\varepsilon \lambda },\mathbf{v}_{\varepsilon \lambda }-\mathbf{v}_{\lambda }\rangle +\langle B\mathbf{v}_{\varepsilon \lambda },\mathbf{v} _{\lambda }-\mathbf{w}\rangle )\\ &= \lim \inf_{\varepsilon \to 0}\langle B\mathbf{v} _{\varepsilon \lambda },\mathbf{v}_{\lambda }-\mathbf{w}\rangle =\langle \mathbf{g},\mathbf{v}_{\lambda }-\mathbf{w}\rangle \\ &\geq \langle B\mathbf{v}_{\lambda },\mathbf{v}_{\lambda }-\mathbf{w} \rangle \end{align*} and so \begin{equation*} \langle \mathbf{g,v}_{\lambda }-\mathbf{w}\rangle \geq \langle B\mathbf{v}_{\lambda },\mathbf{v}_{\lambda }-\mathbf{w} \rangle \end{equation*} and since $\mathbf{w}$ was arbitrary, this shows $\mathbf{g}=B\mathbf{v} _{\lambda }$. It follows there exists a solution to \begin{equation*} \mathbf{v}_{\lambda }'+B\mathbf{v}_{\lambda }+e^{-\lambda ( \cdot )}Qe^{\lambda (\cdot )}\mathbf{v}_{\lambda }=e^{-\lambda (\cdot )}\mathbf{f}\text{ in }\mathcal{V}',\quad \mathbf{v}_{\lambda }(0)=\mathbf{v}_{0} \end{equation*} which implies there exists a solution to \begin{equation*} \mathbf{v}'+\Sigma \mathbf{v}+P(\mathbf{u})+Q( \mathbf{v})\ni \mathbf{f},\quad\mathbf{v}(0)=\mathbf{v}_{0}. \end{equation*} This proves the theorem. \end{proof} \begin{thebibliography}{00} \bibitem{ada75} Adams R., \textit{Sobolev Spaces,} Academic Press, New York (1975). \bibitem{and97} Andrews A., Kuttler K. and Shillor M.; On the Dynamic behavior of a Themoviscoelastic Body in Frictional Contact with a rigid obstacle. \textit{European Journal of Applied Mathematics} (1997 ), vol.8, pp. 417-436. \bibitem{duv76} Duvaut G. and Lions J.L.; \textit{Inequalities in Mechanics and Physics}, Springer-Verlag, Berlin Heidelberg New York\textit{\ (1976).} \bibitem{fig95} I. 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