# Global Existence and Exponential Stability for a Nonlinear Thermoelastic
Kirchhoff-Love Plate

Irena Lasiecka Department of Mathematical Sciences, University of Memphis,
Memphis, TN<EMAIL_ADDRESS>Michael Pokojovy Department of Mathematics,
Karlsruhe Institute of Technology, Karlsruhe, Germany<EMAIL_ADDRESS>Xiang Wan University of Virginia, Charlottesville, VA<EMAIL_ADDRESS>
(August 27, 2024)

###### Abstract

We study an initial-boundary-value problem for a quasilinear thermoelastic
plate of Kirchhoff & Love-type with parabolic heat conduction due to Fourier,
mechanically simply supported and held at the reference temperature on the
boundary. For this problem, we show the short-time existence and uniqueness of
classical solutions under appropriate regularity and compatibility assumptions
on the data. Further, we use barrier techniques to prove the global existence
and exponential stability of solutions under a smallness condition on the
initial data. It is the first result of this kind established for a
quasilinear non-parabolic thermoelastic Kirchhoff & Love plate in multiple
dimensions.

Key words: |  Kirchhoff-Love plates; nonlinear thermoelasticity; Hyperbolic-Parabolic PDE systems; global well-posedness; classical solutions; exponential stability
---|---
MSC (2010): |  Primary 35M33, 35Q74, 74B20, 74F05, 74K20;
|  Secondary 35A01, 35A02, 35A09, 35B40

##  1 Introduction

Let $\Omega\subset\mathbb{R}^{d}$ ($d=2\mbox{ or }3$) be a bounded domain with
a smooth boundary representing the mid-plane of a thermoelastic plate. With
$w$ and $\theta$ denoting the vertical deflection and an appropriately
weighted thermal moment with respect to the plate thickness, both depending on
a scaled time variable $t>0$ and the space variable $(x_{1},x_{2})\in\Omega$,
the nonlinear Kirchhoff & Love thermoelastic plate system reads as

$\displaystyle w_{tt}-\gamma\triangle w_{tt}+a(-\triangle
w)\triangle^{2}w+\alpha\triangle\theta$ $\displaystyle=f(-\triangle
w,-\nabla\triangle w)\phantom{0}\text{ in }(0,\infty)\times\Omega,$ (1.1a)
$\displaystyle\beta\theta_{t}-\eta\triangle\theta+\sigma\theta-\alpha\triangle
w_{t}$ $\displaystyle=0\phantom{f(-\triangle w,-\nabla\triangle w)}\text{ in
}(0,\infty)\times\Omega$ (1.1b) along with the boundary conditions (hinged
mechanical/Dirichlet thermal) $w=\triangle w=\theta=0\text{ in
}(0,\infty)\times\Omega$ (1.1c) and the initial conditions
$w(0,\cdot)=w^{0},\quad
w_{t}(0,\cdot)=w^{1},\quad\theta(0,\cdot)=\theta^{0}\text{ in }\Omega.$ (1.1d)

Here, $\alpha,\beta,\gamma,\eta,\sigma$ are positive constants and
$a\colon\mathbb{R}\to(0,\infty)$ as well as
$f\colon\mathbb{R}\times\mathbb{R}^{d}\to\mathbb{R}$ are smooth functions. For
thin plates, $\gamma$ behaves like $h^{2}$ as $h\to 0$ (cf. [11, Equation
(2.16), p. 13]) and is, therefore, neglected in some literature. In Section 2
below, we present a short physical deduction of Equations (1.1a)–(1.1d).

Lasiecka et al. [12] studied a quasilinear PDE system similar to (1.1a)–(1.1d)
in a smooth, bounded domain $\Omega$ of $\mathbb{R}^{d}$ with $d\leq 3$ given
by a Kirchhoff & Love plate with parabolic heat conduction

$\displaystyle
w_{tt}+\triangle^{2}w-\triangle\theta+a\triangle\big{(}(\triangle
w)^{3}\big{)}$ $\displaystyle=0\text{ in }(0,T)\times\Omega,$ (1.2a)
$\displaystyle\theta_{t}-\triangle\theta+\triangle w_{t}$
$\displaystyle=0\text{ in }(0,T)\times\Omega$ (1.2b)

together with boundary conditions (1.1c) and initial conditions (1.1d) for an
arbitrary $T>0$. For the initial-boundary-value problem (1.2a)–(1.2b),
(1.1c)–(1.1d), they proved the global existence of weak solutions $(w,\theta)$
and their uniform decay in the norm of

$\Big{(}W^{1,\infty}\big{(}0,T;L^{2}(\Omega)\big{)}\cap
L^{\infty}(0,T;W^{2,4}(\Omega)\big{)}\Big{)}\times
L^{\infty}\big{(}0,T;W^{1,2}(\Omega)\big{)}.$

The existence proof was based on a Galerkin approximation and compactness
theorems, while the uniform stability was obtained with the aid of energy
techniques.

In their monograph [3], Chueshov and Lasiecka give an extensive study on the
von Kármán plate system both in pure elastic and thermoelastic cases. With
$w\colon\Omega\to\mathbb{R}$ denoting the vertical displacement and
$v\colon\Omega\to\mathbb{R}$ standing for the Airy stress function of a plate
with its mid-plane occupying in the reference configuration a domain
$\Omega\subset\mathbb{R}^{2}$, the pure elastic version of Kármán plate system
reads as

$\displaystyle w_{tt}-\alpha\triangle w_{tt}+\triangle^{2}u-[u,v+F_{0}]+Lu$
$\displaystyle=p\text{ in }(0,\infty)\times\Omega,$ (1.3a)
$\displaystyle\triangle^{2}v+[u,u]$ $\displaystyle=0\text{ in
}(0,\infty)\times\Omega,$ (1.3b)

where
$[v,w]:=v_{x_{1}x_{1}}w_{x_{2}x_{2}}+v_{x_{2}x_{2}}w_{x_{1}x_{1}}-2v_{x_{1}x_{2}}w_{x_{1}x_{2}}$,
$L$ is a first-order differential operator and
$F_{0},p\colon\Omega\to\mathbb{R}$ are given “force” functions. Imposing
standard initial conditions, under various sets of boundary conditions,
Chueshov and Lasiecka proved Equations (1.3a)–(1.3b) possess a unique
generalized, weak or strong solution depending on the data regularity. The
proof was based on a nonlinear Galerkin-type approximation. Further, they
studied the semiflow associated with the solution to Equations (1.3a)–(1.3b),
in particular, they analyzed its long-time behavior and the existence of
attracting sets. Various damping mechanisms, thermoelastic effects,
structurally coupled systems such as acoustic chambers or gas flow past a
plate were studied. An extremely detailed and comprehensive literature
overview was also given.

Denk et al. [4] considered a linearization of (1.2a)–(1.2b), which corresponds
to letting $a\equiv 0$, in a bounded or exterior $C^{4}$-domain of
$\mathbb{R}^{d}$ for $d\geq 2$ subject to the initial conditions from Equation
(1.1d) and the boundary conditions

$w=\partial_{\nu}w=\theta=0\text{ on }(0,T)\times\partial\Omega,$ (1.4)

where $\partial_{\nu}=(\nabla\cdot)^{T}\nu$ and $\nu$ denotes the outer unit
normal vector to $\Omega$ on $\partial\Omega$. By proving a resolvent estimate
both in the whole space and in the half-space and employing localization
techniques, they showed that the $C_{0}$-semigroup for $(w,w_{t},\theta)$ on
the space

$W^{2,p}_{D}(\Omega)\times L^{p}(\Omega)\times L^{p}(\Omega)\text{ with
}W^{2,p}_{D}(\Omega)=\\{u\in W^{2,p}(\Omega)\,|\,u=\partial_{\nu}u=0\text{ on
}\partial\Omega\\}$

is analytic. In case $\Omega$ is bounded, they also proved an exponential
stability result for the semigroup.

Lasiecka and Wilke [13] presented an $L^{p}$-space treatment of Equations
(1.2a)–(1.2b), (1.1c)–(1.1d) in bounded $C^{2}$-domains $\Omega$ of
$\mathbb{R}^{d}$. By proving the maximal $L^{p}$-regularity for the linearized
problem, they adopted the classical approach to prove the existence and
uniquess of strong solutions satisfying

$(\triangle
w,w_{t},\theta)\in\Big{(}L^{p}_{\mu}\big{(}0,T;W^{2,p}(\Omega)\big{)}\cap
W^{1,p}_{\mu}\big{(}0,T;L^{p}(\Omega)\big{)}\cap
BUC\big{(}0,T;W^{2\mu-2/p,p}(\Omega)\big{)}\Big{)}^{3}$

for $p>1+\tfrac{d}{2}$, where $L^{p}_{\mu}(\Omega)$ is the space of strongly
measurable functions $u$ for which $t\mapsto t^{1-\mu}u(t)$ lies in
$L^{p}(\Omega)$ and $W^{1,p}_{\mu}(\Omega)$ stands for the space of weakly
differentiable functions from $L^{p}_{\mu}(\Omega)$ whose first-order weak
derivatives also lie in $L^{p}_{\mu}(\Omega)$. For $T\leq\infty$, they showed
a global strong solvability result for sufficiently small initial data in the
interpolation space

$(\triangle
w^{0},w^{1},\theta^{0})\in\Big{(}\big{(}L^{p}(\Omega),W^{2,p}(\Omega)\cap
W^{1,p}_{0}(\Omega)\big{)}_{\mu-1/p,p}\Big{)}^{3}.$

They pointed out that similar arguments can be used to obtain a short-time
existence for arbitrarily large initial data. Finally, they studied the first-
and higher-order differentiability as well as analyticity of solutions under
appropriate assumptions on the data.

Recently, Denk and Schnaubelt [5] considered a structurally damped elastic
plate equation

$w_{tt}+\triangle^{2}w-\rho\triangle w_{t}=f\text{ in }(0,\infty)\times\Omega$
(1.5)

in a domain $\Omega\subset\mathbb{R}^{d}$, being either the whole space, a
half-space or a bounded $C^{4}$-domain, subject to inhomogeneous Dirichlet-
Neumann boundary conditions

$w=g_{0},\quad\partial_{\nu}w=g_{1}\text{ on }(0,\infty)\times\Omega$

and the initial conditions

$w(0,\cdot)=w^{0},\quad w_{t}(0,\cdot)=w^{1}\text{ in }\Omega,$

with the data coming from appropriate $L^{p}$-Sobolev spaces for
$p\in(1,\infty)\backslash\\{3/2,3\\}$. By showing the
$\mathcal{R}$-sectoriality of the operator driving the flow
$t\mapsto\big{(}w(t),w_{t}(t)\big{)}$ both in the whole space and the half-
space scenarios, they proved the $L^{p}$-maximum regularity for the generator
on any finite time horizon $T>0$. In case of bounded $C^{4}$-domains, a
standard localization technique was adopted to deduce the maximum
$L^{p}$-regularity for any time horizon $T\in(0,\infty]$.

In the present article, we study the quasilinear PDE system associated with
Equations (1.1a)–(1.1d). In contrast to earlier works, dealing with a
quasilinear system without maximal $L^{p}$-regularity property, it is
technically beneficial to look for classical rather than weak or strong
solutions. The necessity of studying smooth solutions results in a much higher
complexity of the existence and uniqueness proof as it has to be carried out
at a higher energy level, which, in turn, is based on a Kato-type
approximation procedure rather than a Galerkin scheme. The paper is structured
as follows. After a short introduction Section 1, we present in Section 2 a
brief physical deduction of the Kirchhoff & Love plate from Equations
(1.1a)–(1.1d). In Section 4, an existence and uniquess result for Equations
(1.1a)–(1.1d) in the class of classical solutions is shown. The long-time
behavior of Equations (1.1a)–(1.1d) is studied in Section 5. Under a smallness
assumption on the initial data, the global existence and uniquess of solution
is proved using energy estimates and the barrier method. Further, this global
solution is shown to decay at an exponential rate to the zero equilibrium
state. Finally, in the appendix Section A, we present a well-posedness theory
along with higher energy estimates for a linear wave equation with time- and
space-dependent coefficients as well as the homogeneus isotropic heat
equation.

##  2 Model Description $(d=2)$

Consider a prismatic solid plate of uniform thickness $h>0$ and constant
material density $\rho>0$ occupying in a reference configuration the domain
$\mathcal{B}_{h}:=\Omega\times(-\tfrac{h}{2},\tfrac{h}{2})$ of
$\mathbb{R}^{3}$, where $\Omega\subset\mathbb{R}^{2}$ is bounded. The
underlying material is assumed to be elastically and thermally isotropic.
Further, we restrict ourselves to the case of infinitesimal thermoelasticity
with both stresses/strains and temperature gradient/heat flux being small.
Additionally, we assume the strains linearly decompose into elastic and
thermal ones. Despite of these linearity assumptions, a nonlinear
(hypo)elastic law will be postulated allowing for materials with genuinely
nonlinear response such as rubber, liquid crystal elastomers, etc. Figure 1
below (adopted from [15, Chapter 1]) displays a prismatic plate together with
its mid-plane in the reference configuration.

$\mathcal{B}_{r}$$\Omega$$h$$x_{2}$$x_{1}$$x_{3}$ Figure 1: Prismatic plate

We start by interpreting the plate as a 3D body. Let
$\mathbf{U}=(U_{1},U_{2},U_{3})^{T}$ be the displacement vector in Lagrangian
coordinates, $T$ stand for the absolute temperature and
$\mathbf{q}=(q_{1},q_{2},q_{3})^{T}$ be the associated heat flux. Denote by
$T_{0}>0$ a reference temperature for which the body occupies the reference
configuration and is free of any stresses or strains. Further, let $S$ denote
the entropy and

$\boldsymbol{\sigma}=(\sigma_{ij})_{i=1,2,3}^{j=1,2,3}\mbox{\quad and
\quad}\boldsymbol{\varepsilon}=\tfrac{1}{2}\big{(}\nabla\mathbf{U}+(\nabla\mathbf{U})^{T}\big{)}$

stand for the first Piola & Kirchhoff stress tensor and the infinitesimal
Cauchy strain tensor. We assume the total stress tensor decomposes into
elastic and thermal stresses according to

$\boldsymbol{\sigma}=\boldsymbol{\sigma}^{\mathrm{elast}}-\boldsymbol{\sigma}^{\mathrm{therm}}.$
(2.1)

In the absense of external body forces and heat sources, the momentum and
energy balance equations (cf. [1, p. 142] and [11, Chapter 1]) read then as

$\displaystyle\rho\mathbf{U_{tt}}+\operatorname{div}\boldsymbol{\sigma}$
$\displaystyle=0\text{ in }(0,\infty)\times\mathcal{B}_{h},$ (2.2a)
$\displaystyle TS_{t}+\operatorname{div}\mathbf{q}$ $\displaystyle=0\text{ in
}(0,\infty)\times\mathcal{B}_{h}.$ (2.2b)

Similar to Ilyushin [7, p. 42], we define the elastic strain intensity
$\varepsilon_{\mathrm{int}}$ as a properly scaled second invariant of the
elastic strain deviator tensor by means of

$\varepsilon_{\mathrm{int}}^{\mathrm{elast}}=\tfrac{\sqrt{2}}{3}\Big{(}(\operatorname{tr}\boldsymbol{\varepsilon}^{\mathrm{elast}})^{2}-\operatorname{tr}\big{(}(\boldsymbol{\varepsilon}^{\mathrm{elast}})^{2}\big{)}\Big{)}.$

Similarly, we can define the elastic stress intensity via

$\sigma_{\mathrm{int}}^{\mathrm{elast}}=\tfrac{\sqrt{2}}{3}\Big{(}(\operatorname{tr}\boldsymbol{\sigma}^{\mathrm{elast}})^{2}-\operatorname{tr}\big{(}(\boldsymbol{\sigma}^{\mathrm{elast}})^{2}\big{)}\Big{)}.$

Within the classical hypoelasticity, we need to postulate a relation between
these two quantities. Here, we consider a general material law given by

$\sigma_{\mathrm{int}}=\kappa(\varepsilon_{\mathrm{int}}),$ (2.3)

which generalizes power-law-type materials considered by Ambartsumian et al.
in [1, Equation (6)]. For the thermal stresses and strains, we select a linear
material law

$\boldsymbol{\sigma}^{\mathrm{therm}}=\tfrac{E}{1-2\mu}\boldsymbol{\varepsilon}^{\mathrm{therm}},$
(2.4)

where $E$ and $\mu$ play the role of Young’s modulus and Poisson’s ratio and
can be reconstructed from the Hooke’s law resulting from linearizing Equation
(2.3) around zero.

With $\tau=T-T_{0}$ denoting the relative temperature, the thermal linearity
and isotropy assumptions imply

$\boldsymbol{\varepsilon}^{\mathrm{therm}}=\alpha\tau\mathbf{I}_{3\times 3},$
(2.5)

where $\alpha>0$ is the thermal expansion coefficient (cf. [11, p. 29]).
According to Nowicki [14, Chapter 1], a linear approximation for the entropy
reads as

$S=\gamma\operatorname{tr}\big{(}\varepsilon^{\mathrm{elast}}\big{)}+\tfrac{\rho
c}{T_{0}}\tau,$ (2.6)

where $c>0$ is the heat capacity and $\gamma=\frac{E\alpha}{1-2\mu}$. Plugging
Equations (2.1), (2.4), (2.5) and (2.6) into Equations (2.2a)–(2.2b) and
linearizing with respect to $\tau$ around zero, we get

$\displaystyle\rho\mathbf{U}_{tt}+\operatorname{div}\boldsymbol{\sigma}^{\mathrm{elastic}}+\gamma\nabla\tau$
$\displaystyle=0\text{ in }(0,\infty)\times\mathcal{B}_{h},$ (2.7a)
$\displaystyle\rho c\tau_{t}-\lambda_{0}\triangle\tau+\gamma
T_{0}\operatorname{tr}\big{(}\boldsymbol{\varepsilon}^{\mathrm{elast}}_{t}\big{)}$
$\displaystyle=0\text{ in }(0,\infty)\times\mathcal{B}_{h}.$ (2.7b)

Together with Equation (2.3), Equations (2.7a)–(2.7b) constitute the PDE
system of 3D thermoelasticity. In the following, we exploit these equations to
deduce our thermoelastic plate model.

As it is typical for most plate theories, we postulate the hypothesis of
undeformable normals, i.e., the linear filaments being perpendicular to the
mid-plane before deformation should also remain linear after the deformation.
Since we are interested in obtaining a Kirchhoff & Love-type plate model, we
additionally assume these deformed filaments remain perpendicular to the
deformed mid-plane. The in-plane displacements are assumed negligible.
Mathematically, these structural assumptions can be written as

$\begin{split}U_{1}(x_{1},x_{2},x_{3})&=-x_{3}w_{x_{1}}(x_{1},x_{2}),\quad
U_{2}(x_{1},x_{2},x_{3})=-x_{3}w_{x_{2}}(x_{1},x_{2}),\\\
U_{3}(x_{1},x_{2},x_{3})&=\phantom{-x_{3}}w(x_{1},x_{2}),\end{split}$ (2.8)

where $w$ is referred to as the bending component or the vertical
displacement. Thus, the elastic behavior of our plate can fully be described
merely by $w$. Figure 2 is self-describing and illustrates these structural
assumptions.

$\Omega$$x_{1}$$x_{2}$$x_{1}$$w(x_{1},x_{2})$cross-section
$x_{2}=0$$w_{x_{1}}$ Figure 2: Mid-plane of a plate as well as plate cross-
sections $x_{2}=0$ before and after the deformation

As for the thermal part of the system, a properly weighted momentum of the
relative temperature $\tau$ with respect to $x_{3}$ given by

$\theta(x_{1},x_{2})=\frac{12\alpha}{h^{3}}\int_{-h/2}^{h/2}x_{3}\tau(x_{1},x_{2},x_{3})\mathrm{d}x_{3}$

will play a crucial role. Proceeding as Lagnese and Lions [11, pp. 29–31],
Equation (2.7b) can be reduced to

$\rho c\theta_{t}-\lambda_{0}\triangle\theta+\tfrac{12\lambda_{0}}{\rho
ch^{2}}\big{(}\tfrac{h\lambda_{1}}{2}+1\big{)}\theta+\tfrac{\alpha\gamma}{\rho
c}\triangle w_{t}=0\text{ in }(0,\infty)\times\Omega,$

where $\lambda_{1}\geq 0$ is the parameter from the Newton’s cooling law
applied to the lower and upper faces of the plate.

Returning to the elastic part and assuming for a moment the material response
$\kappa(\cdot)$ from Equation (2.3) is an analytic function possessing a
Taylor expansion with the vanishing constant term

$\kappa(s)=\sum_{m=1}a_{m}s^{m}\text{ for some }a_{m}\in\mathbb{R},$

we combine the approaches of Ambartsumian et al. [1] and Lagnese & Lions [11,
Chapter 1] to deduce

$\rho hw_{tt}-\tfrac{\rho h^{3}}{12}\triangle w_{tt}+\triangle K(\triangle
w)+D\tfrac{1+\mu}{2}\triangle\theta=0\text{ in }(0,\infty)\times\Omega,$

where $D=\frac{Eh^{3}}{12(1-\mu^{2})}$ denotes the flexural rigidity and
$K(\cdot)$ is obtained from $\kappa(\cdot)$ by means of

$K(s)=\sum_{n=1}^{\infty}\big{(}\tfrac{2}{\sqrt{3}}\big{)}^{n+1}\tfrac{h^{n+1}}{n+2}a_{n}s^{n}.$

In contrast to [1], the $\triangle w_{tt}$-term is not neglected here allowing
for an adequate description of thicker plates than those accounted for by the
standard theory. Obviously, $K(\cdot)$ is also analytic and its Taylor series
has the same absolute convergence region as $\kappa(\cdot)$. Taking into
account

$\displaystyle K(s)$
$\displaystyle=\sum_{n=1}^{\infty}\big{(}\tfrac{2}{\sqrt{3}}\big{)}^{n+1}\tfrac{h^{n+2}}{n+2}a_{n}s^{n-1}=s^{-3}\sum_{n=1}^{\infty}\big{(}\tfrac{2}{\sqrt{3}}\big{)}^{n+1}\tfrac{h^{n+2}}{n+2}a_{n}s^{n+2}$
$\displaystyle=s^{-3}\sum_{m=1}^{\infty}\big{(}\tfrac{2}{\sqrt{3}}\big{)}^{-1}\tfrac{\big{(}\tfrac{2}{\sqrt{3}}hs\big{)}^{n+2}}{n+2}a_{n}s^{n}=h^{3}\big{(}\tfrac{2}{\sqrt{3}}hs\big{)}^{-3}\big{(}\tfrac{2}{\sqrt{3}}\big{)}^{2}\sum_{n=1}^{\infty}a_{n}\tfrac{\big{(}\tfrac{2}{\sqrt{3}}hs\big{)}^{n+2}}{n+2}$
$\displaystyle=\tfrac{4h^{3}}{3}\big{(}\tfrac{2}{\sqrt{3}}hs\big{)}^{-3}\sum_{n=1}^{\infty}a_{n}\tfrac{\big{(}\tfrac{2}{\sqrt{3}}hs\big{)}^{n+2}}{n+2}=\tfrac{4h^{3}}{3}\sum_{n=1}^{\infty}a_{n}\big{[}I(\cdot)^{n}\big{]}\big{(}\tfrac{2}{\sqrt{3}}hs\big{)}$

with the linear operator

$\big{(}If\big{)}(s)=s^{-3}\int_{0}^{s}\xi f(\xi)\mathrm{d}\xi\text{ for
}s\in\mathbb{R}\backslash\\{0\\},$

the function $K$ can equivalently be written as

$K(s)=\tfrac{4}{3}h^{3}[I\kappa]\big{(}\tfrac{2}{\sqrt{3}}hs\big{)}\text{ for
}s\in\mathbb{R}\backslash\\{0\\}.$

By density and continuity, $I$ can uniquely be extended to a mapping from the
set of continuous functions differentiable and vanishing in 0 with the
following norm being bounded

$\|f\|=\max\big{\\{}\sup_{x\in\mathbb{R}}|f(x)|,|f^{\prime}(0)|\big{\\}}$

into the space of continuous functions vanishing at $0$.

Summarizing, our thermoelastic plate system reads as

$\displaystyle\rho hw_{tt}-\tfrac{\rho h^{3}}{12}\triangle w_{tt}+\triangle
K(\triangle w)+D\tfrac{1+\mu}{2}\triangle\theta$ $\displaystyle=0\text{ in
}(0,\infty)\times\Omega,$ (2.9a) $\displaystyle\rho
c\partial_{t}\theta-\lambda_{0}\triangle\theta+\tfrac{12\lambda_{0}}{\rho
ch^{2}}\big{(}\tfrac{h\lambda_{1}}{2}+1\big{)}\theta+\tfrac{\alpha\gamma\lambda_{0}}{\rho
c}\triangle w_{t}$ $\displaystyle=0\text{ in }(0,\infty)\times\Omega.$ (2.9b)

Various boundary conditions can be adopted. We refer to [2, Chapter 2], [7,
Chapter 4] and [11, Chapter 1] for further details. Here, we consider a simply
supported plate held at the reference temperature at the boundary:

$w=\triangle w=\theta=0\text{ in }(0,\infty)\times\partial\Omega.$

##  3 The Main Results

In this section, we state the main results on the well-posedness and long-time
behavior of Equations (1.1a)–(1.1d). While the local result assumes smoothness
of the boundary of the domain, regularity of the initial data and
nonlinearities as well as certain compatibility conditions, the global results
rely additionally and critically on some further smallness assumption on the
initial data. Recall $\Omega\subset\mathbb{R}^{d}$ ($d=2$ or $3$) is a bounded
domain throughout this paper.

######  Definition 3.1.

Let $s\geq 2$. Under a classical solution to Equations (1.1a)–(1.1d) on
$[0,T]$ at the energy level $s$, we understand a function pair
$(w,\theta)\colon[0,T]\times\bar{\Omega}\to\mathbb{R}\times\mathbb{R}$
satisfying

$\displaystyle w$
$\displaystyle\in\Big{(}\bigcap_{m=0}^{s-1}C^{m}\big{(}[0,T],H^{s+2-m}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}\Big{)}\cap C^{s}\big{(}[0,T],H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)},$ $\displaystyle\theta$
$\displaystyle\in\Big{(}\bigcap_{k=0}^{s-2}C^{k}\big{(}[0,T],H^{s+1-k}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}\Big{)}\cap
C^{s-1}\big{(}[0,T],H^{1}_{0}(\Omega)\big{)}$

and, being plugged into Equations (1.1a)–(1.1d), turns them into tautology.
Classical solutions on $[0,T)$ and $[0,\infty)$ are defined correspondingly.

######  Definition 3.2.

Let $w^{m}$, $m\geq 2$, and $\theta^{k}$, $k\geq 1$, denote the “initial
values” for $\partial_{t}^{m}w$ and $\partial_{t}^{k}\theta$ formally and
recursively computed in terms of $w^{0},w^{1}$ and $\theta^{0}$ based on
Equations (4.2a)–(4.2d) (cf. [8, p. 96])

To proceed with our well-posedness result, we require the following
assumptions.

######  Assumption 3.3.

Let $s\geq 3$ be an integer and let $\partial\Omega\in C^{s}$.

  1. 1.

Let $a\in C^{s-1}(\mathbb{R},\mathbb{R})$.

  2. 2.

Let $f\in C^{s-1}(\mathbb{R}\times\mathbb{R}^{2},\mathbb{R})$.

  3. 3.

Let the initial data satisfy the regularity

$w^{0},\triangle w^{0}\in H^{s}(\Omega)\cap H^{1}_{0}(\Omega),\quad
w^{1},\triangle w^{1}\in H^{s-1}(\Omega)\cap
H^{1}_{0}(\Omega),\quad\theta^{0}\in H^{s+1}(\Omega)\cap H^{1}_{0}(\Omega)$

as well as compatibility conditions

$\begin{split}w^{m},\triangle w^{m}&\in H^{s-m}(\Omega)\cap
H^{1}_{0}(\Omega)\text{ for }m=2,\dots,s-1\text{ and }w^{s}\in
H^{2}(\Omega)\cap H^{1}_{0}(\Omega),\\\ \theta^{k}&\in H^{s+1-k}(\Omega)\cap
H^{1}_{0}(\Omega)\text{ for }k=1,\dots,s-2\text{ and }\theta^{s-1}\in
H^{1}_{0}(\Omega).\end{split}$

  4. 4.

Further, assume the “initial ellipticity” condition for $a(\triangle w^{0})A$,
i.e.,

$\min_{x\in\bar{\Omega}}a\big{(}\triangle w^{0}(x)\big{)}>0,\quad\text{where
}\triangle w^{0}\in C^{0}(\bar{\Omega})\text{ by virtue of Sobolev's imbedding
theorem.}$

Now, we can formulate our local well-posedness result.

######  Theorem 3.4 (Local Well-Posedness).

If Assumption 4.4 is satisfied for some $s\geq 3$, Equations (1.1a)–(1.1d)
possess a unique classical solution $(w,\theta)$ at the energy level $s$ on a
maximal interval $[0,T_{\mathrm{max}})\neq\emptyset$ additionally satisfying

$\partial_{t}^{s-1}\theta\in L^{2}\big{(}0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}\text{ and }\partial_{t}^{s}\theta\in
L^{2}\big{(}0,T;L^{2}(\Omega)\big{)}$

along with

$\min\limits_{x\in\bar{\Omega}}a\big{(}\triangle w(t,x)\big{)}>0\text{ for any
}t\in[0,T_{\mathrm{max}}).$

Unless $T_{\mathrm{max}}=\infty$, we have
$\min\limits_{x\in\bar{\Omega}}a\big{(}\triangle w(t,x)\big{)}\to 0$ as
$\nearrow T_{\mathrm{max}}$ or/and

$\sum_{k=0}^{s}\big{\|}\partial_{t}^{k}w(t,\cdot)\big{\|}_{H^{s+2-k}(\Omega)}^{2}+\sum_{k=0}^{s-2}\big{\|}\partial_{t}^{k}\theta(t,\cdot)\big{\|}_{H^{s+1-k}(\Omega)}^{2}+\big{\|}\partial_{t}^{s-1}\theta(t,\cdot)\big{\|}_{H^{1}(\Omega)}^{2}\to\infty\text{
as }t\nearrow T_{\mathrm{max}}.$

Let
$\|(w,\theta)\|_{{\mathcal{Z}}_{s}\times{\mathcal{T}}_{s}}\equiv\|(\partial_{t}^{\leq
s}w,\partial_{t}^{\leq
s-1}\theta)\|_{{\mathcal{Z}}_{s}\times{\mathcal{T}}_{s}}$ with
$\partial_{t}^{\leq k}=(1,\partial_{t},\dots,\partial_{t}^{k})$ denote the
standard norm associated with the solution space in Definition 3.1 (a precise
definition is given in Equation (5.3)). We now present our global results:

######  Theorem 3.5 (Global Well-Posedness).

Let Assumption 3.3 be satisfied for some $s\geq 3$. Then, there exists a
positive number $\epsilon$ (defined in Theorem 5.6 of Section 5) such that for
any initial data $(w^{0},w^{1},\theta^{0})$ satisfying
$\|(w^{0},w^{1},\theta^{0})\|_{{\mathcal{Z}}_{s}\times{\mathcal{T}}_{s}}\equiv\|(w^{0},w^{1},\dots,w^{s},\theta^{0},\dots,\theta^{s-1})\|_{{\mathcal{Z}}_{s}\times{\mathcal{T}}_{s}}<\epsilon$
(which roughly means the smallness of
$\|w^{0}\|_{H^{5}(\Omega)}^{2}+\|w^{1}\|_{H^{4}(\Omega)}^{2}+\|\theta^{0}\|_{H^{4}(\Omega)}^{2}$
when $s=3$), the unique local solution of system from Theorem 3.4 exists
globally, i.e., $T_{\mathrm{max}}=\infty$.

######  Theorem 3.6 (Uniform Stability).

Under the conditions of Theorem 3.5 and given additionally
$\|(w^{0},w^{1},\theta^{0})\|_{{\mathcal{Z}}_{s}\times{\mathcal{T}}_{s}}\equiv\|(w^{0},w^{1},\dots,w^{s},\theta^{0},\dots,\theta^{s-1})\|_{{\mathcal{Z}}_{s}\times{\mathcal{T}}_{s}}<\tilde{\epsilon}$
for some small positive $\tilde{\epsilon}$ (to be defined in Corollary 5.7 of
Section 5), there exist positive constants $C$ and $k$ such that

$\big{\|}(\partial_{t}^{\leq s}w,\partial_{t}^{\leq
s-1}\theta)(t,\cdot)\big{\|}_{{\mathcal{Z}}_{s}\times{\mathcal{T}}_{s}}\leq
Ce^{-kt}\|(w^{0},w^{1},\theta^{0})\|_{{\mathcal{Z}}_{s}\times{\mathcal{T}}_{s}}\text{
for }t\geq 0.$

##  4 Proof of Theorem 3.4: Local Well-Posedness

To facilitate the analytical treatment of (1.1a)–(1.1d), we first reduce the
order in space from four to two. To this end, let $A$ denote the
$L^{2}(\Omega)$-realization of the negative Dirichlet-Laplacian, i.e.,

$A:=-\triangle,\quad D(A):=\big{\\{}u\in H^{1}_{0}(\Omega)\,|\,\triangle u\in
L^{2}(\Omega)\big{\\}}.$

Assuming $\partial\Omega$ is of class $C^{2}$, the elliptic regularity theory
yields $D(A)=H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$. Moreover, $A$ is an
isomorphism between $D(A)$ and $L^{2}(\Omega)$, $A^{-1}$ is a compact self-
adjoint operator and $(-\infty,0]$ is contained both in the resolvent set of
$A$ and $A^{-1}$. Letting

$z:=Aw=-\triangle w,$ (4.1)

Equations (1.1a)–(1.1d) rewrite as an initial-boundary value problem for a
system of partial (pseudo-)differential equations given by

$\displaystyle\big{(}A^{-1}+\gamma\big{)}z_{tt}+a(z)Az-\alpha A\theta$
$\displaystyle=f(z,\nabla z)\phantom{0}\text{ in }(0,\infty)\times\Omega,$
(4.2a) $\displaystyle\beta\theta_{t}+\eta A\theta+\sigma\theta+\alpha z_{t}$
$\displaystyle=0\phantom{f(z,\nabla z)}\text{ in }(0,\infty)\times\Omega,$
(4.2b) $\displaystyle z=\theta$ $\displaystyle=0\phantom{f(z,\nabla z)}\text{
in }(0,\infty)\times\partial\Omega,$ (4.2c) $\displaystyle
z(0,\cdot)=z^{0},\quad z_{t}(0,\cdot)=z^{1},\quad\theta(0,\cdot)$
$\displaystyle=\theta^{0}\phantom{f(z,\nabla z}\text{ in }\Omega,$ (4.2d)

where $z^{0}:=-\triangle w^{0}$ and $z^{1}:=-\triangle w^{1}$. Note that, for
any $s\geq 0$, the operator $A^{-1}+\gamma$ restricted onto $H^{s}(\Omega)$ is
an automorphism of $H^{s}(\Omega)$. Therefore, Definition 3.1 is equivalent
with the following one in the new variable $z$:

######  Definition 4.1.

Let $s\geq 2$. Under a classical solution to Equations (4.2a)–(4.2d) on
$[0,T]$ at the energy level $s$, we understand a function pair
$(z,\theta)\colon[0,T]\times\bar{\Omega}\to\mathbb{R}\times\mathbb{R}$
satisfying

$\displaystyle z$
$\displaystyle\in\Big{(}\bigcap_{m=0}^{s-1}C^{m}\big{(}[0,T],H^{s-m}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}\Big{)}\cap C^{s}\big{(}[0,T],L^{2}(\Omega)\big{)},$
$\displaystyle\theta$
$\displaystyle\in\Big{(}\bigcap_{k=0}^{s-2}C^{k}\big{(}[0,T],H^{s+1-k}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}\Big{)}\cap
C^{s-1}\big{(}[0,T],H^{1}_{0}(\Omega)\big{)}$

and, being plugged into Equations (4.2a)–(4.2d), turns them into tautology.
Classical solutions on $[0,T)$ and $[0,\infty)$ are defined correspondingly.

######  Remark 4.2.

The choice $s=2$ in Definition 4.1 is standard for the linear situation, i.e.,
when $a(\cdot)$ is constant and the function $f(\cdot,\cdot)$ is linear. In
this case, by virtue of the standard semigroup theory, for any initial data
$(z^{0},z^{1},\theta^{0})\in\big{(}H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}\times H^{1}_{0}(\Omega)\times\big{(}H^{3}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}$ with $\triangle\theta^{0}\in H^{1}_{0}(\Omega)$,
there exists a unique classical solution at the energy level $s=2$.

On the contrary, if $a(\cdot)$ and $f(\cdot,\cdot)$ are both genuinely
nonlinear, one usually can not expect obtaining a classical solution for the
initial data at the energy level $s=2$ (cf. [9, Remark 14.4]). Therefore,
taking a higher energy level is inevitable to obtain classical solutions in
the general nonlinear case. Unfortunately, this not only amounts to putting an
additional Sobolev regularity assumption on the initial data and smoothness
conditions on $a(\cdot)$ and $f(\cdot,\cdot)$, but also makes it necessary to
postulate appropriate compatibility conditions.

To better understand the nature of compatibility conditions, we make the
following observation. Assuming there exists a classical solution at an energy
level $s\geq 2$, we can use the smoothness in $t=0$ and Equations
(4.2a)–(4.2b) to compute

$\begin{split}z_{tt}&=\big{(}A^{-1}+\gamma\big{)}^{-1}\Big{(}f(z,\nabla
z)-a(z)Az+\alpha A\theta\Big{)},\\\ \theta_{t}&=-\tfrac{1}{\beta}\big{(}\eta
A\theta+\sigma\theta+\alpha z_{t}\big{)}.\end{split}$ (4.3)

Evaluating these equations at $t=0$, we obtain

$\begin{split}z_{tt}(0,\cdot)&=\big{(}A^{-1}+\gamma\big{)}^{-1}\Big{(}f(z^{0},\nabla
z^{0})-a(z^{0})Az^{0}+\alpha A\theta^{0}\Big{)},\\\
\theta_{t}(0,\cdot)&=-\tfrac{1}{\beta}\big{(}\eta
A\theta^{0}+\sigma\theta^{0}+\alpha z^{1}\big{)}.\end{split}$

Assuming both $a(\cdot)$ and $f(\cdot,\cdot)$ are sufficiently smooth, we can
differentiate Equation (4.3) with respect to $t$ and repeat the procedure to
explicitely evaluate $\partial_{t}^{m}z(0,\cdot)$ or $\partial_{t}^{k}\theta$
for $m=2,\dots,s$ or $k=1,\dots,s-1$, respectively. Thus, Definition 3.2 and
Assumption 3.3 are equivalent to the following ones:

######  Definition 4.3.

Let $z^{m}$, $m\geq 2$, and $\theta^{k}$, $k\geq 1$, denote the “initial
values” for $\partial_{t}^{m}z$ and $\partial_{t}^{k}\theta$ formally and
recursively computed in terms of $z^{0},z^{1}$ and $\theta^{0}$ based on
Equations (4.2a)–(4.2d) (cf. [8, p. 96]).

######  Assumption 4.4.

Let $s\geq 3$ be an integer and let $\partial\Omega\in C^{s}$.

  1. 1.

Let $a\in C^{s-1}(\mathbb{R},\mathbb{R})$.

  2. 2.

Let $f\in C^{s-1}(\mathbb{R}\times\mathbb{R}^{2},\mathbb{R})$.

  3. 3.

Let the initial data satisfy the regularity

$z^{0}\in H^{s}(\Omega)\cap H^{1}_{0}(\Omega),\quad z^{1}\in
H^{s-1}(\Omega)\cap H^{1}_{0}(\Omega),\quad\theta^{0}\in H^{s+1}(\Omega)\cap
H^{1}_{0}(\Omega)$

as well as compatibility conditions

$\begin{split}z^{m}&\in H^{s-m}(\Omega)\cap H^{1}_{0}(\Omega)\text{ for
}m=2,\dots,s-1\text{ and }z^{s}\in L^{2}(\Omega),\\\ \theta^{k}&\in
H^{s+1-k}(\Omega)\cap H^{1}_{0}(\Omega)\text{ for }k=1,\dots,s-2\text{ and
}\theta^{s-1}\in H^{1}_{0}(\Omega).\end{split}$

  4. 4.

Further, assume the “initial ellipticity” condition for $a(z^{0})A$, i.e.,

$\min_{x\in\bar{\Omega}}a\big{(}z_{0}(x)\big{)}>0,\quad\text{where }z^{0}\in
C^{0}(\bar{\Omega})\text{ by virtue of Sobolev's imbedding theorem.}$

We can reformulate our local well-posedness result Theorem 3.4 in term of $z$
as follows:

######  Theorem 4.5.

If Assumption 4.4 is satisfied for some $s\geq 3$, Equations (4.2a)–(4.2d)
possess a unique classical solution $(z,\theta)$ at the energy level $s$ on a
maximal interval $[0,T_{\mathrm{max}})\neq\emptyset$ additionally satisfying

$\partial_{t}^{s-1}\theta\in L^{2}\big{(}0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}\text{ and }\partial_{t}^{s}\theta\in
L^{2}\big{(}0,T;L^{2}(\Omega)\big{)}$

along with

$\min\limits_{x\in\bar{\Omega}}a\big{(}z(t,x)\big{)}>0\text{ for any
}t\in[0,T_{\mathrm{max}}).$

Unless $T_{\mathrm{max}}=\infty$, we have

$\min\limits_{x\in\bar{\Omega}}a\big{(}z(t,x)\big{)}\to 0\text{ as }t\nearrow
T_{\mathrm{max}}$ (4.4)

or/and

$\sum_{k=0}^{s}\big{\|}\partial_{t}^{k}z(t,\cdot)\big{\|}_{H^{s-k}(\Omega)}^{2}+\sum_{k=0}^{s-2}\big{\|}\partial_{t}^{k}\theta(t,\cdot)\big{\|}_{H^{s+1-k}(\Omega)}^{2}+\big{\|}\partial_{t}^{s-1}\theta(t,\cdot)\big{\|}_{H^{1}(\Omega)}^{2}\to\infty\text{
as }t\nearrow T_{\mathrm{max}}.$ (4.5)

###### Proof.

First, exploiting the second Hilbert’s resolvent identity

$\big{(}A^{-1}+\gamma\big{)}^{-1}=\tfrac{1}{\gamma}-\tfrac{1}{\gamma}A^{-1}\big{(}A^{-1}+\gamma\big{)}^{-1},$

we rewrite Equations (4.2a)–(4.2d) as

$\displaystyle z_{tt}+\tfrac{1}{\gamma}a(z)Az-\tfrac{\alpha}{\gamma}A\theta$
$\displaystyle=\phantom{-}F(z,\theta)$ $\displaystyle\text{ in
}(0,\infty)\times\Omega,$ (4.6a)
$\displaystyle\theta_{t}+\tfrac{\eta}{\beta}A\theta$
$\displaystyle=-\tfrac{1}{\beta}\big{(}\alpha z_{t}+\sigma\theta\big{)}$
$\displaystyle\text{ in }(0,\infty)\times\Omega,$ (4.6b) $\displaystyle
z=\theta$ $\displaystyle=0$ $\displaystyle\text{ on
}(0,\infty)\times\partial\Omega,$ (4.6c) $\displaystyle z(0,\cdot)=z^{0},\quad
z_{t}(t,\cdot)=z^{1},\quad\theta(0,\cdot)$ $\displaystyle=\theta^{0}$
$\displaystyle\text{ in }\Omega,$ (4.6d)

where the nonlinear operator $F$ is given by

$F(z,\theta)=\tfrac{1}{\gamma}(1-K)f(z,\nabla
z)+\tfrac{1}{\gamma}K\big{(}a(z)Az\big{)}-\tfrac{\alpha}{\gamma}KA\theta$

with the compact linear operator

$K:=A^{-1}\big{(}A^{-1}+\gamma\big{)}^{-1}$

continuously mapping $H^{s}(\Omega)$ to $H^{s+2}(\Omega)\cap
H^{1}_{0}(\Omega)$ for any $s\geq 0$ (cf. proof of Theorem A.6). Now,
Equations (4.6a)–(4.6d) are a pseudo-differential perturbation of a second-
order hyperbolic-parabolic system constituted by a quasi-linear wave equation
coupled to a linear heat equation.
Step 1: Modify the nonlinearity $a(\cdot)$. Since no global positivity
assumption is imposed on the nonlinearity $a(\cdot)$, the elipticity condition
for $a(z)A$ can be violated at any time $t>0$. To (preliminarily) rule out
this possible degeneracy, the following construction is performed.

Taking into account the continuity of $z^{0}$ and the connectedness of
$\Omega$, we have

$z^{0}(\bar{\Omega})=\big{[}\min_{x\in\bar{\Omega}}z^{0}(x),\max_{x\in\bar{\Omega}}z^{0}(x)\big{]}=:J_{0}.$
(4.7)

By Assumption 4.4.4, $a(\cdot)$ is strictly positive on $J_{0}$. Consider an
arbitrary closed set $J$ such that

$J_{0}\subset\operatorname{int}(J)\text{ and }a(z)>0\text{ for }z\in J,$ (4.8)

which must exist due to the continuity of $a(\cdot)$. By standard continuation
arguments, there exists a $C^{s}$-function $\hat{a}(\cdot)$ such that

$\hat{a}(\zeta)=a(\zeta)\text{ for }\zeta\in J\text{ and }\inf_{\zeta\not\in
J}\hat{a}(\zeta)>0.$

Now, we replace Equation (4.6a) with

$\displaystyle
z_{tt}+\tfrac{1}{\gamma}\hat{a}(z)Az-\tfrac{\alpha}{\gamma}A\theta=F(z,\theta)\quad\text{
in }(0,\infty)\times\Omega$ (4.9)

and first consider Equations (4.9), (4.6b)–(4.6d). To solve this new problem,
we tranform it to a fixed-point problem and use the Banach fixed-point
theorem. Our proof will be reminiscent of that one by Jiang and Racke [8,
Theorem 5.2] carried out for the quasilinear system of thermoelasticity.
Step 2: Define the fixed-point mapping. Here and in the sequel,
$H^{0}_{0}(\Omega)\equiv H^{0}(\Omega):=L^{2}(\Omega)$. For $N>0$ and $T>0$,
let $X(N,T)$ denote the set of all regular distributions $(z,\theta)$ such
that $(z,\theta)$ together with their weak derivatives satisfy the regularity
conditions

$\displaystyle\partial_{t}^{m}z$ $\displaystyle\in
C^{0}\big{(}[0,T],H^{s-m}(\Omega)\big{)}\text{ for }m=0,1,\dots,s,$
$\displaystyle\partial_{t}^{k}\theta$ $\displaystyle\in
C^{0}\big{(}[0,T],H^{s+1-k}(\Omega)\big{)}\text{ for
}k=0,1,\dots,s-2,\quad\partial_{t}^{s-1}\theta\in
C^{0}\big{(}[0,T],H^{1}_{0}(\Omega)\big{)},$
$\displaystyle\partial_{t}^{s-1}\theta$ $\displaystyle\in
L^{2}\big{(}0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\big{)}\text{ and
}\partial_{t}^{s}\theta\in L^{2}\big{(}0,T;L^{2}(\Omega)\big{)}$

the boundary

$\partial_{t}^{m}z=\partial_{t}^{k}\theta=0\text{ in
}[0,T]\times\partial\Omega\text{ for }m,k=0,1,\dots,s-1$

and the initial conditions

$\partial_{t}^{m}z(0,\cdot)=z^{m}\text{ for }m=0,1,\dots,s\text{ and
}\partial_{t}^{k}\theta(0,\cdot)=\theta^{k}\text{ for }k=0,1,\dots,s-1\text{
in }\Omega$ (4.10)

as well as the energy inequality

$\begin{split}\max_{0\leq t\leq
T}\|\bar{D}^{s}z(t,\cdot)\|_{L^{2}(\Omega)}^{2}&+\sum_{k=0}^{s-2}\max_{0\leq
t\leq
T}\|\partial_{t}^{k}\theta(t,\cdot)\|_{H^{s+1-k}(\Omega)}^{2}+\max_{0\leq
t\leq T}\|\partial_{t}^{s-1}\theta(t,\cdot)\|_{H^{1}(\Omega)}^{2}\\\
&+\int_{0}^{T}\big{(}\|\triangle\partial_{t}^{s-1}\theta(t,\cdot)\|_{L^{2}(\Omega)}^{2}+\|\partial_{t}^{s}\theta(t,\cdot)\|_{L^{2}(\Omega)}^{2}\big{)}\mathrm{d}t\leq
N^{2}.\end{split}$ (4.11)

Here, for $n\geq 0$, we let

$\bar{D}^{n}:=\big{(}(\partial_{t},\nabla)^{\alpha}\,|\,0\leq|\alpha|\leq
n\big{)}.$

For any $T_{0}>0$ and sufficiently large $N>0$, the set $X(N,T)$ is not empty
for any $T\in(0,T_{0}]$. Indeed, if $N$ is sufficiently large, any pair
$(z,\theta)$ of Taylor polynomials

$z(t,\cdot)=\sum_{k=0}^{s}\frac{z^{k}t^{k}}{k!}+P_{z}(t,\cdot)t^{s+1},\quad\theta(t,\cdot)=\sum_{k=0}^{s-1}\frac{\theta^{k}t^{k}}{k!}+P_{\theta}(t,\cdot)t^{s},$

is contained in $X(N,T)$, where $P_{z}$, $P_{\theta}$ are arbitrary
$C^{\infty}_{0}\big{(}\Omega)$-valued polynomials w.r.t. $t$.

For $(\bar{z},\bar{\theta})\in X(N,T)$, consider the linear operator
$\mathscr{F}$ mapping $(\bar{z},\bar{\theta})$ to a function pair $(z,\theta)$
such that $\theta$ is the unique classical solution to the linear heat
equation

$\begin{split}\theta_{t}(t,x)-\tfrac{\eta}{\beta}\triangle\theta(t,x)&=\bar{g}(t,x)\phantom{0}\text{
for }(t,x)\in(0,T)\times\Omega,\\\ \theta&=0\phantom{\bar{g}(t,x)}\text{ for
}(t,x)\in(0,T)\times\partial\Omega,\\\
\theta(0,\cdot)&=\theta^{0}(x)\;\;\text{ for }x\in\Omega\end{split}$ (4.12)

with

$\bar{g}(t,x)=-\tfrac{1}{\beta}\big{(}\alpha\bar{z}_{t}(t,x)+\sigma\bar{\theta}(t,x)\big{)}\text{
for }(t,x)\in[0,T]\times\bar{\Omega}$ (4.13)

and, subsequently, define $z$ to be the unique classical solution to the
linear wave equation

$\begin{split}z_{tt}(t,x)-\bar{a}_{ij}(t,x)\triangle
z(t,x)&=\bar{f}(t,x)\phantom{0}\text{ for }(t,x)\in(0,T)\times\Omega,\\\
z(t,x)&=0\phantom{\bar{f}(t,x)}\text{ for
}(t,x)\in(0,T)\times\partial\Omega,\\\ z(0,x)=z^{0}(x),\quad
z_{t}(0,x)&=z^{1}(x)\phantom{00}\text{ for }x\in\Omega\end{split}$ (4.14)

with

$\begin{split}\bar{a}_{ij}(t,x)&:=\tfrac{1}{\gamma}\hat{a}\big{(}\bar{z}(t,x)\big{)}\delta_{ij}\text{
and }\\\
\bar{f}(t,x)&:=\tfrac{1}{\gamma}\big{(}(1-K)f(\bar{z},\nabla\bar{z})\big{)}(t,x)+\tfrac{1}{\gamma}\big{(}K\big{(}\hat{a}(\bar{z})A\bar{z}\big{)}\big{)}(t,x)-\tfrac{\alpha}{\gamma}\big{(}(1-K)A\theta\big{)}(t,x)\end{split}$
(4.15)

for $(t,x)\in[0,T]\times\Omega$. Note that the right-hand side $\bar{f}$
depends on $A\theta$ and not $A\bar{\theta}$ as the standard procedure would
suggest.

We prove $\mathscr{F}$ is well-defined. By the definition of $\bar{g}$ in
Equation (4.13) and the regularity of $(\bar{z},\bar{g})\in X(N,T)$, we
trivially have

$\partial_{t}^{k}\bar{g}\in C^{0}\big{(}[0,T],H^{s-1-k}(\Omega)\big{)}\text{
for }k=0,1,\dots,s-1.$

By virtue of Theorem A.6, Equation (4.12) possesses a unique classical
solution $\theta$ satisfying

$\displaystyle\partial_{t}^{k}\theta$ $\displaystyle\in
C^{0}\big{(}[0,T],H^{s+1-k}(\Omega)\cap H^{1}_{0}(\Omega)\big{)}\text{ for
}k=0,1,\dots,s-2,$ $\displaystyle\partial_{t}^{s-1}\theta$ $\displaystyle\in
C^{0}\big{(}[0,T],H^{1}_{0}(\Omega)\big{)}\cap
L^{2}\big{(}0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\big{)}\text{ and
}\partial_{t}^{s}\theta\in L^{2}\big{(}0,T;L^{2}(\Omega)\big{)}.$

Now, taking into account the regularity of $\bar{z}$ and $\theta$, exploiting
Assumption 4.4 and applying Sobolev’s imbedding theorem, we can verify that
Assumption A.2 is satisfied with

$\gamma_{i}=\max_{0\leq t\leq
T}\bar{\gamma}_{i}\big{(}\big{\|}\bar{z}(t,\cdot)\big{\|}_{H^{s-1}(\Omega)}\big{)}\text{
for }i=0,1,$ (4.16)

where $\gamma_{0},\gamma_{1}\colon[0,\infty)\to(0,\infty)$ are continuous
functions. Here, we used the Sobolev imbedding $\nabla\bar{z}(t,\cdot)\in
H^{2}(\Omega)\hookrightarrow L^{\infty}(\Omega)$ along with the estimate

$\big{\|}K\big{(}\hat{a}(\bar{z})A\bar{z}\big{)}\big{\|}_{H^{m+2}(\Omega)}\leq
C\big{\|}\hat{a}(\bar{z})A\bar{z}\big{\|}_{H^{m}(\Omega)}\text{ for
}m=0,1,\dots,s-2.$

Here and in the following, $C>0$ denotes a generic constant. Hence, by Theorem
A.3, Equation (4.14) possesses a unique classical solution

$z\in\bigcap_{m=0}^{s-1}C^{m}\big{(}[0,T],H^{s-m}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}\cap C^{s}\big{(}[0,T],L^{2}(\Omega)\big{)}$

implying $(z,\theta)\in X(N,T)$. Therefore, the mapping $\mathscr{F}$ is well-
defined.
Step 3: Show the self-mapping property. We prove that $\mathscr{F}$ maps
$X(N,T)$ into itself provided $N$ is sufficiently large and $T$ is
sufficiently small. We define

$E_{0}(T):=\sum_{m=0}^{s}\|z^{m}\|_{H^{s-m}(\Omega)}^{2}+\sum_{k=0}^{s-2}\|\theta^{k}\|_{H^{s+1-k}(\Omega)}^{2}+\|\theta^{s-1}\|_{H^{1}(\Omega)}^{2}.$

Recalling the definition of $\bar{g}$ in Equation (4.13), applying Theorem A.6
and using Equation (4.11), we can estimate

$\begin{split}\sum_{k=0}^{s-2}\max_{0\leq t\leq
T}\|\partial_{t}^{k}\theta(t,\cdot)&\|_{H^{s+1-k}(\Omega)}^{2}+\max_{0\leq
t\leq T}\|\partial_{t}^{s-1}\theta(t,\cdot)\|_{H^{1}(\Omega)}^{2}\\\
&+\int_{0}^{T}\big{(}\|\triangle\partial_{t}^{s-1}\theta(t,\cdot)\|_{L^{2}(\Omega)}^{2}+\|\partial_{t}^{s}\theta(t,\cdot)\|_{L^{2}(\Omega)}^{2}\big{)}\mathrm{d}t\leq
CN^{2}+CE_{0}.\end{split}$ (4.17)

Further, taking into account Equations (4.10), (4.11) and (4.15) and applying
Sobolev imbedding theorem and [8, Theorem B.6], we obtain

$\int_{0}^{T}\|\partial_{t}^{s-1}\bar{f}(t,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}t\leq
C(N)(1+T)$ (4.18)

and

$\displaystyle\sum_{m=0}^{s-2}$ $\displaystyle\max_{0\leq t\leq
T}\|\partial_{t}^{m}\bar{f}(t,\cdot)\|_{H^{s-2-m}(\Omega)}^{2}$
$\displaystyle\leq\sum_{m=0}^{s-2}\max_{0\leq t\leq
T}\big{\|}\partial_{t}^{m}\big{(}\tfrac{1}{\gamma}(1-K)f(\bar{z},\nabla\bar{z})+\tfrac{1}{\gamma}K\big{(}\hat{a}(\bar{z})A\bar{z}\big{)}-\tfrac{\alpha}{\gamma}(1-K)A\theta\big{)}(t,\cdot)\big{\|}_{H^{s-2-m}(\Omega)}^{2}$
$\displaystyle\leq\sum_{m=0}^{s-2}\big{\|}\partial_{t}^{m}\big{(}\tfrac{1}{\gamma}(1-K)f(\bar{z},\nabla\bar{z})+\tfrac{1}{\gamma}K\big{(}\hat{a}(\bar{z})A\bar{z}\big{)}-\tfrac{\alpha}{\gamma}(1-K)A\theta\big{)}(0,\cdot)\big{\|}_{H^{s-2-m}(\Omega)}^{2}$
$\displaystyle+\sum_{m=0}^{s-2}\int_{0}^{T}\big{\|}\partial_{t}^{m}\big{(}\tfrac{1}{\gamma}(1-K)f(\bar{z},\nabla\bar{z})+\tfrac{1}{\gamma}K\big{(}\hat{a}(\bar{z})A\bar{z}\big{)}-\tfrac{\alpha}{\gamma}(1-K)A\theta\big{)}(t,\cdot)\big{\|}_{H^{s-2-m}(\Omega)}^{2}\mathrm{d}t$
$\displaystyle\leq C(E_{0})+C(N)(1+T),$ (4.19)

where the fundamental theorem of calculus was employed. Plugging Equations
(4.18) and (4.19) into the energy estimate in Theorem A.3, we obtain

$\max_{0\leq t\leq
T}\|\bar{D}^{s}z(t,\cdot)\|_{L^{2}(\Omega)}^{2}\leq\bar{K}(E_{0},\gamma_{0},\gamma_{1})\zeta(N,T)$
(4.20)

with positive constants $\gamma_{0},\gamma_{1}$ defined in Equation (4.16), a
positive constant $K$ being a continuous function of its variables and

$\zeta(N,T)=\Big{(}1+C(N)T^{1/2}\sum_{i=0}^{5}T^{i/2}\Big{)}\exp\big{(}T^{1/2}C(N)(1+T^{1/2}+T+T^{3/2})\big{)}.$

Combining the estimates in Equations (4.17) and (4.20), we obtain

$\begin{split}\max_{0\leq t\leq
T}&\|\bar{D}^{s}z(t,\cdot)\|_{L^{2}(\Omega)}^{2}+\sum_{k=0}^{s-2}\max_{0\leq
t\leq
T}\|\partial_{t}^{k}\theta(t,\cdot)\|_{H^{s+1-k}(\Omega)}^{2}+\max_{0\leq
t\leq T}\|\partial_{t}^{s-1}\theta(t,\cdot)\|_{H^{1}(\Omega)}^{2}\\\
&+\int_{0}^{T}\big{(}\|\triangle\partial_{t}^{s-1}\theta(t,\cdot)\|_{L^{2}(\Omega)}^{2}+\|\partial_{t}^{s}\theta(t,\cdot)\|_{L^{2}(\Omega)}^{2}\big{)}\mathrm{d}t\leq\bar{K}(E_{0},\gamma_{0},\gamma_{1})\zeta(N,T),\end{split}$
(4.21)

possibly, with an increased constant $\bar{K}$.

We now select $N$ such that

$N^{2}\leq\tfrac{1}{2}\bar{K}(E_{0},\gamma_{0},\gamma_{1}).$

Due to continuity of $\zeta(N,\cdot)$ in $T=0$ and $\zeta(N_{0},0)=1$, there
exists $T>0$ such that $\zeta\big{(}N,(0,T]\big{)}\subset[1,2]$. Hence, the
estimate in Equation (4.21) is satisfied with $N^{2}$ on its right-hand side.
Therefore, $(z,\theta)\in X(N,T)$ and $\mathscr{F}$ maps $X(N,T)$ into itself.
Step 4: Prove the contraction property. We consider the metric space

$Y:=\Big{\\{}(z,\theta)\,\big{|}\,z,z_{t},\nabla z\in
L^{\infty}\big{(}0,T;L^{2}(\Omega)\big{)}\text{ and }\theta\in
L^{\infty}\big{(}0,T;H^{1}(\Omega)\big{)}\Big{\\}}$

equipped with the metric

$\rho\big{(}(z,\theta),(\bar{z},\bar{\theta})\big{)}=\Big{(}\mathop{\operatorname{ess\,sup}}_{0\leq
t\leq
T}\big{\|}\bar{D}^{1}\big{(}z-\bar{z}\big{)}(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\mathop{\operatorname{ess\,sup}}_{0\leq
t\leq
T}\big{\|}(\theta-\bar{\theta})(t,\cdot)\big{\|}_{H^{1}(\Omega)}^{2}\Big{)}^{1/2}$

for $(z,\theta),(\bar{z},\bar{\theta})\in Y$. Obviously, $Y$ is complete.
Further, $X(N,T)\subset Y$. Moreover, $X(N,T)$ is closed in $Y$. Indeed,
consider a sequence $\big{(}(z_{n},\theta_{n})\big{)}_{n\in\mathbb{N}}\subset
X(N,T)$ such that it is a Cauchy sequence in $Y$ and, thus, converges to some
$(z,\theta)\in Y$. With the uniform energy bound in Equation (4.11) being
valid for $\big{(}(z_{n},\theta_{n})\big{)}_{n\in\mathbb{N}}$, it must possess
a subsequence which weakly-$\ast$ converges to some element
$(z^{\ast},\theta^{\ast})\subset X(N,T)$ in respective topologies. Since
strong and weak-$\ast$ limits coincide, we have
$(z,\theta)=(z^{\ast},\theta^{\ast})\in X(N,T)$.

We now prove that $\mathscr{F}\colon X(N,T)\to X(N,T)$ is a contraction
mapping w.r.t. $\rho$. For
$(\bar{z},\bar{\theta}),(\bar{z}^{\ast},\bar{z}^{\ast})\in X(N,T)$, let
$(z,\theta):=\mathscr{F}\big{(}(\bar{z},\bar{\theta})\big{)}$,
$(z^{\ast},\theta^{\ast}):=\mathscr{F}\big{(}(\bar{z}^{\ast},\bar{\theta}^{\ast})\big{)}$.
With $(\bar{z},\bar{\theta})$, $(\bar{z}^{\ast},\bar{\theta}^{\ast})$,
$(z,\theta)$, $(z^{\ast},\theta^{\ast})$ all lying in $X(N,T)$, Equation
(4.11) along with Sobolev imbedding theorem imply

$\mathop{\operatorname{ess\,sup}}_{0\leq t\leq
T}\big{\|}\big{(}\bar{D}^{1}(\bar{z},\bar{z}^{\ast},z,z^{\ast})\big{)}(t,\cdot)\big{\|}_{L^{\infty}(\Omega)}\leq
CN.$ (4.22)

Recalling Equations (4.6a)–(4.6b), we can easily see that
$(\tilde{z},\tilde{\theta}):=(z-z^{\ast},\theta-\theta^{\ast})$ satisfies

$\displaystyle\tilde{z}_{tt}+\tfrac{1}{\gamma}\hat{a}(z)A\tilde{z}-\tfrac{\alpha}{\gamma}A\tilde{\theta}$
$\displaystyle=F(\bar{z},\bar{\theta})-F(\bar{z}^{\ast},\bar{\theta}^{\ast})-\big{(}\hat{a}(\bar{z})-\hat{a}(\bar{z}^{\ast})\big{)}Az^{\ast},$
(4.23) $\displaystyle\tilde{\theta}_{t}+\tfrac{\eta}{\beta}A\tilde{\theta}$
$\displaystyle=-\tfrac{\alpha}{\beta}(\bar{z}_{t}-\bar{z}_{t}^{\ast})-\tfrac{\sigma}{\beta}(\bar{\theta}-\bar{\theta}^{\ast})$
(4.24)

in $(0,T)\times\Omega$. Further, we have

$\displaystyle\tilde{z}=\tilde{\theta}$ $\displaystyle=0$ $\displaystyle\text{
on }(0,\infty)\times\partial\Omega,$ (4.25)
$\displaystyle\tilde{z}(0,\cdot)\equiv 0,\quad\tilde{z}_{t}(t,\cdot)\equiv
0,\quad\tilde{\theta}(0,\cdot)$ $\displaystyle\equiv 0$ $\displaystyle\text{
in }\Omega.$ (4.26)

Multiplying Equation (4.24) in $L^{2}(\Omega)$ with $A\tilde{\theta}$, using
Young’s inequality and integrating w.r.t. $t$, we obtain

$\begin{split}\|A^{1/2}&\tilde{\theta}(t,\cdot)\|_{L^{2}(\Omega)}^{2}+\tfrac{\eta}{\beta}\int_{0}^{t}\|A\tilde{\theta}(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\\\
&\leq\varepsilon\int_{0}^{t}\|A\tilde{\theta}(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau+C_{\varepsilon}T\mathop{\operatorname{ess\,sup}}_{0\leq\tau\leq
t}\Big{(}\big{\|}\bar{D}^{1}\big{(}\bar{z}-\bar{z}^{\ast}\big{)}(\tau,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}(\bar{\theta}-\bar{\theta}^{\ast})(\tau,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}\Big{)}.\end{split}$

Hence, by Poincare-Friedrichs’ inequality, selecting $\varepsilon>0$
sufficiently small, we obtain

$\begin{split}\|\tilde{\theta}(t,\cdot)\|_{H^{1}(\Omega)}^{2}&\leq-\tfrac{\eta}{2\beta}\int_{0}^{t}\|A\tilde{\theta}(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau\\\
&+CT\mathop{\operatorname{ess\,sup}}_{0\leq\tau\leq
t}\Big{(}\big{\|}\bar{D}^{1}\big{(}\bar{z}-\bar{z}^{\ast}\big{)}(\tau,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}(\bar{\theta}-\bar{\theta}^{\ast})(\tau,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}\Big{)}\end{split}$
(4.27)

Similarly, multiplying Equations (4.23) in
$L^{2}\big{(}0,T;L^{2}(\Omega)\big{)}$ with $\tilde{z}_{t}$, applying Green’s
formula, using chain and product rules, taking into account Equation (4.25),
exploiting the local Lipschitz continuity of $\hat{a}(\cdot)$ and
$f(\cdot,\cdot)$, using Equations (4.11) and (4.27) as well as exploiting
Young’s and Poincaré-Friedrichs’s inequalities, we can estimate for any
$t\in[0,T]$

$\begin{split}\big{\|}\bar{D}^{1}\tilde{z}(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}&\leq\tfrac{\eta}{2\beta}\int_{0}^{t}\|A\tilde{\theta}(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau+CT\mathop{\operatorname{ess\,sup}}_{0\leq\tau\leq
t}\big{\|}\big{(}\bar{\theta}-\bar{\theta}^{\ast}\big{)}(\tau,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}\\\
&+C(N)\Big{(}(1+T^{-1/2})\int_{0}^{t}\big{\|}\bar{D}^{1}\tilde{z}(\tau,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}\tau\\\
&+T^{1/2}(1+T)\mathop{\operatorname{ess\,sup}}_{0\leq\tau\leq
t}\big{\|}\bar{D}^{1}\big{(}\bar{z}-\bar{z}^{\ast}\big{)}(\tau,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}\Big{)}\end{split}$
(4.28)

Adding up Equations (4.27)–(4.28), using Gronwall’s inequality, taking into
account Equation (4.26) and selecting $T$ sufficiently small, we can estimate

$\rho\big{(}(z,\theta),(z^{\ast},\theta^{\ast})\big{)}\leq\lambda\rho\big{(}(\bar{z},\bar{\theta}),(\bar{z}^{\ast},\bar{\theta}^{\ast})\big{)}$

for some $\lambda\in(0,1)$. Hence, $\mathscr{F}$ is a contraction on $X(N,T)$
in the metric of space $Y$. With $X(N,T)$ being closed, Banach fixed-point
theorem implies $\mathscr{F}$ has a unique fixed point $(z,\theta)\in X(N,T)$.
Finally, due to the smoothness of $(z,\theta)$, we can easily verify
$(z,\theta)$ is a unique classical solution to Equations (4.9), (4.6b)–(4.6d)
at the energy level $s$.
Step 5: Continuation to the maximal interval. Observing that
$z(T,\cdot),z_{t}(T,\cdot)$ and $\theta(T,\cdot)$ satisfy the regularity and
compatibility assumptions and carrying out the standard continuation argument,
we obtain a maximal interval $[0,T^{\ast}_{J})$ for which the classical
solution (uniquely) exists. Due to the interval’s maximality, unless
$T_{J}^{\ast}=\infty$, we have

$\sum_{k=0}^{s}\big{\|}\partial_{t}^{k}z(t,\cdot)\big{\|}_{H^{s-k}(\Omega)}^{2}+\sum_{k=0}^{s-2}\big{\|}\partial_{t}^{k}\theta(t,\cdot)\big{\|}_{H^{s+1-k}(\Omega)}^{2}+\big{\|}\partial_{t}^{s-1}\theta(t,\cdot)\big{\|}_{H^{1}(\Omega)}^{2}\to\infty\text{
as }t\nearrow T^{\ast}_{J}.$ (4.29)

Step 6: Returning to the original system. By virtue of Sobolev’s imbedding
theorem, the function $a\circ z$ is continuous on
$[0,T^{\ast}_{J})\times\bar{\Omega}$. Hence, the number

$T_{\mathrm{max},J}:=\left\\{\begin{array}[]{cl}T^{\ast}_{J},&\text{if
}\hat{a}\circ z\equiv a\circ z\text{ in }[0,T^{\ast})\times\bar{\Omega},\\\
\min\big{\\{}t\in[0,T^{\ast})\,\big{|}\,a(t,x)\not\in\operatorname{int}(J)\text{
for }x\in\bar{\Omega}\big{\\}},&\text{otherwise}\end{array}\right.$

is well-defined and positive by Equation (4.8). Denote by $(z_{J},\theta_{J})$
the unique classical solution to (4.9), (4.6b)–(4.6d) restricted onto
$[0,T_{\mathrm{max},J})$. Consider now an increasing sequence
$(J_{n})_{n\in\mathbb{N}}$ of closed sets satisfying Equation (4.8) such that

$T_{\mathrm{max},J_{n}}\nearrow
T_{\mathrm{max}}:=\sup\big{\\{}T_{\mathrm{max},J}\,\big{|}\,J\text{ satisfies
Equation }(\ref{EQUATION_SET_J_NONTRIVIAL})\big{\\}}\text{ as }n\to\infty.$
(4.30)

By construction, $(z_{J_{n}},\theta_{J_{n}})$ solves the original problem
(4.6a)–(4.6d) on $[0,T_{\mathrm{max},J_{n}})$ and

$(z_{J_{m}},\theta_{J_{m}})\equiv(z_{J_{n}},\theta_{J_{n}})\text{ on
}[0,T_{\mathrm{max},J_{m}})\text{ for }m,n\in\mathbb{N}\text{ with }m\leq n.$

Hence, letting for $t\in[0,T_{\mathrm{max}})$,

$(z,\theta)(t):=(z_{J_{n}},\theta_{J_{n}})(t)\text{ for any
}n\in\mathbb{N}\text{ such that }T_{\mathrm{max},J_{n}}>t,$

we oberve $(z,\theta)$ uniquely defines a classical solution to (4.6a)–(4.6d)
on $[0,T_{\mathrm{max}})$. Moreover, unless $T_{\mathrm{max}}=\infty$, we have
Equation (4.5) and/or Equation (4.4). Indeed, if neither was the case, we
could redefine $J_{0}$ from Equation (4.7) via

$J_{0}:=\big{[}\min_{x\in\bar{\Omega}}a(z(T_{\mathrm{max}},x)),\max_{x\in\bar{\Omega}}a(z(T_{\mathrm{max}},x))\big{]}$

and repeat Step 5 to obtain a classical solution $(z_{J},\theta_{J})$ existing
beyond $T_{\mathrm{max}}$, which would contradict Equation (4.30). The overall
uniqueness follows similar to Step 4. ∎

######  Remark 4.6.

Equation (4.5) is equivalent to

$\big{\|}z(t,\cdot)\big{\|}_{H^{s}(\Omega)}^{2}+\big{\|}z(t,\cdot)\big{\|}_{H^{s-1}(\Omega)}^{2}\to\infty\text{
as }t\nearrow T_{\mathrm{max}}.$

Indeed, arguing by contradiction, if the norms in Equation (4.5) are bounded,
Equations (4.6a), (4.2b) suggest the derivatives of $z$ and $\theta$ as well
as $\theta$ itself are bounded in respective topologies, which contradicts the
maximality of $T_{\mathrm{max}}$.

##  5 Proof of Theorems 3.5 and 3.6: Global Existence and Long-Time Behavior

In this section, we will show the global existence and the exponential
stability of the local solution to Equations (1.1a)–(1.1d) (or, equivalently,
(4.2a)–(4.2d)) established in Theorem 4.5 provided the initial data are
‘small.’ Motivated by [12] (cf. Equation (1.2a)), we consider here a
particular nonlinearity:

$F(z,\nabla z,Az)\equiv\omega A(z^{3})=\omega\big{(}-3z^{2}Az+6z|\nabla
z|^{2}\big{)}.$ (5.1)

For the sake of convenience, without loss of generality, we assume $\omega=1$.
Hence, the system we study becomes:

$\displaystyle\big{(}A^{-1}+\gamma\big{)}z_{tt}+Az-\alpha A\theta$
$\displaystyle=-3z^{2}Az+6z|\nabla z|^{2}\phantom{0}$ $\displaystyle\text{ in
}(0,\infty)\times\Omega,$ (5.2a) $\displaystyle\beta\theta_{t}+\eta
A\theta+\sigma\theta+\alpha z_{t}$ $\displaystyle=0\phantom{f(z,\nabla z)}$
$\displaystyle\text{ in }(0,\infty)\times\Omega,$ (5.2b) $\displaystyle
z=\theta$ $\displaystyle=0\phantom{f(z,\nabla z)}$ $\displaystyle\text{ in
}(0,\infty)\times\partial\Omega,$ (5.2c) $\displaystyle z(0,\cdot)=z^{0},\quad
z_{t}(0,\cdot)=z^{1},\quad\theta(0,\cdot)$
$\displaystyle=\theta^{0}\phantom{f(z,\nabla z}$ $\displaystyle\text{ in
}\Omega.$ (5.2d)

Under Assumption 4.4, Theorem 4.5 establishes the local existence of a unique
classical solution to Equations (5.2a)–(5.2d) at the energy level $s\geq 3$:

$\displaystyle z$
$\displaystyle\in\Big{(}\bigcap_{m=0}^{s-1}C^{m}\big{(}[0,T_{\max}),H^{s-m}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}\Big{)}\cap
C^{s}\big{(}[0,T_{\mathrm{max}}),L^{2}(\Omega)\big{)}$ (5.3)
$\displaystyle\hskip
216.81pt=:C^{0}\big{(}[0,T_{\mathrm{max}}),{\mathcal{Z}}_{s}\big{)},$
$\displaystyle\theta$
$\displaystyle\in\Big{(}\bigcap_{k=0}^{s-2}C^{k}\big{(}[0,T_{\max}),H^{s+1-k}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}\Big{)}\cap
C^{s-1}\big{(}[0,T_{\mathrm{max}}),H^{1}(\Omega)\big{)}$ $\displaystyle\hskip
216.81pt=:C^{0}\big{(}[0,T_{\mathrm{max}}),{\mathcal{T}}_{s}\big{)},$

where $[0,T_{\max})$ is the maximal existence interval (in time) with
$T_{\max}\leq\infty$. Unless $T_{\mathrm{max}}=\infty$, either the solution
norm explodes or the hyperbolicity of Equation (5.2a) is violated at
$T_{\mathrm{max}}$.

For the solution pair $(z,\theta)$, we introduce the squared norm functionals
$E_{k}(t)$ for $k=1,2,3$ and $0\leq t<T_{\max}$

$\displaystyle E_{1}(t)$
$\displaystyle:=\tfrac{1}{2}\big{\|}A^{-\nicefrac{{1}}{{2}}}z_{t}(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{\gamma}{2}\big{\|}z_{t}(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\big{\|}A^{\nicefrac{{1}}{{2}}}z(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\big{\|}A^{\nicefrac{{1}}{{2}}}\theta(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2},$
(5.4) $\displaystyle E_{2}(t)$
$\displaystyle:=\tfrac{1}{2}\big{\|}z_{t}(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{\gamma}{2}\big{\|}A^{1/2}z_{t}(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\big{\|}Az(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\big{\|}A\theta(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2},$
(5.5) $\displaystyle E_{3}(t)$
$\displaystyle:=\tfrac{1}{2}\big{\|}z_{tt}(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{\gamma}{2}\big{\|}A^{1/2}z_{tt}(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\big{\|}Az_{t}(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\big{\|}A\theta_{t}(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}$
(5.6)

and define the natural energy at level $s=3$ by means of

$X(t)\equiv\big{\|}(z,z_{t},z_{tt},\theta,\theta_{t})(t,\cdot)\big{\|}_{X}^{2}:=E_{2}(t)+E_{3}(t).$
(5.7)

Note that $E_{1}(t)$ represents the basic ‘natural’ energy of the system. For
the sake of brevity – and slighly abusing the notation – we will write in the
following:

$\big{\|}(z,\theta)\big{\|}_{X}\text{ instead of
}\big{\|}(z,z_{t},z_{tt},\theta,\theta_{t})\big{\|}_{X}\text{ and
}\big{\|}(z,\theta)\big{\|}_{\mathcal{Z}_{s}\times\mathcal{T}_{s}}\text{
instead of }\big{\|}(\partial_{t}^{\leq s}z,\partial_{t}^{\leq
s-1}\theta)\big{\|}_{X},$

where $\partial_{t}^{\leq k}:=(1,\partial_{t},\dots,\partial_{t}^{k})$.

######  Remark 5.1.

In order to prove the system is globally well-posed, we first seek for an a
priori estimate for the solution and then prove $T_{\max}=\infty$. To capture
the essential decay of the energy, we work with $\|(z,\theta)\|_{X}$, instead
of $\|(z,\theta)\|_{{\mathcal{Z}}_{s}\times{\mathcal{T}}_{s}}$, for most part
of this section. Although all main results in this section are presented in
terms of $X(t)$, their equivalence with the statements in Section 3, given the
smallness of the initial data, is shown in Lemma 5.8.

We start by an observation that, for small data, $z$ has one extra order of
hidden regularity in space encoded in the definition of $E_{2}$.

######  Lemma 5.2 ($z$-energy boost).

For any $t\in(0,T_{\max})$, if $(z,\theta)$ satisfies

$E_{2}(t)<\epsilon_{1}:=\tfrac{1}{2\sqrt{C^{\prime}}}\text{ for some constant
}C^{\prime}>0\mbox{ (defined in \eqref{def_C'})},$ (5.8)

there holds

$\|z(t)\|^{2}_{H^{3}(\Omega)}\leq C\big{(}X(t)+X^{3}(t)\big{)}\text{ for some
}C>0.$

###### Proof.

From Equation (5.2b), $-\eta A^{3/2}\theta=\alpha A^{1/2}z_{t}+\beta
A^{1/2}\theta_{t}+\sigma A^{1/2}\theta$. Hence,

$\big{\|}A^{3/2}\theta\big{\|}_{L^{2}(\Omega)}^{2}\leq
C\big{(}\big{\|}A^{1/2}z_{t}\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}A^{1/2}\theta_{t}\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}A^{1/2}\theta\big{\|}_{L^{2}(\Omega)}^{2}\big{)}\leq
C\big{(}E_{2}(t)+E_{3}(t)\big{)}.$ (5.9)

Using Hölder’s inequality

$\|a\cdot
b\|_{L^{2}(\Omega)}^{2}\leq\|a\|_{L^{6}(\Omega)}^{2}\cdot\|b\|_{L^{3}(\Omega)}^{2}$

and Sobolev imbedding theorem $H^{1}(\Omega)\hookrightarrow
L^{6}(\Omega)\hookrightarrow L^{4}(\Omega)\hookrightarrow L^{3}(\Omega)$, we
arrive at:

$\displaystyle\big{\|}A^{1/2}$ $\displaystyle(6z|\nabla
z|^{2})\big{\|}_{L^{2}(\Omega)}^{2}$ (5.10) $\displaystyle\leq
C\big{\|}A^{1/2}z\cdot|\nabla
z|^{2}\big{\|}_{L^{2}(\Omega)}^{2}+C\big{\|}z(\nabla z\cdot\nabla
A^{1/2}z)\big{\|}_{L^{2}(\Omega)}^{2}$ $\displaystyle\leq
C\|A^{1/2}z\|_{L^{6}(\Omega)}^{2}\|\nabla
z|\|_{L^{6}(\Omega)}^{4}+C^{\prime}\|Az\|_{{}_{L^{2}(\Omega)}}^{2}\|A^{3/2}z\|_{L^{2}(\Omega)}^{2}\big{\|}Az\big{\|}_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq
CE_{2}^{3}(t)+C^{\prime}E_{2}^{2}(t)\|A^{3/2}z\|_{L^{2}(\Omega)}^{2}$

and

$\displaystyle\big{\|}A^{1/2}(1+3z^{2})\cdot Az\big{\|}_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq C\big{\|}zA^{1/2}z\cdot Az\big{\|}_{L^{2}(\Omega)}^{2}$
(5.11) $\displaystyle\leq
C\|z\|_{H^{2}(\Omega)}^{2}\|A^{1/2}z\|_{H^{2}(\Omega)}^{2}\big{\|}Az\big{\|}_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq
C^{\prime}\|Az\|_{L^{2}(\Omega)}^{4}\|A^{3/2}z\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq
C^{\prime}E_{2}^{2}(t)\|A^{3/2}z\|_{L^{2}(\Omega)}^{2}\text{ for some
}C,C^{\prime}>0.$

Now, to estimate $\|A^{3/2}z\|_{L^{2}(\Omega)}^{2}$, we successively transform
Equation (5.2a) to obtain:

$\displaystyle(1+3z^{2})Az$ $\displaystyle=$
$\displaystyle-\left[A^{-1}z_{tt}+\gamma z_{tt}-\alpha A\theta-6z|\nabla
z|^{2}\right],$ $\displaystyle A^{1/2}[(1+3z^{2})Az]$ $\displaystyle=$
$\displaystyle-A^{1/2}\left[A^{-1}z_{tt}+\gamma z_{tt}-\alpha
A\theta-6z|\nabla z|^{2}\right],$ $\displaystyle A^{1/2}(1+3z^{2})\cdot
Az+(1+3z^{2})A^{3/2}z$ $\displaystyle=$
$\displaystyle-A^{1/2}\left[A^{-1}z_{tt}+\gamma z_{tt}-\alpha
A\theta-6z|\nabla z|^{2}\right],$ $\displaystyle(1+3z^{2})A^{3/2}z$
$\displaystyle=$ $\displaystyle-A^{1/2}\left[A^{-1}z_{tt}+\gamma z_{tt}-\alpha
A\theta-6z|\nabla z|^{2}\right],$ $\displaystyle-A^{1/2}(1+3z^{2})\cdot Az,$
$\displaystyle A^{3/2}z$ $\displaystyle=$
$\displaystyle-\tfrac{1}{1+3z^{2}}\Big{[}A^{-1/2}z_{tt}+A^{1/2}\gamma
z_{tt}-\alpha A^{3/2}\theta$ $\displaystyle-A^{1/2}(6z|\nabla
z|^{2})+A^{1/2}(1+3z^{2})\cdot Az\Big{]}.$

Taking into account $\frac{1}{1+3z^{2}}\leq 1$, Equations (5.9), (5.10) and
(5.11) yield

$\displaystyle\big{\|}A^{3/2}z\big{\|}_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq\left\|\tfrac{1}{1+3z^{2}}\right\|_{L^{\infty}(\Omega)}^{2}\Big{[}\big{\|}A^{-1/2}z_{tt}\big{\|}_{L^{2}(\Omega)}+\gamma\big{\|}A^{1/2}z_{tt}\big{\|}_{L^{2}(\Omega)}+\alpha\big{\|}A^{3/2}\theta\big{\|}_{L^{2}(\Omega)}$
$\displaystyle+\big{\|}A^{1/2}(6z|\nabla
z|^{2})\big{\|}_{L^{2}(\Omega)}+\big{\|}A^{1/2}(1+3z^{2})\cdot
Az\big{\|}_{L^{2}(\Omega)}\Big{]}^{2}$ $\displaystyle\leq
C\Big{[}\big{\|}A^{-1/2}z_{tt}\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}A^{1/2}\gamma
z_{tt}\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}A^{3/2}\theta\big{\|}_{L^{2}(\Omega)}^{2}$
$\displaystyle+\big{\|}A^{1/2}(6z|\nabla
z|^{2})\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}A^{1/2}(1+3z^{2})\cdot
Az\big{\|}_{L^{2}(\Omega)}^{2}\Big{]}$ $\displaystyle\leq
C\Big{[}E_{3}(t)+E_{3}(t)+[E_{2}(t)+E_{3}(t)]$
$\displaystyle+CE_{2}^{3}(t)+C^{\prime}E_{2}^{2}(t)\|A^{3/2}z\|_{L^{2}(\Omega)}^{2}+C^{\prime}E_{2}^{2}(t)\|A^{3/2}z\|_{L^{2}(\Omega)}^{2}\Big{]}.$
(5.12)

Hence,

$\big{\|}A^{3/2}z\big{\|}_{L^{2}(\Omega)}^{2}\leq\tfrac{C}{1-2E_{2}^{2}(t)C^{\prime}}\big{[}E_{2}(t)+E_{3}(t)+E_{2}^{3}(t)\big{]}.$

By the assumption in Equation (5.8), $1-2E_{2}^{2}(t)C^{\prime}>\frac{1}{2}$
and, therefore,

$\big{\|}A^{3/2}z\big{\|}_{L^{2}(\Omega)}^{2}\leq
C\big{[}E_{2}(t)+E_{3}(t)+E_{2}^{3}(t)\big{]}\leq
C\big{[}X(t)+X^{3}(t)\big{]},$

which finishes the proof. ∎

######  Lemma 5.3 (_A priori_ estimate).

Let Assumption 4.4 be satisfied for some $s\geq 3$ and $0<T<T_{\max}$ with
$T_{\max}>0$ denoting the maximal existence time from Theorem 4.5 and let

$X(0)<1.$ (5.13)

Then,

$X_{s}(T)+\int_{0}^{T}X_{s}(t)\mathrm{d}t\leq C_{1}X_{s}(0)+C_{2}\sum_{i\in
I}X_{s}^{\alpha_{i}}(T)+C_{3}\sum_{j\in
J}\int_{0}^{T}X_{s}^{\beta_{j}}(t)\mathrm{d}t$ (5.14)

where $C_{k}\geq 0$, $k=1,2,3$ are constants, both $I,J\subset\mathbb{N}$ are
finite sets, $\alpha_{i}>1$ for any $i\in I$ and $\beta_{j}>1$ for any $j\in
J$.

###### Proof.

The proof mainly consists of energy estimates at the energy levels $s=1,2$ and
$3$. Estimates for higher energy spaces ($s\geq 4$) follow similarly.
Throughout this proof, ${\left\langle\cdot,\cdot\right\rangle}$ denotes the
standard $L^{2}(\Omega)$-inner product. Moreover, without loss of generality,
we assume

$X(t)<1\mbox{ for }t\in[0,T].$ (5.15)

If not, by continuity of $X(t)$, we can find a smaller $T$ so that (5.15)
holds true. Equations (5.13) and (5.15) are critical in Step 4.2 of this
proof. The dimension of the domain is also important by virtue of, for
instance, Equation (5.37).

Step 1: Level 1 energy identity. Multiplying Equation (5.2a) with $z_{t}$ in
$L^{2}(\Omega)$ and integrating by parts, we get

$\displaystyle\tfrac{1}{2}\partial_{t}\big{\|}A^{-1/2}z_{t}\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{\gamma}{2}\partial_{t}\big{\|}z_{t}\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\partial_{t}\big{\|}A^{1/2}z\big{\|}_{L^{2}(\Omega)}^{2}-\alpha\left\langle
A\theta,z_{t}\right\rangle,=\left\langle F,z_{t}\right\rangle$ (5.16)

where $F(z,\nabla z,Az)=-3z^{2}Az+6z|\nabla z|^{2}$ from (5.1). Similar
actions on (5.2b) multiplied by $A\theta$ lead to

$\displaystyle\tfrac{\beta}{2}\partial_{t}\big{\|}A^{1/2}\theta\big{\|}_{L^{2}(\Omega)}^{2}+\eta\big{\|}A\theta\big{\|}_{L^{2}(\Omega)}^{2}+\sigma\big{\|}A^{1/2}\theta\big{\|}_{L^{2}(\Omega)}^{2}+\alpha\left\langle
z_{t},A\theta\right\rangle=0.$ (5.17)

From Equations (5.16) and (5.17), we get the $E_{1}$-identity:

$E_{1}(T)+\int_{0}^{T}\left(\eta\big{\|}A\theta\big{\|}_{L^{2}(\Omega)}^{2}+\sigma\big{\|}A^{1/2}\theta\big{\|}_{L^{2}(\Omega)}^{2}\right)\mathrm{d}t=E_{1}(0)+\int_{0}^{T}\left\langle
F,z_{t}\right\rangle\mathrm{d}t.$ (5.18)

Step 2: Level 2 energy estimate. Recalling from Equation (5.5)

$E_{2}(t)=\tfrac{1}{2}\big{\|}z_{t}(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{\gamma}{2}\big{\|}A^{1/2}z_{t}(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\big{\|}Az(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\big{\|}A\theta(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}$

and using Equation (5.18) along with Poincaré-Friedrichs inequality, we get:

$E_{1}(T)\leq E_{1}(0)+\int_{0}^{T}\left\langle
F,z_{t}\right\rangle\mathrm{d}t\leq C\left[E_{2}(0)+\int_{0}^{T}\left\langle
F,z_{t}\right\rangle\mathrm{d}t\right].$ (5.19)

Similarly,

$\displaystyle\int_{0}^{T}\big{\|}A^{1/2}\theta\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t\leq
C\int_{0}^{T}\big{\|}A\theta\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t\leq
C\left[E_{2}(0)+\int_{0}^{T}\left\langle
F,z_{t}\right\rangle\mathrm{d}t\right],$ (5.20)
$\displaystyle\int_{0}^{T}\big{\|}A\theta\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t\leq
C\int_{0}^{T}\big{\|}A\theta\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t\leq
C\left[E_{2}(0)+\int_{0}^{T}\left\langle
F,z_{t}\right\rangle\mathrm{d}t\right],$ (5.21)
$\displaystyle\max\Big{\\{}\big{\|}z_{t}(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2},\big{\|}A^{1/2}z(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}\big{\\}}\leq
CE_{1}(T)\leq C\left[E_{2}(0)+\int_{0}^{T}\left\langle
F,z_{t}\right\rangle\mathrm{d}t\right].$ (5.22)

In order to estimate $E_{2}(t)$ and
$\displaystyle\int_{0}^{T}E_{2}(t)\mathrm{d}t$, we employ a set of higher
energy multipliers. We first start by multiplying Equation (5.2a) with
$Az_{t}$ and recalling (5.19) to observe that

$\displaystyle\tfrac{1}{2}\big{\|}z_{t}(T)\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{\gamma}{2}\big{\|}A^{1/2}z_{t}(T)\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\big{\|}Az(T)\big{\|}_{L^{2}(\Omega)}^{2}-\alpha\int_{0}^{T}\left\langle
A\theta,Az_{t}\right\rangle\mathrm{d}t$ (5.23) $\displaystyle\leq$
$\displaystyle C\left[E_{2}(0)+\int_{0}^{T}\left\langle
F,z_{t}\right\rangle\mathrm{d}t\right]+\int_{0}^{T}\left\langle
F,Az_{t}\right\rangle\mathrm{d}t.$

Second, in order to estimate $\big{\|}A\theta(T)\big{\|}_{L^{2}(\Omega)}^{2}$,
we multiply (5.2b) by $A\theta_{t}$ to get

$\displaystyle\dfrac{\eta}{2}\big{\|}A\theta(T)\big{\|}_{L^{2}(\Omega)}^{2}+\dfrac{\sigma}{2}\big{\|}A^{1/2}\theta(T)\big{\|}_{L^{2}(\Omega)}^{2}+\beta\int_{0}^{T}\big{\|}A^{1/2}\theta_{t}\big{\|}_{L^{2}(\Omega)}^{2}$
$\displaystyle=\ $
$\displaystyle\dfrac{\eta}{2}\big{\|}A\theta(0)\big{\|}_{L^{2}(\Omega)}^{2}+\dfrac{\sigma}{2}\big{\|}A^{1/2}\theta(0)\big{\|}_{L^{2}(\Omega)}^{2}-\alpha\int_{0}^{T}\langle
A^{1/2}z_{t},A^{1/2}\theta_{t}\rangle.$ (5.24)

We apply Young’s inequality to the inner product term, we get the following
estimate

$\displaystyle\dfrac{\eta}{2}\big{\|}A\theta(T)\big{\|}_{L^{2}(\Omega)}^{2}+\beta\int_{0}^{T}\big{\|}A^{1/2}\theta_{t}\big{\|}_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq\ $ $\displaystyle
E_{2}(0)+\dfrac{\sigma}{2}\big{\|}A^{1/2}\theta(T)\big{\|}_{L^{2}(\Omega)}^{2}+\alpha\dfrac{\beta}{\alpha}\int_{0}^{T}\big{\|}A^{1/2}\theta_{t}\big{\|}_{L^{2}(\Omega)}^{2}+C\int_{0}^{T}\big{\|}A^{1/2}z_{t}\big{\|}_{L^{2}(\Omega)}^{2}$
(5.25)

After performing cancellations, employing (5.19), and rescaling, the estimate
becomes

$\displaystyle C\big{\|}A\theta(T)\big{\|}_{L^{2}(\Omega)}^{2}\leq
C\left[E_{2}(0)+\int_{0}^{T}\left\langle
F,z_{t}\right\rangle\mathrm{d}t\right]+\epsilon\int_{0}^{T}\big{\|}A^{1/2}z_{t}\big{\|}_{L^{2}(\Omega)}^{2}.$
(5.26)

Last, by Equation (5.2a), $z_{tt}=-BAz+\alpha BA\theta+BF$, where
$B=(A^{-1}+\gamma)^{-1}$. Multiplying Equation (5.2b) by $Az_{t}$, we get

$\displaystyle-\eta\int_{0}^{T}\left\langle
A\theta,Az_{t}\right\rangle\mathrm{d}t$
$\displaystyle=\alpha\int_{0}^{T}\big{\|}A^{1/2}z_{t}\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t+\beta\left\langle\theta,Az_{t}\right\rangle\Big{|}_{0}^{T}+\beta\int_{0}^{T}\left\langle
A\theta,BAz\right\rangle\mathrm{d}t$
$\displaystyle-\alpha\beta\int_{0}^{T}\left\langle
A\theta,BA\theta\right\rangle\mathrm{d}t-\beta\int_{0}^{T}\left\langle
A\theta,BF\right\rangle\mathrm{d}t+\sigma\int_{0}^{T}\left\langle\theta,Az_{t}\right\rangle\mathrm{d}t.$
(5.27)

Recall from the proof of Theorem 4.5 that $B$ is a continuous operator.
Therefore, $\|B\|\leq C_{\gamma}$. Adding (5.26) and a multiple of Equation
(5.27) to (5.23), we have:

$\displaystyle\begin{split}E_{2}(T)&+\tfrac{\alpha^{2}}{\eta}\int_{0}^{T}\big{\|}A^{1/2}z_{t}\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t\leq
C\left[E_{2}(0)+\int_{0}^{T}\left\langle
F,z_{t}\right\rangle\mathrm{d}t\right]+\int_{0}^{T}\left\langle
F,Az_{t}\right\rangle\mathrm{d}t\\\
&+C_{\epsilon}\big{\|}A^{1/2}\theta(T)\big{\|}_{L^{2}(\Omega)}^{2}+\epsilon\big{\|}A^{1/2}z_{t}(T)\big{\|}_{L^{2}(\Omega)}^{2}\\\
&+C\int_{0}^{T}\big{\|}A\theta\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t+C_{\epsilon}\int_{0}^{T}\big{\|}A^{1/2}\theta\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t+\epsilon\int_{0}^{T}\big{\|}Az\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t\\\
&+C_{\epsilon}\int_{0}^{T}\big{\|}\theta\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t+\epsilon\int_{0}^{T}\big{\|}Az\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t+\tfrac{\alpha\beta}{\eta}\int_{0}^{T}\left\langle
A\theta,BF\right\rangle\mathrm{d}t.\end{split}$ (5.28)

After merging respective terms and applying Equations (5.19) and (5.20),
(5.28) becomes

$\begin{split}(1-\epsilon)E_{2}(T)&+\tfrac{\alpha^{2}}{\eta}\int_{0}^{T}\big{\|}A^{1/2}z_{t}\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t\\\
&\leq C\left[E_{2}(0)+\int_{0}^{T}\left\langle
F,z_{t}\right\rangle\mathrm{d}t+\int_{0}^{T}\left\langle
F,Az_{t}\right\rangle\mathrm{d}t+\int_{0}^{T}\left\langle
BF,A\theta\right\rangle\right]\mathrm{d}t\\\
&+2\epsilon\int_{0}^{T}E_{2}(t)\mathrm{d}t,\text{ where
}C=C(\epsilon,\alpha,\beta,\gamma,\eta,\sigma).\end{split}$ (5.29)

Third, to estimate
$\displaystyle\int_{0}^{T}\big{\|}Az\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t$,
we multiply(5.2a) by $Az$ and, again, use (5.19) to get

$\begin{split}\int_{0}^{T}\big{\|}Az\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t&\leq\dfrac{\alpha^{2}}{2\eta
C_{\gamma,\Omega}}\left|\int_{0}^{T}\left\langle
F,Az\right\rangle\mathrm{d}t\right|+\epsilon^{\prime}\int_{0}^{T}E_{2}(t)\mathrm{d}t+\epsilon^{\prime}E_{2}(T)\\\
&+C_{\epsilon^{\prime}}\left[E_{2}(0)+\int_{0}^{T}\left\langle
F,z_{t}\right\rangle\mathrm{d}t\right]+\dfrac{\alpha^{2}}{2\eta}\int_{0}^{T}\big{\|}A^{1/2}z_{t}\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t.\end{split}$
(5.30)

Finally, after combining Equations (5.20), (5.29) and (5.30), we arrive at

$\displaystyle E_{2}(T)+C_{1}\int_{0}^{T}E_{2}(t)\mathrm{d}t$
$\displaystyle\leq C_{2}E_{2}(0)+C_{3}\left\\{\left|\int_{0}^{T}\left\langle
F,z_{t}\right\rangle\mathrm{d}t\right|+\left|\int_{0}^{T}\left\langle
F,Az\right\rangle\mathrm{d}t\right|\right.$ (5.31)
$\displaystyle+\left.\left|\int_{0}^{T}\left\langle
F,Az_{t}\right\rangle\mathrm{d}t\right|+\left|\int_{0}^{T}\left\langle
BF,A\theta\right\rangle\mathrm{d}t\right|\right\\}.$

Step 3: Level 3 energy estimate. The 3rd level energy space ($E_{3}$) is one
order higher in time than the 2nd level space ($E_{2}$). Hence, after
differentiating Equation (5.2a)–(5.2b) in time

$\displaystyle A^{-1}z_{ttt}+\gamma z_{ttt}+Az_{t}-\alpha A\theta_{t}$
$\displaystyle=\partial_{t}F(z,\nabla z,\triangle z)$ $\displaystyle\text{ in
}(0,\infty)\times\Omega,$ (5.32a) $\displaystyle\beta\theta_{tt}+\eta
A\theta_{t}-\sigma\theta_{t}+\alpha z_{tt}$ $\displaystyle=0$
$\displaystyle\text{ in }(0,\infty)\times\Omega,$ (5.32b) $\displaystyle
z=z_{t}=\theta=\theta_{t}$ $\displaystyle=0$ $\displaystyle\text{ in
}(0,\infty)\times\partial\Omega,$ (5.32c)

a procedure similar to Step 2 can be employed. Denote the right-hand side of
(5.32a) by

$G(z)\equiv\partial_{t}F=\partial_{t}[-3z^{2}Az+6z|\nabla
z|^{2}]=-6zz_{t}Az-3z^{2}Az_{t}+6z_{t}|\nabla z|^{2}+12z(\nabla z\cdot\nabla
z_{t})$ (5.33)

and calculate

$\displaystyle\partial_{t}G(z)$
$\displaystyle=-6z_{t}^{2}Az-6zz_{tt}Az-12zz_{t}Az_{t}-3z^{2}Az_{tt}$ (5.34)
$\displaystyle+6z_{tt}|\nabla z|^{2}+24z_{t}(\nabla z\cdot\nabla
z_{t})+12z|\nabla z_{t}|^{2}+12z(\nabla z_{t}\cdot\nabla z_{tt}).$

Letting $\widetilde{z}=z_{t}$ and $\widetilde{\theta}=\theta_{t}$, we have

$E_{3}(t)=\tfrac{1}{2}\big{\|}\widetilde{z}_{t}\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{\gamma}{2}\big{\|}A^{1/2}\widetilde{z}_{t}\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\big{\|}A\widetilde{z}\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\big{\|}A\widetilde{\theta}\big{\|}_{L^{2}(\Omega)}^{2}.$

Therefore, in a fashion similar to Equation (5.31), we get the following 3rd
level energy estimate:

$\displaystyle E_{3}(T)+C_{1}\int_{0}^{T}E_{3}(t)\mathrm{d}t$
$\displaystyle\leq C_{2}E_{3}(0)+C_{3}\left\\{\left|\int_{0}^{T}\left\langle
G(z),\widetilde{z}_{t}\right\rangle\mathrm{d}t\right|+\left|\int_{0}^{T}\left\langle
G(z),A\widetilde{z}\right\rangle\mathrm{d}t\right|\right.$ (5.35)
$\displaystyle+\left.\left|\int_{0}^{T}\left\langle
G(z),A\widetilde{z}_{t}\right\rangle\mathrm{d}t\right|+\left|\int_{0}^{T}\left\langle
BG(z),A\widetilde{\theta}\right\rangle\mathrm{d}t\right|\right\\}.$

Recalling $X(t)=E_{2}(t)+E_{3}(t)$, combine (5.31) and (5.35):

$\displaystyle X(T)+C_{1}\int_{0}^{T}X(t)\mathrm{d}t$ $\displaystyle\leq
C_{2}X(0)+C_{3}\left\\{\left|\int_{0}^{T}\left\langle
F,z_{t}\right\rangle\mathrm{d}t\right|+\left|\int_{0}^{T}\left\langle
F,Az\right\rangle\mathrm{d}t\right|\right.$ (5.36)
$\displaystyle+\left|\int_{0}^{T}\left\langle
F,Az_{t}\right\rangle\mathrm{d}t\right|+\left|\int_{0}^{T}\left\langle
BF,A\theta\right\rangle\mathrm{d}t\right|+\left|\int_{0}^{T}\left\langle
G,\widetilde{z}_{t}\right\rangle\mathrm{d}t\right|$
$\displaystyle+\left|\int_{0}^{T}\left\langle
G,A\widetilde{z}\right\rangle\mathrm{d}t\right|\left.+\left|\int_{0}^{T}\left\langle
G,A\widetilde{z}_{t}\right\rangle\mathrm{d}t\right|+\left|\int_{0}^{T}\left\langle
BG,A\widetilde{\theta}\right\rangle\mathrm{d}t\right|\right\\}.$

Step 4: We now need to estimate the integrals on the right-hand side (r.h.s.)
of Equation (5.36) (eight terms in the brackets) to get (5.14) for $s=3$. We
will be using the fact that

$H^{2}(\Omega)\hookrightarrow L^{\infty}(\Omega)\quad\mbox{and}\quad
H^{2}(\Omega)\hookrightarrow W^{1,4}(\Omega)\qquad\text{ for }d\in\\{2,3\\}.$
(5.37)

Step 4.1: The first four terms on the r.h.s. of (5.36). The embeddings in
Equation (5.37) together with Young’s inequality lead to an estimate of the
first term:

$\displaystyle\left|\int_{0}^{T}\left\langle
F,z_{t}\right\rangle\mathrm{d}t\right|$
$\displaystyle\leq\left|\int_{0}^{T}\left\langle
3z^{2}Az,z_{t}\right\rangle\mathrm{d}t\right|+\left|\int_{0}^{T}\left\langle
6z|\nabla z|^{2},z_{t}\right\rangle\mathrm{d}t\right|$ (5.38)
$\displaystyle\leq
C_{\epsilon}\int_{0}^{T}X^{3}(t)\mathrm{d}t+\epsilon\int_{0}^{T}X(t)\mathrm{d}t.$

Here, we noted $H^{2}(\Omega)\hookrightarrow W^{1,4}(\Omega)$. In general,
$W^{2,p}(\Omega)\hookrightarrow W^{1,4}(\Omega)$ for $p>{\frac{4d}{4+d}}$.
Since $d=2$ or $3$, we chose $p=2$. Similar arguments apply to the next two
terms with the following inequalities:

$\displaystyle\left|\int_{0}^{T}\left\langle
F,Az\right\rangle\mathrm{d}t\right|$ $\displaystyle\leq
C\int_{0}^{T}\big{\|}Az\big{\|}_{L^{2}(\Omega)}^{3}\big{\|}Az\big{\|}_{L^{2}(\Omega)}\mathrm{d}t\leq
C\int_{0}^{T}X^{2}(t)\mathrm{d}t,$ (5.39)
$\displaystyle\left|\int_{0}^{T}\left\langle
F,Az_{t}\right\rangle\mathrm{d}t\right|$ $\displaystyle\leq
C_{\epsilon}\int_{0}^{T}X^{3}(t)\mathrm{d}t+\epsilon\int_{0}^{T}X(t)\mathrm{d}t.$
(5.40)

Again, by a similar argument, using the continuity of the operator $B$ and
Equation (5.38), we can estimate the 4th term:

$\displaystyle\left|\int_{0}^{T}\left\langle
BF,A\theta\right\rangle\mathrm{d}t\right|$ $\displaystyle\leq
C\left|\int_{0}^{T}\big{\|}B(3z^{2}Az+6z|\nabla
z|^{2})\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t\right|+C\left|\int_{0}^{T}\big{\|}A\theta\big{\|}_{L^{2}(\Omega)}^{2}\right|$
(5.41) $\displaystyle\leq
C\int_{0}^{T}\big{\|}Az\big{\|}_{L^{2}(\Omega)}^{6}\mathrm{d}t+C\left[E_{2}(0)+\left|\int_{0}^{T}\left\langle
F,z_{t}\right\rangle\mathrm{d}t\right|\right]$ $\displaystyle\leq
CX(0)+C_{\epsilon}\int_{0}^{T}X^{3}(t)\mathrm{d}t+\epsilon\int_{0}^{T}X(t)\mathrm{d}t.$

Step 4.2: The highest order terms on the r.h.s. of (5.36). In order to
estimate the remaining four terms containing $G$, we rewrite $G=G_{1}+G_{2}$,
where

$G_{1}=-6zz_{t}Az-3z^{2}Az_{t}+6z_{t}|\nabla z|^{2}\quad\mbox{and}\quad
G_{2}=12z(\nabla z\cdot\nabla z_{t}).$

Therefore, the 7th term can be bounded as follows (after two integrations by
parts):

$\displaystyle\bigg{|}\int_{0}^{T}$ $\displaystyle\left\langle
G,A\widetilde{z}_{t}\right\rangle\mathrm{d}t\bigg{|}\leq\
\left|\int_{0}^{T}\left\langle
G_{1}(z),Az_{tt}\right\rangle\mathrm{d}t\right|+\left|\int_{0}^{T}\left\langle
G_{2}(z),Az_{tt}\right\rangle\mathrm{d}t\right|$ (5.42) $\displaystyle\leq\
\left|\left\langle
G_{1}(z),Az_{t}\right\rangle\Big{|}_{0}^{T}\right|+\left|\int_{0}^{T}\left\langle\partial_{t}G_{1}(z),Az_{t}\right\rangle\mathrm{d}t\right|+\left|\int_{0}^{T}\left\langle
G_{2}(z),Az_{tt}\right\rangle\mathrm{d}t\right|$
$\displaystyle\leq\tfrac{1}{2}\big{\|}G_{1}(z(0))\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\big{\|}Az_{t}(0)\big{\|}_{L^{2}(\Omega)}^{2}+C_{\epsilon}\big{\|}G_{1}(z(T))\big{\|}_{L^{2}(\Omega)}^{2}$
$\displaystyle+\epsilon\big{\|}Az_{t}(T)\big{\|}_{L^{2}(\Omega)}^{2}+\left|\int_{0}^{T}\left\langle\partial_{t}G_{1}(z),Az_{t}\right\rangle\mathrm{d}t\right|+\left|\int_{0}^{T}\left\langle
G_{2}(z),Az_{tt}\right\rangle\mathrm{d}t\right|.$

Step 4.2.1: The first four terms on the r.h.s. of (5.42).

$\displaystyle\big{\|}$ $\displaystyle
G_{1}(z(0))\big{\|}_{L^{2}(\Omega)}^{2}$ $\displaystyle\leq
6\big{\|}z(0)z_{t}(0)Az(0)\big{\|}_{L^{2}(\Omega)}^{2}+3\big{\|}z(0)^{2}Az_{t}(0)\big{\|}_{L^{2}(\Omega)}^{2}+6\big{\|}z_{t}(0)|\nabla
z(0)|^{2}\big{\|}_{L^{2}(\Omega)}^{2}$ $\displaystyle\leq
C\Big{\\{}\|z(0)\|_{H^{2}(\Omega)}^{6}+\|z_{t}(0)\|_{H^{2}(\Omega)}^{6}+\big{\|}Az(0)\big{\|}_{L^{2}(\Omega)}^{6}+\|z(0)\|_{H^{2}(\Omega)}^{8}$
$\displaystyle+\big{\|}Az_{t}(0)\big{\|}_{L^{2}(\Omega)}^{4}+\|z_{t}(0)\|_{H^{2}(\Omega)}^{4}+\|z(0)\|_{H^{2}(\Omega)}^{8}\Big{\\}}$
(5.43) $\displaystyle\leq
C\Big{\\{}\|z(0)\|_{H^{2}(\Omega)}^{2}+\|z_{t}(0)\|_{H^{2}(\Omega)}^{2}+\big{\|}Az(0)\big{\|}_{L^{2}(\Omega)}^{2}+\|z(0)\|_{H^{2}(\Omega)}^{2}+\big{\|}Az_{t}(0)\big{\|}_{L^{2}(\Omega)}^{2}$
$\displaystyle+\|z_{t}(0)\|_{H^{2}(\Omega)}^{2}+\|z(0)\|_{H^{2}(\Omega)}^{2}\Big{\\}}$
$\displaystyle\leq CX(0).$

Here, we used the ‘smallness’ assumption $X(0)<1$ from Equation (5.13). An
argument similar to Equation (5.43) yields

$\displaystyle C_{\epsilon}\big{\|}G_{1}(z(T))\big{\|}_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq C_{\epsilon}\big{\|}(-6zz_{t}Az-3z^{2}Az_{t}+6z_{t}|\nabla
z|^{2})(T)\big{\|}_{L^{2}(\Omega)}^{2}$ $\displaystyle\leq
C_{\epsilon}\Big{\\{}\|z(T)\|_{H^{2}(\Omega)}^{6}+\|z_{t}(T)\|_{H^{2}(\Omega)}^{6}+\big{\|}Az(T)\big{\|}_{L^{2}(\Omega)}^{6}$
(5.44)
$\displaystyle+\|z(T)\|_{H^{2}(\Omega)}^{8}+\big{\|}Az_{t}(T)\big{\|}_{L^{2}(\Omega)}^{4}+\|z_{t}(T)\|_{H^{2}(\Omega)}^{4}+\|z(T)\|_{H^{2}(\Omega)}^{8}\Big{\\}}$
$\displaystyle\leq\ C_{\epsilon}\big{[}X^{4}(T)+X^{6}(T)+X^{8}(T)\big{]}.$

Trivially,

$\tfrac{1}{2}\big{\|}Az_{t}(0)\big{\|}_{L^{2}(\Omega)}^{2}\leq
X(0)\quad\mbox{and}\quad\epsilon\big{\|}Az_{t}(T)\big{\|}_{L^{2}(\Omega)}^{2}\leq\epsilon
X(T).$ (5.45)

Step 4.2.2: The 5th terms on the r.h.s. of (5.42). Estimating
$\displaystyle\left|\int_{0}^{T}\left\langle\partial_{t}G_{1}(z),Az_{t}\right\rangle\mathrm{d}t\right|$
with $G_{1}=-6zz_{t}Az-3z^{2}Az_{t}+6z_{t}|\nabla z|^{2}$ and

$\partial_{t}G_{1}(z)=-6z_{t}^{2}Az-6zz_{tt}Az-12zz_{t}Az_{t}-3z^{2}Az_{tt}+6z_{tt}|\nabla
z|^{2}+12z_{t}(\nabla z\cdot\nabla z_{t})$ (5.46)

amounts to dealing with each of the respective six terms.

  1. (a)

First, we use Young’s inequality to write

$\left|\int_{0}^{T}\left\langle z_{t}^{2}Az,Az_{t}\right\rangle\right|\leq
C\int_{0}^{T}\|z_{t}\|_{H^{2}(\Omega)}^{2}\left(\big{\|}Az\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}Az_{t}\big{\|}_{L^{2}(\Omega)}^{2}\right)\leq
C\int_{0}^{T}X^{2}(t).$ (5.47)

  2. (b)

By Hölder’s inequality, choose $p=3,q=3/2$, we get

$\|a\cdot
b\|_{L^{2}(\Omega)}^{2}\leq\|a\|_{L^{6}(\Omega)}^{2}\cdot\|b\|_{L^{3}(\Omega)}^{2}.$
(5.48)

In bounded domains of $\mathbb{R}^{d}$ for $d=2,3$, we have
$H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)\hookrightarrow
L^{4}(\Omega)\hookrightarrow L^{3}(\Omega)$, i.e.,

$\|a\|_{L^{6}(\Omega)}\leq
C\|a\|_{H^{1}(\Omega)},\qquad\|a\|_{L^{4}(\Omega)}\leq
C\|a\|_{H^{1}(\Omega)},\qquad\|a\|_{L^{3}(\Omega)}\leq
C\|a\|_{H^{1}(\Omega)}.$

Hence,

$\displaystyle\bigg{|}\int_{0}^{T}$ $\displaystyle\left\langle
zz_{tt}Az,Az_{t}\right\rangle\mathrm{d}t\bigg{|}\leq
C\int_{0}^{T}\|z\|_{H^{2}(\Omega)}\left(\|z_{tt}\|_{H^{1}(\Omega)}^{2}\cdot\|Az\|_{H^{1}(\Omega)}^{2}+\big{\|}Az_{t}\big{\|}_{L^{2}(\Omega)}^{2}\right)\mathrm{d}t$
$\displaystyle\leq
C\int_{0}^{T}\Big{(}\|Az\|_{L^{2}(\Omega)}\|A^{1/2}z_{tt}\|_{L^{2}(\Omega)}^{2}\|A^{3/2}z\|_{L^{2}(\Omega)}^{2}+\|Az\|_{L^{2}(\Omega)}\big{\|}Az_{t}\big{\|}_{L^{2}(\Omega)}^{2}\Big{)}\mathrm{d}t$
$\displaystyle\leq
C\int_{0}^{T}\|Az\|_{L^{2}(\Omega)}^{3}\mathrm{d}t+C\int_{0}^{T}\|A^{1/2}z_{tt}\|_{L^{2}(\Omega)}^{6}\mathrm{d}t+C\int_{0}^{T}\|A^{3/2}z\|_{L^{2}(\Omega)}^{6}\mathrm{d}t$
$\displaystyle+\epsilon\int_{0}^{T}\|Az\|_{L^{2}(\Omega)}^{2}\mathrm{d}t+C_{\epsilon}\int_{0}^{T}\big{\|}Az_{t}\big{\|}_{L^{2}(\Omega)}^{4}\mathrm{d}t$
(5.49) $\displaystyle\leq
C\int_{0}^{T}\|Az\|_{L^{2}(\Omega)}^{3}\mathrm{d}t+C\int_{0}^{T}\|A^{1/2}z_{tt}\|_{L^{2}(\Omega)}^{6}\mathrm{d}t+C\Big{[}X(t)+X^{3}(t)\Big{]}^{3}\mathrm{d}t$
$\displaystyle+\epsilon\int_{0}^{T}\|Az\|_{L^{2}(\Omega)}^{2}\mathrm{d}t+C_{\epsilon}\int_{0}^{T}\big{\|}Az_{t}\big{\|}_{L^{2}(\Omega)}^{4}\mathrm{d}t$
$\displaystyle\leq
C\int_{0}^{T}X^{3/2}(t)\mathrm{d}t+C\int_{0}^{T}X^{3}(t)\mathrm{d}t+\epsilon\int_{0}^{T}X(t)\mathrm{d}t+C_{\epsilon}\int_{0}^{T}X^{2}(t)\mathrm{d}t.$

Here, we used (5) and assumption in Equation (5.15) implying $X^{k}(t)\leq
X(t)$ for $k\geq 1$.

  3. (c)

By Hölder’s inequality,

$\displaystyle\left|\int_{0}^{T}\left\langle
zz_{t}Az_{t},Az_{t}\right\rangle\mathrm{d}t\right|$ $\displaystyle\leq
C\int_{0}^{T}\|z\|_{L^{\infty}(\Omega)}\|z_{t}\|_{L^{\infty}(\Omega)}\big{\|}Az_{t}\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t$
(5.50)
$\displaystyle\leq\epsilon\int_{0}^{T}X(t)\mathrm{d}t+C_{\epsilon}\int_{0}^{T}X^{3}(t)\mathrm{d}t.$

  4. (d)

Since

$\partial_{t}\left\langle z^{2}Az_{t},Az_{t}\right\rangle=\left\langle
2zz_{t}Az_{t},Az_{t}\right\rangle+\left\langle
2z^{2}Az_{t},Az_{tt}\right\rangle,$

we have

$\displaystyle\bigg{|}\int_{0}^{T}$ $\displaystyle\left\langle
z^{2}Az_{tt},Az_{t}\right\rangle\mathrm{d}t\bigg{|}=\left|\tfrac{1}{2}\int_{0}^{T}\partial_{t}\left\langle
z^{2}Az_{t},Az_{t}\right\rangle\mathrm{d}t-\int_{0}^{T}\left\langle
zz_{t}Az_{t},Az_{t}\right\rangle\mathrm{d}t\right|$ (5.51) $\displaystyle\leq
C\left|\left\langle
z^{2}Az_{t},Az_{t}\right\rangle(T)\right|+C\left|\left\langle
z^{2}Az_{t},Az_{t}\right\rangle(0)\right|+C\left|\int_{0}^{T}\left\langle
zz_{t}Az_{t},Az_{t}\right\rangle\mathrm{d}t\right|$ $\displaystyle\leq
C\|z(T)\|_{H^{2}(\Omega)}^{2}\big{\|}Az_{t}(T)\big{\|}_{L^{2}(\Omega)}^{2}+C\|z(0)\|_{H^{2}(\Omega)}^{2}\big{\|}Az_{t}(0)\big{\|}_{L^{2}(\Omega)}^{2}$
$\displaystyle+C\int_{0}^{T}\|z\|_{L^{\infty}(\Omega)}\|z_{t}\|_{L^{\infty}(\Omega)}\big{\|}Az_{t}\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t$
$\displaystyle\leq
CX^{2}(0)+CX^{2}(T)+C\int_{0}^{T}\|Az\|_{2}\big{\|}Az_{t}\big{\|}_{L^{2}(\Omega)}^{3}\mathrm{d}t$
$\displaystyle\leq
CX(0)+CX^{2}(T)+\epsilon\int_{0}^{T}X(t)\mathrm{d}t+C_{\epsilon}\int_{0}^{T}X^{3}(t)\mathrm{d}t.$

Here, $X^{2}(0)\leq X(0)$ by Equation (5.13).

  5. (e)

Again, using Equation (5.48), we get

$\displaystyle\left|\int_{0}^{T}\left\langle z_{tt}|\nabla
z|^{2},Az_{t}\right\rangle\mathrm{d}t\right|$ $\displaystyle\leq
C_{\epsilon}\int_{0}^{T}\|z_{tt}\|_{L^{6}(\Omega)}^{2}\||\nabla
z|^{2}\|_{L^{3}(\Omega)}^{2}+\epsilon\int_{0}^{T}\big{\|}Az_{t}\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t$
(5.52) $\displaystyle\leq
C_{\epsilon}\int_{0}^{T}X^{2}(t)\mathrm{d}t+C_{\epsilon}\int_{0}^{T}X^{4}(t)\mathrm{d}t+\epsilon\int_{0}^{T}X(t)\mathrm{d}t.$

  6. (f)

Similarly,

$\left|\int_{0}^{T}\left\langle z_{t}(\nabla z,\nabla
z_{t}),Az_{t}\right\rangle\mathrm{d}t\right|\leq
C\int_{0}^{T}X^{3/2}(t)\mathrm{d}t+C\int_{0}^{T}X^{3/2}(t)\mathrm{d}t+C\int_{0}^{T}X^{3}(t)\mathrm{d}t.$
(5.53)

Now, collecting Equations (5.47)–(5.53) and recalling (5.46), we get

$\displaystyle\left|\int_{0}^{T}\left\langle\partial_{t}G_{1}(z),Az_{t}\right\rangle\mathrm{d}t\right|\leq$
$\displaystyle CX(0)+CX^{2}(T)+\epsilon\int_{0}^{T}X(t)\mathrm{d}t$ (5.54)
$\displaystyle+C_{\epsilon}\int_{0}^{T}\left[X^{3/2}(t)+X^{2}(t)+X^{3}(t)+X^{4}(t)\right]\mathrm{d}t.$

Step 4.2.3: The 6th (last) term on the r.h.s. of (5.42). The estimate is
produced in a similar fashion to Equation (5.53):

$\displaystyle\left|\int_{0}^{T}\left\langle
G_{2}(z),Az_{tt}\right\rangle\mathrm{d}t\right|$ $\displaystyle\leq
C\left|\int_{0}^{T}\|z\|_{H^{2}(\Omega)}\||\nabla
z|\|_{H^{2}(\Omega)}\left\langle
Az_{t},A^{1/2}z_{tt}\right\rangle\mathrm{d}t\right|$ (5.55) $\displaystyle\leq
C\int_{0}^{T}X^{3/2}(t)\mathrm{d}t+C\int_{0}^{T}X^{3/2}(t)\mathrm{d}t+C\int_{0}^{T}X^{3}(t)\mathrm{d}t.$

Step 4.3: The 5th and 6th term on the r.h.s. of (5.36). These are lower-order
terms compared to those from Step 4.2. Hence, we skip the details and just
state the final results:

$\displaystyle\left|\int_{0}^{T}\left\langle
G,\widetilde{z}_{t}\right\rangle\mathrm{d}t\right|$
$\displaystyle\leq\epsilon\int_{0}^{T}X(t)\mathrm{d}t+C_{\epsilon}\int_{0}^{T}\left[X^{3/2}(t)+X^{3}(t)+X^{4}(t)+X^{6}(t)\right]\mathrm{d}t,$
(5.56) $\displaystyle\left|\int_{0}^{T}\left\langle
G,A\widetilde{z}\right\rangle\mathrm{d}t\right|$
$\displaystyle\leq\epsilon\int_{0}^{T}X(t)\mathrm{d}t+C_{\epsilon}\int_{0}^{T}\left[X^{3/2}(t)+X^{3}(t)+X^{4}(t)+X^{6}(t)\right]\mathrm{d}t.$
(5.57)

Step 4.4: The 8th (last) term on the r.h.s. of (5.36). By an argument similar
to Equation (5.20), we get

$\int_{0}^{T}\big{\|}A\theta_{t}\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t\leq
E_{3}(0)+\int_{0}^{T}\left\langle G,z_{tt}\right\rangle\mathrm{d}t.$ (5.58)

Therefore,

$\displaystyle\bigg{|}$ $\displaystyle\int_{0}^{T}\left\langle
BG,A\widetilde{\theta}\right\rangle\mathrm{d}t\bigg{|}\leq
C\int_{0}^{T}\big{\|}G\big{\|}_{L^{2}(\Omega)}^{2}+CE_{3}(0)+C\int_{0}^{T}\left\langle
G,z_{tt}\right\rangle\mathrm{d}t$ (5.59) $\displaystyle\leq\
\epsilon\int_{0}^{T}X(t)\mathrm{d}t+CX(0)+C_{\epsilon}\int_{0}^{T}\left[X^{3/2}(t)+X^{2}(t)+X^{3}(t)+X^{4}(t)+X^{6}(t)\right]\mathrm{d}t.$

_Step 5:_ Plugging Equations (5.38)–(5.41), (5.55)–(5.57) and (5.59) into
(5.36), we finally estimate

$\displaystyle(1-\epsilon)X(T)$
$\displaystyle+(C_{1}-8\epsilon)\int_{0}^{T}X(t)\mathrm{d}t$
$\displaystyle\leq
C_{\epsilon}X(0)+C_{\epsilon}\int_{0}^{T}\left[X^{3/2}(t)+X^{2}(t)+X^{3}(t)+X^{4}(t)+X^{6}(t)\right]\mathrm{d}t$
$\displaystyle+C_{\epsilon}\left[X^{2}(T)+X^{4}(T)+X^{6}(T)+X^{8}(T)\right],$

that is,

$\displaystyle X(T)$ $\displaystyle+C_{1}\int_{0}^{T}X(t)\mathrm{d}t$ (5.60)
$\displaystyle\leq
C_{2}X(0)+C_{3}\int_{0}^{T}\left[X^{3/2}(t)+X^{2}(t)+X^{3}(t)+X^{4}(t)+X^{6}(t)\right]\mathrm{d}t$
$\displaystyle+C_{4}\left[X^{2}(T)+X^{4}(T)+X^{6}(T)+X^{8}(T)\right],$

which finishes the proof. ∎

######  Remark 5.4.

With Equation (5.60) at hand, we can now apply the standard ‘barrier method’
(cf. [6, Lemma 5.1, p 485]) to deduce the globality of the local solution,
whose existence is guaranteed by Theorem 4.5 (or Theorem 3.4) – _not_ in the
energy space (endowed with $\|\cdot\|_{X}$), but in the phase space (endowed
with $\|\cdot\|_{\mathcal{Z}_{s}\times\mathcal{T}_{s}}$) instead. Apparently,
$\max_{0\leq t\leq T}\|\cdot\|_{X}\leq\max_{0\leq t\leq
T}\|\cdot\|_{{\mathcal{Z}}_{3}\times{\mathcal{T}}_{3}}$ for any
$0<T<T_{\mathrm{max}}$. In Lemma 5.5 below, we will show a ‘reverse’
inequality, which is sufficient for a contradiction proof (see the proof of
Theorem 5.6 below). After the uniform stability of the energy is established,
a second lemma, i.e., Lemma 5.8 below, will be presented to show the
equivalence over the whole time half-line $[0,\infty)$. In the spirit of
Remark 5.1, we have:

######  Lemma 5.5 (Controlling $\max_{0\leq t\leq
T}\|\cdot\|_{{\mathcal{Z}}_{3}\times{\mathcal{T}}_{3}}$ in terms of
$\max_{0\leq t\leq T}\|\cdot\|_{X}$).

Assume a classical solution $(z,\theta)$ to Equations (5.2a)–(5.2d) over a
time interval $[0,T_{\max})$ satisfies the smallness condition
$E_{2}(t)<\epsilon_{1}$ from Equation (5.8), then, there holds for any
$T\in(0,T_{\max})$:

$\max_{0\leq t\leq
T}\|(z,\theta)\|_{{\mathcal{Z}}_{3}\times{\mathcal{T}}_{3}}^{2}\leq
C\max_{0\leq t\leq
T}\big{(}\|(z,\theta)\|_{X}^{2}+\|(z,\theta)\|_{X}^{6}\big{)}\text{ for some
}C=C(T)>0.$

###### Proof.

It suffices to consider the four highest energy terms:
$\big{\|}A^{3/2}z\big{\|}_{L^{2}(\Omega)},\big{\|}z_{ttt}\big{\|}_{L^{2}(\Omega)},\big{\|}A^{2}\theta\big{\|}_{L^{2}(\Omega)}$
and $\big{\|}A^{1/2}\theta_{tt}\big{\|}_{L^{2}(\Omega)}$. The first one, as
shown in Lemma 5.2, is bounded by
$C\big{(}\|(z,\theta)\|_{X}+\|(z,\theta)\|_{X}^{3}\big{)}$ for any $t\geq 0$.
Using Equations (5.2a)–(5.2b), the second and the third terms can be
controlled by appropriate lower order terms. Indeed, applying $\partial_{t}$
and $A$ to Equations (5.2a) and (5.2b), respectively, and exploiting the
bounded invertibility of $(A^{-1}+\gamma)$, we estimate

$\displaystyle\big{\|}z_{ttt}\big{\|}_{L^{2}(\Omega)}$ $\displaystyle\leq
C\big{(}\big{\|}Az_{t}\big{\|}_{L^{2}(\Omega)}+\big{\|}A\theta_{t}\big{\|}_{L^{2}(\Omega)}+\big{\|}F^{\prime}\big{\|}_{L^{2}(\Omega)}\big{)}\leq
C\big{(}\|(z,\theta)\|_{X}+\|(z,\theta)\|_{X}^{3}\big{)},$ (5.61)
$\displaystyle\big{\|}A^{2}\theta\big{\|}_{L^{2}(\Omega)}$ $\displaystyle\leq
C\big{(}\big{\|}A\theta_{t}\big{\|}_{L^{2}(\Omega)}+\big{\|}A\theta\big{\|}_{L^{2}(\Omega)}+\big{\|}Az_{t}\big{\|}_{L^{2}(\Omega)}\big{)}\leq
C\|(z,\theta)\|_{X}$ (5.62)

for any $t\geq 0$, which remains true after passing to supremum. The last term
is treated in the same fashion as in the proof of Theorem A.6. Estimating

$\|z_{ttt}\|_{L^{2}(0,T;L^{2}(\Omega))}^{2}\leq\int_{0}^{T}\big{(}X(t)+X^{3}(t)\big{)}\mathrm{d}t\leq
T\Big{(}1+\max_{0\leq t\leq T}X^{2}(t)\Big{)}\max_{0\leq t\leq T}X(t)$ (5.63)

via (5.61) and exploiting the maximal $L^{2}$-regularity of $A$ on $(0,T)$
applied to Equation (5.2b) differentiated twice in time, the desired estimate
follows. ∎

######  Theorem 5.6 (Global Existence).

Let Assumption 4.4 be satisfied for some $s\geq 3$. Then, there exists a
positive number $\epsilon$ such that for any initial data satisfying
$X(0)<\epsilon$ (which roughly means the smallness of
$\|z^{0}\|_{H^{3}(\Omega)}^{2}+\|z^{1}\|_{H^{2}(\Omega)}^{2}+\|\theta^{0}\|_{H^{4}(\Omega)}^{2}$),
the associated local solution of system (5.2a)–(5.2d) from Theorem 4.5 exists
globally, namely, $T_{\mathrm{max}}=\infty$.

###### Proof.

Without loss of generality, assume $X(0)>0$. Indeed, if $X(0)=0$, the only
solution to (5.2a)–(5.2d) is the trivial one and, therefore, exists globally.

We argue by contradiction. Assume $T_{\max}<\infty$.

_Step 0:_ Let

$\epsilon=\min\\{1,\epsilon_{1},\epsilon_{2}\\},$ (5.64)

where $\epsilon_{1}$ is given in Equation (5.8) in Lemma 5.2 and
$\epsilon_{2}$ is defined in Step 1 below. Note that due to the validness of
Equation (5.8) to be established below (cf. proof of Step 3), the
hyperbolicity of Equation (5.2a) is satisfied for all times and the solution
can only cease to exist if $X(t)$ blows-up.

By Theorem 4.5, there exists a (local) solution on the maximal interval
$[0,T_{\max})$. Define

$\displaystyle k(x)$ $\displaystyle=x-C_{4}(x^{2}+x^{4}+x^{6}+x^{8})\text{ and
}$ $\displaystyle h(x)$
$\displaystyle=C_{1}x-C_{3}\left[x^{3/2}+x^{2}+x^{3}+x^{4}+x^{9/2}+x^{6}+x^{9}\right],$

where $C_{1},C_{3},$ and $C_{4}$ come from Equation (5.60). Since
$X(0)<\epsilon\leq 1$, Equation (5.13) is satisfied. Therefore, the estimate
in Equation (5.14), or (5.60), holds and can be rewritten as

$k\left(X(T)\right)+\int_{0}^{T}h(X(t))\mathrm{d}t\leq C_{2}X(0)\text{ for
}0\leq T<T_{\mathrm{max}}.$ (5.65)

Further, we observe that the (algebraic) equation $k(x)=0$ has a unique
positive solution denoted by $\eta$. There also holds $k(x)>0$ for
$x\in[0,\eta)$. Similarly, $h(x)=0$ has a unique positive solution denoted by
$\xi$. Besides, $h(x)>0$ for $x\in[0,\xi)$.

_Step 1: Small initial data $X(0)$._ Consider the inequality

$k(y)\leq C_{2}X(0).$

Due to the continuity of $k$, for small $X(0)$, the above inequality implies
$y\in[0,\delta_{1}]\cup[\delta_{2},\infty)$, where $\delta_{1}\to 0$ and
$\delta_{2}\to\eta$ as $X(0)\to 0$. Therefore, there exists an
$\epsilon_{2}>0$ such that if $X(0)<\epsilon_{2}$, the following hold true:

$\displaystyle X(0)<\epsilon_{2}$ $\displaystyle<\delta_{2},$ (5.66)
$\displaystyle\delta_{1}$ $\displaystyle<\xi,$ (5.67)
$\displaystyle\delta_{1}$ $\displaystyle<\epsilon_{1}\text{ and }$ (5.68)
$\displaystyle\epsilon_{2}$ $\displaystyle<\xi\mbox{ so that
}h\big{[}X(0)\big{]}>0.$

_Step 2: Barrier method._ We claim

$h(X(t))>0\mbox{ for any }t\geq 0.$ (5.69)

If this is not the case, by the continuity of $h\circ X$, there is a $T^{*}>0$
such that $h\big{[}X(T^{*})\big{]}=0$. Hence,

$X(T^{*})=\xi.$ (5.70)

On the other hand, the above assumption also suggests $h\big{[}X(t)\big{]}\geq
0$ for any $t\in[0,T^{*}]$. After invoking this in Equation (5.65), we arrive
at $k\big{[}X(t)\big{]}\leq C_{2}X(0)$ for any $t\in[0,T^{*}]$. By Step 1,
$X(t)\in[0,\delta_{1}]\cup[\delta_{2},\infty)$ for any $t\in[0,T^{*}]$.
However, by continuity of $X$ and Equation (5.66), we can eliminate the second
interval and reduce the inclusion to

$X(t)\in[0,\delta_{1}]\mbox{ for any }t\in[0,T^{*}].$

More specifically, $X(T^{*})\leq\delta_{1}<\xi$ by Equation (5.67), which
contradicts (5.70).

_Step 3: Uniform boundedness of $X(t)$._ Exploiting Equations (5.69) and
(5.65), we get $k\big{[}X(t)\big{]}\leq C_{2}X(0)$ for any $t\geq 0$, which
leads to the global bound on $X(t)$

$X(t)\leq\delta_{1}\mbox{ for any }t\geq 0$ (5.71)

by a similar argument as above. In particular,
$X(T)\leq\delta_{1}\leq\epsilon_{1}$ by Equation (5.68). Equation (5.71) meets
the condition of Lemma 5.5. Hence, after a possible rescaling, Assumption 4.4
is satisfied by $(z,z_{t},\theta)(T_{\mathrm{max}},\cdot)$. Therefore, Theorem
4.5 implies the solution exists on $[T_{\mathrm{max}},T^{\prime})$ for some
$T^{\prime}>T_{\mathrm{max}}$, which contradicts the maximality of
$[0,T_{\mathrm{max}})$. ∎

######  Corollary 5.7 (Uniform Stability).

Under the assumptions of Theorem 5.6 with $X(0)<\tilde{\epsilon}$ for some
positive number $\tilde{\epsilon}$ (possibly smaller than $\epsilon$ from
Equation (5.64) of Theorem 5.6), there exist positive constants $C$ and $k$
such that

$X(t)\leq e^{-kt}CX(0)\text{ for }t\geq 0.$ (5.72)

###### Proof.

We rewrite Equation (5.60) again as follows:

$\displaystyle X(T)$
$\displaystyle\left\\{1-C_{4}\left[X(T)+X^{3}(T)+X^{5}(T)+X^{7}(T)\right]\right\\}$
(5.73)
$\displaystyle+\int_{0}^{T}X(t)\big{\\{}1-C_{4}\big{[}X^{1/2}(t)+X^{1}(t)+X^{2}(t)+X^{3}(t)$
$\displaystyle+X^{7/2}(t)+X^{5}(t)+X^{8}(t)\big{]}\big{\\}}\mathrm{d}t$
$\displaystyle\leq C_{2}X(0).$

Choosing a bound $\epsilon_{3}$ on $X(0)$ small enough, we can make the global
bound $\delta_{1}$ of $X(t)$ satisfy

$\displaystyle
1-C_{4}\left[\delta_{1}+\delta_{1}^{3}+\delta_{1}^{5}+\delta_{1}^{7}\right]$
$\displaystyle\geq\tfrac{1}{2},$ $\displaystyle
1-C_{4}\left[\delta_{1}^{1/2}+\delta_{1}^{1}+\delta_{1}^{2}+\delta_{1}^{3}+\delta_{1}^{7/2}+\delta_{1}^{5}+\delta_{1}^{8}\right]$
$\displaystyle\geq\tfrac{1}{2}.$

This together with Equation (5.73) implies

$X(T)+\int_{0}^{T}X(t)\mathrm{d}t\leq 2C_{2}X(0),$ (5.74)

which gives $X(T)\leq 2C_{2}X(0)$ for any $T>0$. Now we impose the final
assumption on $X(0)$. Recall the number $\epsilon$ from (5.64) and let

$\tilde{\epsilon}=\min\left\\{\tfrac{\epsilon}{2C_{2}},\epsilon_{3}\right\\}.$
(5.75)

Since $X(0)<\tilde{\epsilon}\leq\tfrac{\epsilon}{2C_{2}}$, then
$X(t)\leq\epsilon$ for any $t>0$. Thus, Equation (5.74) can be extended to

$X(T)+\int_{s}^{T}X(t)\mathrm{d}t\leq 2C_{2}X(s)$ (5.76)

for any $s\in(0,T]$. Hence,

$X(t)\geq\tfrac{1}{2C_{2}}X(T)\text{ for }t\in[0,T].$ (5.77)

Combining Equation (5.77) with (5.74), we get $X(T)+\tfrac{T}{2C_{2}}X(T)\leq
2C_{2}X(0)$. Therefore,

$X(T)\leq\frac{1}{1+\frac{T}{2C_{2}}}X(0)\mbox{ for any }T>0.$ (5.78)

By choosing $T$ large enough, we get

$X(T)\leq\kappa X(0)\mbox{ for some }\kappa<1.$ (5.79)

Repeating the procedure on $[T,2T]$, $[2T,3T]$, etc., we arrive at

$X(t)\leq\kappa^{\lceil t/T\rceil}X(0)\leq\kappa^{t/T}X(0)\leq
e^{-\big{(}|\ln(\kappa)|/T\big{)}t}X(0)\text{ for }t\geq 0,$

which finishes the proof. ∎

We have now proved all main results stated in Section 3 in the energy space
associated with $\sup_{0\leq t<\infty}X(t)$ from Equation (5.7). As announced
in Remark 5.1 and stated in Lemma 5.8 below, the Banach space generated by the
energy (supremum) $\sup_{0\leq t<\infty}X(t)$ is isomorphic to the solution
space in Equation (5.3) from our Theorem 4.5 for $s=3$ when the initial data
are sufficiently small (by virtue of Equation (5.64)).

######  Lemma 5.8 (Equivalence of $\sup_{0\leq
t<\infty}\|\cdot\|_{{\mathcal{Z}}_{3}\times{\mathcal{T}}_{3}}$ and
$\sup_{0\leq t<\infty}\|\cdot\|_{X}$).

If a classical solution $(z,\theta)$ to Equations (5.2a)–(5.2d) is global,
satisfies the smallness condition $E_{2}(t)<\epsilon_{1}$ from Equation (5.8)
and decays exponentially as in Equation (5.72), both norms mentioned above are
equivalent:

$c_{1}\sup_{0\leq
t<\infty}\|(z,\theta)\|_{X}\leq\|(z,\theta)\|_{{\mathcal{Z}}_{3}\times{\mathcal{T}}_{3}}\leq
c_{2}\sup_{0\leq t<\infty}\|(z,\theta)\|_{X}\quad\text{ for some }c_{1}\mbox{
and }c_{2}>0.$ (5.80)

###### Proof.

The former inequality is trivial. For the latter one, in contrast to Lemma
5.5, all superlinear terms are linearly dominated because they are bounded by
1, and we are only left to show Equation (5.63) with a constant independent of
$T$. However, due to the exponential decay of $X(t)$ (Equation (5.72)), (5.63)
becomes

$\|z_{ttt}\|_{L^{2}(0,\infty;L^{2}(\Omega))}^{2}\leq\tilde{C}\int_{0}^{\infty}X(t)\mathrm{d}t\leq\tilde{C}C\int_{0}^{\infty}e^{-kt}X(0)\mathrm{d}t\leq
c_{2}X(0)\leq c_{2}\sup_{0\leq t<\infty}X(t).$ (5.81)

With $A$’s maximal $L^{2}$-regularity on $(0,\infty)$, the estimate for
$\sup_{0\leq t<\infty}\big{\|}A^{1/2}\theta_{tt}\big{\|}_{L^{2}(\Omega)}$
follows. ∎

Since the smallness assumption is satisfied in both Equation (5.64) of Theorem
5.6 and Equation (5.75) of Corollary 5.7, we resubstitute $w=A^{-1}z$ and
conclude with the desired results Theorem 3.5 and 3.6.

##  Appendix A Existence Theory for Linear Evolution Equations

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain with a $C^{s}$-boundary
$\partial\Omega$ for some $s\geq\lfloor\tfrac{d}{2}\rfloor+2$ and let $T>0$ be
arbitrary, but fixed. The following well-posedness results are based on Kato’s
solution theory [9] for abstract time-dependent evolution equations and its
improved version presented by Jiang and Racke in [8, Appendix A] as well as
maximal $L^{p}$-regularity theory (see, e.g., [10]).

Thoughout this appendix and in the proof of Theorem 4.5, we employ the
following notation. For $n\geq 0$, we define

$\bar{D}^{n}:=\big{(}(\partial_{t},\nabla)^{\alpha}\,|\,0\leq|\alpha|\leq
n\big{)}\text{ and }H^{0}_{0}(\Omega)\equiv H^{0}(\Omega):=L^{2}(\Omega).$

Let $\phi_{\delta}\colon\mathbb{R}\to[0,\infty)$ denote the one-dimensional
Friedrichs’ mollifier with a ‘bandwidth’ $\delta>0$. For an $L^{1}$-function
$z\colon[0,T]\times\Omega\to\mathbb{R}$, we let

$z_{\delta}(t,\cdot)=\int_{0}^{T}\phi_{\delta}(t-s)z(s,\cdot)\mathrm{d}s\text{
for }t\in[0,T]\text{ in }\Omega.$

For details on approximation properties of mollifiers, we refer to [16,
Chapters 8 and 9]. The following result is known from [8, Lemma A.12].

######  Lemma A.1.

Let $a\in C^{1}\big{(}[0,T],L^{\infty}(\Omega)\big{)}$, $v\in
C^{0}\big{(}[0,T],L^{2}(\Omega)\big{)}$ and $w\in
L^{2}\big{(}0,T;H^{-1}(\Omega)\big{)}$. Then, for any sufficiently small
$\varepsilon>0$, there holds

$\begin{split}\int_{\varepsilon}^{T-\varepsilon}\big{\|}\partial_{t}\big{(}(av)_{\delta}(t,\cdot)-av_{\delta}(t,\cdot)\big{)}\big{\|}_{L^{2}(\Omega)}^{2}\mathrm{d}t&\to
0\text{ and }\\\
\int_{\varepsilon}^{T-\varepsilon}\|w_{\delta}(t,\cdot)\|_{H^{-1}(\Omega)}^{2}\mathrm{d}t&\to\int_{\varepsilon}^{T-\varepsilon}\|w(t,\cdot)\|_{H^{-1}(\Omega)}^{2}\mathrm{d}t\text{
as }\delta\to 0.\end{split}$

###  A.1 Linear Wave Equation

We consider a general linear wave equation with time- and space-dependent
coefficients:

$\displaystyle
z_{tt}(t,x)-\bar{a}_{ij}(t,x)\partial_{x_{i}}\partial_{x_{j}}z(t,x)$
$\displaystyle=\bar{f}(t,x)\phantom{0}\text{ for }(t,x)\in(0,T)\times\Omega,$
(A.1a) $\displaystyle z(t,x)$ $\displaystyle=0\phantom{\bar{f}(t,x)}\text{ for
}(t,x)\in[0,T]\times\partial\Omega,$ (A.1b) $\displaystyle
z(0,x)=z^{0}(x),\quad z_{t}(0,x)$ $\displaystyle=z^{1}(x)\phantom{00}\text{
for }x\in\Omega.$ (A.1c)

######  Assumption A.2.

Let $s\geq\lfloor\frac{d}{2}\rfloor+2$ be a fixed integer and let
$\gamma_{0},\gamma_{1}$ be positive numbers. Assume the following conditions
are satisfied.

  1. 1.

_Coefficient symmetry_ : $\bar{a}_{ij}(t,x)=\bar{a}_{ji}(t,x)$ for
$(t,x)\in[0,T]\times\bar{\Omega}$.

  2. 2.

_Coefficient regularity_ : $\bar{a}_{ij}\in
C^{0}\big{(}[0,T]\times\bar{\Omega}\big{)}$ and

$\partial_{x_{k}}\bar{a}_{ij}\in
L^{\infty}\big{(}0,T;H^{s-1}(\Omega)\big{)},\quad\partial_{t}^{m}\bar{a}_{ij}\in
L^{\infty}\big{(}0,T;H^{s-1-m}(\Omega)\big{)}$

for $m=1,2,\dots,s-1$.

  3. 3.

_Coercivity_ : For $z\in H^{1}_{0}(\Omega)$ and $t\in[0,T]$,

$\|z\|_{H^{1}(\Omega)}^{2}\leq\gamma_{0}\Big{(}\langle\bar{a}_{ij}\partial_{x_{i}}z,\partial_{x_{j}}z\rangle_{L^{2}(\Omega)}+\|z\|^{2}_{L^{2}(\Omega)}\Big{)}.$

  4. 4.

_Elliptic regularity_ : For $m=0,1,\dots,s-2$, $z(t,\cdot)\in
H^{1}_{0}(\Omega)$ and
$\bar{a}_{ij}(t,\cdot)\partial_{x_{i}}\partial_{x_{j}}z(t,\cdot)\in
H^{m}(\Omega)$ for a.e. $t\in[0,T]$ implies $u(t,\cdot)\in H^{m+2}(\Omega)$
and

$\|z(t,\cdot)\|_{H^{m}(\Omega)}\leq\gamma_{1}\Big{(}\|\bar{a}_{ij}(t,\cdot)\partial_{x_{i}}\partial_{x_{j}}z(t,\cdot)\|_{H^{m}(\Omega)}+\|z(t,\cdot)\|_{L^{2}(\Omega)}\Big{)}\text{
for a.e. }t\in[0,T].$

  5. 5.

_Right-hand side regularity_ : For $m=0,1,\dots,s-2$,

$\partial_{t}^{m}\bar{f}\in
C^{0}\big{(}[0,T],H^{s-2-m}(\Omega)\big{)},\quad\partial_{t}^{s-1}\bar{f}\in
L^{2}(0,T;L^{2}(\Omega)\big{)}.$

  6. 6.

_Compatibility conditions_ : For $m=0,1,\dots,s-1$,

$\bar{z}^{m}\in H^{s-m}(\Omega)\cap H^{1}_{0}(\Omega),\quad\bar{z}^{s}\in
L^{2}(\Omega),$

where $\bar{z}^{m}$ is recursively defined by

$\begin{split}\bar{z}^{0}(x)&=z^{0}(x),\quad\bar{z}^{1}(x)=z^{1}(x),\\\
\bar{z}^{m}(x)&=\Big{(}\sum_{n=0}^{m-2}{m-2\choose
n}\partial_{t}^{n}\bar{a}_{ij}\partial_{x_{i}}\partial_{x_{j}}\bar{z}^{m-2-n}+\partial_{t}^{m-2}\bar{f}_{i}\Big{)}(0,x)\text{
for }m\geq 2\end{split}$

for $x\in\Omega$.

Note that Assumption A.2.2 differs from [8, Assumption A.2.1.1]. This extra
regularity for $\bar{a}_{ij}$ will enable us to prove our a priori estimate at
an energy level which is one order lower than in [8, Theorem A.13].

######  Theorem A.3.

Under Assumption A.2, the initial boundary value problem (A.1a)-(A.1c)
possesses a unique classical solution, which satisfies

$z\in\bigcap_{m=0}^{s-1}C^{m}\big{(}[0,T],H^{s-m}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}\cap C^{s}\big{(}[0,T],L^{2}(\Omega)\big{)}.$

Moreover, for $d\in\\{2,3\\}$, letting

$\begin{split}\phi_{0}&=\|\bar{a}_{ij}(0,\cdot)\|_{L^{\infty}(\Omega)}+\|\partial_{x_{k}}\bar{a}_{ij}(0,\cdot)\|_{H^{s-1}(\Omega)},\\\
\phi&=\sup_{0\leq t\leq
T}\Big{(}\|\bar{a}_{ij}(t,\cdot)\|_{L^{\infty}(\Omega)}+\|\partial_{x_{k}}\bar{a}_{ij}(t,\cdot)\|_{H^{s-1}(\Omega)}+\sum_{m=1}^{s-1}\|\partial_{t}^{m}\bar{a}_{ij}(t,\cdot)\|_{H^{s-1-m}(\Omega)}\Big{)},\end{split}$

there exists a positive number $K_{1}$, which is a continuous function of
$\phi_{0}$, $\gamma_{0}$ and $\gamma_{1}$, and a positive number $K_{2}$,
which continuously depends on $\phi$, $\gamma_{0}$ and $\gamma_{1}$, such that

$\sup_{0\leq t\leq T}\|\bar{D}^{s}z(t,\cdot)\|_{L^{2}(\Omega)}^{2}\leq
K_{1}\Lambda_{0}\exp\big{(}K_{2}T^{1/2}(1+T^{1/2}+T+T^{3/2})\big{)},$

where

$\Lambda_{0}:=\sum_{m=0}^{s}\|\bar{z}\|_{H^{s-m}(\Omega)}^{2}+(1+T)\sup_{0\leq
t\leq
T}\big{\|}\bar{D}^{s-2}\bar{f}(t,\cdot)\big{\|}_{L^{2}(\Omega)}+T^{1/2}\int_{0}^{T}\|\partial_{t}^{s-1}\bar{f}(t,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}t.$

###### Proof.

Our proof is based on an abstract well-posedness and regularity result [8,
Theorems A.3 and A.9].
Existence and uniqueness at basic regularity level. Similar to the proof of
[8, Theorem A.11], we define for $t\in[0,T]$ a bounded linear operator

$A(t):=\begin{pmatrix}0&-1\\\
-\bar{a}_{ij}(t,\cdot)\partial_{x_{i}}\partial_{x_{j}}&0\end{pmatrix}\colon
Y_{1}\longrightarrow X_{0},$ (A.2)

where the Hilbert space $X_{0}:=H^{1}_{0}(\Omega)\times L^{2}(\Omega)$ is
equipped with the standard inner product induced by the product topology,
whereas the inner product on the Hilbert space $Y_{1}:=H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)$ reads as

$\langle
V,\bar{V}\rangle_{t}:=\big{\langle}\bar{a}_{ij}(t,\cdot)\partial_{x_{i}}z,\partial_{x_{j}}\bar{z}\big{\rangle}_{L^{2}(\Omega)}+\langle
y,\bar{y}\rangle_{L^{2}(\Omega)}\quad$ (A.3)

for $V=(z,y)$ and $\bar{V}=(\bar{z},\bar{y})\in X_{0}$. Due to uniform
coercivity of $\bar{a}_{ij}$ and by virtue of Poincaré-Friedrichs’ inequality,
each of the norms induced by $\langle\cdot,\cdot\rangle_{t}$ for any
$t\in[0,T]$ is equivalent to the standard norm on $X_{0}$. With this notation,
letting $V:=(z,\partial_{t}z)$, Equations (A.1a)–(A.1c) can be rewritten as an
abstract Cauchy problem

$\partial_{t}V(t)+A(t)V(t)=F(t)\text{ in }(0,T),\quad V(0)=V^{0}$ (A.4)

with $F=(0,\bar{f})$ and $V^{0}=(z^{0},z^{1})$.

We want to show that the triple $\big{(}A;X_{0},Y_{1}\big{)}$ is a CD-system
in sense of [8, Section A.1]. For $t\in[0,T]$, consider the elliptic problem

$\big{(}A(t)+\lambda\big{)}V=F\text{ with }F\in X_{0}.$

Recalling Assumption A.2.3, Lemma of Lax & Milgram implies the resolvent
estimate

$\big{\|}\big{(}A(t)+\lambda\big{)}^{-1}\big{\|}_{L(X_{0})}\leq\tfrac{1}{\lambda-C}\text{
for }\lambda>\beta\text{ for some constants }\beta,C>0,$ (A.5)

where we used Assumption A.2.2 and Sobolev’s imbedding theorem to deduce

$a_{ij}(t,\cdot)\in W^{1,\infty}(\Omega)\text{ for any }t\in[0,T].$

The continuity of the bilinear form follows similarly. By standard elliptic
regularity theory applied to $A(t)$, which is possible because of Assumption
A.2.2 and A.2.4 as well as $C^{s}$-smoothness of $\partial\Omega$, the maximal
domain of $A(t)$ coincides with $Y_{1}$. Hence, the operator $A(t)$ is closed.
This along with Equation (A.5) implies
$(\beta,\infty)\subset\rho\big{(}A(t)\big{)}$. Therefore,
$\big{(}A(t);t\in[0,T]\big{)}$ is a stable family of infinitesimal negative
generators of $C_{0}$-semigroups on $X_{0}$ with stability constants
$1,\beta$. Taking into account regularity conditions from Assumption A.2.5, we
can apply [8, Theorem A.3], we get a unique classical solution

$V\in C^{0}\big{(}[0,T],Y_{1}\big{)}\cap C^{1}\big{(}[0,T],X_{0}\big{)}$

at the at basic regularity level, which is equivalent to

$z\in C^{2}\big{(}[0,T],L^{2}(\Omega)\big{)}\cap
C^{1}\big{(}[0,T],H^{1}_{0}(\Omega)\big{)}\cap
C^{0}\big{(}[0,T],H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\big{)}.$

Higher regularity. For the proof of higher solution regularity, we consider
the following increasing double scale $(X_{j},Y_{j})$ of Hilbert spaces

$\begin{split}X_{j}&=\big{(}H^{j+1}(\Omega)\cap H^{1}_{0}(\Omega)\big{)}\times
H^{j}(\Omega)\text{ for }j\geq 1,\\\ Y^{j}&=\big{(}H^{j+1}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}\times\big{(}H^{j}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}\text{ for }j\geq 1,\quad Y_{0}=X_{0}.\end{split}$

By virtue of Equation (A.2), the condition

$\partial_{t}A\in\mathrm{Lip}\big{(}[0,T],L(Y_{j+r+1},X_{j})\big{)}\text{ for
}j=0,\dots,s-r-1\text{ and }r=0,\dots,s-2$

is equivalent to

$\partial_{t}^{r}\bar{a}_{ij}(t,\cdot)\partial_{x_{i}}\partial_{x_{j}}\in\mathrm{Lip}\big{(}[0,T],L\big{(}H^{j+r+2}(\Omega)\cap
H^{1}_{0}(\Omega),H^{j}(\Omega)\big{)}\big{)}$ (A.6)

for $j=0,\dots,s-r-1$ and $r=0,\dots,s-2$, while the latter is a direct
consequence of Assumption A.2.2 and Sobolev imbedding theorem due to the fact
$H^{\lfloor d/2\rfloor+1}(\Omega)\hookrightarrow L^{\infty}(\Omega)$.
Similarly, exploiting Assumption A.2.4, one can easily verify for
$j=0,\dots,s-2$ and $\phi\in Y_{1}$ and a.e. $t\in[0,T]$ that $A(t)\phi\in
X_{j}$ implies

$\phi\in Y_{j+1}\text{ and }\|\phi\|_{Y_{j+1}}\leq
K\big{(}\|A(t)\phi\|_{X_{j}}+\|\phi\|_{X_{0}}\big{)}\text{ for some constant
}K>0,$

which does not depend on $\phi$. Further, Assumption A.2.5 yields

$\partial_{t}F\in C^{0}\big{(}[0,T],X_{s-1-k}\big{)}\text{ for
}k=0,\dots,s-2\text{ and }\partial_{t}^{s-1}F\in L^{1}(0,T;X_{0}).$

Finally, Assumption A.2.6 implies compatibility conditions in sense of [8,
Equations (A.8) and (A.9)]. Hence, applying [8, Theorem A.9] at the energy
level $s-1$, we obtain additional regularity for the classical solution
satisfying

$V\in\bigcap_{m=0}^{s-1}C^{m}\big{(}[0,T],Y_{s-1-m}\big{)}.$

Rewriting $z$ in terms of $V$, this yields the desired regularity for $z$.
Energy estimates. For $n=1,\dots,s-1$, applying the
$\partial_{t}^{n-1}$-operator to Equation (A.1a), we obtain a linear wave
equation for $\partial_{t}^{n-1}z$ reading as

$\partial_{t}^{2}\big{(}\partial_{t}^{n-1}z\big{)}-\bar{a}_{ij}\partial_{x_{i}}\partial_{x_{j}}\big{(}\partial_{t}^{n-1}z\big{)}=h^{n-1}\text{
in }(0,\infty)\times\Omega,$ (A.7)

where we used Leibniz’ rule to compute

$h^{n-1}=\partial_{t}^{n-1}\bar{f}+\sum_{m=1}^{n-1}\binom{n-1}{m}\big{(}\partial_{t}^{m}\bar{a}_{ij}\big{)}\partial_{x_{i}}\partial_{x_{j}}\partial_{t}^{n-1-m}z.$
(A.8)

Multiplying Equation (A.7) in $L^{2}(\Omega)$ with $\partial_{t}^{n}z$,
applying Green’s formula and using Young’s inequality, we obtain the estimate

$\begin{split}\tfrac{1}{2}\partial_{t}\big{(}\|\partial_{t}^{n}&z(t,\cdot)\|_{L^{2}(\Omega)}^{2}+\|\bar{a}(t,\cdot)\nabla\partial_{t}^{n-1}z(t,\cdot)\|_{L^{2}(\Omega)}^{2}\big{)}\\\
&\leq\tfrac{1}{2}\big{\|}\big{(}\partial_{x_{i}}\bar{a}_{ij}(t,\cdot)\big{)}\partial_{x_{j}}z(t,\cdot)\|_{L^{2}(\Omega)}^{2}+\|\partial_{t}^{n}z(t,\cdot)\|_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\|h^{n-1}(t,\cdot)\|_{L^{2}(\Omega)}^{2}.\end{split}$

Integrating w.r.t. to $t$ over $[0,T]$, exploiting Assumption A.2.3 and
recalling the definition of $\phi_{0}$, we get

$\begin{split}\|\partial_{t}^{n}z(t,\cdot)\|_{L^{2}(\Omega)}^{2}+&\|\partial_{t}^{n-1}z(t,\cdot)\|_{H^{1}(\Omega)}^{2}\leq
C(\gamma_{0},\phi_{0})\big{(}\|\bar{z}^{n}\|_{L^{2}(\Omega)}^{2}+\|\bar{z}^{n-1}\|_{H^{1}(\Omega)}^{2}\big{)}\\\
&+C(\gamma_{0},\phi)\int_{0}^{t}\big{(}\|\partial_{t}^{n}z(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}+\|\partial_{t}^{n-1}z(\tau,\cdot)\|_{H^{1}(\Omega)}^{2}\big{)}\mathrm{d}\tau\\\
&+C(\gamma_{0})\int_{0}^{t}\|h^{n-1}(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau,\end{split}$
(A.9)

where we used Sobolev imbedding theorem to estimate

$\max_{0\leq t\leq
T}\|\partial_{x_{i}}\bar{a}_{ij}(t,\cdot)\|_{L^{\infty}(\Omega)}\leq
C\max_{0\leq t\leq
T}\|\partial_{x_{i}}\bar{a}_{ij}(t,\cdot)\|_{H^{s-1}(\Omega)}\leq\phi.$

Here and in the sequel, $C$ denotes a positive generic constant which does not
depend on the unknown function $z$.

To derive an estimate for $\partial_{t}^{s}z$ and $\nabla\partial_{t}^{s-1}z$,
we need to employ a molifier technique similar to [8, Section A.2]. First, we
select $0<\delta<\varepsilon<T$. Convolving Equations (A.7) for $n=s-1$ with
$\phi_{\delta}$, we obtain for $t\in[\varepsilon,T-\varepsilon]$

$(\partial_{t}^{s}z)_{\delta}-\bar{a}_{ij}\big{(}\partial_{x_{i}}\partial_{x_{j}}\partial_{t}^{s-2}z\big{)}_{\delta}=(h^{s-2})_{\delta}+\eta^{s}(\cdot,\cdot;\delta)$
(A.10)

with a correction term

$\eta^{s}(t,\cdot;\delta)=\big{(}\bar{a}_{ij}\partial_{t}^{s-2}\partial_{x_{i}}\partial_{x_{j}}z\big{)}_{\delta}-\bar{a}_{ij}\big{(}\partial_{t}^{s-2}\partial_{x_{i}}\partial_{x_{j}}z\big{)}_{\delta}\text{
for }t\in[0,T].$

Differentiating Equation (A.10) w.r.t. $t$

$\partial_{t}(\partial_{t}^{s}z)_{\delta}-\partial_{t}\big{(}\bar{a}_{ij}\big{(}\partial_{x_{i}}\partial_{x_{j}}\partial_{t}^{s-2}z\big{)}_{\delta}\big{)}=\partial_{t}(h^{s-2})_{\delta}+\partial_{t}\eta^{s}(\cdot,\cdot;\delta),$

multiplying the resulting equation in $L^{2}(\Omega)$ with
$\partial_{t}^{s}z_{\delta}$, applying Green’s formula and using Young’s
inequality, we estimate

$\begin{split}\tfrac{1}{2}\partial_{t}\big{(}\|(\partial_{t}^{n}z)_{\delta}&(t,\cdot)\|_{L^{2}(\Omega)}^{2}+\|\bar{a}(t,\cdot)\nabla(\partial_{t}^{n-1}z)_{\delta}(t,\cdot)\|_{L^{2}(\Omega)}^{2}\big{)}\\\
&\leq\tfrac{1}{2}\big{\|}\big{(}\partial_{x_{i}}\bar{a}_{ij}(t,\cdot)\big{)}\partial_{x_{j}}z_{\delta}(t,\cdot)\big{\|}_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}(2+T^{-1/2})\|(\partial_{t}^{n}z)_{\delta}(t,\cdot)\|_{L^{2}(\Omega)}^{2}\\\
&+\tfrac{1}{2}T^{1/2}\|\partial_{t}h^{s-2}_{\delta}(t,\cdot)\|_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\|\eta^{s}(t,\cdot;\delta)\|_{L^{2}(\Omega)}^{2}.\end{split}$
(A.11)

Here, we exploited the fact $(\partial_{t}z)_{\delta}=\partial_{t}z_{\delta}$
if $w$ is once weakly differentiable w.r.t. $t$,
$z_{\delta}|_{\partial\Omega}=z|_{\partial\Omega}$ and
$(\partial_{x_{i}}z)_{\delta}=\partial_{x_{i}}z_{\delta}$ if $w$ is once
weakly differentiable w.r.t. $x_{i}$. Now, integrating Equation (A.11) w.r.t.
$t$ over $[\varepsilon,T-\varepsilon]$, letting $\delta$ and then
$\varepsilon$ go to zero, exploiting the regularity of $z$, applying Lemma A.1
and using Assumption A.2.3 , we get

$\begin{split}\|\partial_{t}^{s}&z(t,\cdot)\|_{L^{2}(\Omega)}^{2}+\|\partial_{t}^{s-1}z(t,\cdot)\|_{H^{1}(\Omega)}^{2}\leq
C(\gamma_{0},\phi_{0})\big{(}\|\bar{z}^{s}\|_{L^{2}(\Omega)}^{2}+\|\bar{z}^{s-1}\|_{H^{1}(\Omega)}^{2}\big{)}\\\
&+C(\gamma_{0},\phi)(1+T^{-1/2})\int_{0}^{t}\big{(}\|\partial_{t}^{n}z(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}+\|\partial_{t}^{n-1}z(\tau,\cdot)\|_{H^{1}(\Omega)}^{2}\big{)}\mathrm{d}\tau\\\
&+C(\gamma_{0})T^{1/2}\int_{0}^{t}\|h^{s-2}(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau.\end{split}$
(A.12)

Combining Equations (A.9) and (A.12) leads to

$\begin{split}\sum_{n=1}^{s}&\big{(}\|\partial_{t}^{n}z(t,\cdot)\|_{L^{2}(\Omega)}^{2}+\|\partial_{t}^{m-1}z(t,\cdot)\|_{H^{1}(\Omega)}^{2}\big{)}\leq
C(\gamma_{0},\phi_{0})\Lambda_{0}\\\
&+C(\gamma_{0},\phi)(1+T^{-1/2})\int_{0}^{t}\|\bar{D}^{s}z(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau\\\
&+C(\gamma_{0})\sum_{n=1}^{s-1}\int_{0}^{t}\|h^{n-1}(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau+C(\gamma_{0})T^{1/2}\int_{0}^{t}\|\partial_{t}h^{s-2}(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau.\end{split}$
(A.13)

Using Sobolev imbedding theorem

$W^{1,2}(\Omega)\hookrightarrow L^{6}(\Omega)\hookrightarrow
L^{4}(\Omega)\text{ for }d\leq 3,$

we can estimate

$\displaystyle\sum_{m=1}^{n-1}$
$\displaystyle\|(\partial_{t}^{m}\bar{a}_{ij})\partial_{t}^{n-1-m}\partial_{x_{i}}\partial_{x_{j}}z\big{\|}_{L^{2}(\Omega)}^{2}\leq\sum_{m=1}^{\min\\{n-1,1\\}}\|\partial_{t}^{m}\bar{a}_{ij}\|_{L^{\infty}(\Omega)}^{2}\|\partial_{t}^{n-1-m}z\|_{H^{2}(\Omega)}^{2}$
$\displaystyle+\sum_{m=\min\\{n,2\\}}^{n-1}\|\partial_{t}^{m}\bar{a}_{ij}\|_{L^{4}(\Omega)}^{2}\|\partial_{t}^{n-1-m}z\|_{W^{2,4}(\Omega)}^{2}$
$\displaystyle\leq
C(\phi)\|\bar{D}^{n}z\|_{L^{2}(\Omega)}^{2}+C\sum_{m=\min\\{n,2\\}}^{n-1}\|\partial_{t}^{m}\bar{a}_{ij}\|_{H^{1}(\Omega)}^{2}\|\partial_{t}^{n-1-m}z\|_{W^{3,2}(\Omega)}^{2}$
$\displaystyle\leq
C(\phi)\|\bar{D}^{n}z\|_{L^{2}(\Omega)}^{2}+C\sum_{m=\min\\{n,2\\}}^{n-1}\|\partial_{t}^{m}\bar{a}_{ij}\|_{H^{s-1-m}(\Omega)}^{2}\|\bar{D}^{n-m+2}z\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq C(\phi)\|\bar{D}^{s-1}z\|_{L^{2}(\Omega)}^{2}.$

Recalling Equation (A.8), we obtain

$\begin{split}\int_{0}^{t}\|h^{n-1}(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau&\leq
t\max_{0\leq\tau\leq
t}\|\partial_{t}^{n-1}\bar{f}(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\\\
&+C(\phi)\int_{0}^{t}\|\bar{D}^{s-1}z(t,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau\end{split}$
(A.14)

for $n=1,\dots,s-1$ and $t\in[0,T]$. Similarly, for $t\in[0,T]$,

$\int_{0}^{t}\|\partial_{t}h^{s-2}(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau\leq\int_{0}^{t}\|\partial_{t}^{s-1}\bar{f}(t,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau+C(\phi)\int_{0}^{t}\|\bar{D}^{s}z(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau.$
(A.15)

Now, combining Equation (A.13) as well as Equations (A.14) and (A.15), we
arrive at

$\begin{split}\sum_{n=1}^{s}\big{(}&\|\partial_{t}^{n}z(t,\cdot)\|_{L^{2}(\Omega)}^{2}+\|\partial_{t}^{n-1}z(t,\cdot)\|_{H^{1}(\Omega)}^{2}\big{)}\\\
&\leq
C(\gamma_{0},\phi_{0})\Lambda_{0}+C(\gamma_{0},\phi)(1+T^{1/2}+T^{-1/2})\int_{0}^{t}\|\bar{D}^{s}z(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau\text{
for }t\in[0,T].\end{split}$ (A.16)

To finish the proof, we need to establish estimates for the remaining
derivatives. For $n=1,\dots,s-1$, consider Equation (A.7). Application of the
elliptic regularity (viz. Assumption A.2.4) with $m=s-n-1$ yields

$\begin{split}\|\partial_{t}^{n-1}z(t,\cdot)\|_{H^{m+2}(\Omega)}^{2}\leq\gamma_{1}\big{(}&\|\partial_{t}^{n+1}z(t,\cdot)\|_{H^{m}(\Omega)}^{2}+\|h^{n-1}(t,\cdot)\|_{H^{m}(\Omega)}^{2}\\\
+&\|\partial_{t}^{n-1}z(t,\cdot)\|_{H^{m}(\Omega)}^{2}\big{)}\text{ for
}t\in[0,T].\end{split}$ (A.17)

Using Assumption A.2.1, Sobolev imbedding theorem and Jensen’s inequality and
applying the fundamental theorem of calculus to the second term on the right-
hand side of Equation (A.17), we obtain

$\begin{split}\|h^{n-1}&(t,\cdot)\|_{H^{m}(\Omega)}^{2}\leq\\\ &\leq
C(\phi_{0})\Lambda_{0}+CT\sum_{k=1}^{n-1}\int_{0}^{t}\big{\|}\partial_{t}\big{(}(\partial_{t}^{k}\bar{a}_{ij})\partial_{t}^{n-1-k}\partial_{x_{i}}\partial_{x_{j}}z\big{)}\big{\|}_{H^{m}(\Omega)}^{2}(\tau,\cdot)\mathrm{d}\tau\\\
&\leq
C(\phi_{0})\Lambda_{0}+C(\phi)T\int_{0}^{t}\|\bar{D}^{s}z(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau.\end{split}$
(A.18)

Note that this estimate is only true if $s\geq 3$, which is trivially
satisfied due to Assumption A.2. Combining Equations (A.16), (A.17) and (A.18)
finally yields

$\begin{split}\|\bar{D}^{s}z(t,\cdot)\|_{L^{2}(\Omega)}^{2}&\leq
C(\gamma_{0},\gamma_{1},\phi_{0})\Lambda_{0}\\\
&+C(\gamma_{0},\gamma_{1},\phi)(1+T^{1/2}+T+T^{-1/2})\int_{0}^{t}\|\bar{D}^{s}z(\tau,\cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau\end{split}$

for any $t\in[0,T]$. The claim is now a direct consequence of Gronwall’s
inequality. ∎

######  Remark A.4.

It should be pointed out that our proof differs from that of Jiang and Racke
[8] as we can carry it out at the energy level $s\geq[\tfrac{d}{2}]+2$ whereas
Jiang and Racke [8] require $s\geq[\tfrac{d}{2}]+3$. This “improvement” is
possible since Theorem A.3 is applied to a quasilinear wave equation with the
quasilinearity depending on the function itself and not its gradient. A
comment on this issue can also be found in [9, Remark 14.4].

###  A.2 Linear Heat Equation

In this appendix section, we consider an initial-boundary-value problem with
Dirichlet boundary conditions for the linear homogeneous isotropic heat
equation reading as

$\displaystyle\theta_{t}(t,x)-a\triangle\theta(t,x)$
$\displaystyle=\bar{g}(t,x)\phantom{0}\text{ for }(t,x)\in(0,T)\times\Omega,$
(A.19a) $\displaystyle\theta(t,x)$
$\displaystyle=0\phantom{\bar{g}(t,x)}\text{ for
}(t,x)\in[0,T]\times\partial\Omega,$ (A.19b) $\displaystyle\theta(0,x)$
$\displaystyle=\theta^{0}(x)\phantom{0}\;\,\text{ for }x\in\Omega.$ (A.19c)

We present a well-posedness result for Equations (A.19a)–(A.19c). In contrast
to [8, Chapter A.3], our proof is based on the operator semigroup theory, in
particular, the maximal $L^{2}$-regularity theory, which is equivalent to
analyticity of the semigroup generated by Dirichlet-Laplacian. A different
technique is employed here to obtain a higher solution regularity needed for
the fixed-point iteration in Theorem 4.5. Besides, the topologies used for the
data $(\theta^{0},\bar{g})$ and the solution $\theta$ differ from those in [8,
Chapter A.3].

######  Assumption A.5.

Let $s\geq 2$ and $a>0$. Assume the following assumptions are satisfied.

  1. 1.

Right-hand side regularity:  For $k=0,1,\dots,s-1$,
$\partial_{t}^{k}\bar{g}\in C^{0}\big{(}[0,T],H^{s-1-k}(\Omega)\big{)}$.
Recall $H^{0}_{0}(\Omega)\equiv H^{0}(\Omega):=L^{2}(\Omega)$.

  2. 2.

Regularity and compatibility conditions:  For $k=0,1,\dots,s-2$, let

$\bar{\theta}^{k}\in H^{s+1-k}(\Omega)\cap H^{1}_{0}(\Omega)\qquad\text{ and
}\qquad\bar{\theta}^{s-1}\in H^{1}_{0}(\Omega),$

where $\bar{\theta}^{k}$’s are given by

$\displaystyle\bar{\theta}^{l}(x)=a^{l}\triangle^{l}\theta^{0}(x)+\sum_{n=0}^{l-1}a^{n}\triangle^{n}\theta^{0}(x)\,\partial_{t}^{l-1-n}\bar{g}(0,x)\text{
for }x\in\Omega\text{ and }l=0,\dots,s-1.$

######  Theorem A.6.

Under Assumption A.5, the initial-boundary-value problem (A.19a)–(A.19c)
possesses a unique classical solution satisfying

$\displaystyle\partial_{t}^{k}\theta$ $\displaystyle\in
C^{0}\big{(}[0,T],H^{s+1-k}(\Omega)\cap H^{1}_{0}(\Omega)\big{)}\text{ for
}k=0,\dots,s-2,$ $\displaystyle\partial_{t}^{s-1}\theta$ $\displaystyle\in
C^{0}\big{(}[0,T],H^{1}_{0}(\Omega)\big{)}\cap
L^{2}\big{(}0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\big{)}\text{ and
}\partial_{t}^{s}\theta\in L^{2}\big{(}0,T;L^{2}(\Omega)\big{)}.$

Moreover, there exists a constant $C>0$ such that

$\displaystyle\sum_{k=0}^{s-2}\max_{0\leq t\leq
T}\|\partial_{t}^{k}\theta(t,\cdot)\|_{H^{s+1-k}(\Omega)}^{2}$
$\displaystyle+\max_{0\leq t\leq
T}\|\partial_{t}^{s-1}\theta(t,\cdot)\|_{H^{1}(\Omega)}^{2}$
$\displaystyle+\int_{0}^{T}\big{(}\|\triangle\partial_{t}^{s-1}\theta(t,\cdot)\|_{L^{2}(\Omega)}^{2}+\|\partial_{t}^{s}\theta(t,\cdot)\|_{L^{2}(\Omega)}^{2}\big{)}\mathrm{d}t\leq
C\Theta_{0},$

where

$\Theta_{0}=(1+T)\Big{(}\sum_{k=0}^{s-2}\|\bar{\theta}^{k}\|_{H^{s+1-k}(\Omega)}^{2}+\|\bar{\theta}^{s-1}\|_{H^{1}(\Omega)}^{2}+\sum_{k=0}^{s-1}\max_{0\leq
t\leq
T}\|\partial_{t}^{k}\bar{g}(t,\cdot)\|_{H^{s-1-k}(\Omega)}^{2}\mathrm{d}t\Big{)}.$

###### Proof.

Let $A:=\triangle_{D}$ denote the $L^{2}$-realization of the Dirichlet-
Laplacian with the domain

$D(A):=\big{\\{}\theta\in H^{1}_{0}(\Omega)\,\big{|}\,\triangle\theta\in
L^{2}(\Omega)\big{\\}}=H^{2}(\Omega)\cap H^{1}_{0}(\Omega),$

where the latter identity follows by standard elliptic regularity theory.
(Note the difference in sign over Sections 4 and 5.) Using Lax & Milgram lemma
to prove the resolvent identity

$\sup_{\lambda\in\mathbb{C}\backslash(-\infty,0]}\big{\|}\lambda(\lambda-A)^{-1}\big{\|}_{L(L^{2}(\Omega))}<\infty,$

we conclude that $aA$ generates a bounded analytic semigroup of angle
$\frac{\pi}{2}$ on $L^{2}(\Omega)$. Due to the Hilbert space structure, [10,
1.7 Corollary] further implies $aA$ has the maximal $L^{p}$-regularity
property.

Consider the solution map $\mathscr{S}$ sending
$(\tilde{\theta}^{0},\tilde{g})$ to the (mild) solution $\tilde{\theta}$ of

$\tilde{\theta}_{t}-aA\tilde{\theta}=\tilde{g}\text{ in
}(0,T),\quad\tilde{\theta}(0,\cdot)=\tilde{\theta}^{0}.$ (A.20)

By classic $C_{0}$-semigroup theory and the maximal $L^{p}$-regularity theory,
we have:

  * •

The mapping

$\mathscr{S}\colon H^{1}_{0}(\Omega)\times
L^{2}\big{(}0,T;L^{2}(\Omega)\big{)}\to
H^{1}\big{(}0,T;L^{2}(\Omega)\big{)}\cap L^{2}\big{(}0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}$ (A.21)

is well-defined as an isomorphism between the two spaces.

  * •

The mapping

$\begin{split}\mathscr{S}\colon\big{(}H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}\times&C^{1}\big{(}[0,T],L^{2}(\Omega)\big{)}\to\\\
&C^{1}\big{(}[0,T],L^{2}(\Omega)\big{)}\cap
C^{0}\big{(}[0,T],H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\big{)}\end{split}$
(A.22)

is well-defined and continuous in respective topologies.

Existence and uniqueness at basic level: On the strength of Assumption A.5, we
(in particular) have $\theta^{0}\in H^{2}_{0}(\Omega)\cap H^{1}_{0}$ and
$\bar{g}\in C^{1}\big{(}[0,T],L^{2}(\Omega)\big{)}$. Hence, there exists a
unique classical solution

$\theta\in C^{1}\big{(}[0,T],L^{2}(\Omega)\big{)}\cap
C^{0}\big{(}[0,T],H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\big{)}$ (A.23)

to Equations (A.19a)–(A.19c).
Higher regularity in time: We argue by induction over $k=1,\dots,s-2$ starting
at $k=1$. Applying $\partial_{t}^{k}$ to Equation (A.19a) and using Assumption
A.5.2, we obtain (in distributional sense)

$\partial_{t}(\partial_{t}^{k}\theta)-aA(\partial_{t}^{k}\theta)=\partial_{t}^{k}\bar{g}\text{
in }(0,T).$ (A.24)

This motivates to consider Equation (A.20) with

$\tilde{\theta}^{0}=\bar{\theta}^{k}\in H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)\text{ and }\tilde{g}=\partial_{t}^{k}\bar{g}\in
C^{1}\big{(}[0,T],L^{2}(\Omega)\big{)}.$ (A.25)

By Equation (A.22),

$\tilde{\theta}\in C^{0}\big{(}[0,T],H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)\big{)}\cap C^{1}\big{(}[0,T],L^{2}(\Omega)\big{)}.$

We now show the function

$\bar{\theta}(t,\cdot)=\sum_{l=0}^{k-1}\frac{t^{l}}{l!}\bar{\theta}^{l}+\int_{0}^{t}\int_{0}^{t_{1}}\dots\int_{0}^{t_{k-1}}\tilde{\theta}(\tau,\cdot)\mathrm{d}\tau\mathrm{d}t_{k-1}\dots\mathrm{d}t_{1}$

coincides with $\partial_{t}^{k}\theta$. By construction, $\bar{\theta}$
satisfies

$\partial_{t}(\partial_{t}^{k}\bar{\theta})-aA(\partial_{t}^{k}\bar{\theta})=\partial_{t}^{k}\bar{g}\text{
in }(0,T).$ (A.26)

Subtracting Equation (A.26) from Equation (A.24), multiplying with
$\partial_{t}^{k-1}(\bar{\theta}-\theta)(t,\cdot)$ in $L^{2}(\Omega)$ and
using Green’s formula, we get

$\tfrac{1}{2}\partial_{t}\big{\|}\partial_{t}^{k-1}(\bar{\theta}-\theta)(t,\cdot)\big{\|}_{L^{2}(\Omega)}+a\big{\|}\nabla\partial_{t}^{k-1}(\bar{\theta}-\theta)(t,\cdot)\big{\|}_{L^{2}(\Omega)}=0.$
(A.27)

This along with the fact
$\partial_{t}^{l}\bar{\theta}(0,\cdot)\equiv\bar{\theta}^{l}\equiv\partial_{t}^{l}\theta(0,\cdot)$
for $l=0,\dots,k-1$ enables us to use the Gronwall’s inequality together with
the fundamental theorem of calculus to deduce $\bar{\theta}\equiv\theta$.
Therefore, we have shown

$\partial_{t}^{k}\theta\equiv\partial_{t}^{k}\bar{\theta}\equiv\tilde{\theta}\in
C^{0}\big{(}[0,T],H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\big{)}\cap