# Well-posedness of evolutionary differential variational-hemivariational
inequalities and applications to frictional contact mechanics

Nadia Skoglund Taki and Kundan Kumar  Center for Modeling of Coupled
Subsurface Dynamics
Department of Mathematics
University of Bergen
Postbox 7800
5020 Bergen
Norway<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the well-posedness of a class of evolutionary
variational-hemivariational inequalities coupled with a nonlinear ordinary
differential equation in Banach spaces. The proof is based on an iterative
approximation scheme showing that the problem has a unique mild solution. In
addition, we established the continuity of the flow map with respect to the
initial data. Under the general framework, we consider two new applications
for modelling of frictional contact for viscoelastic materials. In the first
application, we consider Coulomb friction with normal compliance, and in the
second, normal damped response. The structure of the friction coefficient
$\mu$ is new with motivation from geophysical applications in earth sciences
with dependence on an external state variable $\alpha$ and the slip rate
$|\dot{u}_{\tau}|$.

###### Key words and phrases:

Variational-hemivariational inequality, Well-posedness, Contact problem,
Modelling

###### 2020 Mathematics Subject Classification:

Primary: 47J20, 49K40; Secondary: 74M15

##  1\. Introduction

This work concerns the study of an evolutionary differential variational-
hemivariational inequality modelling mathematical problems from contact
mechanics. These systems are relevant for many physical phenomena ranging from
engineering to biology (see, e.g., [6, 9, 33] and the references therein). We
are interested in frictional contact phenomena for viscoelastic materials with
the linearized strain tensor, which have been studied intensively, see, e.g.,
some of the relevant books [9, 23, 33].
Let $V$ and $Y$ be two Banach spaces, and for $T>0$, we let $[0,T]$ be the
time interval of interest. Then, the Cauchy problem under consideration reads

(1.1a) $\displaystyle\dot{\alpha}(t)=\mathcal{G}(t,\alpha(t),Mw(t)),$ (1.1b)
$\displaystyle\langle{\dot{w}(t)+A(t,w(t))-f(t)+\mathcal{R}w(t)},{v-w(t)}\rangle+\varphi(t,\alpha(t),\mathcal{S}_{\varphi}w(t),Mw(t),Kv)$
$\displaystyle-\varphi(t,\alpha(t),\mathcal{S}_{\varphi}w(t),Mw(t),Kw(t))+j^{\circ}(t,\alpha(t),\mathcal{S}_{j}w(t),Nw(t);Nv-
Nw(t))\geq 0,$ for all $v\in V$, a.e. $t\in(0,T)$ with (1.1c) $w(0)=w_{0},\ \
\alpha(0)=\alpha_{0}.$

Here, $A$ and $\mathcal{R}$ are nonlinear operators related to the
viscoelastic constitutive laws. Further, $j^{\circ}$ is a generalized
directional derivative of a functional $j$. The functionals $\varphi$ and $j$
are determined by contact boundary conditions. We require $\varphi$ to be
convex in its last argument, while $j$ may be nonconvex with appropriate
structures given later (see Section 3). The operators $\mathcal{S}_{\varphi}$
and $\mathcal{S}_{j}$ relate to the contact conditions, and $\mathcal{G}$ is
assumed to be a nonlinear operator related to the change in the external state
variable $\alpha$. The data $f$ is related to the given body forces and
surface traction, and $w_{0}$ and $\alpha_{0}$ represent the initial data.
Lastly, $M$, $N$, and $K$ are bounded linear operators related to the
tangential and normal trace operators. The Cauchy problem (1.1b)-(1.1c) is
called a hemivariational inequality if $\varphi\equiv 0$ and variational
inequality if $j^{\circ}\equiv 0$. Moreover, a solution to (1.1) is understood
in the mild sense.

######  Definition 1.1.

A pair of functions $(w,\alpha)$, where $\alpha\in C([0,T];Y)$ and
$w:[0,T]\rightarrow V$ measurable, is said to be a mild solution of (1.1) if
$\alpha$ and $w$, respectively, satisfy

$\alpha(t)=\alpha_{0}+\int_{0}^{t}\mathcal{G}(s,\alpha(s),Mw(s))ds$

and (1.1b)-(1.1c).

The main purpose of this paper is to extend the results from [19, 29] to prove
well-posedness of (1.1) with applications to rate-and-state frictional contact
problems. We prove that the pair $(w,\alpha)$ is a solution to (1.1) in the
sense of Definition 1.1 and that the flow map depends continuously on the
initial data. The problem setting is motivated by [26, 33, 29, 35], and the
techniques have taken inspiration from [20].

###  1.1. Former well-posedness results

Special cases of (1.1) have been investigated in literature. The recent work
[19] is closest to our setting. They prove well-posedness for an ordinary
differential equation coupled with a variational-hemivariational inequality
with applications to viscoplastic material and viscoelasticity with adhesion.
In fact, if we let $\varphi$ be independent of $Mw$ in its third argument and
relax the more generalized structure of $\varphi$ (see Remark 3.2 for more
details), then (1.1) reduces to the problem studied in [19]. However, keeping
the dependence of $Mw$ in $\varphi$ and a generalized structure of $\varphi$
(see Remark 3.2) allows us to include applications with a new structure of the
friction coefficient. On the other hand, neglecting $\alpha$ and $Mw$ in
$\varphi$ and $\alpha$ in $j^{\circ}$, existence and uniqueness is provided in
[35, Section 10.3]. If we let $\varphi\equiv 0$ and $j^{\circ}$ be independent
of $\alpha$ and $\mathcal{S}_{j}w$, existence and uniqueness was proved in
[14, Section 6].

In the quasi-static case tackled in [29] with $j^{\circ}\equiv 0$ and a
simplified structure of $\varphi$ (see Remark 3.2), they proved existence and
uniqueness of the solution pair by an implicit method, where they rewrite
(1.1a) to only depend on $w$. However, the setting of [29] is not applicable
in our case as the inertial term restricts the space-time regularity for $w$.
We refer to [19, p.2] for further discussion.

###  1.2. Physical setting

A mathematical model in contact mechanics needs several relations: a
constitutive law, a balance equation, boundary conditions, interface laws, and
initial conditions. The constitutive laws help us describe the material’s
mechanical reactions (stress-strain type). In most cases, constitutive laws
originate from experiments, though they are verified to satisfy certain
invariance principles. We refer to [13, Chapter 6] for a general description
of several diagnostic experiments which provide the needed information to
construct constitutive laws for specific materials. The interface laws are
prescribed on the possible contact surface. We refer to the interface laws in
tangential direction as friction laws and in normal direction as contact
conditions. The mathematical treatment of these problems gives rise to the
variational-hemivariational inequalities of the form (1.1b)-(1.1c) where we
put appropriate constraints on the operators to fit the applications of
interest.
We are mainly interested in studying frictional problems with the following
dependencies:

(1.2) $\displaystyle\mu$ $\displaystyle=\mu(|\dot{u}_{\tau}(t)|,\alpha(t)),$
$\displaystyle\dot{\alpha}(t)=G(\alpha(t),|\dot{u}_{\tau}(t)|).$

One application with the dependencies seen in (1.2) is a memory-dependent
friction coefficient (see, e.g., [24, Section 5.3]), which is also referred to
as rate-and-state friction law. This is modelled via an ODE, where the _state
variable_ $\alpha$ tracks information of the contact surface using the slip
rate $|\dot{u}_{\tau}(t)|$ found from solving (1.1b)-(1.1c) and then updates
the friction coefficient. Under certain constraints we may consider $\alpha$
as the surface temperature or humidity on the contact surface.

In [29], they assume that $\mu(|\dot{u}_{\tau}(t)|,\alpha(t))$ is bounded and
Lipschitz with respect to both arguments. Additionally to the boundeness
assumption on the friction coefficient, [19] considers applications in the
frictionless setting and $\mu=\mu(u_{\nu}(t))$. Our framework is therefore an
extension of the frameworks in [29, 19], where we need to use different
techniques to prove the well-posedness of (1.1) in the sense of Definition
1.1. Lastly, $\mu=\mu(|\dot{u}_{\tau}|)$ is covered in [23, Section 6.3 and
8.1] (see also [14, p.185-187]). A discussion on many different friction
models can be found in [39].

###  1.3. Contributions and outline

The novelties of this paper are:

  * •

Well-posedness of (1.1) in the sense of Definition 1.1, where the proof is
based on an iterative decoupling approach that directly gives rise to a
numerical method.

  * •

A more complicated structure of $\varphi$ (see Remark 3.2) that allows for a
larger set of conditions on the contact surface.

  * •

Two new applications with Coulomb’s friction. One contact problem is with
normal compliance, and the latter is with normal damped response. The general
framework allows a new structure for the friction coefficient that can be
unbounded.

The paper is organized as follows. In Section 2, we introduce the function
spaces and some basics of nonsmooth analysis in order to better understand the
problem setting. In Section 3, we present our problem statement and the
assumptions on the data. The section ends with our main result, Theorem 3.3,
that summarizes the well-posedness of (1.1) in the sense of Definition 1.1.
The proof of the theorem is presented in Section 5 utilizing a preliminary
result stated in Section 4. Next, two applications fitting our framework will
be introduced in Section 6.1-6.2. In Section 6.3, we introduce an application
motivated by earth sciences. Lastly, in Appendix A, we include remarks on the
assumptions needed in the proof of Theorem 3.3, 6.3, and 6.8. Appendix B-G
contains proofs of results that are similar to ones found elsewhere but needed
throughout the paper.

###  1.4. Notation

We now present some notations that will be used in this paper.

  * •

Let $0<T<\infty$ be the maximal time.

  * •

Let $d$ denote the dimension. In the applications, $d=2,3$.

  * •

A point in $\mathbb{R}^{d}$ is denoted by $x=(x_{i})$, $1\leq i\leq d$.

  * •

$\mathbb{S}^{d}$ denotes the space of second order symmetric tensors on
$\mathbb{R}^{d}$.

  * •

We denote $|\cdot|$ as the Euclidean norm.

  * •

$\Omega\subset\mathbb{R}^{d}$ is a bounded open connected subset with a
Lipschitz boundary $\Gamma=\partial\Omega$. We split $\Gamma$ into three
disjoint parts; $\Gamma_{D}$, $\Gamma_{N}$, and $\Gamma_{C}$ with
$\mathrm{meas}(\Gamma_{D})>0$, $\mathrm{meas}(\Gamma_{C})>0$, i.e., nonzero
Lebesgue measure, but $\Gamma_{N}$ is allowed to be empty.

  * •

In the applications, $\Omega$ is the reference configuration of a viscoelastic
deformable body sliding on a foundation. Moreover, $\Gamma_{D}$ denotes the
Dirichlet boundary, $\Gamma_{N}$ the Neumann boundary, and $\Gamma_{C}$ is the
contact boundary.

  * •

$\nu$ denotes the outward normal on $\Gamma$.

  * •

We denote $\bar{\Omega}=\Omega\cup\Gamma$.

  * •

$L^{p}(\Omega)$ denotes the space of Lebesgue $p$-integrable functions
equipped with the norm $\lVert
v\rVert_{L^{p}(\Omega)}=\big{(}\int_{\Omega}|v|^{p}dx\big{)}^{1/p}$ for $1\leq
p<\infty$. With the usual modifications for $L^{\infty}(\Omega)$.

  * •

$C^{\infty}_{c}(\Omega)$ denotes the space of infinitely differentiable
functions with compact support.

  * •

We will denote $c$ as a positive constant, which might change from line to
line.

  * •

Let $h(t)$ be a function, then we denote the time derivative of $h(t)$ by
$\dot{h}(t)$ and the double time derivative as $\ddot{h}(t)$. Assuming here
that $h(t)$ has enough regularity such that it makes sense to take the time
derivative of it twice.

  * •

$(\tilde{X},\lVert\cdot\rVert_{\tilde{X}})$ and
$(\tilde{Y},\lVert\cdot\rVert_{\tilde{Y}})$ denote arbitrary (separable
reflexive) Banach spaces. For the convenience of the reader, this notation
will be used when introducing general theory. If the theory is only needed for
a specific (separable reflexive) Banach space, we write the specific space.

  * •

The dual space of $\tilde{X}$ will be denoted by $\tilde{X}^{\ast}$.

  * •

$2^{\tilde{X}^{\ast}}$ denotes the set containing all subsets of
$\tilde{X}^{\ast}$.

  * •

The dual product between the spaces $\tilde{X}$ and $\tilde{X}^{\ast}$ will be
denoted by $\langle{\cdot},{\cdot}\rangle_{\tilde{X}^{\ast}\times\tilde{X}}$.

  * •

In the particular case, when $\tilde{X}$ is an inner product space, we denote
$\big{(}{\cdot},{\cdot}\big{)}_{\tilde{X}}$ as its inner product.

  * •

$U$, $V$, $X$, $Y$, and $Z$ denote real separable reflexive Banach spaces, and
$H$ is a real Hilbert space.

  * •

The embedding $V\subset H\subset V^{\ast}$ is referred to as an evolution
triple. Here, the embedding $V\subset H$ is continuous and $V$ dense in $H$.
It follows that $H\subset V^{\ast}$ is continuously embedded (see, e.g., [31,
Section 7.2]). Moreover, $V\subset H$ is compactly embedded, leading to
$H\subset V^{\ast}$ being compactly embedded (see, e.g., [7, Remark 3.4.4]).

  * •

For simplicity of notation, the dual product between $V^{\ast}$ and $V$ is
denoted by
$\langle{\cdot},{\cdot}\rangle=\langle{\cdot},{\cdot}\rangle_{V^{\ast}\times
V}$.

  * •

$\mathcal{L}(\tilde{X},\tilde{Y})$ denotes the set of all bounded linear maps
from $\tilde{X}$ into $\tilde{Y}$.

  * •

We denote the operator norm of the operators $M:V\rightarrow U$,
$N:V\rightarrow X$, and $K:V\rightarrow Z$ as $\lVert M\rVert=\lVert
M\rVert_{\mathcal{L}(V,U)}$, $\lVert N\rVert=\lVert
N\rVert_{\mathcal{L}(V,X)}$, and $\lVert K\rVert=\lVert
K\rVert_{\mathcal{L}(V,Z)}$, respectively.

###### Contents

  1. 1 Introduction
  2. 2 Function spaces and basics of nonsmooth analysis
  3. 3 Problem statement and main result
  4. 4 Preliminary result
  5. 5 Proof of Theorem 3.3
  6. 6 Viscoelastic frictional contact problems
  7. Acknowledgements
  8. A Comments on assumptions
  9. B Proof of Proposition 5.2
  10. C Proof of Lemma 5.3
  11. D Proof of Corollary 5.6
  12. E Proof of Lemma 5.7
  13. F Proof of Lemma 6.7
  14. G Proof of Lemma 6.12

##  2\. Function spaces and basics of nonsmooth analysis

In this section, we present the function spaces and fundamental results. For
further information, we refer to standard textbooks, e.g., [4, 10, 25, 31].

###  2.1. Sobolev spaces

This section defines the solution spaces and the usual Sobolev spaces, which
will become useful in Section 3 and in the applications, i.e., Section 6. We
define

(2.1) $\displaystyle L^{p}(\Omega;\mathbb{R}^{d})$
$\displaystyle=\\{v=(v_{i}):v_{i}\in L^{p}(\Omega),\ 1\leq i\leq d\\},$ (2.2)
$\displaystyle L^{p}(\Omega;\mathbb{S}^{d})$
$\displaystyle=\\{\tau=(\tau_{ik}):\tau_{ik}=\tau_{ki}\in L^{p}(\Omega),\
1\leq i,k\leq d\\},$

for $1\leq p\leq\infty$. The associated norms will be denoted
$\lVert\cdot\rVert_{L^{p}(\Omega;\mathbb{R}^{d})}$ and
$\lVert\cdot\rVert_{L^{p}(\Omega;\mathbb{S}^{d})}$, respectively. For $p=2$,
(2.1)-(2.2) are Hilbert spaces with the canonical inner products

$\displaystyle\big{(}{u},{v}\big{)}_{L^{2}(\Omega;\mathbb{R}^{d})}$
$\displaystyle=\int_{\Omega}u_{i}v_{i}dx=\int_{\Omega}u\cdot vdx,$
$\displaystyle\big{(}{\sigma},{\tau}\big{)}_{L^{2}(\Omega;\mathbb{S}^{d})}$
$\displaystyle=\int_{\Omega}\sigma_{ik}\tau_{ik}dx=\int_{\Omega}\sigma:\tau
dx$

for all $u,v\in L^{2}(\Omega;\mathbb{R}^{d})$, $\sigma,\tau\in
L^{2}(\Omega;\mathbb{S}^{d})$. Moreover, we define the spaces

$H^{1}(\Omega)=\\{v\in L^{2}(\Omega):\text{ the weak derivatives
}\frac{\partial v}{\partial x_{i}}\text{ exist in }L^{2}(\Omega),\ 1\leq i\leq
d\\}$

and

$H^{1}(\Omega;\mathbb{R}^{d})=\\{v=(v_{i}):v_{i}\in H^{1}(\Omega),\ 1\leq
i\leq d\\}.$

With abuse of notation, the trace of functions $v\in
H^{1}(\Omega;\mathbb{R}^{d})$ on $\Gamma$ will still be denoted by $v$. For
the displacement, we use the space

(2.3) $V=\\{v\in H^{1}(\Omega;\mathbb{R}^{d}):v=0\ \text{ on }\Gamma_{D}\\}.$

As a consequence of $\mathrm{meas}(\Gamma_{D})>0$, it follows by Korn’s
inequality, i.e.,
$\lVert\varepsilon(v)\rVert_{L^{2}(\Omega;\mathbb{S}^{d})}\geq c\lVert
v\rVert_{H^{1}(\Omega;\mathbb{R}^{d})}$ (see, e.g., [17, Lemma 6.2]), that $V$
is a Hilbert space with the canonical inner product

$\big{(}{u},{v}\big{)}_{V}=\int_{\Omega}\varepsilon(u):\varepsilon(v)dx.$

Here, $\varepsilon:H^{1}(\Omega;\mathbb{R}^{d})\rightarrow
L^{2}(\Omega;\mathbb{S}^{d})$ is the deformation operator defined by

$\varepsilon(u)=(\varepsilon_{ik}(u)),\ \ \
\varepsilon_{ik}(u)=\frac{1}{2}\Big{(}\frac{\partial u_{k}}{\partial
x_{i}}+\frac{\partial u_{i}}{\partial x_{k}}\Big{)}.$

We denote the associated norm on $V$ by $\lVert\cdot\rVert_{V}$. Moreover, if
$\sigma$ is a regular function, say $\sigma\in
C^{1}(\bar{\Omega};\mathbb{S}^{d})$, the following Green’s formula holds

(2.4)
$\displaystyle\int_{\Omega}(\nabla\cdot\sigma)vdx=\int_{\Gamma}\sigma\nu\cdot
vda-\int_{\Omega}\sigma:\varepsilon(v)dx\ \text{ for all }v\in
H^{1}(\Omega;\mathbb{R}^{d}).$

We also need the following trace theorem from [1, Theorem 4.12].

######  Theorem 2.1.

Let $\Omega\subset\mathbb{R}^{d}$ be bounded with Lipschitz boundary $\Gamma$
and $\Gamma_{C}\subset\Gamma$ be such that $\mathrm{meas}(\Gamma_{C})>0$. Then
there exists a linear continuous operator $\gamma:V\rightarrow
L^{q}(\Gamma_{C};\mathbb{R}^{d})$ satisfying

$\displaystyle\lVert\gamma v\rVert_{L^{q}(\Gamma_{C};\mathbb{R}^{d})}$
$\displaystyle\leq\lVert\gamma\rVert_{\mathcal{L}(V,L^{q}(\Gamma_{C};\mathbb{R}^{d}))}\lVert
v\rVert_{V}$

for all $v\in V$. If $d=2$, then $2\leq q<\infty$. And $d>2$, $2\leq q\leq
p^{\ast}=\frac{4}{d-2}$.

We denote $\mathbb{X}=\prod_{i=1}^{\ell}\tilde{X}_{i}$ as a Cartesian product
space, for some $\ell\in\mathbb{Z}_{+}$, where
$(\tilde{X}_{i},\lVert\cdot\rVert_{\tilde{X}_{i}})$ are normed spaces for
$i=1,\dots,\ell$. Then, $\mathbb{X}$ is equipped with the norm

$\lVert(v_{1},\dots,v_{\ell})\rVert_{\mathbb{X}}=\sum_{i=1}^{\ell}\lVert
v_{i}\rVert_{\tilde{X}_{i}}\ \ \text{ for all $v_{i}\in\tilde{X}_{i}$,
$i=1,\dots,\ell$}.$

Equivalently, we may equip $\mathbb{X}$ with the norm

$\lVert(v_{1},\dots,v_{\ell})\rVert_{\mathbb{X}}^{2}=\sum_{i=1}^{\ell}\lVert
v_{i}\rVert_{\tilde{X}_{i}}^{2}\ \ \text{ for all $v_{i}\in\tilde{X}_{i}$,
$i=1,\dots,\ell$}.$

###  2.2. Time-dependent spaces

Let $(V,H,V^{\ast})$ be an evolution triple.

######  Definition 2.2.

Let $\tilde{X}$ be a Banach space, and $T>0$. The space
$L^{p}(0,T;\tilde{X})$, $1\leq p\leq\infty$, consists of all measurable
functions $v:[0,T]\rightarrow\tilde{X}$ such that

$\displaystyle\int_{0}^{T}\lVert v(t)\rVert_{\tilde{X}}^{p}dt<\infty.$

With the usual modifications for $L^{\infty}(0,T;\tilde{X})$. For brevity, we
use the standard short-hand notation

$L^{p}_{t}\tilde{X}=L^{p}(0,t;\tilde{X})$

for all $t\in[0,T]$ and $1\leq p\leq\infty$.

We also introduce the solution space

(2.5) $\displaystyle\mathcal{W}^{1,2}_{T}=\\{w\in L^{2}_{T}V:\dot{w}\in
L^{2}_{T}V^{\ast}\\},$

equipped with the norm $\lVert w\rVert_{\mathcal{W}^{1,2}_{T}}^{2}=\lVert
w\rVert_{L^{2}_{T}V}^{2}+\lVert\dot{w}\rVert_{L^{2}_{T}V^{\ast}}^{2}$. The
duality pairing between $L^{2}_{T}V^{\ast}$ and $L^{2}_{T}V$ is denoted by

$\langle{\tilde{v}},{v}\rangle_{L^{2}_{T}V^{\ast}\times
L^{2}_{T}V}=\int_{0}^{T}\langle{\tilde{v}(s)},{v(s)}\rangle ds\ \ \text{ for
all }\tilde{v}\in L^{2}_{T}V^{\ast},v\in L^{2}_{T}V.$

We denote the space of continuous functions defined on $[0,T]$ with values in
$\tilde{X}$ by

$C([0,T];\tilde{X})=\\{h:[0,T]\rightarrow\tilde{X}:h\text{ is continuous and
}\lVert h\rVert_{L^{\infty}_{T}{\tilde{X}}}<\infty\\}.$

The next proposition can be found in, e.g., [7, Proposition 3.4.14] or [31,
Lemma 7.3] and will help us provide estimates.

######  Proposition 2.3.

Let $(V,H,V^{\ast})$ be an evolution triple in space, and $0<T<\infty$. Then,
for any $v_{1},v_{2}\in\mathcal{W}^{1,2}_{T}$ (defined in (2.5)), and for all
$0\leq s\leq t\leq T$, the following integration by parts formula holds:

$\big{(}{v_{1}(t)},{v_{2}(t)}\big{)}_{H}-\big{(}{v_{1}(s)},{v_{2}(s)}\big{)}_{H}=\int_{s}^{t}\big{[}\langle{\dot{v}_{1}(\tau)},{v_{2}(\tau)}\rangle+\langle{\dot{v}_{2}(\tau)},{v_{1}(\tau)}\rangle\big{]}d\tau.$

In addition, the embedding $\mathcal{W}^{1,2}_{T}\subset C([0,T];H)$ is
continuous.

For more on evolution spaces and other time-dependent spaces, which are also
referred to as Bochner spaces, see, e.g., [31, Chapter 7], [7, Section 3.4],
[37, Chapter 23], [36, Chapter V, Section 5], or [10, Section 5.9.2 and
Appendix B.2].
We lastly introduce the following Bochner space needed in the applications,
i.e., in Section 6:

######  Definition 2.4.

Let $V$ be a real Banach space, then the Bochner space $W^{1,2}(0,T;V)$
consists of all functions $u\in L^{2}_{T}V$ such that $\dot{u}$ exists in the
weak sense and belongs to $L^{2}_{T}V$. The space $W^{1,2}(0,T;V)$ is equipped
with the norm

$\displaystyle\lVert u\rVert_{W^{1,2}(0,T;V)}^{2}=\lVert
u\rVert_{L^{2}_{T}V}^{2}+\lVert\dot{u}\rVert_{L^{2}_{T}V}^{2}.$

###  2.3. Generalized gradients

Let $\tilde{X}$ be a reflexive Banach space. In contact mechanics, we are
often interested in contact conditions of the form $\zeta_{\nu}\in\partial
h(u_{\nu})$, where $\zeta_{\nu}$ represents an interface force,
$u_{\nu}=u\cdot\nu$ the normal component of the displacement, and $\partial
h(u_{\nu})$ being the Clarke subdifferential of $h$ defined as below.

######  Definition 2.5.

Let $h:\tilde{X}\rightarrow\mathbb{R}$ be a locally Lipschitz function. The
generalized (Clarke) directional derivative of $h$ at $\tilde{x}\in\tilde{X}$
in the direction $v\in\tilde{X}$, denoted $h^{\circ}(\tilde{x};v)$, is defined
by

$h^{\circ}(\tilde{x};v)=\limsup_{\tilde{y}\rightarrow\tilde{x},\
\tilde{\epsilon}\downarrow
0}\frac{h(\tilde{y}+\tilde{\epsilon}v)-h(\tilde{y})}{\tilde{\epsilon}}.$

Moreover, the subdifferential in the sense of Clarke of $h$ at $\tilde{x}$,
denoted $\partial h(\tilde{x})$, is a subset of $\tilde{X}^{\ast}$ of the form

$\partial
h(\tilde{x})=\\{\zeta\in\tilde{X}^{\ast}:h^{\circ}(\tilde{x};v)\geq\langle{\zeta},{v}\rangle_{\tilde{X}^{\ast}\times\tilde{X}}\
\ \text{ for all }v\in\tilde{X}\\}.$

######  Remark 2.6.

To say that a function $h:\tilde{X}\rightarrow\mathbb{R}$ is locally Lipschitz
on $\tilde{X}$ means that $h(\tilde{x})$ is Lipschitz continuous in a
neighborhood of $\tilde{x}\in\tilde{X}$.

######  Remark 2.7.

We refer the reader to, e.g., [14, p.185-187] and [23, Section 6.3] for
examples related to contact mechanics. Other examples may be found in, e.g.,
[5, 8].

######  Proposition 2.8.

Let $\tilde{X}$ be a Banach space and $h:\tilde{X}\rightarrow\mathbb{R}$ be
locally Lipschitz on $\tilde{X}$. Then $(\tilde{x},v)\mapsto
h^{\circ}(\tilde{x};v)$ is upper semicontinuous.

######  Proposition 2.9.

Let $\tilde{X}$ be a Banach space. If
$h:\tilde{X}\rightarrow\mathbb{R}\cup\\{\infty\\}$ is proper, convex, and
lower semicontinuous, then $h$ is locally Lipschitz on the interior of the
domain of $h$.

Proposition 2.8 can be found in [8, Proposition 5.6.6], and Proposition 2.9
can be found in [8, Proposition 5.2.10].

######  Definition 2.10.

Let $h:\tilde{X}\rightarrow\mathbb{R}\cup\\{\infty\\}$ be a proper and convex
function. The (generally multivalued) mapping
$\partial_{c}h:\tilde{X}\rightarrow 2^{\tilde{X}^{\ast}}$, written

$\partial_{c}h(\tilde{x})=\\{x^{\ast}\in\tilde{X}^{\ast}:h(v)-h(\tilde{x})\geq\langle{x^{\ast}},{v-\tilde{x}}\rangle_{\tilde{X}^{\ast}\times\tilde{X}}\
\text{ for all }v\in\tilde{X}\\}$

is called the convex subdifferential of $h$ at $\tilde{x}\in\tilde{X}$.

######  Proposition 2.11.

Let $h:\tilde{X}\rightarrow\mathbb{R}$ be locally Lipschitz on $\tilde{X}$. If
$h$ is convex, then $\partial h$ coincides with $\partial_{c}h$.

The proof of the above proposition is found in [5, Proposition 2.2.7]. Lastly,
to show that $(w,\alpha)$ is indeed a solution to Problem 1 (see Section 3),
we require the following result found in [23, Lemma 3.43].

######  Lemma 2.12.

Let $\tilde{X}$ and $\tilde{Y}$ be two Banach spaces and
$h:\tilde{X}\times\tilde{Y}\rightarrow\mathbb{R}$ be such that

  1. (1)

$h(\cdot,\tilde{y})$ is continuous on $\tilde{X}$ for all
$\tilde{y}\in\tilde{Y}$.

  2. (2)

$h(\tilde{x},\cdot)$ is locally Lipschitz on $\tilde{Y}$ for all
$\tilde{x}\in\tilde{X}$.

  3. (3)

There exists $c>0$ such that for all $\zeta\in\partial h(\tilde{x},\tilde{y})$
we have

$\displaystyle\lVert\zeta\rVert_{\tilde{Y}^{\ast}}\leq
c(1+\lVert\tilde{x}\rVert_{\tilde{X}}+\lVert\tilde{y}\rVert_{\tilde{Y}}),$

where $\partial h$ denotes the generalized gradient of $h(\tilde{x},\cdot)$.

Then $h$ is continuous on $\tilde{X}\times\tilde{Y}$.

To read more on the generalized directional derivatives, subdifferential, and
nonsmooth analysis see, e.g., [5, Chapter 2], [8, Chapter 5], and [16, Chapter
1-3].

##  3\. Problem statement and main result

In this section, we first introduce the problem and then present the main
result.

###  3.1. Problem statement

Let $(V,H,V^{\ast})$ be an evolution triple, and $U$, $X$, $Y$, $Z$ real
separable reflexive Banach spaces, with the other function spaces defined in
Section 2.2. We only seek a solution of (1.1) in the sense of Definition 1.1.
We are therefore interested in the following evolutionary differential
variational-hemivariational inequality:

######  Problem 1.

Find $w\in\mathcal{W}^{1,2}_{T}$ and $\alpha\in C([0,T];Y)$ such that

(3.1a)
$\displaystyle\alpha(t)=\alpha_{0}+\int_{0}^{t}\mathcal{G}(s,\alpha(s),Mw(s))ds,$
(3.1b)
$\displaystyle\langle{\dot{w}(t)+A(t,w(t))-f(t)+\mathcal{R}w(t)},{v-w(t)}\rangle+\varphi(t,\alpha(t),\mathcal{S}_{\varphi}w(t),Mw(t),Kv)$
$\displaystyle-\varphi(t,\alpha(t),\mathcal{S}_{\varphi}w(t),Mw(t),Kw(t))+j^{\circ}(t,\alpha(t),\mathcal{S}_{j}w(t),Nw(t);Nv-
Nw(t))\geq 0$ for all $v\in V$, a.e. $t\in(0,T)$ with (3.1c) $w(0)=w_{0}.$

We require the following assumptions on the operators and data:
$\underline{H(A)}$: $A:(0,T)\times V\rightarrow V^{\ast}\text{ is such that }$

  1. (i)

$A(\cdot,v)$ is measurable on $(0,T)$ for all $v\in V$.

  2. (ii)

$A(t,\cdot)\text{ is demicontinuous on }V\text{ for a.e. }t\in(0,T)$, i.e., if
$v^{n}\rightarrow v$ strongly in $V$, then $A(t,v^{n})\rightarrow A(t,v)$
weakly in $V^{\ast}$ as $n\rightarrow\infty$ for a.e. $t\in(0,T)$.

  3. (iii)

$\lVert A(t,v)\rVert_{V^{\ast}}\leq a_{0}(t)+a_{1}\lVert v\rVert_{V}$ for all
$v\in V$, a.e. $t\in(0,T)$ with $a_{0}\in L^{2}(0,T)$, $a_{0}\geq 0,\
a_{1}\geq 0$.

  4. (iv)

There is a $m_{A}>0$ such that
$\langle{A(t,v_{1})-A(t,v_{2})},{v_{1}-v_{2}}\rangle\geq m_{A}\lVert
v_{1}-v_{2}\rVert_{V}^{2}$ for all $v_{i}\in V$, $i=1,2$, a.e. $t\in(0,T)$.

$\underline{H(\varphi)}$: $\varphi:(0,T)\times Y\times X\times U\times
Z\rightarrow\mathbb{R}\text{ is such that }$

  1. (i)

$\varphi(\cdot,y,z,\tilde{w},\tilde{v})\text{ is measurable on }(0,T)\text{
for all }y\in Y,\ z\in X,\ \tilde{w}\in U,\ \tilde{v}\in Z$.

  2. (ii)

$\varphi(t,\cdot,\cdot,\cdot,\tilde{v})\text{ is continuous on }Y\times
X\times U\text{ for all }\tilde{v}\in Z,\text{ a.e. }t\in(0,T)$.

  3. (iii)

$\varphi(t,y,z,\tilde{w},\cdot)\text{ is convex and lower semicontinuous on
}Z\text{ for all}$ $y\in Y$, $z\in X$, $\tilde{w}\in U$, a.e. $t\in(0,T)$.

  4. (iv)

$\lVert\partial_{c}\varphi(t,y,z,\tilde{w},\tilde{v})\rVert_{Z^{\ast}}\leq
c_{0\varphi}(t)+c_{1\varphi}\lVert y\rVert_{Y}+c_{2\varphi}\lVert
z\rVert_{X}+c_{3\varphi}\lVert\tilde{w}\rVert_{U}+c_{4\varphi}\lVert\tilde{v}\rVert_{Z}$
for all  $y\in Y,\ z\in X,\ \tilde{w}\in U,\ \tilde{v}\in Z,\text{ a.e.
}t\in(0,T)$, $\text{ with }c_{0\varphi}\in L^{2}(0,T),\text{ and
}c_{0\varphi},c_{1\varphi},c_{2\varphi},c_{3\varphi},c_{4\varphi}\geq 0.$

  5. (v)

There are $\beta_{i\varphi}\geq 0$ for $i=1,\dots,7$ such that

$\displaystyle\varphi(t,y_{1},z_{1},\tilde{w}_{1},\tilde{v}_{2})-\varphi(t,y_{1},z_{1},\tilde{w}_{1},\tilde{v}_{1})+\varphi(t,y_{2},z_{2},\tilde{w}_{2},\tilde{v}_{1})-\varphi(t,y_{2},z_{2},\tilde{w}_{2},\tilde{v}_{2})$
$\displaystyle\leq\beta_{1\varphi}\lVert\tilde{w}_{1}\rVert_{U}\lVert
y_{1}-y_{2}\rVert_{Y}\lVert\tilde{v}_{1}-\tilde{v}_{2}\rVert_{Z}+\beta_{2\varphi}\lVert
y_{1}-y_{2}\rVert_{Y}\lVert\tilde{v}_{1}-\tilde{v}_{2}\rVert_{Z}$
$\displaystyle+\beta_{3\varphi}\lVert
z_{1}-z_{2}\rVert_{X}\lVert\tilde{v}_{1}-\tilde{v}_{2}\rVert_{Z}+\beta_{4\varphi}\lVert\tilde{w}_{1}-\tilde{w}_{2}\rVert_{U}\lVert\tilde{v}_{1}-\tilde{v}_{2}\rVert_{Z}$
$\displaystyle+\beta_{5\varphi}\lVert
y_{2}\rVert_{Y}\lVert\tilde{w}_{1}-\tilde{w}_{2}\rVert_{U}\lVert\tilde{v}_{1}-\tilde{v}_{2}\rVert_{Z}+\beta_{6\varphi}\lVert\tilde{w}_{1}\rVert_{U}\lVert
z_{1}-z_{2}\rVert_{X}\lVert\tilde{v}_{1}-\tilde{v}_{2}\rVert_{Z}$
$\displaystyle+\beta_{7\varphi}\lVert y_{1}\rVert_{Y}\lVert
z_{1}-z_{2}\rVert_{X}\lVert\tilde{v}_{1}-\tilde{v}_{2}\rVert_{Z}$

$\text{for all }y_{i}\in Y$, $z_{i}\in X$, $\tilde{w}_{i}\in U$,
$\tilde{v}_{i}\in Z$, $i=1,2$, a.e. $t\in(0,T)$.

$\underline{H(j)}$: $j:(0,T)\times Y\times X\times
X\rightarrow\mathbb{R}\text{ is such that }$

  1. (i)

$j(\cdot,y,z,\tilde{v})\text{ is measurable on }(0,T)\text{ for all }y\in
Y,z,\tilde{v}\in X$.

  2. (ii)

$j(t,y,z,\cdot)\text{ is locally Lipschitz on }X\text{ for all }y\in Y,z\in
X,\text{ a.e. }t\in(0,T)$.

  3. (iii)

For $\\{y_{n}\\}_{n\geq 1}\subset Y$ such that $y_{n}\rightarrow y$ strongly
in $Y$, $\\{z_{n}\\}_{n\geq 1}\subset X$ such that $z_{n}\rightarrow z$
strongly in $X$, $\\{\tilde{v}_{n}\\}_{n\geq 1}\subset X$ satisfying
$\tilde{v}_{n}\rightarrow\tilde{v}$ strongly in $X$, and
$\\{\tilde{u}_{n}\\}_{n\geq 1}\subset X$ such that
$\tilde{u}_{n}\rightarrow\tilde{u}$ strongly in $X$, when
$n\rightarrow\infty$, we have

$\limsup_{n\rightarrow\infty}j^{\circ}(t,y_{n},z_{n},\tilde{v}_{n};\tilde{u}_{n})\leq
j^{\circ}(t,y,z,\tilde{v};\tilde{u})$

for a.e. $t\in(0,T)$.

  4. (iv)

$\lVert\partial j(t,y,z,\tilde{v})\rVert_{X^{\ast}}\leq c_{0j}(t)+c_{1j}\lVert
y\rVert_{Y}+c_{2j}\lVert z\rVert_{X}+c_{3j}\lVert\tilde{v}\rVert_{X}\text{ for
all }y\in Y,z,\tilde{v}\in X$, $\text{ a.e. }t\in(0,T)$ with $c_{0j}\in
L^{2}(0,T)$ and $c_{0j},c_{1j},c_{2j},c_{3j}\geq 0$.

  5. (v)

There are $m_{j},\bar{m}_{j}\geq 0$ such that
$j^{\circ}(t,y_{1},z_{1},\tilde{v}_{1};\tilde{v}_{2}-\tilde{v}_{1})+j^{\circ}(t,y_{2},z_{2},\tilde{v}_{2};\tilde{v}_{1}-\tilde{v}_{2})\leq
m_{j}\lVert\tilde{v}_{1}-\tilde{v}_{2}\rVert_{X}^{2}+\bar{m}_{j}(\lVert
y_{1}-y_{2}\rVert_{Y}+\lVert
z_{1}-z_{2}\rVert_{X})\lVert\tilde{v}_{1}-\tilde{v}_{2}\rVert_{X}$ $\text{for
all }y_{i}\in Y,\ z_{i},\tilde{v}_{i}\in X,\ i=1,2,\text{ a.e. }t\in(0,T)$.

We note that hypothesis $H(j)$(v) is equivalent to

(3.2)
$\displaystyle\langle{z^{\ast}_{1}-z^{\ast}_{2}},{\tilde{v}_{1}-\tilde{v}_{2}}\rangle_{X^{\ast}\times
X}$ $\displaystyle\geq-
m_{j}\lVert\tilde{v}_{1}-\tilde{v}_{2}\rVert_{X}-\bar{m}_{j}(\lVert
y_{1}-y_{2}\rVert_{Y}+\lVert
z_{1}-z_{2}\rVert_{X})\lVert\tilde{v}_{1}-\tilde{v}_{2}\rVert_{X}$

for all $z^{\ast}_{i}\in\partial j(t,y_{i},z_{i},\tilde{v}_{i})$, $y_{i}\in
Y$, $z_{i},\tilde{v}_{i}\in X$, $i=1,2$, a.e. $t\in(0,T)$ with $m_{j}\geq 0$,
$\bar{m}_{j}\geq 0$. With small modifications, the equivalence follows by the
proof in [35, Lemma 7, p.124].

$\underline{H(\mathcal{R})}$: $\mathcal{R}:L^{2}_{T}V\rightarrow
L^{2}_{T}V^{\ast}$ is such that

  1. (i)

$\mathcal{R}\text{ is a history-dependent operator, i.e., }$

$\displaystyle\lVert\mathcal{R}v_{1}(t)-\mathcal{R}v_{2}(t)\rVert_{V^{\ast}}\leq
c_{\mathcal{R}}\int_{0}^{t}\lVert v_{1}(s)-v_{2}(s)\rVert_{V}ds$

$\text{ for all }v_{i}\in L^{2}_{T}V$, $i=1,2$, a.e. $t\in(0,T)\text{ with
}c_{\mathcal{R}}>0$.

  2. (ii)

$\mathcal{R}0$ belongs to a bounded subset of $L^{2}_{T}V^{\ast}$.

$\underline{H(\mathcal{S}_{\varphi})}$:
$\mathcal{S}_{\varphi}:L^{2}_{T}V\rightarrow L^{2}_{T}X$ is such that

  1. (i)

$\mathcal{S}_{\varphi}\text{ is a history-dependent operator, i.e., }$

$\displaystyle\lVert\mathcal{S}_{\varphi}v_{1}(t)-\mathcal{S}_{\varphi}v_{2}(t)\rVert_{X}\leq
c_{\mathcal{S}_{\varphi}}\int_{0}^{t}\lVert v_{1}(s)-v_{2}(s)\rVert_{V}ds$

$\text{ for all }v_{i}\in L^{2}_{T}V$, $i=1,2$, a.e. $t\in(0,T)\text{ with
}c_{\mathcal{S}_{\varphi}}>0$.

  2. (ii)

$\mathcal{S}_{\varphi}0$ belongs to a bounded subset of $L^{2}_{T}X$.

$\underline{H(\mathcal{S}_{j})}$: $\mathcal{S}_{j}:L^{2}_{T}V\rightarrow
L^{2}_{T}X$ is such that

  1. (i)

$\mathcal{S}_{j}\text{ is a history-dependent operator, i.e., }$

$\displaystyle\lVert\mathcal{S}_{j}v_{1}(t)-\mathcal{S}_{j}v_{2}(t)\rVert_{X}\leq
c_{\mathcal{S}_{j}}\int_{0}^{t}\lVert v_{1}(s)-v_{2}(s)\rVert_{V}ds$

$\text{ for all }v_{i}\in L^{2}_{T}V$, $i=1,2$, a.e. $t\in(0,T)\text{ with
}c_{\mathcal{S}_{j}}>0$.

  2. (ii)

$\mathcal{S}_{j}0$ belongs to a bounded subset of $L^{2}_{T}X$.

$\underline{H(\mathcal{G})}$: $\mathcal{G}:(0,T)\times Y\times U\rightarrow
Y\text{ is such that }$

  1. (i)

$\mathcal{G}(\cdot,\alpha,\tilde{v})\text{ is measurable on }(0,T)\text{ for
all }\alpha\in Y,\tilde{v}\in U$.

  2. (ii)

There exists an $L_{\mathcal{G}}>0$ such that

$\displaystyle\lVert\mathcal{G}(t,\alpha_{1}(t),\tilde{v}_{1}(t))-\mathcal{G}(t,\alpha_{2}(t),\tilde{v}_{2}(t))\rVert_{Y}\leq
L_{\mathcal{G}}\Big{(}\lVert\alpha_{1}(t)-\alpha_{2}(t)\rVert_{Y}+\lVert\tilde{v}_{1}(t)-\tilde{v}_{2}(t)\rVert_{U}\Big{)}$

$\text{for all }\alpha_{i}\in Y$, $\tilde{v}_{i}\in U$, $i=1,2$, a.e.
$t\in(0,T).$

  3. (iii)

$\lVert\mathcal{G}(\cdot,0,0)\rVert_{L^{\infty}_{T}Y}<\infty$.

$\underline{H(MNK)}$: $M\in\mathcal{L}(V,U),\ \ N\in\mathcal{L}(V,X),\ \
K\in\mathcal{L}(V,Z)$.

We also assume the following regularity on the source term and initial data:

(3.3a) $\displaystyle f\in L^{2}_{T}V^{\ast},$ $\displaystyle\ \ \ \ w_{0}\in
V,$ (3.3b) $\displaystyle\alpha_{0}$ $\displaystyle\in Y.$

Lastly, we require the following smallness-condition:

(3.4) $m_{A}>m_{j}\lVert
N\rVert^{2}+\sqrt{2}(\beta_{4\varphi}+\beta_{5\varphi}\lVert\alpha_{0}\rVert_{Y})\lVert
K\rVert\lVert M\rVert.$

######  Remark 3.1.

Similar assumptions can be found in, e.g., [29, 19, 20, 33, 35]. The same type
of condition as $H(j)$(iii) is found in, e.g., [21, 22]. If $j^{0}$ is
independent of $\alpha$ and $\mathcal{S}_{j}w$ in (1.1b), we may relax the
assumption $H(j)$(iii), and $H(j)$(ii) and Proposition 2.8 are enough.

######  Remark 3.2.

If
$\beta_{1\varphi}=\beta_{4\varphi}=\beta_{5\varphi}=\beta_{6\varphi}=\beta_{7\varphi}=0$
in $H(\varphi)$(v), then the friction coefficient $\mu$ is bounded.
Consequently, Problem 1 reduces to the one found in [19]. In the quasi-static
setting, taking
$\beta_{1\varphi}=\beta_{5\varphi}=\beta_{6\varphi}=\beta_{7\varphi}=0$, the
problem is covered in [29]. Lastly, taking
$\beta_{3\varphi}=\beta_{6\varphi}=\beta_{7\varphi}=0$ reduces Problem 1 to
the one found in [28, 27] for the first-order approximation of (6.24a) and
(6.23a) introduced in Section 6.3.

We make a brief remark on the assumptions in Appendix A.

###  3.2. Main result

We will now state the main result, i.e., Theorem 3.3; the first part is an
existence and uniqueness result, and the latter provides that the flow map
depends continuously on the initial data. The proof of Theorem 3.3 is deferred
to Section 5 after the preparation in Section 4.

######  Theorem 3.3.

Assume $H(A)$, $H(\varphi)$, $H(j)$, $H(\mathcal{R})$,
$H(\mathcal{S}_{\varphi})$, $H(\mathcal{S}_{j})$, $H(\mathcal{G})$, $H(MNK)$,
and (3.3)-(3.4) holds.

  1. (a)

Then there exists a $T>0$ satisfying

(3.5a) $\displaystyle
T=T(\lVert(f,w_{0},\alpha_{0})\rVert_{L^{2}_{T}V^{\ast}\times V\times Y})$ and
(3.5b) $\displaystyle T(a)>T(b)\quad\text{if }\quad a<b$

so that $w\in\mathcal{W}^{1,2}_{T}\subset C([0,T];H)$ and $\alpha\in
C([0,T];Y)$ is a unqiue solution to Problem 1.

  2. (b)

Moreover, there exists a neighborhood around $(w_{0},\alpha_{0})$ so that the
flow map $F:V\times Y\rightarrow L^{2}_{T/2}V\times C([0,T/2];Y)$ defined by
$(w_{0},\alpha_{0})\mapsto(w,\alpha)$ is continuous.

  3. (c)

If
$\beta_{1\varphi}=\beta_{4\varphi}=\beta_{5\varphi}=\beta_{6\varphi}=\beta_{7\varphi}=0$
in $H(\varphi)$(v), we obtain global time of existence, i.e., the existence of
a solution holds for any finite time $T>0$.

######  Remark 3.4.

The theorem can easily be extended to include more than three history-
dependent operators without needing any additional assumptions other than the
once put on $\mathcal{R}$, $\mathcal{S}_{\varphi}$, and $\mathcal{S}_{j}$,
i.e., $H(\mathcal{R})$, $H(\mathcal{S}_{\varphi})$, and $H(\mathcal{S}_{j})$,
respectively.

###  3.3. Strategy of the proof of Theorem 3.3

The proof of the theorem is divided into six steps. In the first step, we
introduce an auxiliary problem to Problem 1, calling this Problem 3.
Specifically, we fix five of the functions in (3.1b) and leave (3.1a) intact.
We recast the auxiliary problem as a differential inclusion (introduced in
Section 4) and use existing results to prove that Problem 3 has a unique
solution (Step 5-5). Next, we define an iterative scheme for Problem 1 using
Problem 3. This iterative scheme decouples (3.1a) and Problem 3 at each step.
Then, we study the difference between two successive iterates and show that
these iterates are Cauchy sequences. We then pass to the limit to show that
the iterative scheme converges to Problem 1 (Step 5). Finally, we show that
the flow map continuously depends on the initial data (Step 5).

##  4\. Preliminary result

Before proving Theorem 3.3, we present an existence and uniqueness result for
a differential inclusion problem see, e.g., [2]. The forthcoming result will
be used to prove existence of a solution to an auxiliary problem of
(3.1b)-(3.1c) in Problem 1. To utilize this result, we need to introduce a
differential inclusion which we relate to the auxiliary problem of
(3.1b)-(3.1c). This will be made clear in Step 5-5 in the proof of Theorem
3.3.
We begin by introducing the inclusion problem.

######  Problem 2.

Find $w\in\mathcal{W}^{1,2}_{T}$ such that

$\displaystyle\dot{w}(t)$ $\displaystyle+A(t,w(t))+\partial\psi(t,w(t))\ni
f(t)\ \text{ for a.e. }\ t\in(0,T),$ $\displaystyle w(0)$
$\displaystyle=w_{0}.$

For clarity on how to work with the preceding problem, we include the
following definition:

######  Definition 4.1.

A function $w\in\mathcal{W}^{1,2}_{T}$ is called a solution to Problem 2 if
there exists $w^{\ast}\in L^{2}_{T}V^{\ast}$ such that

$\displaystyle\dot{w}(t)+A(t,w(t))+w^{\ast}(t)=f(t)\ \ \text{ for a.e. }\
t\in(0,T),$ $\displaystyle w^{\ast}(t)\in\partial\psi(t,w(t))$

for a.e. $t\in(0,T)$ with

$w(0)=w_{0}.$

In the preliminary existence and uniqueness result, we consider the following
assumptions:
$\underline{H(\psi)}$: $\psi:(0,T)\times V\rightarrow\mathbb{R}\text{ is such
that }$

  1. (i)

$\psi(\cdot,v)$ is measurable on $(0,T)$ for all $v\in V$.

  2. (ii)

$\psi(t,\cdot)\text{ is locally Lipschitz on }V\text{ for a.e. }t\in(0,T)$.

  3. (iii)

$\lVert\partial\psi(t,v)\rVert_{V^{\ast}}\leq c_{0}(t)+c_{1}\lVert
v\rVert_{V}\text{ for all }v\in V$, $\text{ a.e. }t\in(0,T)\text{ with
}c_{0}\in L^{2}(0,T)$, $c_{0}\geq 0,\ c_{1}\geq 0$.

  4. (iv)

There is a $m_{\psi}\geq 0$ such that
$\langle{z_{1}-z_{2}},{v_{1}-v_{2}}\rangle\geq-m_{\psi}\lVert
v_{1}-v_{2}\rVert_{V}^{2}$ $\text{ for all }z_{i}\in\partial\psi(t,v_{i})$,
$z_{i}\in V^{\ast},\ v_{i}\in V,\ i=1,2,\text{ a.e. }t\in(0,T)$.

We further assume that the operator $A:(0,T)\times V\rightarrow V^{\ast}$
satisfies $H(A)$, and the source term $f$ and the initial data $w_{0}$ satisfy
(3.3a). Additionally, we assume that the following smallness-condition holds:

(4.1) $m_{A}>m_{\psi}.$

######  Theorem 4.2.

Assume that $H(A)$, $H(\psi)$, (3.3a), and (4.1) hold. Then Problem 2 has a
unique solution $w\in\mathcal{W}^{1,2}_{T}$ in the sense of Definition 4.1 for
any $T>0$.

The theorem was proved in [20, Theorem 3]. This result will be used in Step 5
of the proof of Theorem 3.3.

##  5\. Proof of Theorem 3.3

With the preparation in Section 2-4, we proceed to the proof of Theorem 3.3.
For the convenience of the reader, the proof is established in several steps,
and some of the proofs have been moved to the appendix. We recall that the
function spaces are defined in Section 2.2.
Step 5 (Auxiliary problem to the evolutionary hemivariational-variational
inequality (3.1b)-(3.1c)). Let $(\alpha,\xi,\eta,g,\chi)\in C([0,T];Y)\times
L^{2}_{T}V^{\ast}\times L^{2}_{T}X\times L^{2}_{T}V\times L^{2}_{T}X$ be
given, then we define an auxiliary problem to (3.1b)-(3.1c) in Problem 1.

######  Problem 3.

Find $w_{\alpha\xi\eta g\chi}\in\mathcal{W}^{1,2}_{T}$ corresponding to
$(\alpha,\xi,\eta,g,\chi)\in C([0,T];Y)\times L^{2}_{T}V^{\ast}\times
L^{2}_{T}X\times L^{2}_{T}V\times L^{2}_{T}X$ such that

$\displaystyle\langle{\dot{w}_{\alpha\xi\eta g\chi}(t)+A(t,w_{\alpha\xi\eta
g\chi}(t))-f(t)+\xi(t)},{v-w_{\alpha\xi\eta
g\chi}(t)}\rangle+\varphi(t,\alpha(t),\eta(t),Mg(t),Kv)$
$\displaystyle-\varphi(t,\alpha(t),\eta(t),Mg(t),Kw_{\alpha\xi\eta
g\chi}(t))+j^{\circ}(t,\alpha(t),\chi(t),Nw_{\alpha\xi\eta g\chi}(t);Nv-
Nw_{\alpha\xi\eta g\chi}(t))\geq 0$

for all $v\in V$, a.e. $t\in(0,T)$ with

$w_{\alpha\xi\eta g\chi}(0)=w_{0}.$

######  Remark 5.1.

A glance at Problem 3 and (3.1b) lets us see that the auxiliary problem keeps
$\xi=\mathcal{R}w$, $\alpha$ (still denoted by $\alpha$),
$\eta=\mathcal{S}_{\varphi}w$, $g=w$, and $\chi=\mathcal{S}_{j}w$ known in
contrast to (3.1b). We find it worth mentioning that we use the subscripts on
$w$ to emphasize that a solution $w_{\alpha\xi\eta g\chi}$ to Problem 3
corresponds to $(\alpha,\xi,\eta,g,\chi)\in C([0,T];Y)\times
L^{2}_{T}V^{\ast}\times L^{2}_{T}X\times L^{2}_{T}V\times L^{2}_{T}X$. This
also helps to distinguish between a solution to Problem 1 and a solution to
Problem 3.

Step 5 (Existence of a solution to Problem 3). Let
$(\alpha,\xi,\eta,g,\chi)\in C([0,T];Y)\times L^{2}_{T}V^{\ast}\times
L^{2}_{T}X\times L^{2}_{T}V\times L^{2}_{T}X$ be given. We wish to utilize
Theorem 4.2 in order to prove that Problem 3 has a solution. We therefore
define the functional $\psi_{\alpha\xi\eta g\chi}:(0,T)\times
V\rightarrow\mathbb{R}$ by

(5.1) $\displaystyle\psi_{\alpha\xi\eta g\chi}(t,v)$
$\displaystyle=\langle{\xi(t)},{v}\rangle+\varphi(t,\alpha(t),\eta(t),Mg(t),Kv)+j(t,\alpha(t),\chi(t),Nv)$

for all $v\in V$, a.e. $t\in(0,T)$. Verification of the hypothesis of Theorem
4.2 follows from the same approach as the first part of the proof in [20,
Theorem 5]. We investigate the assumption $H(\psi)$ and the smallness-
condition (4.1), as there are some modifications in comparison to [20, Theorem
5]. Keeping Proposition 2.9 in mind, we only comment on the changes and leave
the reader to visit [20, Theorem 5] for a detailed verification. Using (3.2),
we find that $H(\psi)$ holds with
$c_{0}(t)=\lVert\xi(t)\rVert_{V^{\ast}}+c_{0j}(t)\lVert N\rVert+c_{2j}\lVert
N\rVert\lVert\chi(t)\rVert_{X}+c_{0\varphi}(t)\lVert
K\rVert+c_{2\varphi}\lVert K\rVert\lVert\eta(t)\rVert_{X}+(c_{1j}\lVert
N\rVert+c_{1\varphi}\lVert
K\rVert)\lVert\alpha(t)\rVert_{Y}+c_{3\varphi}\lVert K\rVert\lVert
M\rVert\lVert g(t)\rVert_{V}$, $c_{1}=c_{3j}\lVert
N\rVert^{2}+c_{4\varphi}\lVert K\rVert^{2}$, and $m_{\psi}=m_{j}\lVert
N\rVert^{2}$. This, together with the smallness-condition (3.4) leads to
(4.1). Thus, we conclude by Theorem 4.2 that there exists a solution
$w_{\alpha\xi\eta g\chi}\in\mathcal{W}^{1,2}_{T}$ of Problem 2 with
$\psi_{\alpha\xi\eta g\chi}$ defined in (5.1). It remains to show that the
existence of a solution to Problem 2 implies the existence of a solution to
Problem 3. This is a consequence of Definition 2.5 and 2.10, and basic results
of the generalized gradients, see, e.g., [14, Theorem 3.7, Proposition
3.10-3.12], where they have summarized these properties, and [35, Lemma 7,
p.124]. A more detailed approach to this part can be found in, e.g., [14,
Section 6] or [35, p.190-192].

Step 5 (Uniqueness of a solution to Problem 3). Uniqueness is immediate from
the proof in [35, Theorem 98] with $\alpha_{j}=m_{j}\lVert N\rVert^{2}$ and
the smallness-condition (3.4), i.e., $m_{A}>m_{j}\lVert N\rVert^{2}$.

Step 5 (Estimate on the solution to Problem 3, that is, $w_{\alpha\xi\eta
g\chi}\in\mathcal{W}^{1,2}_{T}\subset C([0,T];H)$). We now find an estimate on
the solution to Problem 3, which will come in handy later:

######  Proposition 5.2.

Under the assumptions of Theorem 3.3, for given $(\alpha,\xi,\eta,g,\chi)\in$
$C([0,T];Y)\times L^{2}_{T}V^{\ast}\times L^{2}_{T}X\times L^{2}_{T}V\times
L^{2}_{T}X$, let $w_{\alpha\xi\eta g\chi}$ be a solution to Problem 3. Then,
there exists a constant $c>0$ independent of $w_{\alpha\xi\eta g\chi}$ such
that

(5.2a) $\displaystyle\lVert w_{\alpha\xi\eta
g\chi}\rVert_{L^{\infty}_{T}H}^{2}+\lVert w_{\alpha\xi\eta
g\chi}\rVert_{L^{2}_{T}V}^{2}$ $\displaystyle\leq c(1+\lVert
w_{0}\rVert_{V}^{2}+\lVert
f\rVert_{L^{2}_{T}V^{\ast}}^{2}+\lVert\xi\rVert_{L^{2}_{T}V^{\ast}}^{2}+\lVert\alpha\rVert_{L^{\infty}_{T}Y}^{2}+\lVert\eta\rVert_{L^{2}_{T}X}^{2}+\lVert\chi\rVert_{L^{2}_{T}X}^{2})$
$\displaystyle+\frac{\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha\rVert_{L^{\infty}_{T}Y}}{m_{A}-m_{j}\lVert
N\rVert^{2}}\lVert g\rVert_{L^{2}_{T}V}^{2}$
$\displaystyle+\frac{\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha\rVert_{L^{\infty}_{T}Y}}{m_{A}-m_{j}\lVert
N\rVert^{2}}\lVert w_{\alpha\xi\eta g\chi}\rVert_{L^{2}_{T}V}^{2}$ and (5.2b)
$\displaystyle\lVert\dot{w}_{\alpha\xi\eta
g\chi}\rVert_{L^{2}_{T}V^{\ast}}^{2}$ $\displaystyle\leq c(1+\lVert
f\rVert_{L^{2}_{T}V^{\ast}}^{2}+\lVert\xi\rVert_{L^{2}_{T}V^{\ast}}^{2}+\lVert\chi\rVert_{L^{2}_{T}X}^{2}+\lVert\alpha\rVert_{L^{\infty}_{T}Y}^{2}+\lVert\eta\rVert_{L^{2}_{T}X}^{2}$
$\displaystyle+\lVert g\rVert_{L^{2}_{T}V}^{2}+\lVert w_{\alpha\xi\eta
g\chi}\rVert_{L^{2}_{T}V}^{2})+2\beta_{5\varphi}^{2}\lVert K\rVert^{2}\lVert
M\rVert^{2}\lVert\alpha\rVert_{L^{\infty}_{T}Y}^{2}\lVert
g\rVert_{L^{2}_{T}V}^{2}.$

The proof of Proposition 5.2 is postponed to Appendix B.

Step 5 (Scheme for the approximated solution to Problem 1). For
$n\in\mathbb{Z}_{+}$, let $\alpha^{n-1}\in C([0,T];Y)$, and
$w^{n-1}\in\mathcal{W}^{1,2}_{T}$ be known. We construct the approximated
solutions $\\\ \\{(w^{n},\alpha^{n})\\}_{n\geq
1}\subset\mathcal{W}^{1,2}_{T}\times C([0,T];Y)$ to Problem 1, where
$(w^{n},\alpha^{n})$ is a solution of the scheme:

(5.3a)
$\displaystyle\langle{\dot{w}^{n}(t)+A(t,w^{n}(t))-f(t)+\mathcal{R}w^{n-1}(t)},{v-w^{n}(t)}\rangle$
$\displaystyle\hskip
31.2982pt+\varphi(t,\alpha^{n-1}(t),\mathcal{S}_{\varphi}w^{n-1}(t),Mw^{n-1}(t),Kv)$
$\displaystyle\hskip
31.2982pt-\varphi(t,\alpha^{n-1}(t),\mathcal{S}_{\varphi}w^{n-1}(t),Mw^{n-1}(t),Kw^{n}(t))$
$\displaystyle\hskip
31.2982pt+j^{\circ}(t,\alpha^{n-1}(t),\mathcal{S}_{j}w^{n-1}(t),Nw^{n}(t);Nv-
Nw^{n}(t))\geq 0$ for all $v\in V$, a.e. $t\in(0,T)$, and (5.3b)
$w^{n}(0)=w_{0}.$ (5.3c)
$\alpha^{n}(t)=\alpha_{0}+\int_{0}^{t}\mathcal{G}(s,\alpha^{n}(s),Mw^{n}(s))ds\
\ \text{ for a.e. }t\in(0,T),\\\ $

with $w^{0}=w_{0}\in V$ and $\alpha^{0}=\alpha_{0}\in Y$.

Step 5.5 (Existence and uniqueness of
$(w^{n},\alpha^{n})\in\mathcal{W}^{1,2}_{T}\times C([0,T];Y)$ to (5.3a)-(5.3c)
for all $n\in\mathbb{N}$). We establish existence and uniqueness by induction
on $n$. First, applying Minkowski’s inequality, Young’s inequality,
integrating over the time interval $(0,t^{\prime})\subset(0,T)$, and lastly
applying the Cauchy-Schwarz inequality to hypothesis $H(\mathcal{R})$,
$H(\mathcal{S}_{\varphi})$, and $H(\mathcal{S}_{j})$, respectively, yield

(5.4)
$\displaystyle\int_{0}^{t^{\prime}}\lVert\mathcal{R}w^{n-1}(t)\rVert_{V^{\ast}}^{2}dt$
$\displaystyle\leq
2\int_{0}^{t^{\prime}}\lVert\mathcal{R}w^{n-1}(t)-\mathcal{R}0(t)\rVert_{V^{\ast}}^{2}dt+2\int_{0}^{t^{\prime}}\lVert\mathcal{R}0(t)\rVert_{V^{\ast}}^{2}dt$
$\displaystyle\leq 2T^{2}c_{\mathcal{R}}^{2}\int_{0}^{t^{\prime}}\lVert
w^{n-1}(t)\rVert_{V}^{2}dt+2\lVert\mathcal{R}0\rVert_{L^{2}_{t^{\prime}}V^{\ast}}^{2}$
(5.5)
$\displaystyle\int_{0}^{t^{\prime}}\lVert\mathcal{S}_{\varphi}w^{n-1}(t)\rVert_{X}^{2}dt$
$\displaystyle\leq
2T^{2}c_{\mathcal{S}_{\varphi}}^{2}\int_{0}^{t^{\prime}}\lVert
w^{n-1}(t)\rVert_{V}^{2}dt+2\lVert\mathcal{S}_{\varphi}0\rVert_{L^{2}_{t^{\prime}}X}^{2}$
(5.6)
$\displaystyle\int_{0}^{t^{\prime}}\lVert\mathcal{S}_{j}w^{n-1}(t)\rVert_{X}^{2}dt$
$\displaystyle\leq 2T^{2}c_{\mathcal{S}_{j}}^{2}\int_{0}^{t^{\prime}}\lVert
w^{n-1}(t)\rVert_{V}^{2}dt+2\lVert\mathcal{S}_{j}0\rVert_{L^{2}_{t^{\prime}}X}^{2}$

for all $t^{\prime}\in[0,T]$. We combine (5.4)-(5.6) with the estimates
(5.2a)-(5.2b) in Proposition 5.2 for $w_{\alpha\xi\eta g\chi}=w^{n}$,
$\dot{w}_{\alpha\xi\eta g\chi}=\dot{w}^{n}$, $\xi=\mathcal{R}w^{n-1}$,
$\alpha=\alpha^{n-1}$, $\eta=\mathcal{S}_{\varphi}w^{n-1}$, $g=w^{n-1}$, and
$\chi=\mathcal{S}_{j}w^{n-1}$. This implies

(5.7a) $\displaystyle\lVert w^{n}\rVert_{L^{\infty}_{T}H}^{2}$
$\displaystyle+\lVert w^{n}\rVert_{\mathcal{W}^{1,2}_{T}}^{2}$
$\displaystyle\leq c(1+\lVert f\rVert_{L^{2}_{T}V^{\ast}}^{2}+\lVert
w_{0}\rVert_{V}^{2}+\lVert\alpha^{n-1}\rVert_{L^{\infty}_{T}Y}^{2})$
$\displaystyle+\frac{\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha^{n-1}\rVert_{L^{\infty}_{T}Y}}{m_{A}-m_{j}\lVert
N\rVert^{2}}\lVert w^{n-1}\rVert_{L^{2}_{T}V}^{2}$
$\displaystyle+\frac{\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha^{n-1}\rVert_{L^{\infty}_{T}Y}}{m_{A}-m_{j}\lVert
N\rVert^{2}}\lVert w^{n}\rVert_{L^{2}_{T}V}^{2},$ (5.7b)
$\displaystyle\lVert\dot{w}^{n}\rVert_{L^{2}_{T}V^{\ast}}^{2}$
$\displaystyle\leq c(1+\lVert f\rVert_{L^{2}_{T}V^{\ast}}^{2}+\lVert
w^{n-1}\rVert_{L^{2}_{T}V}^{2}+\lVert\alpha^{n-1}\rVert_{L^{\infty}_{T}Y}^{2}+\lVert
w^{n}\rVert_{L^{2}_{T}V}^{2})$
$\displaystyle+\big{(}\sqrt{2}\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha^{n-1}\rVert_{L^{\infty}_{T}Y}\big{)}^{2}\lVert
w^{n-1}\rVert_{L^{2}_{T}V}^{2}.$

for all $n\in\mathbb{Z}_{+}$. Applying Minkowski’s inequality to (5.3c) reads

$\displaystyle\lVert\alpha^{n}(t)\rVert_{Y}\leq\lVert\alpha_{0}\rVert_{Y}+\int_{0}^{t}\lVert\mathcal{G}(s,\alpha^{n}(s),Mw^{n}(s))\rVert_{Y}ds$

for a.e. $t\in(0,T)$. We observe that by Minkowski’s inequality,
$H(\mathcal{G})$(ii) and $H(MNK)$

(5.8) $\displaystyle\lVert\mathcal{G}(s,\alpha^{n}(s),Mw^{n}(s))\rVert_{Y}$
$\displaystyle\leq\lVert\mathcal{G}(s,\alpha^{n}(s),Mw^{n}(s))-\mathcal{G}(s,0,0)\rVert_{Y}+\lVert\mathcal{G}(s,0,0)\rVert_{Y}$
$\displaystyle\leq
L_{\mathcal{G}}\lVert\alpha^{n}(s)\rVert_{Y}+L_{\mathcal{G}}\lVert
M\rVert\lVert w^{n}(s)\rVert_{V}+\lVert\mathcal{G}(s,0,0)\rVert_{Y}$

for a.e. $s\in(0,t)\subset(0,T)$. Accordingly, the Cauchy-Schwarz inequality
and $H(\mathcal{G})$(iii) implies

$\displaystyle\lVert\alpha^{n}(t)\rVert_{Y}$ $\displaystyle\leq
c(\lVert\alpha_{0}\rVert_{Y}+T\lVert\mathcal{G}(\cdot,0,0)\rVert_{L^{\infty}_{T}Y}+T^{1/2}\lVert
w^{n}\rVert_{L^{2}_{T}V})+c\int_{0}^{t}\lVert\alpha^{n}(s)\rVert_{Y}ds$

for a.e $t\in(0,T)$. By Grönwall’s inequality (see, e.g., [10]), we have

(5.9a) $\displaystyle\lVert\alpha^{n}\rVert_{L^{\infty}_{T}Y}\leq
c(\lVert\alpha_{0}\rVert_{Y}$
$\displaystyle+T^{k}\lVert\mathcal{G}(\cdot,0,0)\rVert_{L^{\infty}_{T}Y}+T^{k}\lVert
w^{n}\rVert_{L^{2}_{T}V})(1+cT\mathrm{e}^{cT}),$ and from Young’s inequality
(5.9b) $\displaystyle\lVert\alpha^{n}\rVert_{L^{\infty}_{T}Y}^{2}\leq
c(\lVert\alpha_{0}\rVert_{Y}^{2}$
$\displaystyle+T^{k}(\lVert\mathcal{G}(\cdot,0,0)\rVert_{L^{\infty}_{T}Y}^{2}+T^{k}\lVert
w^{n}\rVert_{L^{2}_{T}V}^{2}))(1+cT^{2}\mathrm{e}^{2ct})$

for $k\geq 1/2$. We will show the uniform bound by induction on $n$.
For $n=1$, with the initial guesses; $w^{0}=w_{0}$ and
$\alpha^{0}=\alpha_{0}$, (5.7a)-(5.7b) becomes

$\displaystyle\lVert w^{1}\rVert_{L^{\infty}_{T}H}^{2}+\lVert
w^{1}\rVert_{L^{2}_{T}V}^{2}$ $\displaystyle\leq c(1+\lVert
w_{0}\rVert_{V}^{2}+\lVert
f\rVert_{L^{2}_{T}V^{\ast}}^{2}+\lVert\alpha_{0}\rVert_{Y}^{2})$
$\displaystyle+T\frac{\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha_{0}\rVert_{Y}}{m_{A}-m_{j}\lVert N\rVert^{2}}\lVert
w_{0}\rVert_{V}^{2}$ $\displaystyle+\frac{\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha_{0}\rVert_{Y}}{m_{A}-m_{j}\lVert N\rVert^{2}}\lVert
w^{1}\rVert_{L^{2}_{T}V}^{2},$
$\displaystyle\lVert\dot{w}^{1}\rVert_{L^{2}_{T}V^{\ast}}^{2}$
$\displaystyle\leq c(1+\lVert f\rVert_{L^{2}_{T}V^{\ast}}^{2}+\lVert
w_{0}\rVert_{V}^{2}+\lVert\alpha_{0}\rVert_{Y}^{2}+\lVert
w^{1}\rVert_{L^{2}_{T}V}^{2})$
$\displaystyle+T\big{(}\sqrt{2}\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha_{0}\rVert_{Y}\big{)}^{2}\lVert w_{0}\rVert_{V}^{2}.$

From the smallness-assumption (3.4), it follows that

(5.10) $\displaystyle\lVert w^{1}\rVert_{L^{\infty}_{T}H}^{2}+\lVert
w^{1}\rVert_{\mathcal{W}^{1,2}_{T}}^{2}$ $\displaystyle\leq c(1+\lVert
f\rVert_{L^{2}_{T}V^{\ast}}^{2}+\lVert
w_{0}\rVert_{V}^{2}+\lVert\alpha_{0}\rVert_{Y}^{2}).$

We next define the complete metric space

$X_{T}(a)=\\{h\in C([0,T];Y):\lVert h\rVert_{L^{\infty}_{T}Y}\leq a,\
\text{with }a\in\mathbb{R}_{+}\\}$

and the operator $\Lambda:X_{T}(a)\rightarrow X_{T}(a)$ by

(5.11)
$\displaystyle\Lambda\alpha^{1}(t)=\alpha_{0}+\int_{0}^{t}\mathcal{G}(s,\alpha^{1}(s),Mw^{1}(s))ds$

for $w^{1}\in\mathcal{W}^{1,2}_{T}$. We verify that $\alpha^{1}$ is indeed a
solution to (5.3c) for $n=1$ in the next lemma.

######  Lemma 5.3.

Let $w^{1}\in\mathcal{W}^{1,2}_{T}$ be a solution of (5.3a)-(5.3b) with
$w^{0}=w_{0}\in V$ and $\alpha^{0}=\alpha_{0}\in Y$. Under the assumptions of
Theorem 3.3, the operator $\Lambda:X_{T}(a)\rightarrow X_{T}(a)$, defined by
(5.11), has a unique fixed-point, i.e., there exists a constant $0\leq L<1$
such that

$\lVert\Lambda\alpha^{1}_{1}-\Lambda\alpha^{1}_{2}\rVert_{X_{T}(a)}\leq
L\lVert\alpha^{1}_{1}-\alpha^{1}_{2}\rVert_{X_{T}(a)}.$

The proof of Lemma 5.3 is moved to Appendix C as it follows from the standard
ODE arguments combined with the estimate (5.10), and the assumptions
$H(\mathcal{G})$ and $H(MNK)$. Next, investigating $n=2$. This implies

$\displaystyle\lVert w^{2}\rVert_{L^{\infty}_{T}H}^{2}+\lVert
w^{2}\rVert_{L^{2}_{T}V}^{2}$ $\displaystyle\leq c(1+\lVert
w_{0}\rVert_{V}^{2}+\lVert
f\rVert_{L^{2}_{T}V^{\ast}}^{2}+\lVert\alpha^{1}\rVert_{L^{\infty}_{T}Y}^{2})$
$\displaystyle+\frac{\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha^{1}\rVert_{L^{\infty}_{T}Y}}{m_{A}-m_{j}\lVert
N\rVert^{2}}\lVert w^{1}\rVert_{L^{2}_{T}V}^{2}$
$\displaystyle+\frac{\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha^{1}\rVert_{L^{\infty}_{T}Y}}{m_{A}-m_{j}\lVert
N\rVert^{2}}\lVert w^{2}\rVert_{L^{2}_{T}V}^{2},$
$\displaystyle\lVert\dot{w}^{2}\rVert_{L^{2}_{T}V^{\ast}}^{2}$
$\displaystyle\leq c(1+\lVert f\rVert_{L^{2}_{T}V^{\ast}}^{2}+\lVert
w^{1}\rVert_{L^{2}_{T}V}^{2}+\lVert\alpha^{1}\rVert_{L^{\infty}_{T}Y}^{2}+\lVert
w^{2}\rVert_{L^{2}_{T}V}^{2})$
$\displaystyle+\big{(}\sqrt{2}\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha^{1}\rVert_{L^{\infty}_{T}Y}\big{)}^{2}\lVert
w^{1}\rVert_{L^{2}_{T}V}^{2}.$

From the estimate (5.9a) for $n=1$, we have that

$\displaystyle\lVert w^{2}\rVert_{L^{\infty}_{T}H}^{2}+$ $\displaystyle\lVert
w^{2}\rVert_{L^{2}_{T}V}^{2}\leq c(1+\lVert w_{0}\rVert_{V}^{2}+\lVert
f\rVert_{L^{2}_{T}V^{\ast}}^{2}+\lVert\alpha^{1}\rVert_{L^{\infty}_{T}Y}^{2})$
$\displaystyle\hskip 5.69046pt+\frac{\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert(\lVert\alpha_{0}\rVert_{Y}+T^{k}c(\lVert(f,w_{0},\alpha_{0})\rVert_{L^{2}_{T}V^{\ast}\times
V\times Y}))}{m_{A}-m_{j}\lVert N\rVert^{2}}\lVert
w^{1}\rVert_{L^{2}_{T}V}^{2}$ $\displaystyle\hskip
5.69046pt+\frac{\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert(\lVert\alpha_{0}\rVert_{Y}+T^{k}c(\lVert(f,w_{0},\alpha_{0})\rVert_{L^{2}_{T}V^{\ast}\times
V\times Y}))}{m_{A}-m_{j}\lVert N\rVert^{2}}\lVert
w^{2}\rVert_{L^{2}_{T}V}^{2},$
$\displaystyle\lVert\dot{w}^{2}\rVert_{L^{2}_{T}V^{\ast}}^{2}$
$\displaystyle\leq c(1+\lVert f\rVert_{L^{2}_{T}V^{\ast}}^{2}+\lVert
w^{1}\rVert_{L^{2}_{T}V}^{2}+\lVert\alpha^{1}\rVert_{L^{\infty}_{T}Y}^{2}+\lVert
w^{2}\rVert_{L^{2}_{T}V}^{2})$
$\displaystyle+\big{(}\sqrt{2}\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert(\lVert\alpha_{0}\rVert_{Y}+T^{k}c(\lVert(f,w_{0},\alpha_{0})\rVert_{L^{2}_{T}V^{\ast}\times
V\times Y})))\big{)}^{2}\lVert w^{1}\rVert_{L^{2}_{T}V}^{2}$

for $k\geq 1/2$. Choosing $T>0$ such that

(5.12) $\displaystyle T^{k}$
$\displaystyle\sim\frac{1}{c(\lVert(f,w_{0},\alpha_{0})\rVert_{L^{2}_{T}V^{\ast}\times
V\times Y})}$

small enough for some $k\geq 1/2$. From the smallness-condition (3.4) and the
choice of $T>0$, we have that

$\displaystyle\frac{\sqrt{2}\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert}{m_{A}-m_{j}\lVert N\rVert^{2}-\sqrt{2}\beta_{4\varphi}\lVert
K\rVert\lVert
M\rVert}(\lVert\alpha_{0}\rVert_{Y}+T^{k}c(\lVert(f,w_{0},\alpha_{0})\rVert_{L^{2}_{T}V^{\ast}\times
V\times Y}))$ $\displaystyle<1,$ $\displaystyle\sqrt{2}\beta_{5\varphi}\lVert
K\rVert\lVert
M\rVert(\lVert\alpha_{0}\rVert_{Y}+T^{k}c(\lVert(f,w_{0},\alpha_{0})\rVert_{L^{2}_{T}V^{\ast}\times
V\times Y}))$ $\displaystyle<m_{A}.$

Gathering the above and using (5.10) implies

$\displaystyle\lVert w^{2}\rVert_{L^{\infty}_{T}H}^{2}+\lVert
w^{2}\rVert_{\mathcal{W}^{1,2}_{T}}^{2}\leq c(1+\lVert
f\rVert_{L^{2}_{T}V^{\ast}}^{2}+\lVert
w_{0}\rVert_{V}^{2}+\lVert\alpha_{0}\rVert_{Y}^{2}).$

Moreover, verifying that $\alpha^{2}\in C([0,T];Y)$ is indeed a solution to
(5.3c) for $n=2$ follows by the same approach as for $n=1$.
The induction step follows the same procedure as for $n=2$. Consequently,
$(w^{n},\alpha^{n})\in\mathcal{W}^{1,2}_{T}\times C([0,T];Y)$ is the
approximated solution of (5.3a)-(5.3c). Further, we obtain the following
uniform bound

(5.13) $\displaystyle\lVert w^{n}\rVert_{L^{\infty}_{T}H}^{2}+\lVert
w^{n}\rVert_{\mathcal{W}^{1,2}_{T}}^{2}+\lVert\alpha^{n}\rVert_{L^{\infty}_{T}Y}^{2}\leq
c(1+\lVert f\rVert_{L^{2}_{T}V^{\ast}}^{2}+\lVert
w_{0}\rVert_{V}^{2}+\lVert\alpha_{0}\rVert_{Y}^{2})$

for all $n\in\mathbb{Z}_{+}$. Additionally, it follows from Proposition 2.3
that $\\{w^{n}\\}_{n\geq 1}\subset C([0,T];H)$.
Step 5.5 (Convergence of the approximated solution). We first show that
$\\{(w^{n},\alpha^{n})\\}_{n\geq 1}$ is a Cauchy sequence in $L^{2}_{T}V\times
C([0,T];Y)$. This is summarized in the proposition below.

######  Proposition 5.4.

Let $w^{0}=w_{0}\in V$ and $\alpha^{0}=\alpha_{0}\in Y$. Under the hypothesis
of Theorem 3.3, let $\\{(w^{n},\alpha^{n})\\}_{n\geq
1}\subset\mathcal{W}^{1,2}_{T}\times C([0,T];Y)$ be the solution of
(5.3a)-(5.3c). Then, $\\{(w^{n},\alpha^{n})\\}_{n\geq 1}$ is a Cauchy sequence
in $L^{2}_{T}V\times C([0,T];Y)$. In addition, $\\{\mathcal{R}w^{n}\\}_{n\geq
1}$, $\\{\mathcal{S}_{\varphi}w^{n}\\}_{n\geq 1}$, and
$\\{\mathcal{S}_{j}w^{n}\\}_{n\geq 1}$ are Cauchy sequences in
$L^{2}_{T}V^{\ast}$, $L^{2}_{T}X$ and $L^{2}_{T}X$, respectively.

######  Remark 5.5.

To cover the case where we obtain global time of existence when
$\beta_{1\varphi}=\beta_{4\varphi}=\beta_{5\varphi}=\beta_{6\varphi}=\beta_{7\varphi}=0$,
the proof needs to be slightly modified. This case is included in Corollary
5.6.

###### Proof.

Let $e_{\dot{w}}^{n}=\dot{w}^{n}-\dot{w}^{n-1}$, $e_{w}^{n}=w^{n}-w^{n-1}$,
and $e_{\alpha}^{n}=\alpha^{n}-\alpha^{n-1}$. To begin with, we add (5.3a) for
two iterations at the levels $n$ and $n-1$. Then, choosing $v=w^{n-1}$ and
$v=w^{n}$ for the levels $n$ and $n-1$, respectively, implies

$\displaystyle\langle{e^{n}_{\dot{w}}(t)},{e^{n}_{w}(t)}\rangle$
$\displaystyle+\langle{A(t,w^{n}(t))-A(t,w^{n-1}(t))},{w^{n}(t)-w^{n-1}(t)}\rangle$
$\displaystyle\leq\langle{\mathcal{R}w^{n-2}(t)-\mathcal{R}w^{n-1}(t)},{e_{w}^{n}(t)}\rangle$
$\displaystyle+\varphi(t,\alpha^{n-1}(t),\mathcal{S}_{\varphi}w^{n-1}(t),Mw^{n-1}(t),Kw^{n-1}(t))$
$\displaystyle-\varphi(t,\alpha^{n-1}(t),\mathcal{S}_{\varphi}w^{n-1}(t),Mw^{n-1}(t),Kw^{n}(t))$
$\displaystyle+\varphi(t,\alpha^{n-2}(t),\mathcal{S}_{\varphi}w^{n-2}(t),Mw^{n-2}(t),Kw^{n}(t))$
$\displaystyle-\varphi(t,\alpha^{n-2}(t),\mathcal{S}_{\varphi}w^{n-2}(t),Mw^{n-2}(t),Kw^{n-1}(t))$
$\displaystyle+j^{\circ}(t,\alpha^{n-1}(t),\mathcal{S}_{j}w^{n-1}(t),Nw^{n}(t);-Ne_{w}^{n}(t))$
$\displaystyle+j^{\circ}(t,\alpha^{n-2}(t),\mathcal{S}_{j}w^{n-2}(t),Nw^{n-1}(t);Ne_{w}^{n}(t))$

for a.e. $t\in(0,T)$ with $w^{n}(0)=w^{n-1}(0)=w_{0}$. We deduce from
hypotheses $H(A)$(iv), $H(\varphi)$(v), $H(j)$(v), $H(MNK)$, and the Cauchy-
Schwarz inequality that

$\displaystyle\langle{e_{\dot{w}}^{n}(t)},{e_{w}^{n}(t)}\rangle+(m_{A}-m_{j}\lVert
N\rVert^{2})\lVert e_{w}^{n}(t)\rVert_{V}^{2}$
$\displaystyle\leq\lVert\mathcal{R}w^{n-1}(t)-\mathcal{R}w^{n-2}(t)\rVert_{V^{\ast}}\lVert
e_{w}^{n}(t)\rVert_{V}$ $\displaystyle+(\bar{m}_{j}\lVert
N\rVert+\beta_{2\varphi}\lVert K\rVert)\lVert
e_{\alpha}^{n-1}(t)\rVert_{Y}\lVert e_{w}^{n}(t)\rVert_{V}$
$\displaystyle+\beta_{1\varphi}\lVert K\rVert\lVert M\rVert\lVert
w^{n-1}(t)\rVert_{V}\lVert e_{\alpha}^{n-1}(t)\rVert_{Y}\lVert
e_{w}^{n}(t)\rVert_{V}$ $\displaystyle+\beta_{3\varphi}\lVert
K\rVert\lVert\mathcal{S}_{\varphi}w^{n-1}(t)-\mathcal{S}_{\varphi}w^{n-2}(t)\rVert_{X}\lVert
e_{w}^{n}(t)\rVert_{V}$ $\displaystyle+\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert\lVert e_{w}^{n-1}(t)\rVert_{V}\lVert e_{w}^{n}(t)\rVert_{V}$
$\displaystyle+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha^{n-2}(t)\rVert_{Y}\lVert e_{w}^{n-1}(t)\rVert_{V}\lVert
e_{w}^{n}(t)\rVert_{V}$ $\displaystyle+\beta_{6\varphi}\lVert K\rVert\lVert
M\rVert\lVert
w^{n-1}(t)\rVert_{V}\lVert\mathcal{S}_{\varphi}w^{n-1}(t)-\mathcal{S}_{\varphi}w^{n-2}(t)\rVert_{X}\lVert
e_{w}^{n}(t)\rVert_{V}$ $\displaystyle+\beta_{7\varphi}\lVert
K\rVert\lVert\alpha^{n-1}(t)\rVert_{Y}\lVert\mathcal{S}_{\varphi}w^{n-1}(t)-\mathcal{S}_{\varphi}w^{n-2}(t)\rVert_{X}\lVert
e_{w}^{n}(t)\rVert_{V}$ $\displaystyle+\bar{m}_{j}\lVert
N\rVert\lVert\mathcal{S}_{j}w^{n-1}(t)-\mathcal{S}_{j}w^{n-2}(t)\rVert_{X}\lVert
e_{w}^{n}(t)\rVert_{V}$

for a.e. $t\in(0,T)$. Integrating over the time interval
$(0,t^{\prime})\subset(0,T)$, we observe after using the integration by parts
formula in Proposition 2.3 (with $v_{1}=v_{2}=e_{w}^{n}(t)$ for a.e.
$t\in(0,T)$) and the Cauchy-Schwarz inequality that

$\displaystyle I_{1}:=\frac{1}{2}\lVert
e_{w}^{n}(t^{\prime})\rVert_{H}^{2}-\frac{1}{2}\lVert
e_{w}^{n}(0)\rVert_{H}^{2}+(m_{A}-m_{j}\lVert
N\rVert^{2})\int_{0}^{t^{\prime}}\lVert e_{w}^{n}(t)\rVert_{V}^{2}dt$
$\displaystyle\leq\int_{0}^{t^{\prime}}\lVert\mathcal{R}w^{n-1}(t)-\mathcal{R}w^{n-2}(t)\rVert_{V^{\ast}}\lVert
e_{w}^{n}(t)\rVert_{V}dt$ $\displaystyle+(\bar{m}_{j}\lVert
N\rVert+\beta_{2\varphi}\lVert K\rVert)\int_{0}^{t^{\prime}}\lVert
e_{\alpha}^{n-1}(t)\rVert_{Y}\lVert e_{w}^{n}(t)\rVert_{V}dt$
$\displaystyle+\beta_{1\varphi}\lVert K\rVert\lVert
M\rVert\int_{0}^{t^{\prime}}\lVert w^{n-1}(t)\rVert_{V}\lVert
e_{\alpha}^{n-1}(t)\rVert_{Y}\lVert e_{w}^{n}(t)\rVert_{V}dt$
$\displaystyle+\beta_{3\varphi}\lVert
K\rVert\int_{0}^{t^{\prime}}\lVert\mathcal{S}_{\varphi}w^{n-1}(t)-\mathcal{S}_{\varphi}w^{n-2}(t)\rVert_{X}\lVert
e_{w}^{n}(t)\rVert_{V}dt$ $\displaystyle+\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert\int_{0}^{t^{\prime}}\lVert e_{w}^{n-1}(t)\rVert_{V}\lVert
e_{w}^{n}(t)\rVert_{V}dt$ $\displaystyle+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\int_{0}^{t^{\prime}}\lVert\alpha^{n-2}(t)\rVert_{Y}\lVert
e_{w}^{n-1}(t)\rVert_{V}\lVert e_{w}^{n}(t)\rVert_{V}dt$
$\displaystyle+\beta_{6\varphi}\lVert K\rVert\lVert
M\rVert\int_{0}^{t^{\prime}}\lVert
w^{n-1}(t)\rVert_{V}\lVert\mathcal{S}_{\varphi}w^{n-1}(t)-\mathcal{S}_{\varphi}w^{n-2}(t)\rVert_{X}\lVert
e_{w}^{n}(t)\rVert_{V}dt$ $\displaystyle+\beta_{7\varphi}\lVert
K\rVert\int_{0}^{t^{\prime}}\lVert\alpha^{n-1}(t)\rVert_{Y}\lVert\mathcal{S}_{\varphi}w^{n-1}(t)-\mathcal{S}_{\varphi}w^{n-2}(t)\rVert_{X}\lVert
e_{w}^{n}(t)\rVert_{V}dt$ $\displaystyle+\bar{m}_{j}\lVert
N\rVert\int_{0}^{t^{\prime}}\lVert\mathcal{S}_{j}w^{n-1}(t)-\mathcal{S}_{j}w^{n-2}(t)\rVert_{X}\lVert
e_{w}^{n}(t)\rVert_{V}dt$
$\displaystyle=:I_{1,1}+I_{1,2}+I_{1,3}+I_{1,4}+I_{1,5}+I_{1,6}+I_{1,7}+I_{1,8}+I_{1,9}$

for a.e. $t^{\prime}\in(0,T)$. We may apply the Cauchy-Schwarz inequality to
$I_{1,1}$, $I_{1,2}$, $I_{1,4}$ , $I_{1,5}$, and $I_{1,9}$. For $I_{1,3}$ and
$I_{1,6}+I_{1,8}$, we respectively apply Hölder’s inequality with
$\frac{1}{2}+\frac{1}{\infty}+\frac{1}{2}=1$ and
$\frac{1}{\infty}+\frac{1}{2}+\frac{1}{2}=1$. To treat $I_{1,7}$, we observe
by the Cauchy-Schwarz inequality and $H(\mathcal{S}_{\varphi})$(i) that

(5.14)
$\displaystyle\lVert\mathcal{S}_{\varphi}w^{n-1}(t)-\mathcal{S}_{\varphi}w^{n-2}(t)\rVert_{X}^{2}$
$\displaystyle\leq c_{\mathcal{S}_{\varphi}}^{2}T\int_{0}^{t}\lVert
e_{w}^{n-1}(s)\rVert_{V}^{2}ds$

for a.e. $t\in(0,T)$. Therefore, we may use Hölder’s inequality with
$\frac{1}{2}+\frac{1}{\infty}+\frac{1}{2}=1$. Similarly, by
$H(\mathcal{R})$(i) and $H(\mathcal{S}_{j})$(i), respectively, we obtain

(5.15)
$\displaystyle\lVert\mathcal{R}w^{n-1}(t)-\mathcal{R}w^{n-2}(t)\rVert_{V^{\ast}}^{2}$
$\displaystyle\leq c_{\mathcal{R}}^{2}T\int_{0}^{t}\lVert
e_{w}^{n-1}(s)\rVert_{V}^{2}ds$ (5.16)
$\displaystyle\lVert\mathcal{S}_{j}w^{n-1}(t)-\mathcal{S}_{j}w^{n-2}(t)\rVert_{X}^{2}$
$\displaystyle\leq c_{\mathcal{S}_{j}}^{2}T\int_{0}^{t}\lVert
e_{w}^{n-1}(s)\rVert_{V}^{2}ds$

for a.e. $t\in(0,T)$. In addition, using that $w^{n}(0)=w^{n-1}(0)$ and the
smallness-assumption (3.4), then

$\displaystyle(m_{A}-m_{j}\lVert N\rVert^{2})\int_{0}^{t^{\prime}}\lVert
e_{w}^{n}(t)\rVert_{V}^{2}dt\leq I_{1}$

for all $t^{\prime}\in[0,T]$. Gathering the above and dividing by $\lVert
e_{w}^{n}\rVert_{L^{2}_{T}V}$, we have

(5.17) $\displaystyle I_{2}$ $\displaystyle:=(m_{A}-m_{j}\lVert
N\rVert^{2})\bigg{[}\int_{0}^{t^{\prime}}\lVert
e_{w}^{n}(t)\rVert^{2}_{V}dt\bigg{]}^{1/2}$ $\displaystyle\leq
T^{k}(c_{\mathcal{R}}+\beta_{3\varphi}\lVert K\rVert
c_{\mathcal{S}_{\varphi}}+\bar{m}_{j}\lVert N\rVert
c_{\mathcal{S}_{j}}+\beta_{7\varphi}\lVert K\rVert
c_{\mathcal{S}_{\varphi}}\lVert\alpha^{n-1}\rVert_{L^{\infty}_{T}Y})$
$\displaystyle\times\bigg{[}\int_{0}^{t^{\prime}}\int_{0}^{t}\lVert
e_{w}^{n-1}(s)\rVert_{V}^{2}dsdt\bigg{]}^{1/2}$
$\displaystyle+(\bar{m}_{j}\lVert N\rVert+\beta_{2\varphi}\lVert
K\rVert)\bigg{[}\int_{0}^{t^{\prime}}\lVert
e_{\alpha}^{n-1}(t)\rVert_{Y}^{2}dt\bigg{]}^{1/2}$
$\displaystyle+\beta_{1\varphi}\lVert K\rVert\lVert M\rVert\lVert
w^{n-1}\rVert_{L^{2}_{T}V}\lVert
e_{\alpha}^{n-1}\rVert_{L^{\infty}_{t^{\prime}}Y}$
$\displaystyle+(\beta_{4\varphi}+\beta_{5\varphi}\lVert\alpha^{n-2}\rVert_{L^{\infty}_{T}Y})\lVert
K\rVert\lVert M\rVert\bigg{[}\int_{0}^{t^{\prime}}\lVert
e_{w}^{n-1}(t)\rVert^{2}_{V}dt\bigg{]}^{1/2}$
$\displaystyle+T^{k}\beta_{6\varphi}c_{\mathcal{S}_{\varphi}}\lVert
K\rVert\lVert M\rVert\lVert
w^{n-1}\rVert_{L^{2}_{T}V}\bigg{[}\int_{0}^{t^{\prime}}\lVert
e_{w}^{n-1}(t)\rVert_{V}^{2}dt\bigg{]}^{1/2}$
$\displaystyle=:I_{2,1}+I_{2,2}+I_{2,3}+I_{2,4}+I_{2,5}$

for all $t^{\prime}\in[0,T]$ and $k\geq 1/2$. Next, subtracting (5.3c) for two
iterations at the levels $n-1$ and $n-2$. Utilizing Minkowski’s inequality and
$H(\mathcal{G})$(ii) yields

(5.18) $\displaystyle\lVert e_{\alpha}^{n-1}(t)\rVert_{Y}$ $\displaystyle\leq
L_{\mathcal{G}}\int_{0}^{t}\lVert
e_{\alpha}^{n-1}(s)\rVert_{Y}ds+L_{\mathcal{G}}\lVert
M\rVert\int_{0}^{t}\lVert e_{w}^{n-1}(s)\rVert_{V}ds$

for a.e. $t\in(0,T)$. Applying a standard Grönwall argument and the Cauchy-
Schwarz inequality reads

(5.19) $\displaystyle\lVert e_{\alpha}^{n-1}(t)\rVert_{Y}$ $\displaystyle\leq
cT^{k}(1+cT\mathrm{e}^{cT})\Big{[}\int_{0}^{t}\lVert
e_{w}^{n-1}(s)\rVert^{2}_{V}ds\Big{]}^{1/2}$

for a.e. $t\in(0,T)$ and $k\geq 1/2$. From (5.13), $I_{2,4}$ becomes

$\displaystyle I_{2,4}$
$\displaystyle\leq(\beta_{4\varphi}+\beta_{5\varphi}\lVert\alpha_{0}\rVert_{Y})\lVert
K\rVert\lVert M\rVert\bigg{[}\int_{0}^{t^{\prime}}\lVert
e_{w}^{n-1}(t)\rVert^{2}_{V}dt\bigg{]}^{1/2}$
$\displaystyle+T^{k}c(\lVert(f,w_{0},\alpha_{0})\rVert_{L^{2}_{T}V^{\ast}\times
V\times Y})\bigg{[}\int_{0}^{t^{\prime}}\lVert
e_{w}^{n-1}(t)\rVert^{2}_{V}dt\bigg{]}^{1/2}$
$\displaystyle=:I_{2,4,1}+I_{2,4,2}.$

for some $k\geq 1/2$. Combining $I_{2,4,2}$ and $I_{2,5}$, we define

$\displaystyle I_{2,6}:=T^{k}(\beta_{6\varphi}c_{\mathcal{S}_{\varphi}}\lVert
w^{n-1}\rVert_{L^{2}_{T}V}+c(\lVert(f,w_{0},\alpha_{0})\rVert_{L^{2}_{T}V^{\ast}\times
V\times Y}))\lVert K\rVert\lVert M\rVert\bigg{[}\int_{0}^{t^{\prime}}\lVert
e_{w}^{n-1}(t)\rVert^{2}_{V}dt\bigg{]}^{1/2}$

for some $k\geq 1/2$. Accordingly, we have that

$\displaystyle I_{2}\leq I_{2,1}+I_{2,2}+I_{2,3}+I_{2,4,1}+I_{2,6}.$

Applying Young’s inequality to $I_{2,4,1}$ and
$I_{2,1}+I_{2,2}+I_{2,3}+I_{2,6}$ and then the arithmetic-quadratic mean
inequality to the latter term, we obtain

$\displaystyle(I_{2})^{2}$
$\displaystyle\leq(I_{2,1}+I_{2,2}+I_{2,3}+I_{2,4,1}+I_{2,6})^{2}$
$\displaystyle\leq 2(I_{2,4,1})^{2}+2(I_{2,1}+I_{2,2}+I_{2,3}+I_{2,6})^{2}$
$\displaystyle\leq
2(I_{2,4,1})^{2}+8\Big{[}(I_{2,1})^{2}+(I_{2,2})^{2}+(I_{2,3})^{2}+(I_{2,6})^{2}\Big{]}.$

From (D.1) and Young’s inequality, the terms $(I_{2,2})^{2}$ and
$(I_{2,3})^{2}$ becomes

$\displaystyle(I_{2,2})^{2}$ $\displaystyle\leq
T^{k}c(1+cT\mathrm{e}^{2cT})\int_{0}^{t^{\prime}}\lVert
e_{w}^{n-1}(t)\rVert_{V}^{2}dt$ $\displaystyle(I_{2,3})^{2}$
$\displaystyle\leq T^{k}c(1+cT^{2}\mathrm{e}^{2cT})\int_{0}^{t^{\prime}}\lVert
e_{w}^{n-1}(t)\rVert_{V}^{2}dt$

for some $k\geq 1/2$. Gathering the above estimates and noting that
$m_{A}-m_{j}\lVert N\rVert^{2}>0$ by the smallness-assumption (3.4) reads

(5.20) $\displaystyle I_{3}$ $\displaystyle:=\int_{0}^{t^{\prime}}\lVert
e_{w}^{n}(t)\rVert_{V}^{2}dt$ $\displaystyle\leq
T^{k}\frac{c}{(m_{A}-m_{j}\lVert
N\rVert^{2})^{2}}\int_{0}^{t^{\prime}}\int_{0}^{t}\lVert
e_{w}^{n-1}(s)\rVert_{V}^{2}dsdt$
$\displaystyle+\frac{2(\beta_{4\varphi}+\beta_{5\varphi}\lVert\alpha_{0}\rVert_{Y})^{2}\lVert
K\rVert^{2}\lVert M\rVert^{2}}{(m_{A}-m_{j}\lVert
N\rVert^{2})^{2}}\int_{0}^{t^{\prime}}\lVert e_{w}^{n-1}(t)\rVert^{2}_{V}dt$
$\displaystyle+T^{k}\frac{c(1+cT^{2}\mathrm{e}^{2cT})}{(m_{A}-m_{j}\lVert
N\rVert^{2})^{2}}\int_{0}^{t^{\prime}}\lVert e_{w}^{n-1}(t)\rVert^{2}_{V}dt$
$\displaystyle=:\Big{(}T^{k}R+\frac{2(\beta_{4\varphi}+\beta_{5\varphi}\lVert\alpha_{0}\rVert_{Y})^{2}\lVert
K\rVert^{2}\lVert M\rVert^{2}}{(m_{A}-m_{j}\lVert
N\rVert^{2})^{2}}\Big{)}\int_{0}^{t^{\prime}}\lVert
e_{w}^{n-1}(t)\rVert^{2}_{V}dt$

for all $t^{\prime}\in[0,T]$ and some $k\geq 1/2$. Here,

$\displaystyle R$
$\displaystyle=\frac{c(\lVert(f,w_{0},\alpha_{0})\rVert_{L^{2}_{T}V^{\ast}\times
V\times Y})}{(m_{A}-m_{j}\lVert N\rVert^{2})^{2}}.$

The aim is to get the right-hand side of (5.20) to go to zero as
$n\rightarrow\infty$. We will combine the smallness condition (3.4) and the
assumption on the final time $T$.
Iterating over $n\in\mathbb{Z}_{+}$, we have that

$\displaystyle I_{3}\leq
c\bigg{(}T^{k}R+\frac{2(\beta_{4\varphi}+\beta_{5\varphi}\lVert\alpha_{0}\rVert_{Y})^{2}\lVert
K\rVert^{2}\lVert M\rVert^{2}}{(m_{A}-m_{j}\lVert
N\rVert^{2})^{2}}\bigg{)}^{n}(\lVert w^{1}\rVert_{L^{2}_{T}V}^{2}+\lVert
w_{0}\rVert_{V}^{2})$

for some $k\geq 1/2$. Choosing $T>0$ such that

$\displaystyle T^{k}\sim\frac{1}{R}$

small enough for some $k\geq 1/2$, the smallness-assumption (3.4) implies that

(5.21) $\displaystyle
T^{k}R+\frac{2(\beta_{4\varphi}+\beta_{5\varphi}\lVert\alpha_{0}\rVert_{Y})^{2}\lVert
K\rVert^{2}\lVert M\rVert^{2}}{(m_{A}-m_{j}\lVert N\rVert^{2})^{2}}<1.$

Hence, (5.10) and then passing the limit $n\rightarrow\infty$ gives us

$\displaystyle\lim_{n\rightarrow\infty}c\bigg{(}T^{k}R+\frac{2(\beta_{4\varphi}+\beta_{5\varphi}\lVert\alpha_{0}\rVert_{Y})^{2}\lVert
K\rVert^{2}\lVert M\rVert^{2}}{(m_{A}-m_{j}\lVert
N\rVert^{2})^{2}}\bigg{)}^{n}=0$

as desired.
Consequently, iterating over $n\in\mathbb{Z}_{+}$ in (5.20), and then passing
the limit $n\rightarrow\infty$ gives us that $\\{w^{n}\\}_{n\geq 1}$ is a
Cauchy sequence in $L^{2}_{T}V$, and (D.1) implies that
$\\{\alpha^{n}\\}_{n\geq 1}$ is a Cauchy sequence in $C([0,T];Y)$. Moreover,
$\\{\mathcal{R}w^{n}\\}_{n\geq 1}$ is a Cauchy sequence in $L^{2}_{T}V^{\ast}$
by (5.15). Similarly, $\\{\mathcal{S}_{\varphi}w^{n}\\}_{n\geq 1}$ is a Cauchy
sequence in $L^{2}_{T}X$ by (5.14), and $\\{\mathcal{S}_{j}w^{n}\\}_{n\geq 1}$
is a Cauchy sequence in $L^{2}_{T}X$ by (5.16). Concluding the proof. ∎

######  Corollary 5.6.

Let $w^{0}=w_{0}\in V$ and $\alpha^{0}=\alpha_{0}\in Y$. Under the hypothesis
of Theorem 3.3 with
$\beta_{1\varphi}=\beta_{4\varphi}=\beta_{5\varphi}=\beta_{6\varphi}=\beta_{7\varphi}=0$,
let $\\{(w^{n},\alpha^{n})\\}_{n\geq 1}\subset\mathcal{W}^{1,2}_{T}\times
C([0,T];Y)$ be the solution of (5.3a)-(5.3c) for any time $T>0$. Then,
$\\{(w^{n},\alpha^{n})\\}_{n\geq 1}$ is a Cauchy sequence in $L^{2}_{T}V\times
C([0,T];Y)$. In addition, $\\{\mathcal{R}w^{n}\\}_{n\geq 1}$,
$\\{\mathcal{S}_{\varphi}w^{n}\\}_{n\geq 1}$, and
$\\{\mathcal{S}_{j}w^{n}\\}_{n\geq 1}$ are Cauchy sequences in
$L^{2}_{T}V^{\ast}$, $L^{2}_{T}X$ and $L^{2}_{T}X$, respectively.

The proof of Corollary 5.6 can be found in Appendix D.

Step 5.5.6 (Passing the limit in (5.3a)-(5.3c)). From Proposition 5.4, it
follows as $n\rightarrow\infty$ that

(5.22a) $\displaystyle w^{n}\rightarrow w\text{ strongly in }$ $\displaystyle
L^{2}_{T}V,$ $\displaystyle\alpha^{n}\rightarrow\alpha\text{ strongly in
}C([0,T];Y),$ (5.22b)
$\displaystyle\mathcal{S}_{\varphi}w^{n}\rightarrow\mathcal{S}_{\varphi}w\text{
strongly in }$ $\displaystyle L^{2}_{T}X,$
$\displaystyle\mathcal{S}_{j}w^{n}\rightarrow\mathcal{S}_{j}w\text{ strongly
in }L^{2}_{T}X,$ (5.22c)
$\displaystyle\mathcal{R}w^{n}\rightarrow\mathcal{R}w\text{ strongly in }$
$\displaystyle L^{2}_{T}V^{\ast}.$

We are now in a position to pass the limit in (5.3a)-(5.3c). First, by (5.13),
we have that $\\{w^{n}\\}_{n\geq 1}$ and $\\{\dot{w}^{n}\\}_{n\geq 1}$ are
uniformly bounded in $L^{2}_{T}V$ and $L^{2}_{T}V^{\ast}$, respectively. Then,
by the Eberlein–Šmulian theorem, as $L^{2}_{T}V$ is a reflexive Banach space,
we have, upon passing to a subsequence, that

$\displaystyle w^{n}\rightarrow w$ $\displaystyle\ \ \text{ weakly in
}L^{2}_{T}V,$ $\displaystyle\dot{w}^{n}\rightarrow\dot{w}\ \ \text{ weakly in
}L^{2}_{T}V^{\ast}.$

Now, with (5.22a)-(5.22c) in mind, we find by a proof of contradiction
(assuming that the below does not hold thereby obtaining a contradiction with
(5.22a)-(5.22c)) that

(5.23a) $\displaystyle w^{n}(t)\rightarrow w(t)\text{ strongly in }$
$\displaystyle V\ \ \text{ for a.e. }t\in(0,T),$ (5.23b)
$\displaystyle\alpha^{n}(t)\rightarrow\alpha(t)\text{ strongly in }$
$\displaystyle Y\ \ \text{ for a.e. }t\in(0,T),$ (5.23c)
$\displaystyle\mathcal{S}_{\varphi}w^{n}(t)\rightarrow\mathcal{S}_{\varphi}w(t)\text{
strongly in }$ $\displaystyle X\ \ \text{ for a.e. }t\in(0,T),$ (5.23d)
$\displaystyle\mathcal{S}_{j}w^{n}(t)\rightarrow\mathcal{S}_{j}w(t)\text{
strongly in }$ $\displaystyle X\ \ \text{ for a.e. }t\in(0,T),$ (5.23e)
$\displaystyle\mathcal{R}w^{n}(t)\rightarrow\mathcal{R}w(t)\text{ strongly in
}$ $\displaystyle V^{\ast}\ \ \text{ for a.e. }t\in(0,T),$ as
$n\rightarrow\infty$. By similar arguments, we find by (5.13) that
$\\{w^{n}(t)\\}_{n\geq 1}$ for a.e. $t\in(0,T)$ is uniformly bounded in $V$
(see, e.g., the first part of the proof of [38, Lemma 13]). Since $V$ is a
reflexive Banach space, it follows by the Eberlein–Šmulian theorem, up to a
subsequence, that $w^{n}(t)\rightarrow\tilde{w}(t)$ weakly in $V$ for a.e.
$t\in(0,T)$. By uniqueness of limits, we have by (5.23a) that
$\tilde{w}(t)=w(t)$ for a.e. $t\in(0,T)$. Following the same reasoning shows
that $\\{\dot{w}^{n}(t)\\}_{n\geq 1}$ for a.e. $t\in(0,T)$ is uniformly
bounded in $V^{\ast}$. Thus, as $n\rightarrow\infty$ (5.23f)
$\displaystyle\dot{w}^{n}(t)\rightarrow\dot{w}(t)\text{ weakly in }$
$\displaystyle V^{\ast}\ \ \text{ for a.e. }t\in(0,T).$

From (5.23a) and $H(A)$(ii), we get

(5.24)
$\lim_{n\rightarrow\infty}\langle{A(t,w^{n}(t))},{v}\rangle=\langle{A(t,w(t))},{v}\rangle\
\ \text{ for all }v\in V,\text{ a.e. }t\in(0,T).$

Moreover, we utilize $H(j)$(iii), (5.23a)-(5.23b), (5.23d), and $H(MNK)$ to
obtain

$\displaystyle\limsup_{n\rightarrow\infty}\ $ $\displaystyle
j^{\circ}(t,\alpha^{n-1}(t),\mathcal{S}_{j}w^{n-1}(t),Nw^{n}(t);Nv-Nw^{n}(t))$
$\displaystyle\leq j^{\circ}(t,\alpha(t),\mathcal{S}_{j}w(t),Nw(t);Nv-Nw(t))$

for all $v\in V$, a.e. $t\in(0,T)$. Next, by (5.23a), (5.23e)-(5.23f), (5.24),
and the Cauchy-Schwarz inequality, we can find that

$\displaystyle\lim_{n\rightarrow\infty}\langle{A(t,w^{n}(t))},{v-w^{n}(t)}\rangle$
$\displaystyle=\langle{A(t,w(t))},{v-w(t)}\rangle,$
$\displaystyle\lim_{n\rightarrow\infty}\langle{\dot{w}^{n}(t)},{v-w^{n}(t)}\rangle$
$\displaystyle=\langle{\dot{w}(t)},{v-w(t)}\rangle,$
$\displaystyle\lim_{n\rightarrow\infty}\langle{\mathcal{R}w^{n}(t)},{v-w^{n}(t)}\rangle$
$\displaystyle=\langle{\mathcal{R}w(t)},{v-w(t)}\rangle,$
$\displaystyle\lim_{n\rightarrow\infty}\langle{f(t)},{v-w^{n}(t)}\rangle$
$\displaystyle=\langle{f(t)},{v-w(t)}\rangle$

for all $v\in V$, a.e. $t\in(0,T)$. Furthermore, let $\mathbb{E}=Y\times
X\times U$ be equipped with the norm $\lVert(x,y,z)\rVert_{\mathbb{E}}=\lVert
x\rVert_{Y}+\lVert y\rVert_{X}+\lVert z\rVert_{U}$. We wish to deduce that
$\varphi(t,\cdot,\cdot,\cdot,\cdot)$ is continuous on $\mathbb{E}\times Z$ for
a.e. $t\in(0,T)$ by applying Lemma 2.12. The conditions (1) and (3) are
directly obtained by $H(\varphi)$(ii),(iv). Lastly, we find that condition (2)
holds by Proposition 2.9. Indeed, as $\varphi$ is lower semicontinuous and
convex in its last argument, by $H(\varphi)$(iii), and the fact that $\varphi$
is finite (does not take the values $\pm\infty$), it then follows by
(5.23a)-(5.23c) and $H(MNK)$ that

$\displaystyle\lim_{n\rightarrow\infty}\big{[}\varphi(t,\alpha^{n-1}(t),\mathcal{S}_{\varphi}w^{n-1}(t),Mw^{n-1}(t),Kv)$
$\displaystyle\qquad-\varphi(t,\alpha^{n-1}(t),\mathcal{S}_{\varphi}w^{n-1}(t),Mw^{n-1}(t),Kw^{n-1}(t))\big{]}$
$\displaystyle\qquad=\varphi(t,\alpha(t),\mathcal{S}_{\varphi}w(t),Mw(t),Kv)-\varphi(t,\alpha(t),\mathcal{S}_{\varphi}w(t),Mw(t),Kw(t))$

for all $v\in V$, a.e. $t\in(0,T)$. Next, we have by $H(\mathcal{G})$(ii),
that $\mathcal{G}(t,\cdot,\cdot)$ is continuous on $Y\times U$ for a.e.
$t\in(0,T)$. From (5.8), $H(\mathcal{G})$(iii), and (5.13), we obtain the
desired bound, i.e., integrable and independent of $n$. Combining (5.23a),
(5.23b), and $H(MNK)$, we may apply the dominated convergence theorem to
conclude

$\displaystyle\lim_{n\rightarrow\infty}\int_{0}^{t}\mathcal{G}(s,\alpha^{n}(s),Mw^{n}(s))ds=\int_{0}^{t}\mathcal{G}(s,\alpha(s),Mw(s))ds$

for all $t\in[0,T]$. Thus, passing the upper limit $n\rightarrow\infty$ in
(5.3a)-(5.3c) gives us that $(w,\alpha)\in\mathcal{W}^{1,2}_{T}\times
C([0,T];Y)$ is indeed a solution to Problem 1.

Step 5 (Continuous dependence on initial data). For simplicity of notation, we
define $\mathbb{U}=L^{2}_{T/2}V\times C([0,T/2];Y)$ equipped with the norm
$\lVert(x,y)\rVert_{\mathbb{U}}^{2}=\lVert x\rVert_{L^{2}_{T/2}V}^{2}+\lVert
y\rVert_{L^{\infty}_{T/2}Y}^{2}$. Consider two sets of initial data
$(w_{01},\alpha_{01}),(w_{02},\alpha_{02})\in V\times Y$, we aim to prove that
for all $\lambda>0$ there exists a $\delta>0$, which will be fixed later, such
that

(5.25a)
$\displaystyle\lVert(w_{01},\alpha_{01})-(w_{02},\alpha_{02})\rVert_{V\times
Y}<\delta$ implies (5.25b)
$\displaystyle\lVert(w_{1},\alpha_{1})-(w_{2},\alpha_{2})\rVert_{\mathbb{U}}<\lambda.$

We consider the time interval $(0,T/2)$ with $T$ as in (3.5) to guarantee that
$\lVert(w_{01},\alpha_{01})-(w_{02},\alpha_{02})\rVert_{V\times Y}<\delta$.
Here, $(w_{1},\alpha_{1}),(w_{2},\alpha_{2})\in\mathcal{W}^{1,2}_{T/2}\times
C([0,T/2];Y)$ are two solutions to Problem 1 corresponding to
$(w_{01},\alpha_{01}),(w_{02},\alpha_{02})$. That is, for $i=1,2$,
$(w_{i},\alpha_{i})\in\mathcal{W}^{1,2}_{T/2}\times C([0,T/2];Y)$ is the
solution to

$\displaystyle\alpha_{i}(t)=\alpha_{0i}+\int_{0}^{t}\mathcal{G}(s,\alpha_{i}(s),Mw_{i}(s))ds,$
$\displaystyle\langle{\dot{w}_{i}(t)+A(t,w_{i}(t))-f(t)+\mathcal{R}w_{i}(t)},{v-w_{i}(t)}\rangle$
$\displaystyle+\varphi(t,\alpha_{i}(t),\mathcal{S}_{\varphi}w_{i}(t),Mw_{i}(t),Kv)-\varphi(t,\alpha_{i}(t),\mathcal{S}_{\varphi}w_{i}(t),Mw_{i}(t),Kw_{i}(t))$
$\displaystyle+j^{\circ}(t,\alpha_{i}(t),\mathcal{S}_{j}w_{i}(t),Nw_{i}(t);Nv-
Nw_{i}(t))\geq 0$ for all $v\in V$, a.e. $t\in(0,T/2)$ with $\displaystyle
w_{i}(0)=w_{0i}.$

Let $\\{(w_{i}^{n},\alpha^{n}_{i})\\}_{n\geq
1}\subset\mathcal{W}_{T/2}^{1,2}\times C([0,T/2];Y)$ be two solutions of
(5.3a)-(5.3c) corresponding to the data $(w_{0i},\alpha_{0i})\in V\times Y$
for $i=1,2$. Then observe that

$\displaystyle\lVert(w_{1},\alpha_{1})-(w_{2},\alpha_{2})\rVert_{\mathbb{U}}$
$\displaystyle\leq\lVert(w_{1}^{n},\alpha_{1}^{n})-(w_{1},\alpha_{1})\rVert_{\mathbb{U}}+\lVert(w_{2}^{n},\alpha_{2}^{n})-(w_{2},\alpha_{2})\rVert_{\mathbb{U}}+\lVert(w_{1}^{n},\alpha_{1}^{n})-(w_{2}^{n},\alpha_{2}^{n})\rVert_{\mathbb{U}}$
$\displaystyle=:B_{1}+B_{2}+B_{3}.$

By Step 5.5, we have that $(w^{n},\alpha^{n})\rightarrow(w,\alpha)$ strongly
in $\mathbb{U}$ when $n\rightarrow\infty$. Consequently, $B_{1}$ and $B_{2}$
must at least satisfy the estimate

$\displaystyle B_{1}+B_{2}<\delta.$

While for $B_{3}$, we need the continuity of the flow map with respect to
(5.3a)-(5.3c). We add the two inequalities and choose $v=w^{n}_{k}(t)$ for
a.e. $t\in(0,T/2)$ and $k=1,2$, $i\neq k$. Let
$(w^{n-1}_{1},\alpha^{n-1}_{1}),(w^{n-1}_{2},\alpha^{n-1}_{2})\in\mathcal{W}^{1,2}_{T/2}\times
C([0,T/2];Y)$ be given, and $W^{0}=W_{0}:=w_{01}-w_{02}\in V$ and
$\Sigma^{0}=\Sigma_{0}:=\alpha_{01}-\alpha_{02}\in Y$. Then
$W^{n}=w^{n}_{1}-w^{n}_{2}$ and $\Sigma^{n}=\alpha_{1}^{n}-\alpha_{2}^{n}$
solves

(5.27a)
$\displaystyle\langle{\dot{W}^{n}(t)+A(t,w^{n}_{1}(t))-A(t,w^{n}_{2}(t))+\mathcal{R}w^{n-1}_{1}(t)-\mathcal{R}w^{n-1}_{2}(t)},{W^{n}(t)}\rangle$
$\displaystyle+\varphi(t,\alpha^{n-1}_{1}(t),\mathcal{S}_{\varphi}w^{n-1}_{1}(t),Mw^{n-1}_{1}(t),Kw^{n}_{2}(t))$
$\displaystyle-\varphi(t,\alpha^{n-1}_{1}(t),\mathcal{S}_{\varphi}w^{n-1}_{1}(t),Mw^{n-1}_{1}(t),Kw^{n}_{1}(t))$
$\displaystyle+\varphi(t,\alpha^{n-1}_{2}(t),\mathcal{S}_{\varphi}w^{n-1}_{2}(t),Mw^{n-1}_{2}(t),Kw^{n}_{1}(t))$
$\displaystyle-\varphi(t,\alpha^{n-1}_{2}(t),\mathcal{S}_{\varphi}w^{n-1}_{2}(t),Mw^{n-1}_{2}(t),Kw^{n}_{2}(t))$
$\displaystyle+j^{\circ}(t,\alpha^{n-1}_{1}(t),\mathcal{S}_{j}w^{n-1}_{1}(t),Nw^{n}_{1}(t);-W^{n}(t))$
$\displaystyle+j^{\circ}(t,\alpha^{n-1}_{2}(t),\mathcal{S}_{j}w^{n-1}_{2}(t),Nw^{n}_{2}(t);W^{n}(t))\geq
0$ for a.e. $t\in(0,T/2)$ with (5.27b) $W^{n}(0)=W_{0},$ and (5.27c)
$\Sigma^{n}(t)=\Sigma_{0}+\int_{0}^{t}[\mathcal{G}(s,\alpha^{n}_{1}(s),Mw^{n}_{1}(s))-\mathcal{G}(s,\alpha^{n}_{2}(s),Mw^{n}_{2}(s))]ds$

for a.e. $t\in(0,T/2)$. To find the desired estimates, we need the following
lemma.

######  Lemma 5.7.

Let $w^{0}_{i}=w_{0i}\in V$ and $\alpha^{0}_{i}=\alpha_{0i}\in Y$ for $i=1,2$.
Under the assumptions of Theorem 3.3, let $\\{(W^{n},\Sigma^{n})\\}_{n\geq
1}\subset\mathcal{W}_{T/2}^{1,2}\times C([0,T/2];Y)$ be the solution to
(5.27a)-(5.27c). Then

(5.28) $\displaystyle\lVert W^{n}\rVert_{L^{2}_{T/2}V}^{2}$ $\displaystyle\leq
c(\lVert
W_{0}\rVert_{V}^{2}+\lVert\Sigma^{n-1}\rVert_{L^{\infty}_{T/2}Y}^{2}+\lVert
w^{n-1}_{1}\rVert_{L^{2}_{T/2}Y}^{2}\lVert\Sigma^{n-1}\rVert_{L^{\infty}_{T/2}Y}^{2}$
$\displaystyle+T^{k}\lVert w_{1}^{n-1}\rVert_{L^{2}_{T/2}V}^{2}\lVert
W^{n-1}\rVert_{L^{2}_{T/2}V}^{2}+T^{k}\lVert\alpha_{1}^{n-1}\rVert_{L^{\infty}_{T/2}Y}^{2}\lVert
W^{n-1}\rVert_{L^{2}_{T/2}V}^{2})$ (5.29)
$\displaystyle+\frac{\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha_{2}^{n-1}\rVert_{L^{\infty}_{T/2}Y}}{m_{A}-m_{j}\lVert
N\rVert^{2}}\lVert W^{n-1}\rVert_{L^{2}_{T/2}V}^{2}$
$\displaystyle+\frac{\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha_{2}^{n-1}\rVert_{L^{\infty}_{T/2}Y}}{m_{A}-m_{j}\lVert
N\rVert^{2}}\lVert W^{n}\rVert_{L^{2}_{T/2}V}^{2}$

for all $n\in\mathbb{Z}_{+}$ and some $k\geq 1/2$.

The proof of Lemma 5.7 is postponed to Appendix E. From (5.12) and (5.13), the
estimate (5.28) becomes

$\displaystyle\lVert W^{n}\rVert_{L^{2}_{T/2}V}^{2}$ $\displaystyle\leq
c(\lVert
W_{0}\rVert_{V}^{2}+\lVert\Sigma^{n-1}\rVert_{L^{\infty}_{T/2}Y}^{2})$
$\displaystyle+\frac{\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha_{2}^{n-1}\rVert_{L^{\infty}_{T/2}Y}}{m_{A}-m_{j}\lVert
N\rVert^{2}}\lVert W^{n-1}\rVert_{L^{2}_{T/2}V}^{2}$
$\displaystyle+\frac{\beta_{4\varphi}\lVert K\rVert\lVert
M\rVert+\beta_{5\varphi}\lVert K\rVert\lVert
M\rVert\lVert\alpha_{2}^{n-1}\rVert_{L^{\infty}_{T/2}Y}}{m_{A}-m_{j}\lVert
N\rVert^{2}}\lVert W^{n}\rVert_{L^{2}_{T/2}V}^{2}$

In a similar manner as we obtained (5.9b), we see from (5.27c), a standard
Grönwall argument, the Cauchy-Schwarz inequality, and Young’s inequality that

$\displaystyle\lVert\Sigma^{n-1}\rVert_{L^{\infty}_{T/2}Y}^{2}\leq
c(\lVert\Sigma_{0}\rVert_{Y}^{2}+T^{k}\lVert
W^{n-1}\rVert_{L^{2}_{T/2}V}^{2}).$

By induction of $n$, it follows the same procedure as in Step 5.5 that

$\displaystyle\lVert(W^{n},\Sigma^{n})\rVert_{L^{2}_{T/2}V\times
L^{\infty}_{T/2}Y}^{2}$ $\displaystyle\leq
c(\lVert(W_{0},\Sigma_{0})\rVert_{V\times Y}^{2})$

for all $n\in\mathbb{Z}_{+}$. Consequently, we may choose a $\delta>0$ in
(5.25a) to obtain (5.25b).
Step 5 (Proof of Theorem 3.3). We now have all the tools to prove the main
theorem.

###### Proof of Theorem 3.3.

Combining Step 5-5 gives us well-posedness of Problem 1. ∎

##  6\. Viscoelastic frictional contact problems

We will present two applications to frictional contact; the first considering
contact with normal compliance, and the second contact with normal damped
response. Moreover, in Section 6.3, we introduce a first-order approximation
of a rate-and-state friction law that is covered by our framework. Let
$u:\Omega\times[0,T]\rightarrow\mathbb{R}^{d}$ denote the displacement,
$\sigma:\Omega\times[0,T]\rightarrow\mathbb{S}^{d}$ the stress tensor, and
$\alpha:\Gamma_{C}\times[0,T]\rightarrow\mathbb{R}$ the external state
variable. In addition, $f_{0}$ denotes the body forces, $f_{N}$ the surface
traction, and $\rho$ the density.

Figure 1. A standard illustration of a sliding block.

We let the spaces $H=L^{2}(\Omega;\mathbb{R}^{d})$,
$Q=L^{2}(\Omega;\mathbb{S}^{d})$, $V$, and $\mathcal{W}^{1,2}_{T}$ be defined
by (2.1), (2.2), (2.3), and (2.5), respectively. We refer to Section 2.1-2.2
for further definitions of the function spaces. Further, let
$X=L^{4}(\Gamma_{C})$ and $U=L^{4}(\Gamma_{C};\mathbb{R}^{d})$. Let
$\gamma_{\nu}:V\rightarrow X$ denote the normal trace operator, and
$\gamma_{\tau}:V\rightarrow U$ denote the tangential trace operator. It then
follows by Theorem 2.1 that $\gamma_{\tau}$ and $\gamma_{\nu}$ are well-
defined for $d=2,3$. For all $v\in V$, we let
$v_{\nu}=\gamma_{\nu}v=v\cdot\nu$ denote the normal components on $\Gamma$,
and $v_{\tau}=\gamma_{\tau}v=v-v_{\nu}\nu$ the tangential components on
$\Gamma$. Similarly, let $\sigma_{\nu}=(\sigma\nu)\cdot\nu$, and
$\sigma_{\tau}=\sigma\nu-\sigma_{\nu}\nu$ be the normal and tangential
components of the tensor $\sigma$ on $\Gamma$, respectively.

###  6.1. Dynamic frictional contact problem with normal compliance

In this section, we present a system of equations describing the evolution of
a viscoelastic body in frictional contact with a foundation. Viscoelastic
contact problems with normal compliance and friction are discussed in, e.g.,
[33, Section 8.3]. The normal compliance condition is used as an approximation
of the Signorini non-penetration condition. More on this can be found in [13,
Chapter 5], [17, Chapter 11] and [34]. We wish to study the following problem:

######  Problem 4.

Find the displacement $u:\Omega\times[0,T]\rightarrow\mathbb{R}^{d}$ and the
external state variable $\alpha:\Gamma_{C}\times[0,T]\rightarrow\mathbb{R}$
such that

(6.1a) $\displaystyle\sigma(t)$
$\displaystyle=\mathcal{A}\varepsilon(\dot{u}(t))+\mathcal{B}\varepsilon(u(t))+\int_{0}^{t}\mathcal{C}(t-s,\varepsilon(\dot{u}(s)))ds$
$\displaystyle\text{ on }\Omega\times(0,T)$ (6.1b)
$\displaystyle\rho\ddot{u}(t)$ $\displaystyle=\nabla\cdot\sigma(t)+f_{0}(t)$
$\displaystyle\text{ on }\Omega\times(0,T)$ (6.1c) $\displaystyle u(t)$
$\displaystyle=0$ $\displaystyle\text{ on }\Gamma_{D}\times(0,T)$ (6.1d)
$\displaystyle\sigma(t){\nu}$ $\displaystyle=f_{N}(t)$ $\displaystyle\text{ on
}\Gamma_{N}\times(0,T)$ (6.1e) $\displaystyle-\sigma_{\nu}(t)$
$\displaystyle=p(u_{\nu}(t))$ $\displaystyle\text{ on }\Gamma_{C}\times(0,T)$
(6.1f) $\displaystyle|\sigma_{\tau}(t)|$
$\displaystyle\leq\mu(|\dot{u}_{\tau}(t)|,\alpha(t))|\sigma_{\nu}(t)|$
$\displaystyle\text{ on }\Gamma_{C}\times(0,T)$ (6.1g)
$\displaystyle-\sigma_{\tau}(t)$
$\displaystyle=\mu(|\dot{u}_{\tau}(t)|,\alpha(t))|\sigma_{\nu}(t)|\frac{\dot{u}_{\tau}(t)}{|\dot{u}_{\tau}(t)|},\
\ \text{ if }\dot{u}_{\tau}(t)\neq 0$ $\displaystyle\text{ on
}\Gamma_{C}\times(0,T)$ (6.1h) $\displaystyle\dot{\alpha}(t)$
$\displaystyle=G(\alpha(t),|\dot{u}_{\tau}(t)|)$ $\displaystyle\text{ on
}\Gamma_{C}\times(0,T)$ with the initial conditions (6.1i) $\displaystyle
u(0)$ $\displaystyle=u_{0},\ \ \dot{u}(0)=w_{0}$ $\displaystyle\text{ on
}\Omega$ (6.1j) $\displaystyle\alpha(0)$ $\displaystyle=\alpha_{0}$
$\displaystyle\text{ on }\Gamma_{C}.$

In the above problem, (6.1a) is a general viscoelastic constitutive law, where
$\mathcal{A}$ is a viscosity operator, $\mathcal{B}$ an elasticity operator,
and $\mathcal{C}$ is referred to as a relaxation tensor. We note that
$\mathcal{A}\varepsilon(\dot{u})$ and $\mathcal{B}\varepsilon(u)$ are short-
hand notation for $\mathcal{A}(x,\varepsilon(\dot{u}))$ and
$\mathcal{B}(x,\varepsilon(u))$, respectively. Moreover, (6.1b) is a momentum
balance equation, (6.1c) denotes the Dirichlet boundary conditions, and (6.1d)
the traction applied to the surface. The equation (6.1e) is a contact
condition, where $p$ is a prescribed function describing the penetration
condition. Next, (6.1f)-(6.1g) denotes a generalized Coulomb’s friction law,
and (6.1h) describes the evolution of the external state variable, see Section
1.2 for a discussion on this equation. Lastly, (6.1i)-(6.1j) are initial
conditions. We wish to investigate (6.1a)-(6.1j) under the following
assumptions:
$\underline{H(\mathcal{A})}$:
$\mathcal{A}:\Omega\times\mathbb{S}^{d}\rightarrow\mathbb{S}^{d}\text{ is such
that }$

  1. (i)

For any $\varepsilon\in\mathbb{S}^{d},\ x\mapsto\mathcal{A}(x,\varepsilon)$ is
measurable on $\Omega$.

  2. (ii)

There exists $L_{\mathcal{A}}>0$ such that
$|\mathcal{A}(x,\varepsilon_{1})-\mathcal{A}(x,\varepsilon_{2})|\leq
L_{\mathcal{A}}|\varepsilon_{1}-\varepsilon_{2}|$ for all
$\varepsilon_{1},\varepsilon_{2}\in\mathbb{S}^{d}$, a.e. $x\in\Omega$.

  3. (iii)

There exists $m_{\mathcal{A}}>0$ such that
$(\mathcal{A}(x,\varepsilon_{1})-\mathcal{A}(x,\varepsilon_{2})):(\varepsilon_{1}-\varepsilon_{2})\geq
m_{\mathcal{A}}|\varepsilon_{1}-\varepsilon_{2}|^{2}$, for all
$\varepsilon_{1},\varepsilon_{2}\in\mathbb{S}^{d}$, a.e. $x\in\Omega$.

  4. (iv)

$\mathcal{A}(x,0)=0$ for a.e. $x\in\Omega$.

$\underline{H(\mathcal{B})}$:
$\mathcal{B}:\Omega\times\mathbb{S}^{d}\rightarrow\mathbb{S}^{d}\text{ is such
that }$

  1. (i)

For any $\varepsilon\in\mathbb{S}^{d},\ x\mapsto\mathcal{B}(x,\varepsilon)$ is
measurable on $\Omega$.

  2. (ii)

There exists $L_{\mathcal{B}}>0$ such that
$|\mathcal{B}(x,\varepsilon_{1})-\mathcal{B}(x,\varepsilon_{2})|\leq
L_{\mathcal{B}}|\varepsilon_{1}-\varepsilon_{2}|$ for all
$\varepsilon_{1},\varepsilon_{2}\in\mathbb{S}^{d}$, a.e. $x\in\Omega$.

  3. (iii)

$\lVert\mathcal{B}(\cdot,0)\rVert_{Q}<\infty$.

$\underline{H(\mu)}$:
$\mu:\Gamma_{C}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\text{ is
such that }$

  1. (i)

The mapping $x\mapsto\mu(x,r,y)$ is measurable on $\Gamma_{C}$ for all
$r,y\in\mathbb{R}$.

  2. (ii)

There exist $L_{1\mu},L_{2\mu},L_{3\mu}\geq 0$ such that
$|\mu(x,r_{1},y_{1})-\mu(x,r_{2},y_{2})|\leq(L_{1\mu}+L_{2\mu}|y_{2}|)|r_{1}-r_{2}|+L_{3\mu}|r_{1}||y_{1}-y_{2}|$
for all $r_{1},r_{2}\in\mathbb{R}$, $y_{1},y_{2}\in\mathbb{R}$, a.e.
$x\in\Gamma_{C}$.

  3. (iii)

There exist $\kappa_{1},\kappa_{2},\kappa_{3}\geq 0$ such that
$|\mu(x,r,y)|\leq\kappa_{1}+\kappa_{2}|y|+\kappa_{3}|r|$ for all
$r\in\mathbb{R}$, $y\in\mathbb{R}$, a.e. $x\in\Gamma_{C}$.

$\underline{H(p)}$: $p:\Gamma_{C}\times\mathbb{R}\rightarrow\mathbb{R}\text{
is such that }$

  1. (i)

The mapping $x\mapsto p(x,r)$ is measurable on $\Gamma_{C}$ for all
$r\in\mathbb{R}$.

  2. (ii)

There exists $L_{p}>0$ such that $|p(x,r_{1})-p(x,r_{2})|\leq
L_{p}|r_{1}-r_{2}|$ for all $r_{1},r_{2}\in\mathbb{R}$, a.e. $x\in\Gamma_{C}$.

  3. (iii)

There exists $p^{\ast}>0$ such that $p(x,r)\leq p^{\ast}$ for all
$r\in\mathbb{R}$, a.e. $x\in\Gamma_{C}$.

$\underline{H(G)}$:
$G:\Gamma_{C}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\text{ is
such that }$

  1. (i)

The mapping $x\mapsto G(x,\alpha,r)$ is measurable on $\Omega$ for all
$\alpha,r\in\mathbb{R}$.

  2. (ii)

There exists $L_{G}>0$ such that
$|G(x,\alpha_{1},r_{1})-G(x,\alpha_{2},r_{2})|\leq
L_{G}(|\alpha_{1}-\alpha_{2}|+|r_{1}-r_{2}|)$ for all
$\alpha_{1},\alpha_{2},r_{1},r_{2}\in\mathbb{R}$, a.e. $x\in\Gamma_{C}$.

  3. (iii)

$\lVert G(\cdot,0,0)\rVert_{L^{2}(\Gamma_{C})}<\infty$.

$\underline{H(\mathcal{C})}$:
$\mathcal{C}:\Omega\times(0,T)\times\mathbb{S}^{d}\rightarrow\mathbb{S}\text{
is such that }$

  1. (i)

$\mathcal{C}(x,t,\varepsilon)=(c(x,t)\varepsilon)$ for all
$\varepsilon\in\mathbb{S}^{d}$, a.e. $(x,t)\in\Omega\times(0,T)$.

  2. (ii)

$c(x,t)=(c_{ijkl}(x,t))$ with $c_{ijkl}=c_{jikl}=c_{lkij}\in
L^{\infty}_{T}L^{\infty}(\Omega)$.

(6.2) $\text{The mass density is assumed to be a positive constant
}\rho>0\text{ and}$ (6.3) $f_{0}\in L^{2}_{T}H,\ \ f_{N}\in
L^{2}_{T}L^{2}(\Gamma_{N};\mathbb{R}^{d})$

with the initial data satisfying

(6.4a) $\displaystyle u_{0},\ $ $\displaystyle w_{0}\in V,$ (6.4b)
$\displaystyle\alpha_{0}$ $\displaystyle\in L^{2}(\Gamma_{C}).$

######  Remark 6.1.

Similar assumptions on the operators and data are found in, e.g., [19, 29,
35]. In comparison, the assumptions $H(\mu)$(ii)-(iii) are generalized. In
particular, we relaxed the boundedness assumption on $\mu$.

We refer the reader to Appendix A for a discussion on applications under these
assumptions.

####  6.1.1. Variational formulation

We find a formal derivation of the variational formulation of Problem 4, i.e.,
assuming sufficiently regular functions, as we only are interested in a mild
solution (see Definition 1.1). We refer to, e.g., [33, Section 5.2] for a more
detailed derivation, especially how to deal with the contact conditions.
Inserting (6.1b) into Green’s formula (2.4) yields

$\displaystyle\int_{\Omega}\rho\ddot{u}(t)\cdot\big{[}v-\dot{u}(t)\big{]}dx+\int_{\Omega}\sigma(t):\big{[}\varepsilon(v)-\varepsilon(\dot{u}(t))\big{]}dx$
$\displaystyle=\int_{\Omega}f_{0}(t)\cdot\big{[}v-\dot{u}(t)\big{]}dx$
$\displaystyle+\int_{\Gamma}\sigma(t)\nu\cdot\big{[}v-\dot{u}(t)\big{]}da$

for all $v\in V$, a.e. $t\in(0,T)$. From the terms on $\Gamma_{C}$, we deduce

$\displaystyle\int_{\Gamma_{C}}\sigma_{\tau}(t)\cdot\big{[}v_{\tau}-\dot{u}_{\tau}(t)\big{]}da+\int_{\Gamma_{C}}\sigma_{\nu}(t)\big{[}v_{\nu}-\dot{u}_{\nu}(t)\big{]}da$
$\displaystyle\geq\int_{\Gamma_{C}}\mu(|\dot{u}_{\tau}(t)|,\alpha(t))p(u_{\nu}(t))\big{[}|\dot{u}_{\tau}(t)|-|v_{\tau}|\big{]}da+\int_{\Gamma_{C}}p(u_{\nu}(t))\big{[}\dot{u}_{\nu}(t)-v_{\nu}\big{]}da$

for all $v\in V$, a.e. $t\in(0,T)$. Combining the above reads

$\displaystyle\int_{\Omega}\rho\ddot{u}(t)\cdot\big{[}v-\dot{u}(t)\big{]}dx+\int_{\Omega}\sigma(t):\big{[}\varepsilon(v)-\varepsilon(\dot{u}(t))\big{]}dx$
$\displaystyle+\int_{\Gamma_{C}}\mu(|\dot{u}_{\tau}(t)|,\alpha(t))p(u_{\nu}(t))\big{[}|v_{\tau}|-|\dot{u}_{\tau}(t)|\big{]}da+\int_{\Gamma_{C}}p(u_{\nu}(t))\big{[}v_{\nu}-\dot{u}_{\nu}(t)\big{]}da$
$\displaystyle\geq\int_{\Gamma_{N}}f_{N}(t)\cdot\big{[}v-\dot{u}(t)\big{]}da+\int_{\Omega}f_{0}(t)\cdot\big{[}v-\dot{u}(t)\big{]}dx$

for all $v\in V$, a.e. $t\in(0,T)$. We write the above inequality slightly
more compactly. We observe that the map
$t\mapsto\int_{\Gamma_{N}}f_{N}(t)\cdot vda+\int_{\Omega}f_{0}(t)\cdot vdx$ is
linear and bounded in $V$. Consequently, the Riesz representation theorem
implies the existence of $f(t)\in V^{\ast}$ such that

(6.5)
$\displaystyle\langle{f(t)},{v}\rangle=\big{(}{f_{N}(t)},{v}\big{)}_{L^{2}(\Gamma_{N};\mathbb{R}^{d})}+\big{(}{f_{0}(t)},{v}\big{)}_{H}\
\ \text{ for all }v\in V,\ \ \text{ a.e. }t\in(0,T).$

As mentioned, we are interested in a mild solution of (6.1h) (see Definition
1.1), so we integrate (6.1h) over the time interval $(0,t)$ and use the
initial condition (6.1j) to obtain this equation on the desired form. We may
now formulate a variational inequality of Problem 4.

######  Problem 5.

Find $u:\Omega\times[0,T]\rightarrow\mathbb{R}^{d}$ and
$\alpha:\Gamma_{C}\times[0,T]\rightarrow\mathbb{R}$ such that

(6.6a)
$\displaystyle\alpha(t)=\alpha_{0}+\int_{0}^{t}G(\alpha(s),|\dot{u}_{\tau}(s)|)ds,$
(6.6b)
$\displaystyle\int_{\Gamma_{C}}\mu(|\dot{u}_{\tau}(t)|,\alpha(t))p(u_{\nu}(t))\big{[}|v_{\tau}|-|\dot{u}_{\tau}(t)|\big{]}da+\int_{\Gamma_{C}}p(u_{\nu}(t))\big{[}v_{\nu}-\dot{u}_{\nu}(t)\big{]}da$
$\displaystyle+\int_{\Omega}\rho\ddot{u}(t)\cdot\big{[}v-\dot{u}(t)\big{]}dx+\big{(}{\mathcal{A}\varepsilon(\dot{u}(t))},{\varepsilon(v)-\varepsilon(\dot{u}(t))}\big{)}_{Q}$
$\displaystyle+\big{(}{\mathcal{B}\varepsilon(u(t))+\int_{0}^{t}\mathcal{C}(t-s,\varepsilon(\dot{u}(s)))ds},{\varepsilon(v)-\varepsilon(\dot{u}(t))}\big{)}_{Q}\geq\langle{f(t)},{v-\dot{u}(t)}\rangle$

for all $v\in V$, a.e. $t\in(0,T)$ with

$u(0)=u_{0},\ \ \dot{u}(0)=w_{0}.$

######  Remark 6.2.

Conversely, under the assumption of sufficient regularity, showing that (6.6a)
is equivalent to (6.1h) together with (6.1j) is an application of the
fundamental theorem of calculus. Moreover, choosing the test functions
$v=\dot{u}\pm\tilde{v}$ with $\tilde{v}\in C^{\infty}_{c}(\Omega)$ in (6.6b)
implies that Problem 5 is indeed equivalent to Problem 4, see, e.g., [26,
Section 2.6] or [33, Section 5.2].

The well-posedness result for Problem 5 is summarized below.

######  Theorem 6.3.

Assume that $H(\mathcal{A})$, $H(\mathcal{B})$, $H(\mathcal{C})$, $H(\mu)$,
$H(p)$, $H(G)$, and (6.2)-(6.4) holds. Then, there exists a $T>0$ satisfying
(3.5) such that Problem 5 has at most one solution $(u,\alpha)$ under the
smallness-assumption

(6.7) $\displaystyle m_{\mathcal{A}}>\sqrt{2}\>p^{\ast}$
$\displaystyle(L_{1\mu}\sqrt{\mathrm{meas}(\Gamma_{C})}+L_{2\mu}\lVert\alpha_{0}\rVert_{Y})$
$\displaystyle\times\lVert(\gamma_{\tau},\gamma_{\nu})\rVert_{\mathcal{L}(V,L^{4}(\Gamma_{C};\mathbb{R}^{d})\times
L^{4}(\Gamma_{C}))}\lVert\gamma_{\tau}\rVert_{\mathcal{L}(V,L^{4}(\Gamma_{C};\mathbb{R}^{d}))}.$

In addition $(u,\alpha)$ has the following regularity:

$u\in W^{1,2}(0,T;V),\ \ \dot{u}\in\mathcal{W}^{1,2}_{T}\subset C([0,T];H),\ \
\alpha\in C([0,T];L^{2}(\Gamma_{C})).$

Moreover, there exists a neighborhood around $(u_{0},w_{0},\alpha_{0})$ so
that the flow map $\tilde{F}:V\times V\times L^{2}(\Gamma_{C})\rightarrow
W^{1,2}(0,T/2;V)\times C([0,T/2];L^{2}(\Gamma_{C}))$ defined by
$(u_{0},w_{0},\alpha_{0})\mapsto(u,\alpha)$ is continuous.

######  Remark 6.4.

A similar smallness-assumption (6.7) is used in, e.g., [29].

######  Remark 6.5.

We will show that there exists a solution for $\rho\equiv 1$. For $\rho>0$, we
may take $w(t)=w(\rho t)$.

####  6.1.2. Proof of Theorem 6.3

Our aim is to use Theorem 3.3 to prove Theorem 6.3. Our first task is to
rewrite Problem 5 in the same form as Problem 1. Then, we will verify the
hypothesis of Theorem 3.3.
Let $Y=L^{2}(\Gamma_{C})$ and $Z=U\times X$. We then define the operators
$A:(0,T)\times V\rightarrow V^{\ast}$, $\mathcal{R}:L^{2}_{T}V\rightarrow
L^{2}_{T}V^{\ast}$, $\mathcal{S}_{\varphi}:L^{2}_{T}V\rightarrow L^{2}_{T}X$,
$\mathcal{G}:(0,T)\times Y\times U\rightarrow Y$, $M:V\rightarrow U$, and
$N:V\rightarrow X$, respectively, by

(6.8a) $\displaystyle\langle{A(t,w)},{v}\rangle$
$\displaystyle=\int_{\Omega}\mathcal{A}\varepsilon(w):\varepsilon(v)dx,\
\text{ for }w,v\in V,\text{ a.e. }t\in(0,T),$ (6.8b)
$\displaystyle\langle{\mathcal{R}w(t)},{v}\rangle$
$\displaystyle=\int_{\Omega}\mathcal{B}(\varepsilon(\int_{0}^{t}w(s)ds+u_{0})):\varepsilon(v)dx$
$\displaystyle+\int_{\Omega}\int_{0}^{t}\mathcal{C}(t-s,\varepsilon(w(s)))ds:\varepsilon(v)dx,$
$\displaystyle\text{for }w\in L^{2}_{T}V,\ v\in V,\text{ a.e. }t\in(0,T),$
(6.8c) $\displaystyle\mathcal{S}_{\varphi}w(t)$
$\displaystyle=\gamma_{\nu}\Big{(}\int_{0}^{t}w(s)ds+u_{0}\Big{)},\ \text{ for
}w\in L^{2}_{T}V,\text{ a.e. }t\in(0,T),$ (6.8d)
$\displaystyle\mathcal{G}(t,\alpha,Mw)$
$\displaystyle=G(\alpha,|w_{\tau}|),\text{ for }\alpha\in Y,\ w\in V,\text{
a.e. }t\in(0,T),$ (6.8e) $\displaystyle Mv=$ $\displaystyle v_{\tau},\ \
Nv=v_{\nu},\ \ \text{ for }v\in V.$ We define $K:V\rightarrow Z$ by (6.8f)
$Kv=(Mv,Nv),\ \ \text{ for }v\in V,$ which is linear and bounded in both
arguments, i.e., $K\in\mathcal{L}(V,Z)$. We consider the functional
$\varphi:(0,T)\times Y\times X\times U\times Z\rightarrow\mathbb{R}$ by (6.8g)
$\varphi(t,y,z,\tilde{w},\tilde{v})=\int_{\Gamma_{C}}p(z)v^{(2)}da+\int_{\Gamma_{C}}\mu(|\tilde{w}|,y)p(z)|v^{(1)}|da$
for $y\in Y$, $z\in X$, $\tilde{w}\in U$, $\tilde{v}=(v^{(1)},v^{(2)})\in Z$,
a.e. $t\in(0,T)$. We let $f:(0,T)\rightarrow V^{\ast}$ be defined as in (6.5).

This yields the following generalization of Problem 5:

######  Problem 6.

Find $w\in\mathcal{W}^{1,2}_{T}$ and $\alpha\in C([0,T];Y)$ such that

$\displaystyle\alpha(t)=\alpha_{0}+\int_{0}^{t}\mathcal{G}(s,\alpha(s),Mw(s))ds,$
$\displaystyle\langle{\rho\dot{w}(t)},{v-w(t)}\rangle+\langle{A(t,w(t))},{v-w(t)}\rangle+\langle{\mathcal{R}w(t)},{v-w(t)}\rangle$
$\displaystyle+\varphi(t,\alpha(t),\mathcal{S}_{\varphi}w(t),Mw(t),Kv)-\varphi(t,\alpha(t),\mathcal{S}_{\varphi}w(t),Mw(t),Kw(t))\geq\langle{f(t)},{v-w(t)}\rangle$

for all $v\in V$, a.e. $t\in(0,T)$, with $w(0)=w_{0}$.

######  Remark 6.6.

Since Problem 5 is contained in Problem 6, it suffices to show well-posedness
of Problem 6.

######  Lemma 6.7.

Under the assumptions of Theorem 6.3, the hypothesis of Theorem 3.3 holds for
(6.8a)-(6.8g). Here,
$c_{0\varphi}(t)=\kappa_{1}(\mathrm{meas}(\Gamma_{C}))^{3/2}$,
$c_{1\varphi}=\kappa_{2}\mathrm{meas}(\Gamma_{C})$,
$c_{2\varphi}=L_{p}(\mathrm{meas}(\Gamma_{C}))^{5/4}$,
$c_{3\varphi}=\kappa_{3}(\mathrm{meas}(\Gamma_{C}))^{5/4}$,
$\beta_{1\varphi}=p^{\ast}L_{3\mu}$,
$\beta_{3\varphi}=L_{p}(1+\kappa_{1})\sqrt{\mathrm{meas}(\Gamma_{C})}$,
$\beta_{4\varphi}=p^{\ast}L_{1\mu}\sqrt{\mathrm{meas}(\Gamma_{C})}$,
$\beta_{5\varphi}=p^{\ast}L_{2\mu}$,
$\beta_{6\varphi}=\kappa_{3}L_{p}(\mathrm{meas}(\Gamma_{C}))^{1/4}$,
$\beta_{7\varphi}=\kappa_{2}L_{p}$, and $c_{4\varphi}=\beta_{2\varphi}=0$.

To maintain the flow of the article, the proof of Lemma 6.7 is placed in
Appendix F. We are now ready to prove Theorem 6.3.

###### Proof of Theorem 6.3.

The proof relies on Theorem 3.3 with $j^{\circ}\equiv 0$. In light of Lemma
6.7, the hypotheses of Theorem 3.3 are fulfilled. Consequently,
$(w,\alpha)\in\mathcal{W}^{1,2}_{T}\times C([0,T];Y)$ is a unique solution of
Problem 6. Moreover, we define the function $u:[0,T]\rightarrow V$ by

(6.9) $u(t)=u_{0}+\int_{0}^{t}w(s)ds$

for all $t\in[0,T]$. As a consequence of Bochner space theory, the fact that
$w\in L^{2}_{T}V\subset L^{1}_{T}V$ and (6.9), we have that $u\in C([0,T];V)$
and $\dot{u}=w$. Consequently, $\dot{u}\in\mathcal{W}^{1,2}_{T}$, which
implies $u\in W^{1,2}(0,T;V)$. Furthermore, we define the set

$\mathbb{Y}=V\times V\times Y,$

and show that the flow map $\tilde{F}:\mathbb{Y}\rightarrow
W^{1,2}(0,T/2;V)\times C([0,T/2];Y)$ defined by
$(u_{0},w_{0},\alpha_{0})\mapsto(u,\alpha)$ is continuous. That is, we claim
that for all $\lambda>0$, there exists a $\delta>0$, chosen later, such that

(6.10a)
$\lVert(u_{01},w_{01},\alpha_{01})-(u_{02},w_{02},\alpha_{02})\rVert_{\mathbb{Y}}<\delta$
implies (6.10b)
$\lVert(u_{1},\alpha_{1})-(u_{2},\alpha_{2})\rVert_{W^{1,2}(0,T/2;V)\times
L^{\infty}_{T/2}Y}<\lambda.$

To check this, let us use the continuous dependence result in Theorem 3.3. We
observe that

(6.11)
$\displaystyle\lVert(w_{01},\alpha_{01})-(w_{02},\alpha_{02})\rVert_{V\times
Y}^{2}\leq\lVert(u_{01},w_{01},\alpha_{01})-(u_{02},w_{02},\alpha_{02})\rVert_{\mathbb{Y}}^{2}.$

From (6.10a), (6.11), and Theorem 3.3(b), we obtain

(6.12) $\lVert(w_{1},\alpha_{1})-(w_{2},\alpha_{2})\rVert_{L^{2}_{T/2}V\times
L^{\infty}_{T/2}Y}^{2}<\delta.$

Next, by (6.9), the triangle inequality, Minkowski’s inequality, Young’s
inequality, the Cauchy-Schwarz inequality, and integrating over the time
interval $(0,T/2)$, we have

(6.13) $\displaystyle\lVert u_{1}-u_{2}\rVert_{L^{2}_{T/2}V}^{2}\leq c(T\lVert
u_{01}-u_{02}\rVert_{V}^{2}+T^{2}\lVert
w_{1}-w_{2}\rVert_{L^{2}_{T/2}V}^{2}).$

Combining (6.12)-(6.13), while remembering that $w_{i}=\dot{u}_{i}$ for
$i=1,2$, we may choose a $\delta>0$ so that (6.10b) is acquired. ∎

###  6.2. Dynamic frictional contact problem with normal damped response

In our second application, we consider contact with normal damped response,
i.e., a wet material or some lubrication between the foundation and the
reference configuration of a viscoelastic body (see, e.g., [34]). We present
the problem:

######  Problem 7.

Find the displacement $u:\Omega\times[0,T]\rightarrow\mathbb{R}^{d}$ and the
external state variable $\alpha:\Gamma_{C}\times[0,T]\rightarrow\mathbb{R}$
such that

(6.14a) $\displaystyle\sigma(t)$
$\displaystyle=\mathcal{A}\varepsilon(\dot{u}(t))+\mathcal{B}\varepsilon(u(t))+\int_{0}^{t}\mathcal{C}(t-s,\varepsilon(\dot{u}(s)))ds$
$\displaystyle\text{ on }\Omega\times(0,T)$ (6.14b)
$\displaystyle\rho\ddot{u}(t)$ $\displaystyle=\nabla\cdot\sigma(t)+f_{0}(t)$
$\displaystyle\text{ on }\Omega\times(0,T)$ (6.14c) $\displaystyle u(t)$
$\displaystyle=0$ $\displaystyle\text{ on }\Gamma_{D}\times(0,T)$ (6.14d)
$\displaystyle\sigma(t){\nu}$ $\displaystyle=f_{N}(t)$ $\displaystyle\text{ on
}\Gamma_{N}\times(0,T)$ (6.14e) $\displaystyle-\sigma_{\nu}(t)$
$\displaystyle\in\partial j_{\nu}(\dot{u}_{\nu}(t))$ $\displaystyle\text{ on
}\Gamma_{C}\times(0,T)$ (6.14f) $\displaystyle|\sigma_{\tau}(t)|$
$\displaystyle\leq\mu(|\dot{u}_{\tau}(t)|,\alpha(t))$ $\displaystyle\text{ on
}\Gamma_{C}\times(0,T)$ (6.14g) $\displaystyle-\sigma_{\tau}(t)$
$\displaystyle=\mu(|\dot{u}_{\tau}(t)|,\alpha(t))\frac{\dot{u}_{\tau}(t)}{|\dot{u}_{\tau}(t)|},\
\ \text{ if }\dot{u}_{\tau}(t)\neq 0$ $\displaystyle\text{ on
}\Gamma_{C}\times(0,T)$ (6.14h) $\displaystyle\dot{\alpha}(t)$
$\displaystyle=G(\alpha(t),|\dot{u}_{\tau}(t)|)$ $\displaystyle\text{ on
}\Gamma_{C}\times(0,T)$ with (6.14i) $\displaystyle u(0)$
$\displaystyle=u_{0},\ \ \dot{u}(0)=w_{0}$ $\displaystyle\text{ on }\Omega$
(6.14j) $\displaystyle\alpha(0)$ $\displaystyle=\alpha_{0}$
$\displaystyle\text{ on }\Gamma_{C}.$

We summarized (6.14a)-(6.14d), and (6.14h)-(6.14j) underneath Problem 4.
However, (6.14e) is a general form of the contact condition for normal damped
response describing the contact with a lubricated foundation [23, Section
6.3]. The equations (6.14f)-(6.14g) is a version of Coulomb’s law of dry
friction, where (6.14f)-(6.14g) is a generalization of [35, Problem 68,
p.268]. We investigate Problem 7 under the hypotheses of $H(A)$,
$H(\mathcal{B})$, $H(\mu)$, $H(G)$, $H(\mathcal{C})$, and (6.2)-(6.4). In
addition, we require an assumption on $j_{\nu}$ in (6.14e):

$\underline{H(j_{\nu})}$:
$j_{\nu}:\Gamma_{C}\times\mathbb{R}\rightarrow\mathbb{R}\text{ is such that }$

  1. (i)

$j_{\nu}(\cdot,r)$ is measurable on $\Gamma_{C}$ for all $r\in\mathbb{R}$, and
there exists $\bar{e}\in L^{4}(\Gamma_{C})$ such that
$j_{\nu}(\cdot,\bar{e}(\cdot))\in L^{1}(\Gamma_{C})$.

  2. (ii)

$j_{\nu}(x,\cdot)$ is locally Lipschitz on $\mathbb{R}$ for a.e.
$x\in\Gamma_{C}$.

  3. (iii)

$|\partial j_{\nu}(x,r)|\leq\bar{c}_{0}+\bar{c}_{1}|r|$ for all
$r\in\mathbb{R}$, a.e. $x\in\Gamma_{C}$ with $\bar{c}_{0},\bar{c}_{1}\geq 0$.

  4. (iv)

$j_{\nu}^{0}(x,r_{1};r_{2}-r_{1})+j_{\nu}^{0}(x,r_{2};r_{1}-r_{2})\leq
m_{j_{\nu}}|r_{1}-r_{2}|^{2}$ for all $r_{1},r_{2}\in\mathbb{R}$, a.e.
$x\in\Gamma_{C}$ with $m_{j_{\nu}}\geq 0$.

We refer the reader to Appendix A for a small discussion on the assumptions on
the operators and functions in Problem 7.

####  6.2.1. Variational formulation

We make use of the derivations in Section 6.1, but include the new term for
the normal stress. By definition of the Clarke subgradient (see Definition
2.5) and (6.14e), we have

$\displaystyle-\sigma_{\nu}(t)\tilde{v}_{\nu}\leq
j_{\nu}^{0}(\dot{u}_{\nu}(t);\tilde{v}_{\nu})\ \ \text{ for all }\tilde{v}\in
V,\ \text{a.e. }t\in(0,T).$

Integrating over $\Gamma_{C}$, and choosing
$\tilde{v}_{\nu}=v_{\nu}-\dot{u}_{\nu}(t)$ gives us

(6.15)
$\displaystyle\int_{\Gamma_{C}}\sigma_{\nu}(t)[v_{\nu}-\dot{u}_{\nu}(t)]da\leq\int_{\Gamma_{C}}j_{\nu}^{0}(\dot{u}_{\nu}(t);v_{\nu}-\dot{u}_{\nu}(t))da$

for all $v\in V$, a.e. $t\in(0,T)$. Combining (6.15) with the calculations
from Section 6.1.1, we have the following problem:

######  Problem 8.

Find $u:\Omega\times[0,T]\rightarrow\mathbb{R}^{d}$ and
$\alpha:\Gamma_{C}\times[0,T]\rightarrow\mathbb{R}$ such that

(6.16a)
$\displaystyle\alpha(t)=\alpha_{0}+\int_{0}^{t}G(\alpha(s),|\dot{u}_{\tau}(s)|)ds,$
(6.16b)
$\displaystyle\int_{\Gamma_{C}}\mu(|\dot{u}_{\tau}(t)|,\alpha(t))[|v_{\tau}|-|\dot{u}_{\tau}(t)|]da+\int_{\Gamma_{C}}j^{\circ}_{\nu}(\dot{u}_{\nu}(t);v_{\nu}-\dot{u}_{\nu}(t))da$
$\displaystyle+\int_{\Omega}\rho\ddot{u}(t)\cdot[v-\dot{u}(t)]dx+\big{(}{\mathcal{A}\varepsilon(\dot{u}(t))},{\varepsilon(v)-\varepsilon(\dot{u}(t))}\big{)}_{Q}$
$\displaystyle+\big{(}{\mathcal{B}\varepsilon(u(t))+\int_{0}^{t}\mathcal{C}(t-s,\varepsilon(\dot{u}(s)))ds},{\varepsilon(v)-\varepsilon(\dot{u}(t))}\big{)}_{Q}\geq\langle{f(t)},{v-\dot{u}(t)}\rangle$
for all $v\in V$, a.e. $t\in(0,T)$ with $u(0)=u_{0},\ \ \dot{u}(0)=w_{0}.$

The well-posedness result for Problem 8 is stated in the following theorem.

######  Theorem 6.8.

Assume that $H(\mathcal{A})$, $H(\mathcal{B})$, $H(\mu)$, $H(G)$,
$H(\mathcal{C})$, (6.2)-(6.4), and $H(j_{\nu})$ holds. Then, there exists a
$T>0$ satisyfing (3.5) so that Problem 8 has a unique solution $(u,\alpha)$
under the smallness-condition

(6.17) $\displaystyle m_{\mathcal{A}}$
$\displaystyle>m_{j_{\nu}}\sqrt{\mathrm{meas}(\Gamma_{C})}\lVert\gamma_{\nu}\rVert_{\mathcal{L}(V,L^{4}(\Gamma_{C}))}^{2}$
$\displaystyle+\sqrt{2}\>(L_{1\mu}\sqrt{\mathrm{meas}(\Gamma_{C})}+L_{2\mu}\lVert\alpha_{0}\rVert_{Y})\lVert\gamma_{\tau}\rVert^{2}_{\mathcal{L}(V,L^{4}(\Gamma_{C};\mathbb{R}^{d}))}.$

In addition, we have the regularity:

(6.18) $u\in W^{1,2}(0,T;V),\ \ \ \dot{u}\in\mathcal{W}^{1,2}_{T}\subset
C([0,T];H),\ \ \ \alpha\in C([0,T];L^{2}(\Gamma_{C})).$

Moreover, the flow map depends continuously on the initial data.

######  Remark 6.9.

The constraint (6.17) can also be found in, e.g., [14, Theorem 4.4] for
$\alpha_{\varphi}c_{\alpha}^{2}=\sqrt{2}\>(L_{1\mu}\sqrt{\mathrm{meas}(\Gamma_{C})}+L_{2\mu}\lVert\alpha_{0}\rVert_{Y})\lVert\gamma_{\tau}\rVert^{2}_{\mathcal{L}(V,L^{4}(\Gamma_{C};\mathbb{R}^{d}))}$
and
$\alpha_{j}c_{j}^{2}=m_{j_{\nu}}\sqrt{\mathrm{meas}(\Gamma_{C})}\lVert\gamma_{\nu}\rVert_{\mathcal{L}(V,L^{4}(\Gamma_{C}))}^{2}$.

######  Remark 6.10.

We show that there exists a solution for $\rho\equiv 1$, and then take
$w(t)=w(\rho t)$.

####  6.2.2. Proof of Theorem 6.8

We use the same approach as in Section 6.1, meaning that we will use Theorem
3.3 to prove Theorem 6.8. But to use this theorem, we need to first rewrite
Problem 8 into the same form as Problem 1. Then we will verify the hypothesis
of Theorem 3.3.
First recall that $X=L^{4}(\Gamma_{C})$ and
$U=L^{4}(\Gamma_{C};\mathbb{R}^{d})$. Then take $Y=L^{2}(\Gamma_{C})$ and
$Z=U$. We define $A:(0,T)\times V\rightarrow V^{\ast}$,
$\mathcal{R}:L^{2}_{T}V\rightarrow L^{2}_{T}V^{\ast}$,
$\mathcal{G}:(0,T)\times Y\times U\rightarrow Y$, and $f:(0,T)\rightarrow
V^{\ast}$ as in (6.8a)-(6.8d) and (6.5), respectively. We choose
$M:V\rightarrow U$ and $N:V\rightarrow X$ to be as in (6.8e) and $K\equiv M$.
Moreover, we define the functional $\varphi:(0,T)\times Y\times U\times
Z\rightarrow\mathbb{R}$ by

(6.19a)
$\varphi(t,y,\tilde{w},\tilde{v})=\int_{\Gamma_{C}}\mu(|\tilde{w}|,y)|\tilde{v}|da\
\ \text{ for }y\in Y,\ \tilde{w}\in U,\ \tilde{v}\in Z,\ \text{ a.e.
}t\in(0,T)$ and the functional $j:(0,T)\times X\rightarrow\mathbb{R}$ by
(6.19b) $\displaystyle j(t,\tilde{w})=\int_{\Gamma_{C}}j_{\nu}(\tilde{w})da\ \
\text{ for }\tilde{w}\in X,\ \text{ a.e. }t\in(0,T).$

The problem is then on the following form.

######  Problem 9.

Find $(w,\alpha)\in\mathcal{W}^{1,2}_{T}\times C([0,T];Y)$ such that

$\displaystyle\alpha(t)=\alpha_{0}+\int_{0}^{t}\mathcal{G}(s,\alpha(s),Mw(s))ds,$
$\displaystyle\langle{\rho\dot{w}(t)},{v-w(t)}\rangle+\langle{A(t,w(t))},{v-w(t)}\rangle+\langle{\mathcal{R}w(t)},{v-w(t)}\rangle+j^{\circ}(t,Nw(t);Nv-
Nw(t))$
$\displaystyle+\varphi(t,\alpha(t),Mw(t),Kv)-\varphi(t,\alpha(t),Mw(t),Kw(t))\geq\langle{f(t)},{v-w(t)}\rangle$

for all $v\in V$, a.e. $t\in(0,T)$ with $w(0)=w_{0}$.

To see that it suffices to prove existence of a solution to Problem 9 in order
for Problem 8 to have a solution, we introduce the following result, which is
of a similar form as found in [35, Lemma 8, p.126] (see also [23, Theorem
3.47]). The result will also be useful to prove uniqueness.

######  Corollary 6.11.

Assume that $H(j_{\nu})$ holds. Then, the functional $j$ defined by (6.19b)
has the following properties:

  1. (i)

$j(\cdot,v)$ is measurable on $(0,T)$ for all $v\in X$.

  2. (ii)

$j(t,\cdot)$ is locally Lipschitz on $X$ for a.e. $t\in(0,T)$.

  3. (iii)

For all $\tilde{w},v\in X$, we have
$j^{\circ}(t,\tilde{w};v)\leq\int_{\Gamma_{C}}j_{\nu}^{\circ}(\tilde{w};v)da$.

######  Lemma 6.12.

Under the assumptions of Theorem 6.8, $H(\varphi)$ holds for $\varphi$ defined
by (6.19a) for $c_{0\varphi}(t)=\kappa_{1}(\mathrm{meas}(\Gamma_{C}))^{3/2}$,
$c_{1\varphi}=\kappa_{2}\mathrm{meas}(\Gamma_{C})$,
$c_{3\varphi}=\kappa_{3}(\mathrm{meas}(\Gamma_{C}))^{5/4}$,
$\beta_{1\varphi}=L_{3\mu}$,
$\beta_{4\varphi}=L_{1\mu}\sqrt{\mathrm{meas}(\Gamma_{C})}$,
$\beta_{5\varphi}=L_{2\mu}$, and
$c_{4\varphi}=\beta_{2\varphi}=\beta_{3\varphi}=\beta_{6\varphi}=\beta_{7\varphi}=0$.
Moreover, $j$ defined by (6.19b) satisfies $H(j)$ for
$c_{0j}(t)=2^{1/4}(\mathrm{meas}(\Gamma_{C}))^{3/4}\bar{c}_{0}$,
$c_{3j}=2^{1/4}(\mathrm{meas}(\Gamma_{C}))^{2/3}\bar{c}_{1}$,
$m_{j}=m_{j_{\nu}}\sqrt{\mathrm{meas}(\Gamma_{C})}$, and
$\bar{m}_{j}=c_{1j}=c_{2j}=0$.

The proof of Lemma 6.12 is placed in Appendix G. We may now prove Theorem 6.8.

###### Proof of Theorem 6.8.

We wish to utilize Theorem 3.3. The hypothesis of Theorem 3.3 holds by (6.17),
Lemma 6.7 and 6.12, and Corollary 6.11(i)-(ii). With the help of Corollary
6.11(iii), we may conclude that there exists a solution to Problem 8.
Moreover, the fact that the flow map depends continuously on the initial data
follows by the same approach as in the proof of Theorem 6.3. To obtain
uniqueness, we let $(u_{1},\alpha_{1}),(u_{2},\alpha_{2})\in
W^{1,2}(0,T;V)\times C([0,T];Y)$ be two pairs of solutions to Problem 8.
Choosing the test functions $v=\dot{u}_{2}(t)$ and $v=\dot{u}_{1}(t)$ for a.e.
$t\in(0,T)$, respectively, in (6.16b) yields

(6.20)
$\displaystyle\big{(}{\mathcal{A}\varepsilon(\dot{u}_{1}(t))-\mathcal{A}\varepsilon(\dot{u}_{2}(t))},{\varepsilon(\dot{u}_{1}(t))-\varepsilon(\dot{u}_{2}(t))}\big{)}_{Q}$
$\displaystyle+\int_{\Omega}\rho\big{[}\ddot{u}_{1}(t)-\ddot{u}_{2}(t)\big{]}\cdot\big{[}\dot{u}_{1}(t)-\dot{u}_{2}(t)\big{]}dx$
$\displaystyle\leq\int_{\Gamma_{C}}\big{[}\mu(|\dot{u}_{1\tau}(t)|,\alpha_{1}(t))-\mu(|\dot{u}_{2\tau}(t)|,\alpha_{2}(t))\big{]}\big{[}|\dot{u}_{2\tau}(t)|-|\dot{u}_{1\tau}(t)|\big{]}da$
$\displaystyle+\int_{\Gamma_{C}}[j^{\circ}_{\nu}(\dot{u}_{1\nu}(t);\dot{u}_{2\nu}(t)-\dot{u}_{1\nu}(t))+j^{\circ}_{\nu}(\dot{u}_{2\nu}(t);\dot{u}_{1\nu}(t)-\dot{u}_{2\nu}(t))]da$
$\displaystyle+\big{(}{\mathcal{B}\varepsilon(u_{1}(t))-\mathcal{B}\varepsilon(u_{2}(t))},{\varepsilon(\dot{u}_{2}(t))-\varepsilon(\dot{u}_{1}(t))}\big{)}_{Q}$
$\displaystyle+\big{(}{\int_{0}^{t}[\mathcal{C}(t-s,\varepsilon(\dot{u}_{1}(s)))-\mathcal{C}(t-s,\varepsilon(\dot{u}_{2}(s)))]ds},{\varepsilon(\dot{u}_{2}(t))-\varepsilon(\dot{u}_{1}(t))}\big{)}_{Q}$

for a.e. $t\in(0,T)$. Using $H(G)$(ii), a standard Grönwall argument, the
Cauchy-Schwarz inequality, and Minkowski’s inequality in (6.16a) implies

(6.21) $\displaystyle\int_{\Gamma_{C}}|\alpha_{1}(t)-\alpha_{2}(t)|^{2}dx$
$\displaystyle\leq
c\int_{0}^{t}\int_{\Omega}|\dot{u}_{1}(s)-\dot{u}_{2}(s)|^{2}dxds$

for a.e. $t\in(0,T$). We next utilize $H(\mathcal{A})$(iii),
$H(\mathcal{B})$(ii), $H(\mathcal{C})$, $H(j_{\nu})$(iv), $H(\mu)$(ii), (6.2),
the Cauchy-Schwarz inequality, and Young’s inequality to (6.20):

$\displaystyle
m_{A}\lVert\varepsilon(\dot{u}_{1}(t))-\varepsilon(\dot{u}_{2}(t))\rVert_{Q}^{2}+\int_{\Omega}\rho\big{[}\ddot{u}_{1}(t)-\ddot{u}_{2}(t)\big{]}\cdot\big{[}\dot{u}_{1}(t)-\dot{u}_{2}(t)\big{]}dx$
$\displaystyle\leq
L_{1\mu}\sqrt{\mathrm{meas}(\Gamma_{C})}\lVert\dot{u}_{1\tau}(t)-\dot{u}_{2\tau}(t)\rVert_{L^{4}(\Gamma_{C};\mathbb{R}^{d})}^{2}$
$\displaystyle+L_{2\mu}\lVert\alpha_{1}(t)\rVert_{L^{2}(\Gamma_{C})}\lVert\dot{u}_{1\tau}(t)-\dot{u}_{2\tau}(t)\rVert_{L^{4}(\Gamma_{C};\mathbb{R}^{d})}^{2}$
$\displaystyle+L_{3\mu}\lVert\dot{u}_{2\tau}(t)\rVert_{L^{4}(\Gamma_{C})}\lVert\alpha_{1}(t)-\alpha_{2}(t)\rVert_{L^{2}(\Gamma_{C})}\lVert\dot{u}_{1\tau}(t)-\dot{u}_{2\tau}(t)\rVert_{L^{4}(\Gamma_{C};\mathbb{R}^{d})}$
$\displaystyle+m_{j\nu}\sqrt{\mathrm{meas}(\Gamma_{C})}\lVert\dot{u}_{1\nu}(t)-\dot{u}_{2\nu}(t)\rVert_{L^{4}(\Gamma_{C})}^{2}$
$\displaystyle+L_{\mathcal{B}}\lVert\varepsilon(u_{1}(t))-\varepsilon(u_{2}(t))\rVert_{Q}\lVert\varepsilon(\dot{u}_{1}(t))-\varepsilon(\dot{u}_{2}(t))\rVert_{Q}$
$\displaystyle+\lVert\mathcal{C}\rVert_{L^{\infty}_{T}L^{\infty}(\Omega;\mathbb{S}^{d})}\lVert\int_{0}^{t}[\varepsilon(\dot{u}_{1}(s))-\varepsilon(\dot{u}_{2}(s))]ds\rVert_{Q}\lVert\varepsilon(\dot{u}_{2}(t))-\varepsilon(\dot{u}_{1}(t))\rVert_{Q}$

for a.e. $t\in(0,T)$. Next, we use the fact that if
$\dot{u}\in\mathcal{W}^{1,2}_{T}$, then
$\frac{d}{dt}\lVert\dot{u}(t)\rVert_{H}^{2}=2\langle{\ddot{u}(t)},{\dot{u}(t)}\rangle$
[10, Theorem 3 in Section 5.9.2], (6.9), and integrating over the time
interval $(0,t^{\prime})\subset(0,T)$. We observe by (6.17) that
$m_{A}-m_{j_{\nu}}\sqrt{\mathrm{meas}(\Gamma_{C})}\lVert\gamma_{\nu}\rVert_{\mathcal{L}(V,X)}^{2}>0$.
In addition, we use (6.21) and then apply Hölder’s inequality and Minkowski’s
inequality to obtain

$\displaystyle\lVert\dot{u}_{1}-\dot{u}_{2}\rVert_{L^{2}_{t^{\prime}}V}^{2}$
$\displaystyle\leq\frac{(L_{1\mu}\sqrt{\mathrm{meas}(\Gamma_{C})}+L_{2\mu}\lVert\alpha_{1}\rVert_{L^{\infty}_{T}Y})\lVert\gamma_{\tau}\rVert_{\mathcal{L}(V,U)}^{2}}{m_{A}-m_{j_{\nu}}\sqrt{\mathrm{meas}(\Gamma_{C})}\lVert\gamma_{\nu}\rVert_{\mathcal{L}(V,X)}^{2}}\lVert\dot{u}_{1}-\dot{u}_{2}\rVert_{L^{2}_{t^{\prime}}V}^{2}$
$\displaystyle+c\lVert\dot{u}_{2}\rVert_{L^{2}_{T}V}\lVert\alpha_{1}-\alpha_{2}\rVert_{L^{\infty}_{t^{\prime}}Y}\lVert\dot{u}_{1}-\dot{u}_{2}\rVert_{L^{2}_{t^{\prime}}V}$
$\displaystyle+c\Big{[}\int_{0}^{t^{\prime}}\int_{0}^{t}\lVert\dot{u}_{1}(s)-\dot{u}_{2}(s)\rVert_{V}^{2}dsdt\Big{]}^{1/2}\lVert\dot{u}_{1}-\dot{u}_{2}\rVert_{L^{2}_{t^{\prime}}V}$
$\displaystyle=:I+II+III$

for a.e. $t^{\prime}\in(0,T)$. Using Minkowski’s inequality, $H(G)$(ii), and
the Cauchy-Schwarz inequality implies

$\displaystyle\lVert\alpha_{1}(t)\rVert_{L^{2}(\Gamma_{C})}$
$\displaystyle\leq\lVert\alpha_{0}\rVert_{L^{2}(\Gamma_{C})}+L_{G}\int_{0}^{t}[\lVert\alpha_{1}(s)\rVert_{L^{2}(\Gamma_{C})}+\sqrt{\mathrm{meas}(\Gamma_{C})}\lVert\dot{u}_{1\tau}(s)\rVert_{L^{4}(\Gamma_{C})}]ds$
$\displaystyle\leq\lVert\alpha_{0}\rVert_{L^{2}(\Gamma_{C})}+L_{G}\int_{0}^{t}\lVert\alpha_{1}(s)\rVert_{L^{2}(\Gamma_{C})}ds+cT^{k}\Big{(}\int_{0}^{t}\lVert\dot{u}_{1}(s)\rVert^{2}_{V}ds\Big{)}^{1/2}$

for a.e. $t\in(0,T)$ and $k\geq 1/2$. From Grönwall’s inequality, we deduce

$\displaystyle\lVert\alpha_{1}\rVert_{L^{\infty}_{T}L^{2}(\Gamma_{C})}$
$\displaystyle\leq(\lVert\alpha_{0}\rVert_{L^{2}(\Gamma_{C})}+cT^{k}\lVert\dot{u}_{1}\rVert_{L^{2}_{T}V})(1+cT\mathrm{e}^{cT})$

for some $k\geq 1/2$. Utilizing Young’s inequality to $I$ and $II+III$ and
then the arithmetic-quadratic mean inequality to the latter term while keeping
in mind (6.17)-(6.18) and (6.21), we obtain

$\displaystyle\lVert\dot{u}_{1}-\dot{u}_{2}\rVert_{L^{2}_{t^{\prime}}V}^{2}$
$\displaystyle\leq\frac{2(L_{1\mu}\sqrt{\mathrm{meas}(\Gamma_{C})}+L_{2\mu}(\lVert\alpha_{0}\rVert_{Y}+T^{k}c))^{2}\lVert\gamma_{\tau}\rVert_{\mathcal{L}(V,U)}^{4}}{(m_{A}-m_{j_{\nu}}\sqrt{\mathrm{meas}(\Gamma_{C})}\lVert\gamma_{\nu}\rVert_{\mathcal{L}(V,X)}^{2})^{2}}\lVert\dot{u}_{1}-\dot{u}_{2}\rVert_{L^{2}_{t^{\prime}}V}^{2}$
$\displaystyle+c\int_{0}^{t^{\prime}}\int_{0}^{t}\lVert\dot{u}_{1}(s)-\dot{u}_{1}(s)\rVert_{V}^{2}dsdt$

for all $t^{\prime}\in(0,T)$. A consequence of (6.17) and choosing $T>0$ such
that

$\displaystyle
T^{k}\sim\frac{m_{A}-m_{j_{\nu}}\sqrt{\mathrm{meas}(\Gamma_{C})}\lVert\gamma_{\nu}\rVert_{\mathcal{L}(V,X)}^{2}}{cL_{2\mu}\lVert\gamma_{\tau}\rVert_{\mathcal{L}(V,U)}^{2}}$

small enough for some $k\geq 1/2$ implies

$\frac{\sqrt{2}(L_{1\mu}\sqrt{\mathrm{meas}(\Gamma_{C})}+L_{2\mu}(\lVert\alpha_{0}\rVert_{Y}+T^{k}c))\lVert\gamma_{\tau}\rVert_{\mathcal{L}(V,U)}^{2}}{m_{A}-m_{j_{\nu}}\sqrt{\mathrm{meas}(\Gamma_{C})}\lVert\gamma_{\nu}\rVert_{\mathcal{L}(V,X)}^{2}}<1,$

we obtain

$\displaystyle\lVert\dot{u}_{1}-\dot{u}_{2}\rVert_{L^{2}_{t^{\prime}}V}^{2}\leq
c\int_{0}^{t^{\prime}}\lVert\dot{u}_{1}-\dot{u}_{2}\rVert_{L^{2}_{t}V}^{2}dt$

for all $t^{\prime}\in(0,T)$. Applying a standard Grönwall argument reads

(6.22) $\displaystyle\lVert\dot{u}_{1}-\dot{u}_{2}\rVert_{L^{2}_{T}V}^{2}\leq
0.$

By the definition of $u$, i.e., (6.9), the smallness-condition (6.17) and
(6.21)-(6.22), we conclude that $(u,\alpha)$ is the unique solution to Problem
8. ∎

###  6.3. Application to rate-and-state friction

The coupling (1.2) are standard in geophysical applications in earth sciences,
where the experimentally derived Dieterich-Ruina laws are commonly used. We
refer to [18] for an overview and comparison of some commonly used laws. There
have been physical issues with the standard rate law, e.g.,
$|\dot{u}_{\tau}(t)|\rightarrow 0$ resulting in a negative friction
coefficient. This was repaired by using the regularized or truncated law (see
[26, Section 1.1-1.3] and references therein), which are, respectively, given
by

(6.23a) $\displaystyle\mu(|\dot{u}_{\tau}(t)|,\alpha(t))$
$\displaystyle=a\>\mathrm{arcsinh}\bigg{(}\frac{|\dot{u}_{\tau}(t)|}{2v_{\alpha}(t)}\bigg{)},$
(6.23b) $\displaystyle\mu(|\dot{u}_{\tau}(t)|,\alpha(t))$
$\displaystyle=a\log^{+}\bigg{(}\frac{|\dot{u}_{\tau}(t)|}{v_{\alpha}(t)}\bigg{)}\
\ \text{ with }\log^{+}v=\begin{dcases}\log v,\ \ \text{ if }v\geq 1,\\\ 0,\ \
\text{ otherwise},\end{dcases}$

where $v_{\alpha}(t)=v_{0}e^{-\frac{1}{a}(\mu_{0}+b\alpha(t))}$. The
coefficients $a$, $b$, $v_{0}$, and $\mu_{0}$ are system parameters (see,
e.g., [18, 15],[26, Section 1.2]). Another regularization used in literature
can be found in [32]. The most popular rate-and-state friction laws are the
aging and slip laws, respectively, described by

(6.24a) $\displaystyle\dot{\alpha}(t)$
$\displaystyle=\frac{v_{0}\mathrm{e}^{-\alpha(t)}-|\dot{u}_{\tau}(t)|}{L},$
(6.24b) $\displaystyle\dot{\alpha}(t)$
$\displaystyle=-\frac{|\dot{u}_{\tau}(t)|}{L}\Big{[}\log\Big{(}\frac{|\dot{u}_{\tau}(t)|}{v_{0}}\Big{)}+\alpha(t)\Big{]},$

with $L$ being a system parameter (see, e.g., [15]). In the framework
presented in this paper, we are not able to include (6.23)-(6.24). The main
issue is that $|\dot{u}_{\tau}(t)|\not\in L^{\infty}(\Gamma_{C})$. We
therefore consider a first-order Taylor approximation of
$\mathrm{e}^{\hat{c}\alpha(t)}$ around $\alpha_{0}$ with
$\hat{c}=-1,\frac{b}{a}$, which is used in (6.23a) and (6.24a). This leads to
a first-order approximation of (6.23a) and (6.24a).
The first-order Taylor approximation of $\mathrm{e}^{\hat{c}\alpha(t)}$ around
$\alpha_{0}$ reads

(6.25)
$\displaystyle\mathrm{e}^{\hat{c}\alpha(t)}=\mathrm{e}^{\hat{c}\alpha_{0}}(1+\hat{c}(\alpha(t)-\alpha_{0}))+\mathcal{O}((\alpha(t)-\alpha_{0})^{2}).$

Using the above approximation in (6.23a) and (6.24a) gives us

(6.26a) $\displaystyle G(\alpha(t),|\dot{u}_{\tau}(t)|)$
$\displaystyle=\frac{v_{0}\mathrm{e}^{-\alpha_{0}}(1-\alpha(t)+\alpha_{0})-|\dot{u}_{\tau}(t)|}{L},$
(6.26b) $\displaystyle\mu(|\dot{u}_{\tau}(t)|,\alpha(t))$
$\displaystyle=a\>\mathrm{arcsinh}\bigg{(}\frac{\mathrm{e}^{\frac{\mu_{0}+b\alpha_{0}}{a}}|\dot{u}_{\tau}(t)|(1+\frac{b}{a}(\alpha(t)-\alpha_{0}))}{2v_{0}}\bigg{)}.$

We will make a formal argument to justify that these are first-order
approximations.

###### Formal augmentations of (6.26a)-(6.26b).

For simplicity in notation, we let $y=\alpha(t)$ and $r=\dot{u}_{\tau}(t)$. We
wish to have the same order of error for the approximation of (6.23a) and
(6.24a) as in (6.25). For (6.26a), we are considering $\hat{c}=-1$. We
directly obtain

$\displaystyle\frac{v_{0}}{L}(\mathrm{e}^{-y}-\mathrm{e}^{-\alpha_{0}}(1-(y-\alpha_{0})))=\mathcal{O}((y-\alpha_{0})^{2}).$

Next, considering (6.26b), we are interested in the approximation (6.25) for
$\hat{c}=\frac{b}{a}$. We let
$g=\frac{1}{2v_{0}}\mathrm{e}^{\frac{\mu_{0}+by}{a}}$ and
$f=\frac{1}{2v_{0}}\mathrm{e}^{\frac{\mu_{0}+b\alpha_{0}}{a}}(1+\frac{b}{a}(y-\alpha_{0}))$.
Then, by the mean value theorem

$\displaystyle\mathrm{arcsinh}(|r|g)-\mathrm{arcsinh}(|r|f)$
$\displaystyle=\frac{|r|}{\sqrt{1+z^{2}}}(g-f)$

with $z\in(|r|f,|r|g)$ where $f\leq g$. So, we have

$\displaystyle\mathrm{arcsinh}(|r|g)=\mathrm{arcsinh}(|r|f)+\frac{|r|}{\sqrt{1+z^{2}}}\mathcal{O}((y-\alpha_{0})^{2}).$

We also note that

$\displaystyle
0\leq\frac{|r|}{\sqrt{1+z^{2}}}\leq\frac{|r|}{\sqrt{1+(|r|f)^{2}}}\leq\frac{1}{|f|}=\frac{1}{|\tilde{c}||1+\frac{b}{a}(y-\alpha_{0})|},$

where $f\sim\tilde{c}$ if $y$ is close to $\alpha_{0}$. Consequently, we have
the same order of error as in (6.25) as desired. ∎

######  Remark 6.13.

Above, we gave formal arguments that our model is a first-order expansion
around the initial value $\alpha_{0}$. We will now investigate if our
approximated model has the same qualitative behavior as the original model.
Following [30], the key restriction on $G$ is that when the slip rate
$|\dot{u}_{\tau}|$ is constant, the equation
$\dot{\alpha}=G(\alpha,|\dot{u}_{\tau}|)$ has a stable solution that evolves
monotonically towards a definite value of $\alpha$, denoted
$\alpha^{\ast}=\alpha^{\ast}(|\dot{u}_{\tau}|)$, at which
$G(\alpha^{\ast},|\dot{u}_{\tau}|)=0$. This holds true if $\dfrac{\partial
G}{\partial\alpha}<0$, which is easily verified for the approximation of $G$
given by (6.26a).
For the friction term, again following [30], we seek
$\dfrac{\partial\mu}{\partial|\dot{u}_{\tau}|}>0$ and
$\dfrac{\partial\mu}{\partial\alpha}>0$. The first condition is consistent
with the experimental observations and holds if $\alpha$ is close to
$\alpha_{0}$. The latter condition agrees with the established convention for
the state variable; larger values mean greater strength. This is also
consistent with the usual interpretation of $\alpha$ as a measure of contact
maturity and the fact that more mature contact is stronger. Consequently, this