Daiki Mizuno Division of Mathematics and Informatics,
Graduate School of Science and Engineering, Chiba University,
1-33, Yayoi-cho, Inage-ku, 263-8522, Chiba, Japan<EMAIL_ADDRESS>
Ken Shirakawa Department of Mathematics, Faculty of Education, Chiba
University
1–33 Yayoi-cho, Inage-ku, 263–8522, Chiba, Japan<EMAIL_ADDRESS>
∗ This work is supported by Grant-in-Aid for Scientific Research (C) No.
20K03672, JSPS. HA and KS are also partially supported by NSF grant
DMS-2110263, the Air Force Office of Scientific Research (AFOSR) under Award
NO: FA9550-22-1-0248, and the Office of Naval Research (ONR) under Award NO:
N00014-24-1-2147.
AMS Subject Classification: 35A15, 35G50, 35G61, 35K70, 74N20.
Keywords: KWC-type system, pseudo-parabolic nature, existence, uniqueness,
regularity, continuous-dependence

# Well-posedness of a Pseudo-Parabolic KWC System
in Materials Science 11footnotemark: 1

Harbir Antil  Department of Mathematical Sciences and
the Center for Mathematics and Artificial Intelligence, (CMAI),
George Mason University, Fairfax, VA 22030, USA<EMAIL_ADDRESS>
Abstract. The original KWC-system is widely used in materials science. It was
proposed in [Kobayashi et al, Physica D, 140, 141–150 (2000)] and is based on
the phase field model of planar grain boundary motion. This model suffers from
two key challenges. Firstly, it is difficult to establish its relation to
physics, in particular, a variational model. Secondly, it lacks uniqueness.
The former has been recently studied within the realm of BV-theory. The latter
only holds under various simplifications. This article introduces a pseudo-
parabolic version of the KWC-system. A direct relationship with variational
model (as gradient-flow) and uniqueness are established without making any
unrealistic simplifications. Namely, this is the first KWC-system which is
both physically and mathematically valid. The proposed model overcomes the
well-known open issues.

##  1 Introduction

The Kobayashi–Warren–Carter (KWC) system consists of a set of non-smooth
parabolic PDEs and is widely used in materials science [14, 15]. It is based
on the phase field model of planar grain boundary motion. This model suffers
from two fundamental challenges: (i) _Physics:_ It is difficult to establish
KWC-system as the gradient flow of a variational model; (ii) _Mathematics:_
The solution to KWC-system are known to be unique only under special cases.
Both of these challenges makes it difficult to rigorously use this model in
practice or carry our new material design via optimization [5, 6].

This article aims to overcome both these challenges by introducing a pseudo-
parabolic version of the KWC-system. Well-posedness (both existence and
uniqueness) of the resulting system (S), which arise from gradient flow based
on the _KWC-energy_

$\displaystyle\mathcal{F}:[\eta,\theta]\in[L^{2}(\Omega)]^{2}\mapsto\mathcal{F}(\eta,\theta):=\frac{1}{2}\int_{\Omega}|\nabla\eta|^{2}\,dx+\int_{\Omega}G(\eta)\,dx$
(1.1) $\displaystyle+\int_{\Omega}\alpha(\eta)|D\theta|\in[0,\infty],$

is established. Namely, this article addresses open issues from previous works
that deal with the KWC-system (cf. [18, 19, 22, 20, 21, 17]) and its
regularized versions (cf. [13, 24, 7, 5, 6, 16]).

Next, we describe the system (S). Let $0<T<\infty$ be a fixed final time, and
let $N\in\\{1,2,3\\}$ denote the spatial dimension. Let
$\Omega\subset\mathbb{R}^{N}$ be an open bounded spatial domain with a
boundary $\Gamma:=\partial\Omega$. When $N>1$, $\Gamma$ is assumed to be
sufficiently smooth, with the unit outer normal $n_{\Gamma}$. Besides, we let
$Q:=(0,T)\times\Omega$ and $\Sigma:=(0,T)\times\Gamma$. Then the pseudo-
parabolic system denoted by (S), with two constants $\mu\geq 0$ and $\nu>0$,
is given by

$\displaystyle\mbox{(S)}:~{}~{}$
$\displaystyle\begin{cases}\partial_{t}\eta-\mathit{\Delta}\bigl{(}\eta+\mu^{2}\partial_{t}\eta\bigr{)}+g(\eta)+\alpha^{\prime}(\eta(t))|\nabla\theta|=u(t,x),~{}\mbox{for
$(t,x)\in Q$,}\\\\[2.15277pt]
\nabla\bigl{(}\eta+\mu^{2}\partial_{t}\eta\bigr{)}\cdot
n_{\Gamma}=0,~{}\mbox{on $\Sigma$,}\ \\\\[2.15277pt]
\eta(0,x)=\eta_{0}(x),~{}\mbox{for $x\in\Omega$,}\end{cases}$
$\displaystyle\begin{cases}\displaystyle\alpha_{0}(\eta)\partial_{t}\theta-\mathrm{div}\left(\alpha(\eta)\frac{D\theta}{|D\theta|}+\nu^{2}\nabla\partial_{t}\theta\right)=v(t,x),~{}\mbox{for
$(t,x)\in Q$,}\\\\[4.30554pt]
\bigl{(}\alpha(\eta)\frac{D\theta}{|D\theta|}+\nu^{2}\nabla\partial_{t}\theta\bigr{)}\cdot
n_{\Gamma}=0~{}\mbox{on $\Sigma$,}\\\\[2.15277pt]
\theta(0,x)=\theta_{0}(x),~{}\mbox{for $x\in\Omega$.}\end{cases}$

Here, the unknowns $\eta=\eta(t,x)$ and $\theta=\theta(t,x)$ are order
parameters that indicate the _orientation order_ and _orientation angle_ of
the polycrystal body, respectively. Besides, $\eta_{0}=\eta_{0}(x)$ and
$\theta_{0}=\theta_{0}(x)$ is the _initial data_. Moreover, $u=u(t,x)$ and
$v=v(t,x)$ are the forcing terms. Additionally, $\alpha_{0}=\alpha_{0}(\eta)$
and $\alpha=\alpha(\eta)$ are fixed positive-valued functions to reproduce the
mobilities of grain boundary motions. Finally, $g=g(\eta)$ is a perturbation
for the orientation order $\eta$, having a nonnegative potential $G=G(\eta)$,
i.e. $\frac{d}{d\eta}G(\eta)=g(\eta)$.

A generic form of the “KWC-system” is given by the evolution equation (cf.
[14, 15]):

$\displaystyle-\mathcal{A}_{0}\bigl{(}\eta(t)\bigr{)}\frac{d}{dt}\left[\begin{matrix}\eta(t)\\\\[4.30554pt]
\theta(t)\end{matrix}\right]=\frac{\delta}{\delta[\eta,\theta]}\mathcal{F}(\eta(t),\theta(t))+\left[\begin{matrix}u(t)\\\\[4.30554pt]
v(t)\end{matrix}\right]$ (1.2) in $[L^{2}(\Omega)]^{2}$, for $t\in(0,T)$,

which is motivated by the gradient flow of the free-energy, namely the KWC-
energy (1.1), with a functional derivative
$\frac{\delta}{\delta[\eta,\theta]}\mathcal{F}$, and an unknown-dependent
monotone operator $\mathcal{A}_{0}(\eta)\subset[L^{2}(\Omega)]^{2}$. Here, the
evolution equation (1.2) can be considered as the common root of the original
KWC-system (cf. [14, 15]) and our system (S). Indeed, our system (S) is
derived from the evolution equation (1.2), in the case when:

$\displaystyle\mathcal{A}_{0}(\eta):\left[\begin{matrix}\widetilde{\eta}\\\\[4.30554pt]
\widetilde{\theta}\end{matrix}\right]\in[H^{2}(\Omega)]^{2}\mapsto\mathcal{A}_{0}(\eta)\left[\begin{matrix}\widetilde{\eta}\\\\[4.30554pt]
\widetilde{\theta}\end{matrix}\right]:=\left[\begin{matrix}\widetilde{\eta}-\mu^{2}\mathit{\Delta}\widetilde{\eta}\\\\[4.30554pt]
\alpha_{0}(\eta)\widetilde{\theta}-\nu^{2}\mathit{\Delta}\widetilde{\theta}\end{matrix}\right]\in[L^{2}(\Omega)]^{2},$
(1.3) for each $\eta\in L^{2}(\Omega)$, subject to the zero-Neumann boundary
condition,

while the original KWC-system corresponds to the case when $\mu=\nu=0$.

In recent years, the principal issue has been to clarify the variational
structure (representation) of the functional derivative
$\frac{\delta}{\delta[\eta,\theta]}\mathcal{F}$ of the nonsmooth and nonconvex
energy $\mathcal{F}$ in (1.1). The positive answer to the issue was obtained
in [18, 19, 22], by means of BV-theory (cf. [4, 11, 12, 8, 2, 3]), and this
work has provided a basis of the study of KWC-system, e.g. the existence and
large-time behavior [18, 19], the observations under other boundary conditions
[20, 21], the time-periodic solution [17], and so on.

However, the uniqueness of solutions has been a significant challenge, due to
the velocity term $\alpha_{0}(\eta)\partial_{t}\theta$ and the singular
diffusion flux $\alpha(\eta)\frac{D\theta}{|D\theta|}$, both of which depend
on the unknown-dependent mobilities. Therefore, previous researchers have
implemented the following modifications to the modelling framework (1.1) and
(1.3):

$\bullet$

resetting $\alpha_{0}$ to be a function which is independent of $\eta$
(effectively a constant);

$\bullet$

modifying the free-energy functional to a more relaxed form:

$\displaystyle\mathcal{F}_{\varepsilon}:[\eta,\theta]\in[L^{2}(\Omega)]^{2}\mapsto\mathcal{F}_{\varepsilon}(\eta,\theta)=\mathcal{F}(\eta,\theta)+\frac{\varepsilon^{2}}{2}\int_{\Omega}|\nabla\theta|^{2}\,dx,$
with a small constant $\varepsilon>0$.

These modifications have been pivotal in addressing the uniqueness challenges
(cf. [13, 24]), and several advanced issues, such as the optimal control
problems (see [7, 5, 6, 16]).

In light of this, we can expect that the pseudo-parabolic nature of our system
will effectively address the uniqueness challenge. This is due to the positive
constants $\mu$ and $\nu$ in (1.3), which are expected to bring a smoothing
effect for the regularity of solution. In addition, it is also crucial from a
mathematical perspective to clarify the similarities and differences between
our pseudo-parabolic system and the original parabolic KWC-system.

Consequently, we set the goal to prove the following two Main Theorems,
concerned with the well-posedness of our pseudo-parabolic system (S), i.e. the
evolution equation (1.2) under (1.1) and (1.3).

Main Theorem 1.

Existence and regularity of solution to (S).

Main Theorem 2.

Uniqueness of solution to (S), and continuous dependence with respect to the
initial data and forcings.

These Main Theorems will provide the positive answer to our earlier
expectation regarding the effectiveness of the pseudo-parabolic nature of our
system in resolving the uniqueness issue. Also, the Main Theorems will focus
on two conflicting properties: the singularity in the diffusion flux
$\alpha(\eta)\frac{D\theta}{|D\theta|}$; and the smoothing effect encouraged
by the Laplacian in (1.3). This conflicting situation will be clarified by the
differences in regularity between components: $\eta\in
W^{1,2}(0,T;H^{2}(\Omega))$; and $\theta\in W^{1,2}(0,T;H^{1}(\Omega))\cap
L^{\infty}(0,T;H^{2}(\Omega))$; in the Main Theorem 1. Moreover, the results
of this paper will form a fundamental part of the optimization problem in
grain boundary motion, which will be exploring in a forthcoming paper.

Outline: Preliminaries are given in Section 1, and on this basis, the Main
Theorems are stated in Section 2. For the proofs of Main Theorems, we prepare
Section 3 to set up an approximation method for (S). Based on these, the Main
Theorems are proved in Section 4, by means of the auxiliary results obtained
in Section 3.

##  2 Preliminaries

We begin by prescribing the notations used throughout this paper.

Notations in real analysis. We define:

$\displaystyle r\vee s:=\max\\{r,s\\}~{}\mbox{ and }~{}r\wedge
s:=\min\\{r,s\\},\mbox{ for all $r,s\in[-\infty,\infty]$,}$

and especially, we write:

$\displaystyle[r]^{+}:=r\vee 0~{}\mbox{ and }~{}[r]^{-}:=-(r\wedge 0),\mbox{
for all $r\in[-\infty,\infty]$.}$

Additionally, for any $M>0$, let
$\mathcal{T}_{M}:\mathbb{R}\longrightarrow[-M,M]$ be the truncation operator,
defined as:

$\mathcal{T}_{M}:r\in\mathbb{R}\mapsto(r\vee(-M))\wedge M\in[-M,M].$

Let $d\in\mathbb{N}$ be a fixed dimension. We denote by $|y|$ and $y\cdot z$
the Euclidean norm of $y\in\mathbb{R}^{d}$ and the scalar product of
$y,z\in\mathbb{R}^{d}$, respectively, i.e.,

$\begin{array}[]{c}|y|:=\sqrt{y_{1}^{2}+\cdots+y_{d}^{2}}\mbox{ \ and \
}y\cdot z:=y_{1}z_{1}+\cdots+y_{d}z_{d},\\\\[4.30554pt] \mbox{ for all
$y=[y_{1},\ldots,y_{d}],~{}z=[z_{1},\ldots,z_{d}]\in\mathbb{R}^{d}$.}\end{array}$

Besides, we let:

$\displaystyle\mathbb{B}^{d}:=\left\\{\begin{array}[]{l|l}y\in\mathbb{R}^{d}&|y|<1\end{array}\right\\}~{}\mbox{
and
}~{}\mathbb{S}^{d-1}:=\left\\{\begin{array}[]{l|l}y\in\mathbb{R}^{d}&|y|=1\end{array}\right\\}.$

We denote by $\mathcal{L}^{d}$ the $d$-dimensional Lebesgue measure, and we
denote by $\mathcal{H}^{d}$ the $d$-dimensional Hausdorff measure. In
particular, the measure theoretical phrases, such as “a.e.”, “$dt$”, and
“$dx$”, and so on, are all with respect to the Lebesgue measure in each
corresponding dimension. Also on a Lipschitz-surface $S$, the phrase “a.e.” is
with respect to the Hausdorff measure in each corresponding Hausdorff
dimension. In particular, if $S$ is $C^{1}$-surface, then we simply denote by
$dS$ the area-element of the integration on $S$.

For a Borel set $E\subset\mathbb{R}^{d}$, we denote by
$\chi_{E}:\mathbb{R}^{d}\longrightarrow\\{0,1\\}$ the characteristic function
of $E$. Additionally, for a distribution $\zeta$ on an open set in
$\mathbb{R}^{d}$ and any $i\in\\{1,\dots,d\\}$, let $\partial_{i}\zeta$ be the
distributional differential with respect to $i$-th variable of $\zeta$. As
well as we consider, the differential operators, such as $\nabla,\
\operatorname{div},\ \nabla^{2}$, and so on, in distributional sense.

Abstract notations. (cf. [9, Chapter II]) For an abstract Banach space $X$, we
denote by $|\cdot|_{X}$ the norm of $X$, and denote by
$\langle\cdot,\cdot\rangle_{X}$ the duality pairing between $X$ and its dual
$X^{*}$. In particular, when $X$ is a Hilbert space, we denote by
$(\cdot,\cdot)_{X}$ the inner product of $X$.

For two Banach spaces $X$ and $Y$, let $\mathscr{L}(X;Y)$ be the Banach space
of bounded linear operators from $X$ into $Y$.

For Banach spaces $X_{1},\dots,X_{d}$ with $1<d\in\mathbb{N}$, let
$X_{1}\times\dots\times X_{d}$ be the product Banach space endowed with the
norm $|\cdot|_{X_{1}\times\cdots\times
X_{d}}:=|\cdot|_{X_{1}}+\cdots+|\cdot|_{X_{d}}$. However, when all
$X_{1},\dots,X_{d}$ are Hilbert spaces, $X_{1}\times\dots\times X_{d}$ denotes
the product Hilbert space endowed with the inner product
$(\cdot,\cdot)_{X_{1}\times\cdots\times
X_{d}}:=(\cdot,\cdot)_{X_{1}}+\cdots+(\cdot,\cdot)_{X_{d}}$ and the norm
$|\cdot|_{X_{1}\times\cdots\times
X_{d}}:=\bigl{(}|\cdot|_{X_{1}}^{2}+\cdots+|\cdot|_{X_{d}}^{2}\bigr{)}^{\frac{1}{2}}$.
In particular, when all $X_{1},\dots,X_{d}$ coincide with a Banach space $Y$,
the product space $X_{1}\times\dots\times X_{d}$ is simply denoted by
$[Y]^{d}$.

Basic notations. Let $0<T<\infty$ be a fixed constant of time, and let
$N\in\\{1,2,3\\}$ is a fixed dimension. Let $\Omega\subset\mathbb{R}^{N}$ be a
bounded domain with a boundary $\Gamma:=\partial\Omega$, and when $N>1$,
$\Gamma$ has $C^{\infty}$-regularity with the unit outer normal $n_{\Gamma}$.
Additionally, as notations of base spaces, we let:

$H:=L^{2}(\Omega),\ V:=H^{1}(\Omega),~{}\mathscr{H}:=L^{2}(0,T;H),\mbox{ and
}\mathscr{V}:=L^{2}(0,T;V).$

Let $W_{0}\subset H^{2}(\Omega)$ be the closed linear subspace of $H$, given
by:

$W_{0}:=\\{z\in H^{2}(\Omega)\,|\,\nabla z\cdot n_{\Gamma}=0\mbox{ on
}\Gamma\\}.$

Let $A_{N}$ be a differential operator, defined as:

$A_{N}:\,z\in W_{0}\subset H\mapsto A_{N}z:=-\mathit{\Delta}z\in H.$

It is well-known that $A_{N}\subset H\times H$ is linear, positive, and self-
adjoint, and the domain $W_{0}$ is a Hilbert space, endowed with the inner
product:

$\displaystyle(z_{1},z_{2})_{W_{0}}:=(z_{1},z_{2})_{H}+(A_{N}z_{1},z_{2})_{H}~{}\bigl{(}=(z_{1},z_{2})_{V}\bigr{)},\mbox{
for $z_{k}\in W_{0}$, $k=1,2$.}$

Moreover, there exists a positive constant $C_{0}$ such that:

$\displaystyle|z|_{H^{2}(\Omega)}^{2}\leq
C_{0}\bigl{(}|A_{N}z|_{H}^{2}+|z|_{H}^{2}\bigr{)},~{}\mbox{ for all $z\in
W_{0}$.}$ (2.1)

Notations for the time-discretization. Let $\tau>0$ be a constant of the time
step-size, and let $\\{t_{i}\\}_{i=0}^{\infty}\subset[0,\infty)$ be the time
sequence defined as:

$t_{i}:=i\tau,\ i=0,1,2,\ldots.$

Let $X$ be a Banach space. Then, for any sequence
$\\{[t_{i},z_{i}]\\}_{i=0}^{\infty}\subset[0,\infty)\times X$, we define the
forward time-interpolation $[\overline{z}]_{\tau}\in
L^{\infty}_{\mathrm{loc}}([0,\infty);X)$, the backward time-interpolation
$[\underline{z}]_{\tau}\in L^{\infty}_{\mathrm{loc}}([0,\infty);X)$ and the
linear time-interpolation $[z]_{\tau}\in
W^{1,2}_{\mathrm{loc}}([0,\infty);X)$, by letting:

$\left\\{\begin{aligned}
&[\overline{z}]_{\tau}(t):=\chi_{(-\infty,0]}z_{0}+\sum_{i=1}^{\infty}\chi_{(t_{i-1},t_{i}]}(t)z_{i},\\\
&[\underline{z}]_{\tau}(t):=\sum_{i=0}^{\infty}\chi_{(t_{i},t_{i+1}]}(t)z_{i},\\\
&[z]_{\tau}(t):=\sum_{i=1}^{\infty}\chi_{[t_{i-1},t_{i})}(t)\left(\frac{t-t_{i-1}}{\tau}z_{i}+\frac{t_{i}-t}{\tau}z_{i-1}\right),\end{aligned}\right.~{}{\rm
in}~{}X,\ {\rm for}~{}t\geq 0,$

respectively.

In the meantime, for any $q\in[1,\infty)$ and any $\zeta\in
L_{\mathrm{loc}}^{q}([0,\infty);X)$, we denote by
$\\{\zeta_{i}\\}_{i=0}^{\infty}\subset X$ the sequence of time-discretization
data of $\zeta$, defined as:

$\displaystyle\zeta_{0}:=0\mbox{ in $X$, and
}\zeta_{i}:=\frac{1}{\tau}\int_{t_{i-1}}^{t_{i}}\zeta(\varsigma)\,d\varsigma~{}\mbox{
in $X$, ~{} for $i=1,2,3,\dots$.}$ (2.2a) As is easily checked, the time-
interpolations $[\overline{\zeta}]_{\tau},[\underline{\zeta}]_{\tau}\in
L^{q}_{\mathrm{loc}}([0,\infty);X)$ for the above
$\\{\zeta_{i}\\}_{i=0}^{\infty}$ fulfill that:
$\displaystyle[\overline{\zeta}]_{\tau}\to\zeta\mbox{ and
}[\underline{\zeta}]_{\tau}\to\zeta\mbox{ in
$L^{q}_{\mathrm{loc}}([0,\infty);X)$, as $\tau\downarrow 0$.}$ (2.2b)

##  3 Main results

In this paper, the main assertions are discussed under the following
assumptions.

  * (A1)

$\mu>0$ and $\nu>0$ are fixed constants.

  * (A2)

$g:\mathbb{R}\longrightarrow\mathbb{R}$ is a locally Lipschitz continuous
function with a nonnegative primitive $G\in C^{1}(\mathbb{R})$. Moreover, $g$
satisfies the following condition:

$\liminf_{\xi\downarrow-\infty}g(\xi)=-\infty,\
\limsup_{\xi\uparrow\infty}g(\xi)=\infty.$

  * (A3)

$\alpha_{0}:\mathbb{R}\longrightarrow(0,\infty)$ is a locally Lipschitz
continuous function, and $\alpha:\mathbb{R}\longrightarrow[0,\infty)$ is a
$C^{1}$-class convex function, such that:

$\alpha^{\prime}(0)=0\mbox{ and
}\delta_{\alpha}:=\inf\alpha_{0}(\mathbb{R})>0.$

  * (A4)

$u,v\in L_{\rm loc}^{2}([0,\infty);H)$, and $u\in L^{\infty}(Q)$.

  * (A5)

The initial data $[\eta_{0},\theta_{0}]$ belong to the class
$[W_{0}]^{2}\subset[H^{2}(\Omega)]^{2}$.

Now, the main results are stated as follows.

######  Main Theorem 1 (Existence and regularity).

Under the assumptions (A1)–(A5), the system (S) admits a solution
$[\eta,\theta]\in[\mathscr{H}]^{2}$ in the following sense.

  * (S0)

$[\eta,\theta]\in\bigl{[}W^{1,2}(0,T;W_{0})\cap
L^{\infty}(Q)\bigr{]}\times\bigl{[}W^{1,2}(0,T;V)\times
L^{\infty}(0,T;W_{0})\bigr{]}$.

  * (S1)

$\eta$ solves the following variational identity:

$\displaystyle(\partial_{t}\eta(t),\varphi)_{H}+(\nabla(\eta+\mu^{2}\partial_{t}\eta)(t),\nabla\varphi)_{[H]^{N}}+(g(\eta(t)),\varphi)_{H}$
$\displaystyle+(\alpha^{\prime}(\eta(t))|\nabla\theta(t)|,\varphi)_{H}=(u(t),\varphi)_{H},$
$\displaystyle\mbox{ for any }\varphi\in V,\mbox{ and a.e. }t\in(0,T).$

  * (S2)

$\theta$ solves the following variational inequality:

$\displaystyle\bigl{(}(\alpha_{0}(\eta)\partial_{t}\theta)(t),\theta(t)-\psi\bigr{)}_{H}+\nu^{2}(\nabla\partial_{t}\theta(t),\nabla(\theta(t)-\psi))_{[H]^{N}}$
$\displaystyle+\int_{\Omega}\alpha(\eta(t))|\nabla\theta(t)|\,dx\leq\int_{\Omega}\alpha(\eta(t))|\nabla\psi|\,dx+(v(t),\theta(t)-\psi)_{H},$
$\displaystyle\mbox{ for any }\psi\in V,\mbox{ and a.e. }t\in(0,T).$

  * (S3)

$[\eta(0),\theta(0)]=[\eta_{0},\theta_{0}]$ in $[H]^{2}$.

######  Main Theorem 2 (Uniqueness and continuous dependence).

Under the assumptions (A1)–(A5), let $[\eta^{k},\theta^{k}],\,k=1,2$ be two
solutions to (S) with two initial values $[\eta_{0}^{k},\theta_{0}^{k}]$ and
two forcings $[u^{k},v^{k}]$, $k=1,2$. Then, there exists a constant
$C_{1}=C_{1}(\nu)>0$, depending on $\nu$, such that:

$\displaystyle J(t)\leq
C_{1}(\nu)\bigl{(}J(0)+|u^{1}-u^{2}|_{\mathscr{H}}^{2}+|v^{1}-v^{2}|_{\mathscr{H}}^{2}\bigr{)},$
$\displaystyle\mbox{ for any }t\in[0,T]\mbox{ and any }T>0,$

where

$\displaystyle J(t):=$
$\displaystyle|(\eta^{1}-\eta^{2})(t)|_{H}^{2}+\mu^{2}|\nabla(\eta^{1}-\eta^{2})(t)|_{[H]^{N}}^{2}+|\sqrt{\alpha_{0}(\eta^{1})}(\theta^{1}-\theta^{2})(t)|_{H}^{2}$
$\displaystyle\quad+\nu^{2}|\nabla(\theta^{1}-\theta^{2})(t)|_{[H]^{N}}^{2},\
\mbox{ for any }t\geq 0.$

##  4 Approximating method

In the Main Theorems, the solution to (S) will be obtained by means of the
time-discretization method. In this light, let $\tau\in(0,1)$ be a constant of
the time-step size, and let $\varepsilon\in(0,1)$ be a relaxation constant.
Based on this, we adopt the following time-discretization scheme
(AP)${}_{\tau}^{\varepsilon}$, as our approximating problem of (S).

(AP)${}_{\tau}^{\varepsilon}$: To find
$\\{[\eta_{i},\theta_{i}]\\}_{i=0}^{\infty}\subset[W_{0}]^{2}$ satisfying:

$\displaystyle\frac{\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}}{\tau}+A_{N}\left(\eta_{i}^{\varepsilon}+\frac{\mu^{2}}{\tau}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})\right)+g(\mathcal{T}_{M}\eta_{i}^{\varepsilon})$
(4.1)
$\displaystyle+\alpha^{\prime}(\mathcal{T}_{M}\eta_{i}^{\varepsilon})\gamma_{\varepsilon}(\nabla\theta_{i}^{\varepsilon})=u_{i}\mbox{
in }H,$
$\displaystyle\frac{\alpha_{0}(\mathcal{T}_{M}\eta_{i-1}^{\varepsilon})}{\tau}(\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon})-\operatorname{div}\bigl{(}\widetilde{\alpha}_{M}(\eta)\nabla\gamma_{\varepsilon}(\nabla\theta_{i}^{\varepsilon})\bigr{)}$
(4.2)
$\displaystyle+\frac{\nu^{2}}{\tau}A_{N}(\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon})=v_{i}\mbox{
in }H,$ $\displaystyle\mbox{for }i=1,2,3,\dots,\mbox{ subject to
}[\eta_{0}^{\varepsilon},\theta_{0}^{\varepsilon}]=[\eta_{0},\theta_{0}]\mbox{
in }[H]^{2}.$

In this context, $\gamma_{\varepsilon}\in C^{0,1}(\mathbb{R}^{N})\cap
C^{\infty}(\mathbb{R}^{N})$ is a smooth approximation of the Euclidean norm
$|\cdot|\in C^{0,1}(\mathbb{R}^{N})$, defined as:

$\gamma_{\varepsilon}:y\in\mathbb{R}^{N}\mapsto\gamma_{\varepsilon}(y):=\sqrt{\varepsilon^{2}+|y|^{2}}\in[0,\infty).$

Also, we define an approximating free-energy $\mathcal{F}_{\varepsilon}$ on
$[H]^{2}$, by setting:

$\displaystyle\mathcal{F}_{\varepsilon}:[\eta,\theta]\in[H]^{2}\mapsto\mathcal{F}_{\varepsilon}(\eta,\theta):=\frac{1}{2}\int_{\Omega}|\nabla\eta|^{2}\,dx+\int_{\Omega}\widetilde{G}_{M}(\eta)$
(4.3)
$\displaystyle+\int_{\Omega}\widetilde{\alpha}_{M}(\eta)\gamma_{\varepsilon}(D\theta),\
\mbox{ for any }0<\varepsilon<1.$

where $\widetilde{\alpha}_{M}\in C^{1,1}(\mathbb{R})$ and
$\widetilde{G}_{M}\in C^{1,1}(\mathbb{R})$ are nonnegative primitives of
$\alpha\circ\mathcal{T}_{M}\in W^{1,\infty}(\mathbb{R})$ and
$G\circ\mathcal{T}_{M}\in W^{1,\infty}(\mathbb{R})$, respectively. Finally,
$u_{i},v_{i}$ are given as in (2.2).

The solution to (AP)${}_{\tau}^{\varepsilon}$ is given as follows.

######  Definition 1.

The sequence of functions
$\\{[\eta_{i}^{\varepsilon},\theta_{i}^{\varepsilon}]\\}_{i=0}^{\infty}$ is
called a solution to (AP)${}_{\tau}^{\varepsilon}$ iff.
$\\{[\eta_{i}^{\varepsilon},\theta_{i}^{\varepsilon}]\\}_{i=0}^{\infty}\subset[W_{0}]^{2}$,
and $[\eta_{i}^{\varepsilon},\theta_{i}^{\varepsilon}]$ fulfills (4.1) and
(4.2) for any $i=1,2,3,\dots$.

In this paper, the following theorem will plays an important role for the
proof of Main Theorems.

######  Theorem 1 (Solvability of the approximating problem).

There exists a sufficiently small constant $\tau_{0}\in(0,1)$ such that for
any $\tau\in(0,\tau_{0})$ and $\varepsilon\in(0,1)$,
(AP)${}_{\tau}^{\varepsilon}$ admits a unique solution
$\\{[\eta_{i}^{\varepsilon},\theta_{i}^{\varepsilon}]\\}_{i=0}^{\infty}$.
Additionally, the following energy inequality holds:

$\displaystyle\frac{1}{4\tau}|\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}|_{H}^{2}+\frac{\mu^{2}}{\tau}|\nabla(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})|_{[H]^{N}}^{2}+\frac{\delta_{\alpha}}{2\tau}|\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon}|_{H}^{2}$
(4.4)
$\displaystyle+\frac{\nu^{2}}{\tau}|\nabla(\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon})|_{[H]^{N}}^{2}+\mathcal{F}_{\varepsilon}(\eta_{i}^{\varepsilon},\theta_{i}^{\varepsilon})$
$\displaystyle\leq\mathcal{F}_{\varepsilon}(\eta_{i-1}^{\varepsilon},\theta_{i-1}^{\varepsilon})+\frac{\tau}{2}|u_{i}|_{H}^{2}+\frac{\tau}{2\delta_{\alpha}}|v_{i}|_{H}^{2},\
\mbox{ for any }i=1,2,3,\dots.$

Theorem 1 is proved through several lemmas.

######  Lemma 4.1.

For arbitrary $\widetilde{\theta}\in V$, $\widetilde{\eta}_{0}\in W_{0}$ and
$\widetilde{u}\in H$, we consider the following elliptic problem:

$\frac{1}{\tau}(\eta-\widetilde{\eta}_{0})+A_{N}\left(\eta+\frac{\mu^{2}}{\tau}(\eta-\widetilde{\eta}_{0})\right)+g(\mathcal{T}_{M}\eta)+\alpha^{\prime}(\mathcal{T}_{M}\eta)\gamma_{\varepsilon}(\nabla\widetilde{\theta})=\widetilde{u},\mbox{
in }H.$ (4.5)

Then, there exists a small constant $\tau_{1}\in(0,1)$, depending only on
$|g^{\prime}|_{L^{\infty}(-M,M)}$, and for any $0<\tau<\tau_{1}$, the elliptic
problem (4.5) admits a unique solution $\eta\in W_{0}$.

###### Proof.

First, for any $\eta^{\dagger}\in V$, we define a functional
$\Upsilon:H\longrightarrow(-\infty,\infty]$ as follows:

$\Upsilon:z\in H\mapsto\Upsilon(z):=\left\\{\begin{aligned}
&\frac{1}{2\tau}\int_{\Omega}|z|^{2}\,dx+\frac{1}{2}\int_{\Omega}|\nabla
z|^{2}\,dx+\frac{\mu^{2}}{2\tau}\int_{\Omega}|\nabla(z-\widetilde{\eta}_{0})|^{2}\,dx\\\
&\quad+\int_{\Omega}\widetilde{g}(\mathcal{T}_{M}\eta^{\dagger})z\,dx+\int_{\Omega}\widetilde{\alpha}_{M}(z)\gamma_{\varepsilon}(\nabla\widetilde{\theta})\,dx\\\
&\quad-\int_{\Omega}\widetilde{u}z\,dx,\ \mbox{ if }z\in V,\\\ &\infty,\mbox{
otherwise}.\end{aligned}\right.$

As is easily checked, $\Upsilon$ is proper, l.s.c., strictly convex, and
coercive, and its unique minimizer solves the following elliptic equation:

$\frac{1}{\tau}(\eta-\widetilde{\eta}_{0})+A_{N}\left(\eta+\frac{\mu^{2}}{\tau}(\eta-\widetilde{\eta}_{0})\right)+g(\mathcal{T}_{M}\eta^{\dagger})+\alpha^{\prime}(\mathcal{T}_{M}\eta)\gamma_{\varepsilon}(\nabla\widetilde{\theta})=\widetilde{u},\mbox{
in }H.$ (4.6)

Now, we define an operator $S_{\tau}:V\longrightarrow H^{2}(\Omega)$ which
maps any $\eta^{\dagger}\in V$ to the unique solution to (4.6), and consider
the smallness condition of $\tau$ for $S$ to be contractive. Here, let
$\eta_{k}:=S_{\tau}\eta_{k}^{\dagger}\in H^{2}(\Omega)$, $k=1,2$. By taking
differences of (4.6), multiplying both sides by $\eta_{1}-\eta_{2}$ and
applying Young’s inequality, we see from (A1) and (A2) that:

$\displaystyle\frac{1\wedge\mu^{2}}{2\tau}|\eta_{1}-\eta_{2}|_{V}^{2}\leq\frac{\tau|g^{\prime}|_{L^{\infty}(-M,M)}^{2}}{2}|\eta_{1}^{\dagger}-\eta_{2}^{\dagger}|_{H}^{2}.$

Therefore, if we assume that

$0<\tau<\tau_{1}:=\left(\frac{1\wedge\mu^{2}}{|g|_{L^{\infty}(-M,M)}^{2}}\right)^{\frac{1}{2}},$
(4.7)

then the mapping $S_{\tau}$ becomes a contraction mapping from $V$ into
itself. Therefore, applying Banach’s fixed point theorem, we find a unique
fixed point $\widetilde{\eta}\in V$ of $S_{\tau}$ under the condition (4.7).
The identity $S_{\tau}\widetilde{\eta}=\widetilde{\eta}$ implies that
$\widetilde{\eta}$ is the unique solution to (4.5). ∎

######  Lemma 4.2.

For arbitrary $\widetilde{\eta}\in H^{2}(\Omega)$, $\widetilde{\theta}_{0}\in
W_{0}$ and $\widetilde{v}\in H$, we consider the following elliptic equation:

$\left\\{\begin{aligned}
&\alpha_{0}(\mathcal{T}_{M}\widetilde{\eta})\frac{\theta-\widetilde{\theta}_{0}}{\tau}-\operatorname{div}\left(\widetilde{\alpha}_{M}(\widetilde{\eta})\nabla\gamma_{\varepsilon}(\nabla\theta)+\frac{\nu^{2}}{\tau}\nabla(\theta-\widetilde{\theta}_{0})\right)=\widetilde{v}\mbox{
a.e. in }\Omega,\\\ &\nabla\theta\cdot n_{\Gamma}=0\mbox{ a.e. on
}\Gamma.\end{aligned}\right.$ (4.8)

Then, for any $0<\tau<1$, (4.8) admits a unique solution $\theta\in W_{0}$.

###### Proof.

Let us consider a proper, l.s.c., strictly convex, and coercive function
$\Upsilon_{*}$ defined as follows:

$\Upsilon_{*}:z\in H\mapsto\Upsilon_{*}(z):=\left\\{\begin{aligned}
&\frac{1}{2\tau}\int_{\Omega}\alpha_{0}(\mathcal{T}_{M}\widetilde{\eta})|z-\widetilde{\theta}_{0}|^{2}\,dx+\int_{\Omega}\widetilde{\alpha}_{M}(\widetilde{\eta})\gamma_{\varepsilon}(\nabla
z)\,dx\\\
&\quad+\frac{\nu^{2}}{2\tau}\int_{\Omega}|\nabla(z-\widetilde{\theta}_{0})|^{2}\,dx-\int_{\Omega}\widetilde{v}z\,dx,\
\mbox{ if }z\in V,\\\ &\infty,\mbox{ otherwise. }\end{aligned}\right.$

As is discussed in [1, Theorem 1], the unique minimizer $\widetilde{\theta}$
of $\Upsilon_{*}$ solves (4.8), and $\widetilde{\theta}$ belongs to $W_{0}$. ∎

###### Proof of theorem 1.

Let us fix any $\tau\in(0,\tau_{1})$ and any $\varepsilon\in(0,1)$. Then, for
any $i\in\mathbb{N}$, we can obtain $\theta_{i}^{\varepsilon}\in W_{0}$ by
applying lemma 4.2 in the case that:

$\widetilde{\eta}=\eta_{i-1}^{\varepsilon},\
\widetilde{\theta}_{0}=\theta_{i-1}^{\varepsilon}\mbox{ and
}\widetilde{v}=v_{i}\mbox{ in }H.$

Moreover, for any $i\in\mathbb{N}$, the component $\eta_{i}^{\varepsilon}\in
W_{0}$ can be obtained by applying lemma 4.1 in the case that:

$\widetilde{\theta}=\theta_{i}^{\varepsilon},\
\widetilde{\eta}_{0}:=\eta_{i-1}^{\varepsilon}\mbox{ and
}\widetilde{u}=u_{i}\mbox{ in }H.$

Thus we can find the unique solution
$\\{[\eta_{i}^{\varepsilon},\theta_{i}^{\varepsilon}]\\}_{i=0}^{\infty}\subset[W_{0}]^{2}$
to (AP)${}_{\tau}^{\varepsilon}$.

Next, we verify the inequality (4.4). Multiplying both sides of (4.1) with
$\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}$, we see that:

$\displaystyle\frac{1}{2\tau}|\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}|_{H}^{2}+\frac{\mu^{2}}{\tau}|\nabla(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})|_{[H]^{N}}^{2}+\frac{1}{2}|\nabla\eta_{i}^{\varepsilon}|_{[H]^{N}}^{2}-\frac{1}{2}|\nabla\eta_{i-1}^{\varepsilon}|_{[H]^{N}}^{2}$
(4.9)
$\displaystyle+\bigl{(}g(\mathcal{T}_{M}\eta_{i}^{\varepsilon}),\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}\bigr{)}_{H}+\bigl{(}\alpha^{\prime}(\mathcal{T}_{M}\eta_{i}^{\varepsilon})\gamma_{\varepsilon}(\nabla\theta_{i}^{\varepsilon}),\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}\bigr{)}_{H}\leq\frac{\tau}{2}|u_{i}|_{H}^{2},$
$\displaystyle\mbox{for }i=1,2,3,\dots.$

via the following computations:

$\displaystyle(\nabla\eta_{i}^{\varepsilon},\nabla(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}))_{[H]^{N}}\geq\frac{1}{2}\bigl{(}|\nabla\eta_{i}^{\varepsilon}|_{[H]^{N}}^{2}-|\nabla\eta_{i-1}^{\varepsilon}|_{[H]^{N}}^{2}\bigr{)},$
$\displaystyle\frac{\mu^{2}}{\tau}(\nabla(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}),\nabla(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}))_{[H]^{N}}=\frac{\mu^{2}}{\tau}|\nabla(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})|_{[H]^{N}}^{2},$

and

$\displaystyle(u_{i},\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})_{H}\leq\frac{1}{2\tau}|\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}|_{H}^{2}+\frac{\tau}{2}|u_{i}|_{H}^{2}.$

In addition, by using (A1), it is obtained that:

$\displaystyle\bigl{(}g(\mathcal{T}_{M}\eta_{i}^{\varepsilon}),\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}\bigr{)}_{H}\geq\int_{\Omega}\widetilde{G}_{M}(\eta_{i}^{\varepsilon})\,dx-\int_{\Omega}\widetilde{G}_{M}(\eta_{i-1}^{\varepsilon})\,dx$
(4.10)
$\displaystyle+\bigl{(}g(\mathcal{T}_{M}\eta_{i}^{\varepsilon})-g(\mathcal{T}_{M}\eta_{i-1}^{\varepsilon}),\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}\bigr{)}_{H}-\frac{1}{2}|g^{\prime}|_{L^{\infty}(-M,M)}|\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}|_{H}^{2}$
$\displaystyle\geq\int_{\Omega}\widetilde{G}_{M}(\eta_{i}^{\varepsilon})\,dx-\int_{\Omega}\widetilde{G}_{M}(\eta_{i-1}^{\varepsilon})\,dx-\frac{3}{2}|g^{\prime}|_{L^{\infty}(-M,M)}|\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}|_{H}^{2},$
$\displaystyle\mbox{ for }i=1,2,3,\dots,$

and by the convexity of $\widetilde{\alpha}_{M}$,

$\displaystyle\bigl{(}\alpha^{\prime}(\mathcal{T}_{M}\eta_{i}^{\varepsilon})\gamma_{\varepsilon}(\nabla\theta_{i}^{\varepsilon}),\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}\bigr{)}_{H}\geq\int_{\Omega}\widetilde{\alpha}_{M}(\eta_{i}^{\varepsilon})\gamma_{\varepsilon}(\nabla\theta_{i}^{\varepsilon})\,dx$
(4.11)
$\displaystyle-\int_{\Omega}\widetilde{\alpha}_{M}(\eta_{i-1}^{\varepsilon})\gamma_{\varepsilon}(\nabla\theta_{i}^{\varepsilon})\,dx,\mbox{
for }i=1,2,3,\dots.$

On account of (4.9)–(4.11), it is inferred that:

$\displaystyle\left(\frac{1}{2}-\frac{3\tau}{2}|g^{\prime}|_{L^{\infty}(-M,M)}\right)\frac{1}{\tau}|\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}|_{H}^{2}+\frac{\mu^{2}}{\tau}|\nabla(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})|_{[H]^{N}}^{2}$
(4.12)
$\displaystyle+\frac{1}{2}|\nabla\eta_{i}^{\varepsilon}|_{[H]^{N}}^{2}+\int_{\Omega}\widetilde{G}_{M}(\eta_{i}^{\varepsilon})\,dx+\int_{\Omega}\widetilde{\alpha}_{M}(\eta_{i}^{\varepsilon})\gamma_{\varepsilon}(\nabla\theta_{i}^{\varepsilon})\,dx$
$\displaystyle\leq\frac{1}{2}|\nabla\eta_{i-1}^{\varepsilon}|_{[H]^{N}}^{2}+\int_{\Omega}\widetilde{G}_{M}(\eta_{i-1}^{\varepsilon})\,dx+\int_{\Omega}\widetilde{\alpha}_{M}(\eta_{i-1}^{\varepsilon})\gamma_{\varepsilon}(\nabla\theta_{i}^{\varepsilon})\,dx+\frac{\tau}{2}|u_{i}|_{H}^{2},$
$\displaystyle\mbox{ for }i=1,2,3,\dots.$

On the other hand, by multiplying both sides of (4.2) by
$\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon}$, and using (A3) and the
convexity of $\gamma_{\varepsilon}$, we have

$\displaystyle\frac{\delta_{\alpha}}{2\tau}|\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon}|_{H}^{2}+\frac{\nu^{2}}{\tau}|\nabla(\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon})|_{[H]^{N}}^{2}+\int_{\Omega}\widetilde{\alpha}_{M}(\eta_{i-1}^{\varepsilon})\gamma_{\varepsilon}(\nabla\theta_{i}^{\varepsilon})\,dx$
(4.13)
$\displaystyle\leq\int_{\Omega}\widetilde{\alpha}_{M}(\eta_{i-1}^{\varepsilon})\gamma_{\varepsilon}(\nabla\theta_{i-1}^{\varepsilon})\,dx+\frac{\tau}{2\delta_{\alpha}}|v_{i}|_{H}^{2},\
\mbox{ for }i=1,2,3,\dots,$

via the following computation:

$\displaystyle\frac{1}{\tau}(\alpha_{0}(\mathcal{T}_{M}\eta_{i-1})(\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon}),\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon})_{H}\geq\frac{\delta_{\alpha}}{\tau}|\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon}|_{H}^{2},$
$\displaystyle(\widetilde{\alpha}_{M}(\eta_{i-1}^{\varepsilon})\nabla\gamma_{\varepsilon}(\nabla\theta_{i}^{\varepsilon}),\nabla(\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon}))_{[H]^{N}}\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\geq\int_{\Omega}\widetilde{\alpha}_{M}(\eta_{i-1}^{\varepsilon})\gamma_{\varepsilon}(\nabla\theta_{i}^{\varepsilon})\,dx-\int_{\Omega}\widetilde{\alpha}_{M}(\eta_{i-1}^{\varepsilon})\gamma_{\varepsilon}(\nabla\theta_{i-1}^{\varepsilon})\,dx,$

and

$\displaystyle(u_{i},\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon})_{H}\leq\frac{\delta_{\alpha}}{2\tau}|\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon}|_{H}^{2}+\frac{\tau}{2\delta_{\alpha}}|u_{i}|_{H}^{2}.$

Now, let us set $\tau_{0}$ as

$\displaystyle\tau_{0}:=\min\left\\{\tau_{1},\frac{1}{6|g^{\prime}|_{L^{\infty}(-M,M)}}\right\\},\mbox{
with the constant $\tau_{1}$ as in<EMAIL_ADDRESS>
Then, from (4.12) and (4.13), we obtain that:

$\displaystyle\frac{1}{4\tau}|\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}|_{H}^{2}+\frac{\mu^{2}}{\tau}|\nabla(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})|_{[H]^{N}}^{2}+\frac{\delta_{\alpha}}{2\tau}|\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon}|_{H}^{2}$
$\displaystyle\quad+\frac{\nu^{2}}{\tau}|\nabla(\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon})|_{[H]^{N}}^{2}+\mathcal{F}_{\varepsilon}(\eta_{i}^{\varepsilon},\theta_{i}^{\varepsilon})$
$\displaystyle\quad\leq\mathcal{F}_{\varepsilon}(\eta_{i-1}^{\varepsilon},\theta_{i-1}^{\varepsilon})+\frac{\tau}{2}|u_{i}|_{H}^{2}+\frac{\tau}{2\delta_{\alpha}}|v_{i}|_{H}^{2},\
\mbox{ for }i=1,2,3,\dots.$

Thus, we conclude theorem 1. ∎

##  5 Proofs of Main Theorems

In this section, we will provide proofs of Main Theorems. We set

$n_{\tau}:=\min\\{n\in\mathbb{N}\,|\,n\tau\geq T\\}.$ (5.1)

Additionally, under the notations as in theorem 1, we invoke (2.2a) and
(2.2b), and take a small constant $\tau_{*}\in(0,\tau_{0})$, such that:

$\tau\sum_{i=1}^{n_{\tau}}\bigl{(}|u_{i}|_{H}^{2}+|v_{i}|_{H}^{2}\bigr{)}\leq|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1,\mbox{
whenever }\tau\in(0,\tau_{*}).$

###  5.1 Proof of Main Theorem 1

Before we deal with the proof, we will prepare some lemmas. Hereafter, based
on (A1), (A3), (A4) and (A5), we set the constant $M>0$ of truncation, so
large to satisfy that:

$M\geq|\eta_{0}|_{L^{\infty}(\Omega)},\ g(M)\geq|u|_{L^{\infty}(Q)},\mbox{ and
}g(-M)\leq-|u|_{L^{\infty}(Q)}.$ (5.2)

Then, it immediately follows that:

$\widetilde{G}_{M}(\eta_{0})=G(\eta_{0}),\mbox{ and
}\widetilde{\alpha}_{M}(\eta_{0})=\alpha(\eta_{0}).$ (5.3)

######  Lemma 5.1.

Let $\tau\in(0,\tau_{*})$. Then, there exists a constant $C_{2}>0$,
independent of $\varepsilon$ and $\tau$, such that:

$\displaystyle\frac{1}{\tau}\sum_{i=1}^{n_{\tau}}|\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}|_{H^{2}(\Omega)}^{2}\leq
C_{2}\bigl{(}|\eta_{0}|_{H^{2}(\Omega)}^{2}+|\theta_{0}|_{V}^{2}+|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)}.$
(5.4)

###### Proof.

First, from the definition of $\mathcal{F}_{\varepsilon}$ (4.3), (5.3), and
the embedding $W_{0}\subset L^{\infty}(\Omega)$ under $N\leq 3$, it is seen
that:

$\displaystyle\mathcal{F}_{\varepsilon}(\eta_{0},\theta_{0})$
$\displaystyle=\frac{1}{2}\int_{\Omega}|\nabla\eta_{0}|^{2}\,dx+\int_{\Omega}\widetilde{G}_{M}(\eta_{0})\,dx+\int_{\Omega}\widetilde{\alpha}_{M}(\eta_{0})\gamma_{\varepsilon}(\nabla\theta_{0})\,dx$
$\displaystyle=\frac{1}{2}\int_{\Omega}|\nabla\eta_{0}|^{2}\,dx+\int_{\Omega}G(\eta_{0})\,dx+\int_{\Omega}\alpha(\eta_{0})\gamma_{\varepsilon}(\nabla\theta_{0})\,dx$
$\displaystyle\leq\frac{1}{2}|\eta_{0}|_{V}^{2}+|G(\eta_{0})|_{L^{1}(\Omega)}+|\alpha(\eta_{0})|_{L^{2}(\Omega)}^{2}+\mathcal{L}^{N}(\Omega)+|\theta_{0}|_{V}^{2}.$

Hence,

$C_{F}:=\sup_{\varepsilon\in(0,1)}\mathcal{F}_{\varepsilon}(\eta_{0},\theta_{0})<\infty.$

Also, from (4.4),(5.1), theorem 1, and Hölder’s inequality, it is observed
that:

$\displaystyle|\theta_{i}^{\varepsilon}|_{V}^{2}\leq
2|\theta_{0}|_{V}^{2}+2\left(\sum_{i=1}^{n_{\tau}}|\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon}|_{V}\right)^{2}$
(5.5) $\displaystyle\leq~{}$ $\displaystyle
2|\theta_{0}|_{V}^{2}+2(T+1)\sum_{i=1}^{n_{\tau}}\frac{1}{\tau}|\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon}|_{V}^{2}$
$\displaystyle\leq~{}$ $\displaystyle
2|\theta_{0}|_{V}^{2}+\frac{4(T+1)}{\delta_{\alpha}\wedge\nu^{2}}\left(\mathcal{F}_{\varepsilon}(\eta_{0},\theta_{0})+\frac{1}{2(1\wedge\delta_{\alpha})}\bigl{(}|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)}\right)$
$\displaystyle\leq~{}$
$\displaystyle\frac{4(C_{F}+1)(T+1)}{1\wedge\delta_{\alpha}^{2}\wedge\nu^{4}}\bigl{(}|\theta_{0}|_{V}^{2}+|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)},\
\mbox{ for any }i=1,2,3,\dots,n_{\tau}.$

Next, we verify the estimate (5.4). Multiplying the both side of (4.1) by
$-\mathit{\Delta}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})$ and
applying Young’s inequality, it can be seen that:

$\displaystyle\frac{3\mu^{2}}{4\tau}|\mathit{\Delta}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})|_{H}^{2}\leq\frac{1}{2}\bigl{(}|\mathit{\Delta}\eta_{i-1}^{\varepsilon}|_{H}^{2}-|\mathit{\Delta}\eta_{i}^{\varepsilon}|_{H}^{2}\bigr{)}$
(5.6)
$\displaystyle\quad+(g(\mathcal{T}_{M}\eta_{i}^{\varepsilon}),\mathit{\Delta}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}))_{H}+(\alpha^{\prime}(\mathcal{T}_{M}\eta_{i}^{\varepsilon})\gamma_{\varepsilon}(\nabla\theta_{i}^{\varepsilon}),\mathit{\Delta}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}))_{H}$
$\displaystyle\quad+\frac{\tau}{\mu^{2}}|u_{i}|_{H}^{2},\ \mbox{ for
}i=1,2,3,\dots,n_{\tau},$

via the following calculations:

$\displaystyle(-\mathit{\Delta}\eta_{i}^{\varepsilon},-\mathit{\Delta}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}))_{H}\geq\frac{1}{2}\bigl{(}|\mathit{\Delta}\eta_{i}^{\varepsilon}|_{H}^{2}-|\mathit{\Delta}\eta_{i-1}^{\varepsilon}|_{H}^{2}\bigr{)},$

and

$\displaystyle(u_{i},-\mathit{\Delta}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}))_{H}\leq\frac{\mu^{2}}{4\tau}|\mathit{\Delta}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})|_{H}^{2}+\frac{\tau}{\mu^{2}}|u_{i}|_{H}^{2}.$
$\displaystyle(g(\mathcal{T}_{M}\eta_{i}^{\varepsilon}),\mathit{\Delta}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}))_{H}$
(5.7)
$\displaystyle\quad\leq\frac{\tau}{\mu^{2}}|g|_{L^{\infty}(-M,M)}^{2}\mathcal{L}^{N}(\Omega)+\frac{\mu^{2}}{4\tau}|\mathit{\Delta}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})|_{H}^{2},$

and

$\displaystyle(\alpha^{\prime}(\mathcal{T}_{M}\eta_{i}^{\varepsilon})\gamma_{\varepsilon}(\nabla\theta),\mathit{\Delta}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}))_{H}$
(5.8)
$\displaystyle\quad\leq\frac{\tau}{\mu^{2}}|\alpha^{\prime}|_{L^{\infty}(-M,M)}^{2}\int_{\Omega}\bigl{(}\varepsilon^{2}+|\nabla\theta_{i}^{\varepsilon}|^{2}\bigr{)}\,dx+\frac{\mu^{2}}{4\tau}|\mathit{\Delta}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})|_{H}^{2}$
$\displaystyle\quad\leq\frac{\tau}{\mu^{2}}|\alpha^{\prime}|_{L^{\infty}(-M,M)}^{2}\bigl{(}\mathcal{L}^{N}(\Omega)+|\theta_{i}^{\varepsilon}|_{V}^{2}\bigr{)}+\frac{\mu^{2}}{4\tau}|\mathit{\Delta}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})|_{H}^{2},$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\mbox{ for
}i=1,2,3,\dots,n_{\tau}.$

On account of (5.5)–(5.8), we infer that:

$\displaystyle\frac{\mu^{2}}{4\tau}|\mathit{\Delta}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})|_{H}^{2}$
(5.9)
$\displaystyle\quad\leq\frac{1}{2}\bigl{(}|\mathit{\Delta}\eta_{i-1}^{\varepsilon}|_{H}^{2}-|\mathit{\Delta}\eta_{i}^{\varepsilon}|_{H}^{2}\bigr{)}+\frac{\tau\mathcal{L}^{N}(\Omega)}{\mu^{2}}\bigl{(}|g|_{L^{\infty}(-M,M)}^{2}+|\alpha^{\prime}|_{L^{\infty}(-M,M)}\bigr{)}$
$\displaystyle\qquad+\frac{4\tau(C_{F}+1)(T+1)|\alpha^{\prime}|_{L^{\infty}(-M,M)}^{2}}{\mu^{2}(1\wedge\delta_{\alpha}^{2}\wedge\nu^{4})}\bigl{(}|\theta_{0}|_{V}^{2}+|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)}$
$\displaystyle\quad\leq\frac{1}{2}\bigl{(}|\mathit{\Delta}\eta_{i-1}^{\varepsilon}|_{H}^{2}-|\mathit{\Delta}\eta_{i}^{\varepsilon}|_{H}^{2}\bigr{)}+\tau\widetilde{C}_{3}\bigl{(}|\theta_{0}|_{V}^{2}+|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)},$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\mbox{ for
}i=1,2,3,\dots,n_{\tau},$

with

$\widetilde{C}_{3}:=\frac{4(C_{F}+\mathcal{L}^{N}(\Omega)+1)(T+1)\bigl{(}|g|_{L^{\infty}(-M,M)}^{2}+|\alpha^{\prime}|_{L^{\infty}(-M,M)}^{2}+1\bigr{)}}{\mu^{2}(1\wedge\delta_{\alpha}^{2}\wedge\nu^{4})}.$

Hence, taking the sum of (5.9) with respect to $i=1,2,3,\dots,n_{\tau}$, one
can deduce from (2.1), (4.4), and (5.9) that:

$\displaystyle\frac{1}{\tau}\sum_{i=1}^{n_{\tau}}|\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}|_{H^{2}(\Omega)}^{2}\leq\frac{C_{0}}{\tau}\sum_{i=1}^{n_{\tau}}\bigl{(}|A_{N}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})|_{H}^{2}+|\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}|_{H}^{2}\bigr{)}$
$\displaystyle\quad=\frac{C_{0}}{\tau}\sum_{i=1}^{n_{\tau}}\bigl{(}|\mathit{\Delta}(\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon})|_{H}^{2}+|\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}|_{H}^{2}\bigr{)}$
$\displaystyle\quad\leq\frac{2C_{0}}{\mu^{2}}|\mathit{\Delta}\eta_{0}|_{H}^{2}+\frac{4C_{0}\widetilde{C}_{3}(T+1)}{\mu^{2}}\bigl{(}|\theta_{0}|_{V}^{2}+|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)}$
$\displaystyle\quad\qquad+4C_{0}\left(\mathcal{F}_{\varepsilon}(\eta_{0},\theta_{0})+\frac{1}{2(1\wedge\delta_{\alpha})}\bigl{(}|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)}\right)$
$\displaystyle\quad\leq\frac{2NC_{0}}{\mu^{2}}|\eta_{0}|_{H^{2}(\Omega)}^{2}+\frac{4C_{0}\widetilde{C}_{3}(T+1)}{\mu^{2}}\bigl{(}|\theta_{0}|_{V}^{2}+|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)}$
$\displaystyle\quad\qquad+\frac{4C_{0}(C_{F}+1)}{1\wedge\delta_{\alpha}}\bigl{(}|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)}$
$\displaystyle\quad\leq
C_{2}\bigl{(}|\eta_{0}|_{H^{2}(\Omega)}^{2}+|\theta_{0}|_{V}^{2}+|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)},$

where

$\displaystyle
C_{2}:=\frac{4NC_{0}(T+1)(\widetilde{C}_{3}+C_{F}+1)}{1\wedge\mu^{2}\wedge\delta_{\alpha}}.$

Thus we conclude lemma 5.1. ∎

######  Lemma 5.2.

There exist a small time-step size $\tau_{**}\in(0,\tau_{*})$ and a constant
$C_{4}>0$ such that for any $\tau\in(0,\tau_{**})$, the following estimate
holds:

$\displaystyle|\theta_{i}^{\varepsilon}|_{H^{2}(\Omega)}^{2}\leq
C_{4}\bigl{(}|\eta_{0}|_{H^{2}(\Omega)}^{2}+|\theta_{0}|_{H^{2}(\Omega)}^{2}+|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)}^{2},$
(5.10) $\displaystyle\mbox{for }i=1,2,3,\dots,n_{\tau}.$

We can prove this lemma 5.2 by using the following technical lemma, obtained
in [1].

######  Lemma 5.3.

(cf. [1, Lemma 3.2]) Let us fix $\varepsilon>0$, $w\in W_{0}$, and
$\alpha^{\circ}\in L^{\infty}(\Omega)\cap V$. Then, for any
$L\geq|\alpha^{\circ}|_{L^{\infty}(\Omega)}$, there exists a constant
$C_{5}(L)>0$, depending only on $L$, and being independent of $\varepsilon$
and $w$, such that:

$\bigl{(}\operatorname{div}(\alpha^{\circ}\nabla\gamma_{\varepsilon}(\nabla
w)),\mathit{\Delta}w\bigr{)}_{H}\geq-|\nabla^{2}w|_{[H]^{N\times
N}}^{2}-C_{5}(L)(|\alpha^{\circ}|_{V}^{2}+1)(|w|_{V}^{2}+1).$

###### Proof of lemma 5.2.

First, we note that lemma 5.1 leads to the boundedness of
$\\{\eta_{i}^{\varepsilon}\\}_{i=0}^{n_{\tau}}$ in $H^{2}(\Omega)$, with the
following estimate:

$\displaystyle|\eta_{i}^{\varepsilon}|_{H^{2}(\Omega)}^{2}$ $\displaystyle\leq
2|\eta_{0}|_{H^{2}(\Omega)}^{2}+2\left(\sum_{i=1}^{n_{\tau}}|\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}|_{H^{2}(\Omega)}\right)^{2}$
(5.11) $\displaystyle\leq
2|\eta_{0}|_{H^{2}(\Omega)}^{2}+2(T+1)\sum_{i=1}^{n_{\tau}}\frac{1}{\tau}|\eta_{i}^{\varepsilon}-\eta_{i-1}^{\varepsilon}|_{H^{2}(\Omega)}^{2}$
$\displaystyle\leq
2(T+1)(C_{2}+1)\bigl{(}|\eta_{0}|_{H^{2}(\Omega)}^{2}+|\theta_{0}|_{V}^{2}+|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)},$
$\displaystyle\qquad\qquad\qquad\ \mbox{ for }i=1,2,3,\dots,n_{\tau}.$

Moreover, by (5.11) and continuous embedding from $H^{2}(\Omega)$ to
$L^{\infty}(\Omega)$ under $N\leq 3$, we see that
$\widetilde{\alpha}_{M}(\eta_{i}^{\varepsilon})\in L^{\infty}(\Omega)\cap V$
for any $i=1,2,3,\dots,n_{\tau}$, with the following estimates hold:

$\displaystyle|\widetilde{\alpha}_{M}(\eta_{i}^{\varepsilon})|_{L^{\infty}(\Omega)}^{2}$
$\displaystyle\leq
2\alpha(0)^{2}+2|\alpha^{\prime}|_{L^{\infty}(-M,M)}^{2}|\eta_{i}^{\varepsilon}|_{L^{\infty}(\Omega)}^{2}$
(5.12) $\displaystyle\leq
2\alpha(0)^{2}+2(C_{H^{2}}^{L^{\infty}})^{2}|\alpha^{\prime}|_{L^{\infty}(-M,M)}^{2}|\eta_{i}^{\varepsilon}|_{H^{2}(\Omega)}^{2},$

and

$\displaystyle|\widetilde{\alpha}_{M}(\eta_{i}^{\varepsilon})|_{V}^{2}\leq\mathcal{L}^{N}(\Omega)|\widetilde{\alpha}_{M}(\eta_{i}^{\varepsilon})|_{L^{\infty}(\Omega)}^{2}+|\alpha^{\prime}|_{L^{\infty}(-M,M)}^{2}|\nabla\eta_{i}^{\varepsilon}|_{[H]^{N}}^{2}$
(5.13) $\displaystyle\leq
2\mathcal{L}^{N}(\Omega)\bigl{(}\alpha(0)^{2}+(C_{H^{2}}^{L^{\infty}})^{2}|\alpha^{\prime}|_{L^{\infty}(-M,M)}^{2}|\eta_{i}^{\varepsilon}|_{H^{2}(\Omega)}^{2}\bigr{)}+|\alpha^{\prime}|_{L^{\infty}(-M,M)}^{2}|\eta_{i}^{\varepsilon}|_{H^{2}(\Omega)}^{2}$
$\displaystyle\leq\widetilde{C}_{6}\bigl{(}|\eta_{i}^{\varepsilon}|_{H^{2}(\Omega)}^{2}+1\bigr{)},\
\mbox{ for }i=1,2,3,\dots,n_{\tau},$

where $C_{H^{2}}^{L^{\infty}}$ is a constant of the embedding from
$H^{2}(\Omega)$ to $L^{\infty}(\Omega)$, and

$\widetilde{C}_{6}:=2\bigl{(}\mathcal{L}^{N}(\Omega)\alpha(0)^{2}+\mathcal{L}^{N}(\Omega)(C_{H^{2}}^{L^{\infty}})^{2}+1\bigr{)}\bigl{(}|\alpha^{\prime}|_{L^{\infty}(-M,M)}^{2}+1\bigr{)},$

Next, we verify the inequality (5.10). Let us consider to multiply the both
sides of (4.2) by $-\mathit{\Delta}\theta_{i}^{\varepsilon}$.

By applying Young’s inequality, we have:

$\displaystyle\frac{\nu^{2}}{\tau}\bigl{(}-\mathit{\Delta}(\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon}),-\mathit{\Delta}\theta_{i}^{\varepsilon}\bigr{)}_{H}\geq\frac{\nu^{2}}{2\tau}\bigl{(}|\mathit{\Delta}\theta_{i}^{\varepsilon}|_{H}^{2}-|\mathit{\Delta}\theta_{i-1}^{\varepsilon}|_{H}^{2}\bigr{)},$
(5.14)

and

$\displaystyle(v_{i},-\mathit{\Delta}\theta_{i}^{\varepsilon})_{H}\leq\frac{1}{2}|\mathit{\Delta}\theta_{i}^{\varepsilon}|_{H}^{2}+\frac{1}{2}|v_{i}|_{H}^{2},\
\mbox{ for }i=1,2,3,\dots,n_{\tau}.$ (5.15)

Moreover, from (4.4) and (A3), we see that:

$\displaystyle\frac{1}{\tau}\bigl{(}\alpha_{0}(\mathcal{T}_{M}\eta_{i-1}^{\varepsilon})(\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon}),-\mathit{\Delta}\theta_{i}^{\varepsilon}\bigr{)}_{H}$
(5.16)
$\displaystyle\qquad\geq-\frac{1}{2\tau^{2}}|\alpha_{0}|_{L^{\infty}(-M,M)}^{2}|\theta_{i}^{\varepsilon}-\theta_{i-1}^{\varepsilon}|_{H}^{2}-\frac{1}{2}|\mathit{\Delta}\theta_{i}^{\varepsilon}|_{H}^{2},$
$\displaystyle\qquad\geq\frac{|\alpha_{0}|_{L^{\infty}(-M,M)}^{2}}{2\delta_{\alpha}}\cdot\frac{1}{\tau}(\mathcal{F}_{\varepsilon}(\eta_{i}^{\varepsilon},\theta_{i}^{\varepsilon})-\mathcal{F}_{\varepsilon}(\eta_{i-1}^{\varepsilon},\theta_{i-1}^{\varepsilon}))-\frac{1}{2}|\mathit{\Delta}\theta_{i}^{\varepsilon}|_{H}^{2},$
$\displaystyle\qquad\qquad\qquad\qquad\mbox{ for }i=1,2,3,\dots,n_{\tau}.$

Using (2.1), (5.11)–(5.13), and applying lemma 5.3 to the case that:

$\displaystyle\alpha^{\circ}=\widetilde{\alpha}_{M}(\eta_{i-1}^{\varepsilon})\mbox{
and }w=\theta_{i}^{\varepsilon},\mbox{ for each
}i\in\\{1,2,\ldots,n_{\tau}\\},$

and

$\displaystyle L$
$\displaystyle=2\alpha(0)^{2}+2(C_{H^{2}}^{L^{\infty}})^{2}|\alpha^{\prime}|_{L^{\infty}(-M,M)}^{2}\cdot$
$\displaystyle\cdot
2(T+1)(C_{2}+1)\bigl{(}|\eta_{0}|_{H^{2}(\Omega)}^{2}+|\theta_{0}|_{V}^{2}+|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)},$

it is observed that:

$\displaystyle\quad\bigl{(}\operatorname{div}(\widetilde{\alpha}_{M}(\eta_{i-1}^{\varepsilon})\nabla\gamma_{\varepsilon}(\nabla\theta_{i}^{\varepsilon})),\mathit{\Delta}\theta_{i}^{\varepsilon}\bigr{)}_{H}$
(5.17) $\displaystyle\geq-|\nabla^{2}\theta_{i}^{\varepsilon}|_{[H]^{N\times
N}}^{2}-C_{5}(L)(|\widetilde{\alpha}_{M}(\eta_{i-1}^{\varepsilon})|_{V}^{2}+1)(|\theta_{i}^{\varepsilon}|_{V}^{2}+1)$
$\displaystyle\geq-
C_{0}\bigl{(}|A_{N}\theta_{i}^{\varepsilon}|_{H}^{2}+|\theta_{i}^{\varepsilon}|_{H}^{2}\bigr{)}$
$\displaystyle\qquad-
C_{5}(L)\bigl{(}\widetilde{C}_{6}\bigl{(}|\eta_{i-1}^{\varepsilon}|_{H^{2}(\Omega)}^{2}+1\bigr{)}+1\bigr{)}\bigl{(}|\theta_{i}^{\varepsilon}|_{V}^{2}+1\bigr{)}$
$\displaystyle\geq-
C_{0}|\mathit{\Delta}\theta_{i}^{\varepsilon}|_{H}^{2}-(C_{5}(L)+C_{0})(\widetilde{C}_{6}+1)\bigl{(}|\theta_{i}^{\varepsilon}|_{V}^{2}+1\bigr{)}\bigl{(}|\eta_{i-1}^{\varepsilon}|_{H^{2}(\Omega)}^{2}+1\bigr{)},$
$\displaystyle\qquad\qquad\qquad\qquad\mbox{ for }i=1,2,3,\dots,n_{\tau}.$

Now, by using (5.14)–(5.17), we will obtain that:

$\frac{1}{\tau}(X_{i}-X_{i-1})\leq\widetilde{C}_{7}(X_{i}+F_{i}),\ \mbox{ for
}i=1,2,3,\dots,n_{\tau},$ (5.18)

with

$\displaystyle\begin{cases}\displaystyle
X_{i}:=\nu^{2}|\mathit{\Delta}\theta_{i}^{\varepsilon}|_{H}^{2}+\frac{|\alpha_{0}|_{L^{\infty}(-M,M)}^{2}}{\delta_{\alpha}}\mathcal{F}_{\varepsilon}(\eta_{i}^{\varepsilon},\theta_{i}^{\varepsilon}),\\\\[8.61108pt]
\displaystyle
F_{i}:=\bigl{(}|\theta_{i}^{\varepsilon}|_{V}^{2}+|v_{i}|_{H}^{2}+1\bigr{)}\bigl{(}|\eta_{i-1}^{\varepsilon}|_{H^{2}(\Omega)}^{2}+1\bigr{)},\end{cases}\mbox{
for }i=1,2,3,\dots,n_{\tau},$

and

$\widetilde{C}_{7}:=\frac{4(C_{5}(L)+C_{0}+1)(\widetilde{C}_{6}+1)}{1\wedge\nu^{2}}\geq
4.$

Here, let us take $\tau_{**}\in(0,\tau_{*})$ satisfying:

$\tau_{**}<\min\left\\{\tau_{*},\frac{1}{2\widetilde{C}_{7}}\right\\},\mbox{
and in particular, }1-\tau_{**}\widetilde{C}_{7}>\frac{1}{2}.$

Then, applying the discrete version of Gronwall’s lemma (cf. [10, Section
3.1]) to (5.18), one can see from (5.5), (5.11), and (5.18) that:

$\displaystyle\quad
X_{i}\leq\widetilde{C}_{7}e^{2\widetilde{C}_{7}(T+1)}\left(X_{0}+\tau\sum_{i=1}^{n_{\tau}}F_{i}\right)$
(5.19)
$\displaystyle\leq\widetilde{C}_{7}e^{2\widetilde{C}_{7}(T+1)}\left(\nu^{2}|\mathit{\Delta}\theta_{0}|_{H}^{2}+\frac{|\alpha_{0}|_{L^{\infty}(-M,M)}^{2}C_{F}}{\delta_{\alpha}}\right)$
$\displaystyle\,\quad+\frac{4\widetilde{C}_{7}e^{2\widetilde{C}_{7}(T+1)}(C_{F}+3)(T+1)^{2}}{1\wedge\delta_{\alpha}^{2}\wedge\nu^{4}}\bigl{(}|\theta_{0}|_{V}^{2}+|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)}\cdot$
$\displaystyle\qquad\quad\cdot
2(T+1)(C_{2}+2)\bigl{(}|\eta_{0}|_{H^{2}(\Omega)}^{2}+|\theta_{0}|_{V}^{2}+|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)}$
$\displaystyle\leq\widetilde{C}_{8}\bigl{(}|\eta_{0}|_{H^{2}(\Omega)}^{2}+|\theta_{0}|_{H^{2}(\Omega)}^{2}+|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)}^{2},\
\mbox{ for }i=1,2,3,\dots,n_{\tau},$

where

$\widetilde{C}_{8}:=\frac{8\widetilde{C}_{7}e^{2\widetilde{C}_{7}(T+1)}(C_{F}+3)(T+1)^{3}(C_{2}+N\nu^{2}+|\alpha_{0}|_{L^{\infty}(-M,M)}+2)}{1\wedge\delta_{\alpha}^{2}\wedge\nu^{4}}.$

In the light of (2.1) (5.5), and (5.19), we arrive at:

$\displaystyle\quad|\theta_{i}^{\varepsilon}|_{H^{2}(\Omega)}^{2}\leq
C_{0}\bigl{(}|A_{N}\theta_{i}^{\varepsilon}|_{H}^{2}+|\theta_{i}^{\varepsilon}|_{H}^{2}\bigr{)}=C_{0}\bigl{(}|\mathit{\Delta}\theta_{i}^{\varepsilon}|_{H}^{2}+|\theta_{i}^{\varepsilon}|_{H}^{2}\bigr{)}$
$\displaystyle\leq\frac{C_{0}}{\nu^{2}}X_{i}+\frac{4C_{0}(C_{F}+1)(T+1)}{1\wedge\delta_{\alpha}^{2}\wedge\nu^{4}}\bigl{(}|\theta_{0}|_{V}^{2}+|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)}$
$\displaystyle\leq\frac{4C_{0}(C_{F}+\widetilde{C}_{8}+1)(T+1)}{1\wedge\delta_{\alpha}^{2}\wedge\nu^{4}}\bigl{(}|\eta_{0}|_{H^{2}(\Omega)}^{2}+|\theta_{0}|_{H^{2}(\Omega)}^{2}+|u|_{\mathscr{H}}^{2}+|v|_{\mathscr{H}}^{2}+1\bigr{)}^{2},$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\mbox{ for
}i=1,2,3,\dots,n_{\tau}.$

Thus we conclude lemma 5.2 with the constant:

$C_{4}=\frac{4C_{0}(C_{F}+\widetilde{C}_{8}+1)(T+1)}{1\wedge\delta_{\alpha}^{2}\wedge\nu^{4}}.$

∎

Next, we confirm the comparison principle for single pseudo-parabolic
equation, which will play a key-role in the $L^{\infty}$-estimate of the
component $\eta$.

######  Lemma 5.4.

We assume that $\eta^{1},\eta^{2}\in W^{1,2}(0,T;W_{0})$,
$\eta_{0}^{1},\eta_{0}^{2}\in W_{0}$, $\widetilde{\theta}\in\mathscr{V}$,
$\widetilde{u}\in\mathscr{H}$, and

$\left\\{\begin{aligned}
&(-1)^{i-1}\left(\partial_{t}\eta^{i}-\mathit{\Delta}(\eta^{i}+\mu^{2}\partial_{t}\eta^{i})+g(\mathcal{T}_{M}\eta^{i})+\alpha^{\prime}(\mathcal{T}_{M}\eta^{i})|\nabla\widetilde{\theta}|\right)\\\
&\qquad\leq\widetilde{u},\mbox{ a.e. in }Q,\\\
&\eta^{i}(0)=\eta_{0}^{i},\mbox{ a.e. in }\Omega.\end{aligned}\right.$ (5.20)

Then, there exists a constant $C_{9}>0$ such that:

$\bigl{|}[\eta^{1}-\eta^{2}]^{+}(t)\bigr{|}_{V}^{2}\leq
C_{9}\bigl{|}[\eta_{0}^{1}-\eta_{0}^{2}]^{+}\bigr{|}_{V}^{2},\mbox{ for any
}t\in[0,T].$

###### Proof.

Taking the difference of two inequality (5.20) for $\eta^{i}$, $i=1,2$, and
multiplying the both sides by $[\eta^{1}-\eta^{2}]^{+}(t)$, we see that:

$\displaystyle\frac{1}{2}\frac{d}{dt}\bigl{|}[\eta^{1}-\eta^{2}]^{+}(t)\bigr{|}_{H}^{2}+\frac{\mu^{2}}{2}\frac{d}{dt}\bigl{|}\nabla[\eta^{1}-\eta^{2}]^{+}(t)\bigr{|}_{[H]^{N}}^{2}$
(5.21)
$\displaystyle\qquad+\bigl{|}\nabla[\eta^{1}-\eta^{2}]^{+}(t)\bigr{|}_{[H]^{N}}^{2}+(g(\mathcal{T}_{M}\eta^{1}(t))-g(\mathcal{T}_{M}\eta^{2}(t)),[\eta^{1}-\eta^{2}]^{+}(t))_{H}$
$\displaystyle\qquad+\int_{\Omega}\bigl{(}(\alpha^{\prime}(\mathcal{T}_{M}\eta^{1}(t))-\alpha^{\prime}(\mathcal{T}_{M}\eta^{2}(t))\bigr{)}|\nabla\theta(t)|,[\eta^{1}-\eta^{2}]^{+}(t)\,dx$
$\displaystyle\leq 0,\ \mbox{ for a.e. }t\in(0,T).$

Here, from the assumption (A1), it is deduced that:

$\displaystyle(g(\mathcal{T}_{M}\eta^{1}(t))-g(\mathcal{T}_{M}\eta^{2}(t)),[\eta^{1}-\eta^{2}]^{+}(t))_{H}$
$\displaystyle\qquad\geq-|g^{\prime}|_{L^{\infty}(-M,M)}\bigl{|}[\eta^{1}-\eta^{2}]^{+}(t)\bigr{|}_{H}^{2},\
\mbox{ for a.e. }t\in(0,T).$ (5.22)

Also, by the monotonicity of $\alpha^{\prime}\circ\mathcal{T}_{M}$, we can say
that:

$\bigl{(}\alpha^{\prime}(\mathcal{T}_{M}\eta^{1})-\alpha^{\prime}(\mathcal{T}_{M}\eta^{2})\bigr{)}[\eta^{1}-\eta^{2}]^{+}\geq
0,\ \mbox{ a.e. in }Q.$

Hence one can see that:

$\displaystyle\int_{\Omega}\bigl{(}(\alpha^{\prime}(\mathcal{T}_{M}\eta^{1}(t))-\alpha^{\prime}(\mathcal{T}_{M}\eta^{2}(t))\bigr{)}|\nabla\theta(t)|,[\eta^{1}-\eta^{2}]^{+}(t)\,dx$
$\displaystyle\geq 0,\ \mbox{ for a.e. }t\in(0,T).$ (5.23)

Now, in the light of (5.21)–(5.23), it is deduced that:

$\displaystyle\frac{d}{dt}\left(\bigl{|}[\eta^{1}-\eta^{2}]^{+}(t)\bigr{|}_{H}^{2}+\mu^{2}\bigl{|}\nabla[\eta^{1}-\eta^{2}]^{+}(t)\bigr{|}_{[H]^{N}}^{2}\right)$
$\displaystyle\qquad\leq
2|g^{\prime}|_{L^{\infty}(-M,M)}\bigl{|}[\eta^{1}-\eta^{2}]^{+}(t)\bigr{|}_{H}^{2},\
\mbox{ for a.e. }t\in(0,T).$

Applying Gronwall’s inequality, we arrive at:

$\displaystyle\bigl{|}[\eta^{1}-\eta^{2}]^{+}(t)\bigr{|}_{H}^{2}+\mu^{2}\bigl{|}\nabla[\eta^{1}-\eta^{2}]^{+}(t)\bigr{|}_{[H]^{N}}^{2}$
(5.24) $\displaystyle\ \leq
e^{2T|g^{\prime}|_{L^{\infty}(-M,M)}}\left(\bigl{|}[\eta_{0}^{1}-\eta_{0}^{2}]^{+}\bigr{|}_{H}^{2}+\mu^{2}\bigl{|}\nabla[\eta_{0}^{1}-\eta_{0}^{2}]^{+}\bigr{|}_{[H]^{N}}^{2}\right),$
$\displaystyle\qquad\qquad\qquad\qquad\mbox{ for any }t\in[0,T].$

(5.24) finishes the proof of lemma 5.4 with the constant:

$C_{9}:=\frac{1+\mu^{2}}{1\wedge\mu^{2}}e^{2T|g^{\prime}|_{L^{\infty}(-M,M)}}.$

∎

###### The proof of 1.

Let us take $\varepsilon\in(0,1)$, $\tau\in(0,\tau_{**})$ where $\tau_{**}$ is
given in lemma 5.2. As a consequence of lemmas 5.1, 5.2 and 1, the following
boundedness are derived:

$\bullet$

$\\{[\eta^{\varepsilon}]_{\tau}\,|\,\varepsilon\in(0,1),\tau\in(0,\tau_{**})\\}$
is bounded in $W^{1,2}(0,T;W_{0})$,

$\bullet$

$\\{[\overline{\eta}^{\varepsilon}]_{\tau}\,|\,\varepsilon\in(0,1),\tau\in(0,\tau_{**})\\}$,
$\\{[\underline{\eta}^{\varepsilon}]_{\tau}\,|\,\varepsilon\in(0,1),\tau\in(0,\tau_{**})\\}$
is bounded in $L^{\infty}(0,T;W_{0})$,

$\bullet$

$\\{[\theta^{\varepsilon}]_{\tau}\,|\,\varepsilon\in(0,1),\tau\in(0,\tau_{**})\\}$
is bounded in $L^{\infty}(0,T;W_{0})$ and in $W^{1,2}(0,T;V)$,

$\bullet$

$\\{[\overline{\theta}^{\varepsilon}]_{\tau}\,|\,\varepsilon\in(0,1),\tau\in(0,\tau_{**})\\}$,
$\\{[\underline{\theta}^{\varepsilon}]_{\tau}\,|\,\varepsilon\in(0,1),\tau\in(0,\tau_{**})\\}$
is bounded in $L^{\infty}(0,T;W_{0})$.

Therefore, by applying Aubin’s type compactness theory (cf. [23, Corollary
4]), we can find sequences $\\{\varepsilon_{n}\\}_{n=1}^{\infty}\subset(0,1)$,
$\\{\tau_{n}\\}_{n=1}^{\infty}\subset(0,\tau_{**})$ and a pair of functions
$[\eta,\theta]\in[\mathscr{H}]^{2}$ such that $\varepsilon_{n}\searrow 0$ and
$\tau_{n}\searrow 0$, as $n\to\infty$, we obtain the following convergences as
$n\to\infty$:

$\left\\{\begin{aligned}
&\eta_{n}:=[\eta^{\varepsilon_{n}}]_{\tau_{n}}\to\eta\mbox{ in
}C([0,T];V),\mbox{ and weakly in }W^{1,2}(0,T;W_{0}),\\\
&\theta_{n}:=[\theta^{\varepsilon_{n}}]_{\tau_{n}}\to\theta\mbox{ in
}C([0,T];V),\mbox{ weakly in }W^{1,2}(0,T;V),\\\ &\qquad\mbox{and weakly-$*$
in }L^{\infty}(0,T;W_{0}).\end{aligned}\right.$ (5.25)

Besides, having in mind:

$\left\\{\begin{aligned}
&\max\left\\{|[\overline{\eta}^{\varepsilon_{n}}]_{\tau_{n}}-\eta_{n}|_{V},\,|[\underline{\eta}^{\varepsilon_{n}}]_{\tau_{n}}-\eta_{n}|_{V}\right\\}\leq\int_{\Delta_{i,\tau}}|\partial_{t}\eta_{n}|_{V}\,dt<\infty,\\\
&\max\left\\{|[\overline{\theta}^{\varepsilon_{n}}]_{\tau_{n}}-\theta_{n}|_{V},\,|[\underline{\theta}^{\varepsilon_{n}}]_{\tau_{n}}-\theta_{n}|_{V}\right\\}\leq\int_{\Delta_{i,\tau}}|\partial_{t}\theta_{n}|_{V}\,dt<\infty,\end{aligned}\right.$

we can derive that:

$\left\\{\begin{aligned}
&\overline{\eta}_{n}:=[\overline{\eta}^{\varepsilon_{n}}]_{\tau_{n}}\to\eta\mbox{
and
}\underline{\eta}_{n}:=[\underline{\eta}^{\varepsilon_{n}}]_{\tau_{n}}\to\eta\mbox{
in }L^{\infty}(0,T;V),\mbox{ and }\\\ &\mbox{weakly-$*$ in
}L^{\infty}(0,T;W_{0}),\\\
&\overline{\theta}_{n}:=[\overline{\theta}^{\varepsilon_{n}}]_{\tau_{n}}\to\theta\mbox{
and
}\underline{\theta}_{n}:=[\underline{\theta}^{\varepsilon_{n}}]_{\tau_{n}}\to\theta\mbox{
in }L^{\infty}(0,T;V),\mbox{ and }\\\ &\mbox{weakly-$*$ in
}L^{\infty}(0,T;W_{0}).\end{aligned}\right.$ (5.26)

Now, we verify that the limiting pair $[\eta,\theta]$ satisfies (S0)–(S3). Let
us take an arbitrary open interval $I\subset(0,T)$. Then, in the light of
(4.1), (4.2), and the convexity of $\gamma_{\varepsilon}$, the sequences as in
(5.25) and (5.26) should fulfill the following two variational forms:

$\displaystyle\int_{I}(\partial_{t}\eta_{n}(t),\varphi)\,dt+\int_{I}(\nabla(\overline{\eta}_{n}+\mu^{2}\partial_{t}\eta_{n})(t),\nabla\varphi)_{[H]^{N}}\,dt$
(5.27)
$\displaystyle\quad+\int_{I}(g(\mathcal{T}_{M}\,\overline{\eta}_{n}(t)),+\alpha^{\prime}(\mathcal{T}_{M}\,\overline{\eta}_{n}(t))\gamma_{\varepsilon}(\nabla\overline{\theta}_{n}(t)),\varphi)_{H}\,dt=\int_{I}([\overline{u}]_{\tau_{n}}(t),\varphi)_{H}\,dt,$
for all $\varphi\in V$, and $n=1,2,3,\dots,n_{\tau}$,

and

$\displaystyle\int_{I}\bigl{(}(\alpha_{0}(\mathcal{T}_{M}\underline{\eta}_{n})\partial_{t}\theta_{n})(t),\overline{\theta}_{n}(t)-\psi\bigr{)}_{H}\,dt+\int_{I}\int_{\Omega}\alpha(\underline{\eta}_{n}(t))\gamma_{\varepsilon}(\nabla\overline{\theta}_{n}(t))\,dxdt$
$\displaystyle\quad+\nu^{2}\int_{I}(\nabla\partial_{t}\theta_{n}(t),\nabla(\overline{\theta}_{n}(t)-\psi))_{[H]^{N}}\,dt$
(5.28)
$\displaystyle\quad\leq\int_{I}\int_{\Omega}\alpha(\underline{\eta}_{n}(t))\gamma_{\varepsilon}(\nabla\psi)\,dxdt+\int_{I}([\overline{v}]_{\tau_{n}}(t),\overline{\theta}_{n}(t)-\psi)_{H}\,dt,$
for all $\psi\in V$, and $n=1,2,3,\dots,n_{\tau}$.

On this basis, having in mind (5.25), (A1), (A2), and the fact:

$\gamma_{\varepsilon}\to|\cdot|\mbox{ uniformly on }\mathbb{R}^{N},\mbox{ as
}\varepsilon\to 0,$

letting $n\to\infty$ in (5.27) and (5.28) yields that:

$\displaystyle\int_{I}(\partial_{t}\eta(t),\varphi)\,dt+\int_{I}(\nabla(\eta+\mu^{2}\partial_{t}\eta)(t),\nabla\varphi)_{[H]^{N}}\,dt$
(5.29)
$\displaystyle\quad+\int_{I}(g(\mathcal{T}_{M}\,\eta(t))+\alpha^{\prime}(\mathcal{T}_{M}\,\eta(t))|\nabla\theta(t)|,\varphi)_{H}\,dt=\int_{I}(u(t),\varphi)_{H}\,dt,$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\mbox{ for any }\varphi\in
V,$

and

$\displaystyle\int_{I}\bigl{(}(\alpha_{0}(\mathcal{T}_{M}\eta)\partial_{t}\theta)(t),\theta(t)-\psi\bigr{)}_{H}\,dt+\int_{I}\int_{\Omega}\alpha(\eta(t))|\nabla\theta(t)|\,dxdt$
$\displaystyle\quad+\nu^{2}\int_{I}(\nabla\partial_{t}\theta(t),\nabla(\theta(t)-\psi))_{[H]^{N}}\,dt$
$\displaystyle\quad\leq\int_{I}\int_{\Omega}\alpha(\eta(t))|\nabla\psi|\,dxdt+\int_{I}(v(t),\theta(t)-\psi)_{H}\,dt,\
\mbox{ for any }\psi\in V,$

respectively. Since $I\subset(0,T)$ is arbitrary, $[\eta,\theta]$ should
satisfy (S1) and (S2).

Next, let us verify $\eta\in L^{\infty}(Q)$. By (5.2), the following
inequalities can be obtained:

$\displaystyle\left\\{\begin{aligned}
&\partial_{t}M-\mathit{\Delta}(M+\mu^{2}\partial_{t}M)+g(M)+\alpha^{\prime}(M)|\nabla\theta(t)|\geq
u,\\\
&\partial_{t}(-M)-\mathit{\Delta}((-M)+\mu^{2}\partial_{t}(-M))+g(-M)+\alpha^{\prime}(-M)|\nabla\theta(t)|\leq
u,\end{aligned}\right.$ $\displaystyle\mbox{a.e. in }Q.$

Hence, applying lemma 5.4 to the case when

$[\eta^{1},\eta^{2},\widetilde{\theta},\widetilde{u}]=[\eta,M,\theta,u],~{}~{}[\eta_{0}^{1},\eta_{0}^{2}]=[\eta_{0},M],$

and

$[\eta^{1},\eta^{2},\widetilde{\theta},\widetilde{u}]=[-M,\eta,\theta,u],~{}~{}[\eta_{0}^{1},\eta_{0}^{2}]=[-M,\eta_{0}],$

we arrive at:

$\displaystyle\left\\{\begin{aligned}
&\bigl{|}[\eta-M]^{+}(t)\bigr{|}_{V}^{2}\leq
C_{9}\bigl{|}[\eta_{0}-M]^{+}\bigr{|}_{V}^{2}=0,\\\
&\bigl{|}[-M-\eta]^{+}(t)\bigr{|}_{V}^{2}\leq
C_{9}\bigl{|}[-M-\eta_{0}]^{+}\bigr{|}_{V}^{2}=0,\end{aligned}\right.\mbox{
for any }t\in[0,T],$ (5.30)

respectively. This implies that:

$|\eta(t)|_{L^{\infty}(\Omega)}\leq M,\ \mbox{ for any }t\in[0,T].$

Finally, due to (5.25), the condition (S3) as in 1 is immediately confirmed as
follows:

$\eta(0)=\lim_{n\to\infty}\eta_{n}(0)=\eta_{0}\mbox{ and
}\theta(0)=\lim_{n\to\infty}\theta_{n}(0)=\theta_{0}\mbox{ in }H.$

Thus, we conclude that $[\eta,\theta]$ is a solution to (S).

∎

###  5.2 Proof of 2

Let $[\eta^{k},\theta^{k}],\,k=1,2$, be the solutions to (S) corresponding to
initial values $\eta_{0}^{k},\,\theta_{0}^{k}$ and forcings $u^{k},\,v^{k}$,
$k=1,2$. Let us set

$M_{0}:=|\eta^{1}|_{L^{\infty}(Q)}\vee|\eta^{2}|_{L^{\infty}(Q)},\mbox{ and
}\delta_{*}(\nu):=1\wedge\delta_{\alpha}\wedge\nu^{2},$ (5.31)

and take the difference between the variational formulas for
$\eta^{k},\,k=1,2$, and put $\varphi:=(\eta^{1}-\eta^{2})(t)$. Then, by using
(A1), the monotonicity of $\alpha^{\prime}$, and Young’s inequality, we see
that:

$\displaystyle\frac{1}{2}\frac{d}{dt}\bigl{(}|(\eta^{1}-\eta^{2})(t)|_{H}^{2}+\mu^{2}|\nabla(\eta^{1}-\eta^{2})(t)|_{[H]^{N}}^{2}\bigr{)}$
(5.32)
$\displaystyle\quad\qquad-|g|_{L^{\infty}(-M_{0},M_{0})}|(\eta^{1}-\eta^{2})(t)|_{H}^{2}$
$\displaystyle\quad\leq\int_{\Omega}\alpha^{\prime}(\eta^{1}(t))(\eta^{2}-\eta^{1})(t)|\nabla\theta^{1}(t)|\,dx$
$\displaystyle\quad\qquad+\int_{\Omega}\alpha^{\prime}(\eta^{2}(t))(\eta^{1}-\eta^{2})(t)|\nabla\theta^{2}(t)|\,dx$
$\displaystyle\quad\qquad+\bigl{(}(\eta^{1}-\eta^{2})(t),(u^{1}-u^{2})(t)\bigr{)}_{H}$
$\displaystyle\quad\leq\int_{\Omega}|\alpha(\eta^{1}(t))-\alpha(\eta^{2}(t))||\nabla(\theta^{1}-\theta^{2})(t)|\,dx$
$\displaystyle\quad\qquad+|(\eta^{1}-\eta^{2})(t)|_{H}|(u^{1}-u^{2})(t)|_{H}$
$\displaystyle\quad\leq\frac{|\alpha^{\prime}|_{L^{\infty}(-M_{0},M_{0})}}{2}\left(|(\eta^{1}-\eta^{2})(t)|_{H}^{2}+|\nabla(\theta^{1}-\theta^{2})(t)|_{[H]^{N}}^{2}\right)$
$\displaystyle\quad\qquad+\frac{1}{2}|(\eta^{1}-\eta^{2})(t)|_{H}^{2}+\frac{1}{2}|(u^{1}-u^{2})(t)|_{H}^{2},\
\mbox{ for a.e. }t>0.$

On the other hand, by putting $\psi=\theta^{2}$ in the variational inequality
for $\theta^{1}$, and $\psi=\theta^{1}$ in the one for $\theta^{2}$, and by
taking the sum of two inequalities, we have:

$\displaystyle\bigl{(}\alpha_{0}(\eta^{1})\partial_{t}\theta^{1}(t)-\alpha_{0}(\eta^{2})\partial_{t}\theta^{2}(t),(\theta^{1}-\theta^{2})(t)\bigr{)}_{H}$
(5.33)
$\displaystyle\quad\qquad+\frac{1}{2}\frac{d}{dt}\bigl{(}\nu^{2}|\nabla(\theta^{1}-\theta^{2})(t)|_{[H]^{N}}^{2}\bigr{)}$
$\displaystyle\quad\leq\int_{\Omega}\alpha(\eta^{1}(t))\bigl{(}|\nabla\theta^{2}(t)|-|\nabla\theta^{1}(t)|\bigr{)}\,dx$
$\displaystyle\quad\qquad+\int_{\Omega}\alpha(\eta^{2}(t))\bigl{(}|\nabla\theta^{1}(t)|-|\nabla\theta^{2}(t)|\bigr{)}\,dx$
$\displaystyle\quad\qquad+\bigl{(}(\theta^{1}-\theta^{2})(t),(v^{1}-v^{2})(t)\bigr{)}_{H}$
$\displaystyle\quad\leq\int_{\Omega}|\alpha(\eta^{1}(t))-\alpha(\eta^{2}(t))|\nabla(\theta^{1}-\theta^{2})(t)|\,dx$
$\displaystyle\quad\qquad+|(\theta^{1}-\theta^{2})(t)|_{H}|(v^{1}-v^{2})(t)|_{H}$
$\displaystyle\quad\leq\frac{|\alpha^{\prime}|_{L^{\infty}(-M_{0},M_{0})}}{2}\left(|(\eta^{1}-\eta^{2})(t)|_{H}^{2}+|\nabla(\theta^{1}-\theta^{2})(t)|_{[H]^{N}}^{2}\right)$
$\displaystyle\quad\qquad+\frac{1}{2}|(\theta^{1}-\theta^{2})(t)|_{H}^{2}+\frac{1}{2}|(v^{1}-v^{2})(t)|_{H}^{2},\
\mbox{ a.e. }t>0.$

Here, we can compute the first term in (5.33) as follows:

$\displaystyle\quad\bigl{(}\alpha_{0}(\eta^{1}(t))\partial_{t}\theta^{1}(t)-\alpha_{0}(\eta^{2}(t))\partial_{t}\theta^{2}(t),(\theta^{1}-\theta^{2})(t)\bigr{)}_{H}$
(5.34)
$\displaystyle=\frac{1}{2}\frac{d}{dt}|\sqrt{\alpha_{0}(\eta^{1}(t))}(\theta^{1}-\theta^{2})(t)|_{H}^{2}-\frac{1}{2}\int_{\Omega}\alpha_{0}^{\prime}(\eta^{1}(t))\partial_{t}\eta^{1}(t)|(\theta^{1}-\theta^{2})(t)|^{2}\,dx$
$\displaystyle\quad+\int_{\Omega}(\alpha_{0}(\eta^{1}(t))-\alpha_{0}(\eta^{2}(t)))\partial_{t}\theta^{2}(t)(\theta^{1}-\theta^{2})(t)\,dx.$

Also, by using (5.31), the continuous embedding from $H^{1}(\Omega)$ to
$L^{4}(\Omega)$ under $N\leq 3$, and Young’s inequality, one can see that:

$\displaystyle-\int_{\Omega}\alpha_{0}^{\prime}(\eta^{1}(t))\partial_{t}\eta^{1}(t)|(\theta^{1}-\theta^{2})(t)|^{2}\,dx$
(5.35) $\displaystyle\geq$
$\displaystyle-|\alpha_{0}^{\prime}|_{L^{\infty}(-M_{0},M_{0})}|\partial_{t}\eta^{1}(t)|_{H}|(\theta_{1}-\theta^{2})(t)|_{L^{4}(\Omega)}^{2}$
$\displaystyle\geq$
$\displaystyle-(C_{H^{1}}^{L^{4}})^{2}|\alpha_{0}^{\prime}|_{L^{\infty}(-M_{0},M_{0})}|\partial_{t}\eta^{1}(t)|_{H}|(\theta_{1}-\theta^{2})(t)|_{V}^{2}$
$\displaystyle\geq$
$\displaystyle-\frac{(C_{H^{1}}^{L^{4}})^{2}|\alpha_{0}^{\prime}|_{L^{\infty}(-M_{0},M_{0})}}{\delta_{*}(\nu)}|\partial_{t}\eta^{1}(t)|_{H}\cdot$
$\displaystyle\qquad\cdot\bigl{(}|\sqrt{\alpha_{0}(\eta^{1}(t))}(\theta^{1}-\theta^{2})(t)|_{H}^{2}+\nu^{2}|\nabla(\theta^{1}-\theta^{2})(t)|_{[H]^{N}}^{2}\bigr{)},$

and

$\displaystyle\int_{\Omega}(\alpha_{0}(\eta^{1}(t))-\alpha_{0}(\eta^{2}(t)))\partial_{t}\theta^{2}(t)(\theta^{1}-\theta^{2})(t)\,dx$
(5.36) $\displaystyle\geq$
$\displaystyle-|\alpha_{0}^{\prime}|_{L^{\infty}(-M_{0},M_{0})}|\partial_{t}\theta^{2}(t)|_{L^{4}(\Omega)}|(\eta^{1}-\eta^{2})(t)|_{H}|(\theta^{1}-\theta^{2})(t)|_{L^{4}(\Omega)}$
$\displaystyle\geq$
$\displaystyle-(C_{H^{1}}^{L^{4}})^{2}|\alpha_{0}^{\prime}|_{L^{\infty}(-M_{0},M_{0})}|\partial_{t}\theta^{2}(t)|_{V}|(\eta^{1}-\eta^{2})(t)|_{H}|(\theta^{1}-\theta^{2})(t)|_{V}$
$\displaystyle\geq$
$\displaystyle-\frac{(C_{H^{1}}^{L^{4}})^{2}|\alpha_{0}^{\prime}|_{L^{\infty}(-M_{0},M_{0})}}{2}|\partial_{t}\theta^{2}(t)|_{V}\bigl{(}|(\eta^{1}-\eta^{2})(t)|_{H}^{2}+|(\theta^{1}-\theta^{2})(t)|_{V}^{2}\bigr{)}$
$\displaystyle\geq$
$\displaystyle-\frac{(C_{H^{1}}^{L^{4}})^{2}|\alpha_{0}^{\prime}|_{L^{\infty}(-M_{0},M_{0})}}{2\delta_{*}(\nu)}|\partial_{t}\theta^{2}(t)|_{V}\cdot$
$\displaystyle\quad\cdot\bigl{(}|(\eta^{1}-\eta^{2})(t)|_{H}^{2}+|\sqrt{\alpha_{0}(\eta^{1}(t))}(\theta^{1}-\theta^{2})(t)|_{H}^{2}+\nu^{2}|\nabla(\theta^{1}-\theta^{2})(t)|_{[H]^{N}}^{2}\bigr{)},$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\mbox{ for a.e. }t>0,$

where $C_{H^{1}}^{L^{4}}$ is a constant of the continuous embedding from
$H^{1}(\Omega)$ to $L^{4}(\Omega)$.

Therefore, putting

$\displaystyle J_{1}(t)$
$\displaystyle:=|(u^{1}-u^{2})(t)|_{H}^{2}+|(v^{1}-v^{2})(t)|_{H}^{2},\ \mbox{
for }t\geq 0,$

and

$\widetilde{C}_{10}:=2\bigl{(}|\alpha^{\prime}|_{L^{\infty}(-M_{0},M_{0})}+|g^{\prime}|_{L^{\infty}(-M_{0},M_{0})}+(C_{H^{1}}^{L^{4}})^{2}|\alpha_{0}^{\prime}|_{L^{\infty}(-M_{0},M_{0})}+1\bigr{)},$

it is deduced from (5.32)–(5.36) that:

$\frac{d}{dt}J(t)\leq\frac{\widetilde{C}_{10}}{\delta_{*}(\nu)}\bigl{(}|\partial_{t}\eta^{1}(t)|_{H}+|\partial_{t}\theta^{2}(t)|_{V}+1\bigr{)}J(t)+J_{1}(t),\
\mbox{ a.e. }t>0,$ (5.37)

Applying Gronwall’s lemma in (5.37), it can be obtained that for any $T>0$,:

$\displaystyle J(t)$ $\displaystyle\leq
C_{1}(\nu)\left(J(0)+\int_{0}^{T}J_{1}(s)\,ds\right),\ \mbox{ for any
}t\in[0,T].$

with

$C_{1}(\nu):=\exp\left(\frac{\widetilde{C}_{10}\sqrt{T}}{\delta_{*}(\nu)}\bigl{(}|\partial_{t}\eta^{1}|_{\mathscr{H}}+|\partial_{t}\theta^{2}|_{\mathscr{V}}+\sqrt{T}\bigr{)}\right)$

Thus, we finish the proof of 2. ∎

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