# Initial-boundary value problems for Poiseuille flow of nematic liquid
crystal via full Ericksen-Leslie model

Geng Chen(∗)
(*) Department of Mathematics, University of Kansas, Lawrence, KS, U.S.A.
e-mail<EMAIL_ADDRESS>,
Yanbo Hu(∗∗)
(**) Department of Mathematics, Hangzhou Normal University, Hangzhou, China
e-mail<EMAIL_ADDRESS>and
Qingtian Zhang(∗∗∗)
(***) Department of Mathematics, West Virginia University, Morgantown, WV
26501, U.S.A.
e-mail<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the initial-boundary value problem for the Poiseuille
flow of hyperbolic-parabolic Ericksen-Leslie model of nematic liquid crystals
in one space dimension. Due to the quasilinearity, the solution of this model
in general forms cusp singularity. We prove the global existence of Hölder
continuous solution, which may include cusp singularity, for initial-boundary
value problems with different types of boundary conditions.

KEYWORDS.  Liquid crystal; Ericksen-Leslie; Poiseuille flow; global existence;
Initial-boundary value problem.

AMS subject classifications. 35M33; 35L53; 76D03.

##  1\. Introduction

The hydrodynamic theory of incompressible liquid crystals was established by
Ericksen [14, 15, 16] and Leslie [27]. The Ericksen-Leslie’s system is written
as

$\left\\{\begin{array}[]{ll}\rho\dot{\mathbf{u}}+\nabla
P=\nabla\cdot\sigma-\nabla(\frac{\partial
W}{\partial\nabla\mathbf{n}}\otimes\nabla\mathbf{n}),\\\
\nabla\cdot\mathbf{u}=0,\\\
\nu\ddot{\mathbf{n}}=\lambda\mathbf{n}-\frac{\partial
W}{\partial\mathbf{n}}-\mathbf{g}+\nabla\cdot(\frac{\partial
W}{\partial\nabla\mathbf{n}}),\\\ |\mathbf{n}|=1,\end{array}\right.$ (1.1)

where $\mathbf{u}$ is the velocity, $\mathbf{n}$ is the director field of the
liquid crystal molecules, $\rho$ is the constant density, $P$ is the pressure,
$\nu$ is the inertia coefficient of the director $\mathbf{n}$. $\mathbf{g}$
and $\sigma$ are the kinematic transport and the viscous stress tensor,
respectively, which satisfy

$\displaystyle\mathbf{g}=\gamma_{1}N+\gamma_{2}D\mathbf{n},\quad
N=\dot{\mathbf{n}}-\omega\mathbf{n},$ $\displaystyle
D=\frac{1}{2}(\nabla\mathbf{u}+\nabla^{t}\mathbf{u}),\quad\omega=\frac{1}{2}(\nabla\mathbf{u}-\nabla^{t}\mathbf{u}),$
$\displaystyle\sigma=\alpha_{1}(\mathbf{n}^{t}D\mathbf{n})\mathbf{n}\otimes\mathbf{n}+\alpha_{2}N\otimes\mathbf{n}+\alpha_{3}\mathbf{n}\otimes
N+\alpha_{4}D+\alpha_{5}(D\mathbf{n})\otimes\mathbf{n}+\alpha_{6}\mathbf{n}\otimes(D\mathbf{n}),$

where $\alpha_{1},\cdots,\alpha_{6},\gamma_{1},\gamma_{2}$ are physical
coefficients satisfying (see [27, 8])

$\gamma_{1}=\alpha_{3}-\alpha_{2},\gamma_{2}=\alpha_{6}-\alpha_{5},\alpha_{2}+\alpha_{3}=\alpha_{6}-\alpha_{5},$
$\alpha_{4}>0,2\alpha_{1}+3\alpha_{4}+2\alpha_{5}+2\alpha_{6}>0,\gamma_{1}>0,$
$2\alpha_{4}+\alpha_{5}+\alpha_{6}>0,4\gamma_{1}(2\alpha_{4}+\alpha_{5}+\alpha_{6})>(\alpha_{2}+\alpha_{3}+\gamma_{2})^{2}.$

If the orientation order parameters of nematic materials are treated as a unit
vector ${\mathbf{n}}\in\mathbb{S}^{2}$, the director, then the Oseen-Frank
energy density determines the macrostructure of the crystal structure ([27])

$\displaystyle\begin{split}2W(\mathbf{n},\nabla\mathbf{n})=&K_{1}(\nabla\cdot\mathbf{n})^{2}+K_{2}(\mathbf{n}\cdot(\nabla\times\mathbf{n}))^{2}+K_{3}|\mathbf{n}\times(\nabla\times\mathbf{n})|^{2}\\\
&+(K_{2}+K_{4})[\hbox{tr}(\nabla\mathbf{n})^{2}-(\nabla\cdot\mathbf{n})^{2}],\end{split}$

where $K_{j}$, $j=1,2,3$, are the positive constants representing splay,
twist, and bend effects respectively, with $K_{2}\geq|K_{4}|$, $2K_{1}\geq
K_{2}+K_{4}$.

Successful theories with a wide range of interesting properties have been
established for the equilibrium theory (the elliptic case on $\mathbf{n}$) and
the evolutionary theory (the parabolic case on $\mathbf{n}$) when the inertial
effect in (1.1) is neglected, i.e. when $\nu=0$. See a partial list of
references in [18, 32, 33, 31, 22, 34, 20, 28, 36, 37].

However, there are very few studies on the full Ericksen-Leslie system,
including the inertial effect, i.e. when $\nu=0$. Many fundamental problems
including global wellposedness are still wide open. When one considers a
special Oseen-Frank potential $W=|\nabla\mathbf{n}|^{2}$, which makes the wave
equations on $\mathbf{n}$ essentially semilinear, for small data problem,
Jiang and his collaborators obtained a series of existence results on regular
solutions [24, 26, 12, 23, 25].

In general, the solution of (1.1) might form a finite time singularity even in
one space dimension, due to the quasilinearity in the wave equation of
$\mathbf{n}$. See [8] for the formation of finite time cusp singularity for
the Poiseuille flow, and [13] for the formation of cusp singularity in
multiple space dimension. For solutions in multiple space dimension, Chen-
Huang-Xu in [9] found another type of singularity for (1.1) due to the
geometric effect, similar as the one for the semilinear wave map equation.

In [8], the first large data global existence result on the Cauchy problem of
(1.1) was established by Chen-Huang-Liu in [8] for the 1-d Poiseuille flow,
where the solution may include cusp singularity.

In this paper, we establish some global existence results for the initial-
boundary value problem of the Poiseuille flow for nematic liquid crystals via
the full Ericksen-Leslie model (1.1). More precisely, for the 1-d Poiseuille
flow with $\mathbf{u}=(0,0,u)^{t},\mathbf{n}=(\sin\theta,0,\cos\theta)^{t}$,
the system of $u$ and $\theta$ becomes

$\left\\{\begin{array}[]{ll}\rho
u_{t}=(g(\theta)u_{x}+h(\theta)\theta_{t})_{x},\\\
\nu\theta_{tt}+\gamma_{1}\theta_{t}=c(\theta)(c(\theta)\theta_{x})_{x}-h(\theta)u_{x},\end{array}\right.$
(1.2)

where

$\displaystyle g(\theta)\mathrel{\mathop{\mathchar
58\relax}}=\alpha_{1}\sin^{2}\theta\cos^{2}\theta+\frac{\alpha_{5}-\alpha_{2}}{2}\sin^{2}\theta+\frac{\alpha_{3}+\alpha_{6}}{2}\cos^{2}\theta+\frac{\alpha_{4}}{2},$
$\displaystyle c(\theta)\mathrel{\mathop{\mathchar
58\relax}}=\sqrt{K_{1}\cos^{2}\theta+K_{3}\sin^{2}\theta},$ $\displaystyle
h(\theta)\mathrel{\mathop{\mathchar
58\relax}}=\alpha_{3}\cos^{2}\theta-\alpha_{2}\sin^{2}\theta=\frac{\gamma_{1}+\gamma_{2}\cos(2\theta)}{2}.$

See the derivation of this model in [8]. Notice that
$g(\theta),c(\theta),h(\theta)$ are smooth with respect to $\theta$, and
$g(\theta),c(\theta),h(\theta),g^{\prime}(\theta),c^{\prime}(\theta),c^{\prime\prime}(\theta),h^{\prime}(\theta)$
are uniformly bounded.

There are several physically interesting boundary conditions. For the velocity
$u$, we have the following possible choices. We denote an important quantity

$J=g(\theta)u_{x}+h(\theta)\theta_{t}.$

  * •

Nonslip boundary condition. If the boundary is a solid wall, we can propose
the nonslip boundary condition $u=0$ on the boundary. If the boundary is
moving, we can also propose $u=f(t)$ on the boundary with $f(t)$ is a given
function with $t$. This corresponds to the Dirichlet boundary condition on
$u$.

  * •

Stress-free boundary condition. On the boundary the shear stress is zero.
$(1,0,0)D=(0,0,J)^{t}={\mathbf{0}}$, where $(1,0,0)$ is the normal direction
of the boundary. This boundary condition can be seen in the jet with a free
boundary. Under the assumption of the Poiseuille flow, there is no in-flow or
out-flow, so the boundary is fixed. This corresponds to the Neumann boundary
condition on $u$.

  * •

Navier boundary condition. On the boundary, the shear stress is proportional
to the tangential velocity, that is $J=-\gamma u$. This corresponds to the
Robin boundary condition on $u$.

For the director $\mathbf{n}$, one can propose the following boundary
conditions [35].

  * •

Strong anchoring condition. On the boundary, $\mathbf{n}$ is given. This
corresponds to the Dirichlet boundary condition.

  * •

No anchoring condition. $\frac{\partial W}{\partial n_{i,j}}\nu_{j}=0$.

  * •

Weak anchoring condition. $\frac{\partial W_{F}}{\partial
n_{i,j}}\nu_{j}+\frac{\partial W_{s}}{\partial n_{i}}=\gamma n_{i}$.

In this paper, we will consider all combinations of boundary conditions,
excluding the Navier boundary condition.

###  1.1. Main result

For simplicity, we only consider a special case when $\alpha_{1}=0$,
$\alpha_{5}-\alpha_{2}=\alpha_{3}+\alpha_{6}$, so that $g(\theta)$ is a
constant $\frac{\alpha_{3}+\alpha_{4}+\alpha_{6}}{2}$. Without loss of
generality, we assume $g(\theta)=1$, $\nu=1,\gamma_{1}=2,\gamma_{2}=0$ and
$\rho=1$. So $h(\theta)=1$. Then the system is written as

$\left\\{\begin{array}[]{ll}u_{t}=(u_{x}+\theta_{t})_{x},\\\
\theta_{tt}+2\theta_{t}=c(\theta)(c(\theta)\theta_{x})_{x}-u_{x},\end{array}\right.$
(1.3)

where the function $c(\cdot)$ is a $C^{2}$ function satisfying

$\displaystyle 0<C_{L}\leq c(\cdot)\leq C_{U}<\infty,\ \
|c^{\prime}(\cdot)|\leq C_{1}<\infty,$ (1.4)

for some positive constants $C_{L}$, $C_{U}$ and $C_{1}$. In this case,
$J=u_{x}+\theta_{t}$.

The first equation of (1.3) has constant coefficients. This gives us some
technical advantage than (1.2). In [8], the authors first considered the
simplified system (1.3). Later in [10], the global existence result of Cauchy
problem was extended to the general system (1.2). We conject the result in
this paper on (1.3) (Theorem 1.1) still holds for (1.2).

The following global existence theorem on the initial boundary value problem
of (1.3) on $x\in[0,\pi]$ is our main theorem. For the convenience on
notations, we choose the domain as $x\in[0,\pi]$. The result holds for any
bounded domain $x\in[a,b]$ by the same proof.

######  Theorem 1.1.

Assume initially

$u(x,0)=u_{0}(x)\in H^{1}([0,\pi]),~{}~{}\theta(x,0)=\theta_{0}(x)\in
H^{1}([0,\pi]),~{}~{}\theta_{t}(x,0)=\theta_{1}(x)\in L^{2}([0,\pi]).$ (1.5)

We consider one of the following boundary conditions:

$\displaystyle u(0,t)=u(\pi,t)=0,$ (1.6)
$\displaystyle-\iota_{1}\theta(0,t)+\iota_{2}\theta_{x}(0,t)=\iota_{3}\theta(\pi,t)+\iota_{4}\theta_{x}(\pi,t)=0,$

or

$\displaystyle(u_{x}+\theta_{t})(0,t)=(u_{x}+\theta_{t})(\pi,t)=0,$ (1.7)
$\displaystyle-\iota_{1}\theta(0,t)+\iota_{2}\theta_{x}(0,t)=\iota_{3}\theta(\pi,t)+\iota_{4}\theta_{x}(\pi,t)=0,$

where $\iota_{1}$ to $\iota_{4}$ are nonnegative constants satisfying
$\iota_{1}^{2}+\iota_{2}^{2}>0$ and $\iota_{3}^{2}+\iota_{4}^{2}>0$. The
functions $u_{0}(x)$, $\theta_{0}(x)$ and $\theta_{1}(x)$ satisfy the
corresponding compatibility conditions at $0$ and $\pi$, and two additional
conditions:

$\displaystyle u_{0}^{\prime}(x)+\theta_{1}(x)\in C^{\alpha}([0,\pi]),$ (1.8)

for some $\alpha\in(0,1/4)$, and $\theta_{0}$ are absolutely continuous.

For any given time $T\in(0,\infty)$, the initial-boundary value problem
(1.3)-(1.8) admits a weak solution $(u(x,t),\theta(x,t))$ defined on
$[0,\pi]\times[0,T]$ in the sense of Definition 1.2. Moreover, the associated
energy

$\displaystyle\mathcal{E}(t)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}\int_{0}^{\pi}(\theta_{t}^{2}+c^{2}(\theta)\theta_{x}^{2}+u^{2})(x,t)\
{\rm d}x+B_{\pi}(\theta(\pi,t))+B_{0}(\theta(0,t)),$ (1.9)

is well-defined for $t\in[0,T]$ and satisfies

$\displaystyle\mathcal{E}(t)\leq\mathcal{E}(0)-\int_{0}^{t}\int_{0}^{\pi}((u_{x}+\theta_{t})^{2}+\theta_{t}^{2})(x,t)\
{\rm d}x{\rm d}t,$ (1.10)

where $B_{0}(\theta(0,t))$ and $B_{\pi}(\theta(\pi,t))$ are, respectively, the
boundary energies on $x=0$ and $x=\pi$, defined as

$B_{0}(\theta(0,t))=\left\\{\begin{aligned}
\frac{\iota_{1}}{\iota_{2}}\int_{0}^{\theta(0,t)}c^{2}(s)s\ {\rm d}s,\
&\iota_{2}\neq 0,\\\ 0,\ &\iota_{2}=0,\end{aligned}\right.\
B_{\pi}(\theta(\pi,t))=\left\\{\begin{aligned}
\frac{\iota_{3}}{\iota_{4}}\int_{0}^{\theta(\pi,t)}c^{2}(s)s\ {\rm d}s,\
&\iota_{4}\neq 0,\\\ 0,\ &\iota_{4}=0.\end{aligned}\right.$ (1.11)

Next, we define the weak solution. First, the energy equality for the smooth
solution is

$\displaystyle\frac{\rm d}{{\rm
d}t}\int_{0}^{\pi}\frac{1}{2}u^{2}+\frac{1}{2}\theta_{t}^{2}+\frac{1}{2}c^{2}\theta_{x}^{2}{\rm
d}x+\int_{0}^{\pi}\theta_{t}^{2}+(u_{x}+\theta_{t})^{2}{\rm
d}x-\big{[}(u_{x}+\theta_{t})u+c^{2}\theta_{t}\theta_{x}\big{]}\Big{|}_{x=0}^{x=\pi}=0,$
$\displaystyle\frac{\rm d}{{\rm
d}t}\left[\int_{0}^{\pi}\frac{1}{2}u^{2}+\frac{1}{2}\theta_{t}^{2}+\frac{1}{2}c^{2}\theta_{x}^{2}{\rm
d}x+B_{\pi}(\theta(\pi,t))+B_{0}(\theta(0,t))\right]+\int_{0}^{\pi}\theta_{t}^{2}+(u_{x}+\theta_{t})^{2}{\rm
d}x=0.$ (1.12)

######  Definition 1.2.

For any given time $T>0$, we say $(u(x,t),\theta(x,t))$, defined for all
$(x,t)\in[0,\pi]\times[0,T]$, is a weak solution to the initial-boundary value
problem (1.3)-(1.8) if

(i) there hold

$\displaystyle\int_{0}^{T}\int_{0}^{\pi}\bigg{(}u\psi_{t}-(u_{x}+\theta_{t})\psi_{x}\bigg{)}\
{\rm d}x{\rm
d}t+\int_{0}^{T}[(u_{x}+\theta_{t})\psi]\bigg{|}_{x=0}^{x=\pi}{\rm d}t=0,$
(1.13)

and

$\displaystyle\int_{0}^{T}\int_{0}^{\pi}\bigg{(}\theta_{t}\varphi_{t}-(c(\theta)\varphi)_{x}c(\theta)\theta_{x}-2\theta_{t}\varphi-
u_{x}\varphi\bigg{)}\ {\rm d}x{\rm
d}t+\int_{0}^{T}(c^{2}\varphi\theta_{x})\bigg{|}_{x=0}^{x=\pi}\ {\rm d}t=0,$
(1.14)

for any test functions $\psi,\varphi\in\mathcal{F}$, where

$\displaystyle\mathcal{F}\mathrel{\mathop{\mathchar 58\relax}}=\bigg{\\{}f\in
C^{\infty}((0,\pi)\times(0,T))\mathrel{\mathop{\mathchar 58\relax}}\
\partial_{t}^{i}\partial_{x}^{j}f\bigg{|}_{t=0,T}=0,\ \ \forall\
i,j=0,1,2\cdots\bigg{\\}},$ (1.15) $\displaystyle\theta\in
C^{1/2}([0,\pi]\times[0,T])\cap L^{2}([0,T],H^{1}([0,\pi])),$ (1.16)

and

$\displaystyle u$ $\displaystyle\in L^{\infty}([0,T],H^{1}([0,\pi]))\cap
L^{\infty}([0,\pi]\times[0,T]),$ (1.17) $\displaystyle u_{t}$
$\displaystyle\in L^{2}([0,T],H^{-1}([0,\pi]));$ (1.18)

(ii) the first and second equations for initial conditions in (1.5) are
satisfied pointwise, and the third equation holds in $L^{p}$ for $p\in[1,2)$;

(iii) the boundary conditions in (1.6) or in (1.7) are satisfied in
$L^{2}(0,T)$ sense.

We note that the global existence theory for (1.3), is base on earlier work of
Bressan-Zheng in [5] on the variational wave equation. In fact, before
considering (1.3), a class of simplified 1-d wave models only on $\mathbf{n}$
were first studied. For example, when
$\mathbf{n}=(\cos\theta,0,\sin\theta)(x,t)$, $x\in\mathbb{R}$, the variational
wave equation satisfies

$\theta_{tt}-c(\theta)\bigl{(}c(\theta)\theta_{x}\bigr{)}_{x}=0,\quad\theta(x,0)=\theta_{0}(x)\in
H^{1},\quad\theta_{t}(x,0)=\theta_{1}(x)\in L^{2}.$ (1.19)

It is natural to consider the finite energy ($H^{1}$) initial data. Since
$H^{1}\hookrightarrow C^{1/2}$ in 1-d, which is not Lipschitz, finite time
cusp singularity forms even if initial data are smooth [17]. For the Cauchy
problem (1.19), the existence, uniqueness and Lipschitz continuity of global
$C^{1/2}$ energy conservative solutions was established by [5, 19], [3] and
[2, 1], respectively. Later, these results were extended to a wave system with
$\mathbf{n}\in{\mathbb{S}}^{2}$ when the Oseen-Frank potential takes its
general form in [11, 6, 7, 39, 40], and another general system in [21]. See
existence of dissipative solution of (1.19) for the Cauchy problem at [38, 4].

Our existence result on the initial boundary value problem also applies to the
variational wave equation (1.19), which is missing for many years.

However, one cannot easily extend the result for the variational wave equation
to (1.3) (or (1.2)). This is because $u_{x}$ has the similar regularity as
$\theta_{t}$, which maybe unbounded. There is no direct method to cope with
the variational wave equation with an unbounded source term. The key
observation in [8] which helps solving this issue, is to find
$J=u_{x}+\theta_{t}\in C^{\alpha}\cap L^{2}\cap L^{\infty}$ on $(x,t)$ for
(1.3).

For the initial-boundary value problem, we still use $J$ to rewrite (1.3).
Then, for any given $J\in C^{\alpha}\cap L^{2}\cap L^{\infty}$, we first
establish the global existence result on $\theta(x,t)\in L^{\infty}(H^{1})$,
by solving an initial-boundary value problem of (1.19) with damping and the
force term $J$. In this step, we need to change the equation into a semilinear
system on characteristic coordinates. This method was first used by Bressan-
Zheng in [5], then by Chen-Huang-Liu in [8], for the Cauchy problem. For the
boundary value problem, we need to transform the boundary conditions on
$(u,\theta)$ in the original $(x,t)$-coordinates to conditions on $J$ and
other new dependent variables under the new coordinates. Here we need some new
methods to cope with these additional boundary values (see Subsection 2.5).

The second step is to find a fixed point using the heat equation on $J$ for
the boundary value problem. The difficulty lies in the fact that the source
term of the equation on $J$ is only $H^{-1}$. One cannot directly use the
smoothing effect of the heat equation, but need to first use the wave equation
to change $\theta_{tt}$ into $\theta_{xx}$ and other lower order terms. Then
one can show the enhanced regularity on $J$, since the smoothing effect of
heat equation on the source term $\theta_{xx}$ is much better than
$\theta_{tt}$. Different from the Cauchy problem [8], in this initial-boundary
value problem, the heat kernel is expressed as an infinite series by using the
image method. Finally we used the Duhamel’s principle and the Schauder fixed
point theorem to prove the existence of a fixed point.

###  1.2. Structure of this paper

We will first consider the nonslip (Dirichlet) boundary condition (1.6) on
velocity $u$, in sections 3 and 4. To cope with different types of boundary
conditions on $\theta$, we only need some minor changes in the proof. To avoid
a repeat, we propose to only prove the following mixed boundary conditions.

$\displaystyle u(0,t)=u(\pi,t)=0,$ (1.20)
$\displaystyle\theta(0,t)=\iota\theta(\pi,t)+\theta_{x}(\pi,t)=0,$

where $\iota\geq 0$. This is a special case of (1.6). The functions
$u_{0}(x),\theta_{0}(x)$ and $\theta_{1}(x)$ satisfy the corresponding
compatibility conditions at $0$ and $\pi$

$\displaystyle\theta_{0}(0)=\iota\theta_{0}(\pi)+\theta_{1}(\pi)=0,\quad
u_{0}(0)=u_{0}(\pi)=0.$ (1.21)

Correspondingly, our results can be stated as follows.

######  Theorem 1.3.

Assume all conditions in Theorem 1.1 hold. For any given time
$T\in(0,\infty)$, the initial-boundary value problem (1.3)-(1.8) admits a weak
solution $(u(x,t),\theta(x,t))$ defined on $[0,\pi]\times[0,T]$ in the sense
of Definition 1.2. Moreover, the associated energy

$\displaystyle\mathcal{E}(t)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}\int_{0}^{\pi}(\theta_{t}^{2}+c^{2}(\theta)\theta_{x}^{2}+u^{2})(x,t)\
{\rm d}x+B(\theta(\pi,t)),$ (1.22)

with $B(\theta(\pi,t))=\int_{0}^{\theta(\pi,t)}\iota c^{2}(s)s\ {\rm d}s\geq
0$, is well-defined for $t\in[0,T]$ and satisfies

$\displaystyle\mathcal{E}(t)\leq\mathcal{E}(0)-\int_{0}^{t}\int_{0}^{\pi}((u_{x}+\theta_{t})^{2}+\theta_{t}^{2})(x,t)\
{\rm d}x{\rm d}t.$ (1.23)

In the proof of Theorem 1.3, we include all necessary techniques to cope with
three types of (Dirichlet, Neumann and Robin) boundary conditions on $\theta$
included in (1.6). So all other cases in Theorem 1.1 with the boundary
condition (1.6) can be proved similarly. We leave them to the reader.

In section 5, we will present a proof of Theorem 1.1 with stress-free
(Neumann) boundary condition (1.7).

######  Remark 1.4.

The result is still correct for the general Dirichlet and Neumann boundary on
$u$ and $\theta$, with for example, $u(0,t)=f_{1}(t)$, $u(\pi,t)=f_{2}(t)$,
$\theta(0,t)=g_{1}(t)$ and $\iota\theta(\pi,t)+\theta_{x}(\pi,t)=g_{2}(t)$
with absolutely continuous $f_{1,2}(t),g_{2}(t)$ functions and $C^{1}$
continuous $g_{1}(t)$ function. For this case, when considering the global
existence of the wave equations, we replace the boundary conditions on $L_{0}$
and $L_{\pi}$ in (2.63) by

$\displaystyle\begin{array}[]{ll}\displaystyle\frac{z}{2}=\arctan[2g_{1}^{\prime}(t)-\tan\frac{w}{2}],\
q-\frac{1+[2g_{1}^{\prime}(t)-\tan\frac{w}{2}]^{2}}{1+\tan^{2}\frac{w}{2}}q=0,&{\rm
on}\ L_{0},\\\\[8.0pt]
\displaystyle\frac{w}{2}=\arctan[\tan\frac{z}{2}+2c(\theta)g_{2}(t)-2\iota
c(\theta)\theta],\ p-\frac{1+[\tan\frac{z}{2}+2c(\theta)g_{2}(t)-2\iota
c(\theta)\theta]^{2}}{1+\tan^{2}\frac{z}{2}}q=0,&\text{on
}L_{\pi},\end{array}$

and replace the boundary energy $B(\theta(\pi,t))$ in (1.22) by a
corresponding form. Moreover, instead of (3.1), (3.2), the current initial-
boundary value problem for the variable $J=u_{x}+\theta_{t}$ is

$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle
J_{t}-J_{xx}=c(\theta)(c(\theta)\theta_{x})_{x}-\theta_{t}-J,\\\ \displaystyle
J(x,0)=J_{0}(x),\\\ \displaystyle J_{x}(0,t)=f_{1}^{\prime}(t),\ \
J_{x}(\pi,t)=f_{2}^{\prime}(t).\end{array}\right.$ (1.27)

Furthermore, replaced (3.53), the initial-boundary value problem for the
variable $u$ now is

$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle
u_{t}-u_{xx}=\theta_{tx},\\\ \displaystyle u(x,0)=u_{0}(x),\\\ \displaystyle
u(0,t)=f_{1}(t),\ \ u(\pi,t)=f_{2}(t).\end{array}\right.$ (1.31)

We can introduce some suitable variables for problems (1.27) and (1.31) to
transform them into homogeneous boundary condition problems, and then use the
Green/Neumann functions defined in Section 3 to express the corresponding weak
solutions. The proof is very similar to the cases of the homogeneous boundary
conditions (1.6) or (1.7).

##  2\. Existence of wave equation for any given $J=u_{x}+\theta_{t}$

In this section, we show the global existence of Hölder continuous solutions
to the initial-boundary value problem of the nonlinear wave equation in (1.3)
for any given $J=u_{x}+\theta_{t}$.

###  2.1. The semilinear system in characteristic coordinates

Let $J=u_{x}+\theta_{t}$. The wave equation in (1.3) reads

$\displaystyle\theta_{tt}-c(\theta)(c(\theta)\theta_{x})_{x}+\theta_{t}+J=0.$
(2.1)

Denote

$\displaystyle R\mathrel{\mathop{\mathchar
58\relax}}=\theta_{t}+c(\theta)\theta_{x},\quad S\mathrel{\mathop{\mathchar
58\relax}}=\theta_{t}-c(\theta)\theta_{x},$ (2.2)

so that

$\displaystyle\theta_{t}=\frac{R+S}{2},\quad\theta_{x}=\frac{R-S}{2c(\theta)}.$
(2.3)

By (2.1), the equations in terms of variables $(R,S)$ are

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle
R_{t}-c(\theta)R_{x}=\frac{c^{\prime}(\theta)}{4c(\theta)}(R^{2}-S^{2})-\frac{1}{2}(R+S)-J,\\\\[8.0pt]
\displaystyle
S_{t}+c(\theta)S_{x}=\frac{c^{\prime}(\theta)}{4c(\theta)}(S^{2}-R^{2})-\frac{1}{2}(R+S)-J.\end{array}\right.$
(2.6)

Let $(x,t)$ be any point in $[0,\pi]\times[0,\infty)$. We define the forward
and backward characteristics $x=x_{\pm}(s;x,t)(s\leq t)$ passing through the
point $(x,t)$ as follows

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle\frac{{\rm
d}x_{\pm}(s;x,t)}{{\rm d}s}=\pm c(\theta(x_{\pm}(s;x,t),s)),\\\ \displaystyle
x_{\pm}(t;x,t)=x.\end{array}\right.$ (2.9)

We now define the coordinate transformation $(x,t)\rightarrow(X,Y)$ on
$[0,\pi]\times[0,\infty)$. We first specify this transformation to transform
the lines $x=0$ and $x=\pi$ with $t\geq 0$ into the lines $Y=X$ with $X\geq 0$
and $Y=X-\widetilde{X}$ with $X\geq\widehat{X}$, respectively, where

$\displaystyle\widetilde{X}=\int_{0}^{\pi}(1+R_{0}^{2}(z))\ {\rm
d}z-\int_{\pi}^{0}(1+S_{0}^{2}(z))\ {\rm
d}z,\quad\widehat{X}=\int_{0}^{\pi}(1+R_{0}^{2}(z))\ {\rm d}z.$ (2.10)

Here

$\displaystyle
R_{0}(x)=\theta_{1}(x)+c(\theta_{0}(x))\theta_{0}^{\prime}(x),\quad
S_{0}(x)=\theta_{1}(x)-c(\theta_{0}(x))\theta_{0}^{\prime}(x),\ \ \forall\
x\in[0,\pi].$ (2.11)

Moreover, we set that the segment $t=0$ with $x\in[0,\pi]$ is transformed to a
piece of curve $\Gamma_{0}\mathrel{\mathop{\mathchar
58\relax}}Y=\phi(X)(X\in[0,\widehat{X}])$ defined through a parametric
$x\in[0,\pi]$

$\displaystyle X=\int_{0}^{x}(1+R_{0}^{2}(z))\ {\rm d}z,\quad
Y=\int_{x}^{0}(1+S_{0}^{2}(z))\ {\rm d}z.$ (2.12)

It is observed by the initial data (1.5) that the two functions $X=X(x)$ and
$Y=Y(x)$ with $x\in[0,\pi]$ are well defined and absolutely continuous.
Moreover, $X(x)$ is strictly increasing while $Y(x)$ is strictly decreasing.
Hence the function $Y=\phi(X)$ is continuous and strictly decreasing. Next for
any point $(x,t)\in[0,\pi]\times(0,\infty)$, we draw the backward
characteristic $x_{-}(s;x,t)$ up to a point $P_{1}$ on $x=\pi$, and then draw
the forward characteristic $x_{+}(s;P_{1})$ up to a point $P_{2}$ on $x=0$.
Repeating the above process, since the wave speed $c(\theta)\geq C_{L}$, we
can reach the segment $t=0(x\in[0,\pi])$ through finite steps by (1.4). Assume
that there exists a point $P_{l}$ on $x=0$ or $x=\pi$ such that the backward
characteristic $x_{-}(s;P_{l})$ or the forward characteristic $x_{+}(s;P_{l})$
intersects the segment $t=0(x\in[0,\pi])$ at a point $P^{*}(x_{P^{*}},0)$. It
is clear that the points $P_{i}$ are on $x=\pi$ for odd numbers $i\leq l$, and
on $x=0$ for even numbers $i\leq l$. See Fig. 2.1 (a) for the illustration.
Denote the coordinate of $P_{i}$ by $(\pi,t_{i})$ if $i$ is an odd number, and
by $(0,t_{i})$ if $i$ is even. The numbers $t_{i}(i=1,\cdots,l)$ and
$x_{P^{*}}$ can be determined sequentially as follows

$\displaystyle\left\\{\begin{array}[]{l}x_{-}(t_{1};x,t)=\pi,\\\
x_{+}(t_{2};P_{1})=0,\\\ x_{-}(t_{3};P_{2})=\pi,\\\ \vdots\\\
x_{+}(t_{l};P_{l-1})=0,\ \ x_{P^{*}}=x_{-}(0;P_{l}),\ \ \text{ if $l$ is an
even number},\\\ x_{-}(t_{l};P_{l-1})=\pi,\ \ x_{P^{*}}=x_{+}(0;P_{l}),\ \
\text{ if $l$ is an odd number}.\end{array}\right.$ (2.19)

Then we define the value of $X(x,t)$ by

$\displaystyle X(x,t)=$ $\displaystyle
X(P_{1})=Y(P_{1})+\widetilde{X}=Y(P_{2})+\widetilde{X}=X(P_{2})+\widetilde{X}$
$\displaystyle=$ $\displaystyle
X(P_{3})+\widetilde{X}=(Y(P_{3})+\widetilde{X})+\widetilde{X}=Y(P_{4})+2\widetilde{X}$
$\displaystyle=$
$\displaystyle\cdots=\left\\{\begin{array}[]{l}X(P_{l})+k\widetilde{X},\qquad\quad\
l=2k\\\ Y(P_{l})+(k+1)\widetilde{X},\ \ l=2k+1\end{array}\right.$ (2.22)
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l}\displaystyle\int_{0}^{x_{P^{*}}}(1+R_{0}^{2}(z))\
{\rm d}z+k\widetilde{X},\qquad\ \ l=2k,\\\\[8.0pt]
\displaystyle\int_{x_{P^{*}}}^{0}(1+S_{0}^{2}(z))\ {\rm
d}z+(k+1)\widetilde{X},\ \ l=2k+1.\end{array}\right.$ (2.25)

(a)

(b)

Figure 2.1. Characteristic curves.

To define the value of $Y(x,t)$, we draw the forward characteristic
$x_{+}(s;x,t)$ up to a point $Q_{1}$ on $x=0$, and then draw the backward
characteristic $x_{-}(s;Q_{1})$ up to a point $Q_{2}$ on $x=\pi$. Similarly,
we can reach the segment $t=0(x\in[0,\pi])$ through finite steps by repeating
the above process. Assume that there exists a point $Q_{m}$ on $x=0$ or
$x=\pi$ such that the backward characteristic $x_{-}(s;Q_{m})$ or the forward
characteristic $x_{+}(s;Q_{m})$ intersects the segment $t=0(x\in[0,\pi])$ at a
point $Q^{*}(x_{Q^{*}},0)$. Obviously, the points $Q_{i}$ are on $x=0$ for odd
numbers $i\leq m$, and on $x=\pi$ for even numbers $i\leq m$. See Fig. 2.1 (b)
for the illustration. Denote the coordinate of $Q_{i}$ by $(0,\tilde{t}_{i})$
if $i$ is an odd number, and by $(\pi,\tilde{t}_{i})$ if $i$ is even. We can
determine the numbers $\tilde{t}_{i}(i=1,\cdots,m)$ and $x_{Q^{*}}$
sequentially

$\displaystyle\left\\{\begin{array}[]{l}x_{+}(\tilde{t}_{1};x,t)=0,\\\
x_{-}(\tilde{t}_{2};Q_{1})=\pi,\\\ x_{+}(\tilde{t}_{3};Q_{2})=0,\\\ \vdots\\\
x_{-}(\tilde{t}_{m};Q_{m-1})=\pi,\ \ x_{Q^{*}}=x_{+}(0;Q_{m}),\ \ \text{ if
$m$ is an even number},\\\ x_{+}(\tilde{t}_{m};Q_{m-1})=0,\ \
x_{Q^{*}}=x_{-}(0;Q_{m}),\ \ \text{ if $m$ is an odd
number}.\end{array}\right.$ (2.32)

Thus the value of $Y(x,t)$ can be defined as follows

$\displaystyle Y(x,t)=$ $\displaystyle
Y(Q_{1})=X(Q_{1})=X(Q_{2})=Y(Q_{2})+\widetilde{X}=Y(Q_{3})+\widetilde{X}$
$\displaystyle=$ $\displaystyle
X(Q_{3})+\widetilde{X}=X(Q_{4})+\widetilde{X}=(Y(Q_{4})+\widetilde{X})+\widetilde{X}=Y(Q_{4})+2\widetilde{X}$
$\displaystyle=$
$\displaystyle\cdots=\left\\{\begin{array}[]{l}Y(Q_{m})+k\widetilde{X},\ \
m=2k\\\ X(Q_{m})+k\widetilde{X},\ \ m=2k+1\end{array}\right.$ (2.35)
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l}\displaystyle\int_{x_{Q^{*}}}^{0}(1+S_{0}^{2}(z))\
{\rm d}z+k\widetilde{X},\ \ \ \ m=2k,\\\\[12.0pt]
\displaystyle\int_{0}^{x_{Q^{*}}}(1+R_{0}^{2}(z))\ {\rm d}z+k\widetilde{X},\ \
m=2k+1.\end{array}\right.$ (2.38)

It is easy to note by (2.19) and (2.32) that for the point $(x,t)$ on
$t=0(x\in[0,\pi])$, the transformation defined in (2.1), (2.1) reduces to
(2.12). Furthermore, according to the construction process of the
transformation $(x,t)\rightarrow(X,Y)$, we see that $X$ and $Y$ are constants
along backward and forward characteristic, respectively; that is,

$\displaystyle X_{t}-c(\theta)X_{x}=0,\quad Y_{t}+c(\theta)Y_{x}=0,$ (2.39)

from which one has for any smooth function $f$

$\displaystyle\begin{array}[]{l}f_{t}+c(\theta)f_{x}=(f_{X}X_{t}+f_{Y}Y_{t})+c(\theta)(f_{X}X_{x}+f_{Y}Y_{x})=2cX_{x}f_{X},\\\
f_{t}-c(\theta)f_{x}=(f_{X}X_{t}+f_{Y}Y_{t})-c(\theta)(f_{X}X_{x}+f_{Y}Y_{x})=-2cY_{x}f_{Y}.\end{array}$
(2.42)

For convenience to deal with possibly unbounded values of $R$ and $S$, one can
introduce the variables

$\displaystyle w\mathrel{\mathop{\mathchar 58\relax}}=2\arctan R,\quad
z\mathrel{\mathop{\mathchar 58\relax}}=2\arctan S.$ (2.43)

In order to complete the system, we further introduce two key dependent
variables

$\displaystyle p\mathrel{\mathop{\mathchar
58\relax}}=\frac{1+R^{2}}{X_{x}},\quad q\mathrel{\mathop{\mathchar
58\relax}}=\frac{1+S^{2}}{-Y_{x}}.$ (2.44)

Then by summing (2.6), (2.39)-(2.44), we acquire a semilinear hyperbolic
system with smooth coefficients for the variables $\theta,w,z,p,q$ in terms of
the coordinates $(X,Y)$

$\displaystyle\begin{array}[]{l}\displaystyle\theta_{X}=\frac{\sin
w}{4c}p,\quad\theta_{Y}=\frac{\sin z}{4c}q,\\\\[8.0pt] \displaystyle
w_{Y}=\frac{q}{4c}\bigg{\\{}\frac{c^{\prime}}{c}\bigg{(}\cos^{2}\frac{z}{2}-\cos^{2}\frac{w}{2}\bigg{)}-\sin
w\cos^{2}\frac{z}{2}-\sin
z\cos^{2}\frac{w}{2}-4J\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\bigg{\\}},\\\\[8.0pt]
\displaystyle
z_{X}=\frac{p}{4c}\bigg{\\{}\frac{c^{\prime}}{c}\bigg{(}\cos^{2}\frac{w}{2}-\cos^{2}\frac{z}{2}\bigg{)}-\sin
w\cos^{2}\frac{z}{2}-\sin
z\cos^{2}\frac{w}{2}-4J\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\bigg{\\}},\\\\[8.0pt]
\displaystyle p_{Y}=\frac{pq}{2c}\bigg{\\{}\frac{c^{\prime}}{4c}(\sin z-\sin
w)-\frac{1}{4}\sin w\sin z-\sin^{2}\frac{w}{2}\cos^{2}\frac{z}{2}-J\sin
w\cos^{2}\frac{z}{2}\bigg{\\}},\\\\[8.0pt] \displaystyle
q_{X}=\frac{pq}{2c}\bigg{\\{}\frac{c^{\prime}}{4c}(\sin w-\sin
z)-\frac{1}{4}\sin w\sin z-\sin^{2}\frac{z}{2}\cos^{2}\frac{w}{2}-J\sin
z\cos^{2}\frac{w}{2}\bigg{\\}}.\end{array}$ (2.50)

The detailed derivation of (2.50) can be found in Chen-Huang-Liu [8]. In
addition, one also has

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle
x_{X}=\frac{1}{2X_{x}}=\frac{1+\cos w}{4}p,\\\\[8.0pt] \displaystyle
x_{Y}=\frac{1}{2Y_{x}}=-\frac{1+\cos
z}{4}q,\end{array}\right.\quad\left\\{\begin{array}[]{l}\displaystyle
t_{X}=\frac{1}{2cX_{t}}=\frac{1+\cos w}{4c}p,\\\\[8.0pt] \displaystyle
t_{Y}=-\frac{1}{2cY_{t}}=\frac{1+\cos z}{4c}q,\end{array}\right.$ (2.55)

which can be achieved by setting $f=x$ and $f=t$ in (2.42). It suggests by
(2.55) that

$\displaystyle{\rm d}x{\rm
d}t=\frac{pq}{2c}\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\ {\rm d}X{\rm d}Y.$
(2.56)

###  2.2. The boundary value problem in the $(X,Y)$ coordinates

According to the construction of the coordinate transformation
$(x,t)\rightarrow(X,Y)$, we know that the segment $t=0(x\in[0,\pi])$ is
transformed into a piece of continuous and strictly decreasing curve
$\Gamma_{0}\mathrel{\mathop{\mathchar
58\relax}}Y=\phi(X)(X\in[0,\widehat{X}])$, which defines in (2.12) by a
parametric $x\in[0,\pi]$. Moreover, the lines $x=0(t\geq 0)$ and $x=\pi(t\geq
0)$ are transformed into the lines $L_{0}\mathrel{\mathop{\mathchar
58\relax}}Y=X(X\geq 0)$ and $L_{\pi}\mathrel{\mathop{\mathchar
58\relax}}Y=X-\widetilde{X}(X\geq\widehat{X})$, respectively. See Fig. 2.2.

Figure 2.2. The region in the $(x,Y)$ plane.

Since $\Gamma_{0}$ is parameterized by the parameter $x$, we can thus assign
the boundary data $(\bar{\theta},\bar{w},\bar{z},\bar{p},\bar{q})\in
L^{\infty}$ defined by

$\displaystyle\bar{\theta}=\theta_{0}(x),\ \bar{w}=2\arctan R_{0}(x),\
\bar{z}=2\arctan S_{0}(x),\ \bar{p}=1,\ \bar{q}=1,$ (2.57)

where $R_{0}(x)$ and $S_{0}(x)$ are given in (2.11) and the boundary values
$(p,q)$ come from (2.44). Moreover, recalling (1.20) yields $\theta_{t}=0$ on
$x=0$, which implies by (2.3) and (2.43) that $R+S=0$ and $w+z=0$ on $x=0$. On
$x=\pi$, $\theta_{x}=-\iota\theta$ leads to
$R-S=\tan\frac{w}{2}-\tan\frac{z}{2}=-2\iota c(\theta)\theta$.

Hence the boundary values of $(w,z)$ satisfy

$\displaystyle w+z=0,\ \ {\rm on}\
L_{0},\quad\frac{w}{2}=\arctan[\tan\frac{z}{2}-2\iota c(\theta)\theta],\ \
\text{ on }L_{\pi}.$ (2.58)

Furthermore, in view of (2.55), we see that

${\rm d}x=x_{X}{\rm d}X+x_{Y}{\rm d}Y=\frac{1+\cos w}{4}p{\rm d}X-\frac{1+\cos
z}{4}q{\rm d}Y,$

which indicates by (2.58) and the definitions of $L_{0},L_{\pi}$ that

$0=\frac{1+\cos w}{4}(p-q){\rm d}X,\ \ {\rm on}\ \ L_{0},\quad 0=[(1+\cos
w)p-(1+\cos z)q]{\rm d}X,\ \ {\rm on}\ \ L_{\pi}.$

Thus the boundary values of $(p,q)$ satisfy

$\displaystyle p-q=0,\ \ {\rm on}\ L_{0},\quad
p-\frac{1+(\tan\frac{z}{2}-2\iota
c(\theta)\theta)^{2}}{1+\tan^{2}\frac{z}{2}}q=0,\text{ on }L_{\pi}.$ (2.59)

Summing up (2.57), (2.58) and (2.59), the new boundary value problem in the
$(X,Y)$ coordinate plane is the semilinear system (2.50) supplemented with

$\displaystyle\begin{array}[]{ll}(\theta,w,z,p,q)=(\bar{\theta},\bar{w},\bar{z},\bar{p},\bar{q}),&{\rm
on}\ \Gamma_{0},\\\ w+z=0,\ \ p-q=0,&{\rm on}\ L_{0},\\\
\displaystyle\frac{w}{2}=\arctan[\tan\frac{z}{2}-2\iota c(\theta)\theta],\ \
p-\frac{1+(\tan\frac{z}{2}-2\iota
c(\theta)\theta)^{2}}{1+\tan^{2}\frac{z}{2}}q=0,&\text{on }L_{\pi}\end{array}$
(2.63)

We use $\Omega$ to denote the region bounded by $t=0,x=0$ and $x=\pi$ in the
$(x,t)$ plane, and use $\widetilde{\Omega}$ to represent its image in the
$(X,Y)$ plane which is bounded by $\Gamma_{0},L_{0}$ and $L_{\pi}$.

###  2.3. Local existence of the boundary value problem to the semilinear
system

We use the level lines of $X$ and $Y$ to divide the region
$\widetilde{\Omega}$ into a series of subregions

$\widetilde{\Omega}=\bigcup_{n=0}^{\infty}\Omega^{n},$

where $\Omega^{0}=\Omega_{1}^{0}\cup\Omega_{2}^{0}\cup\Omega_{3}^{0}$ with

$\displaystyle\begin{array}[]{l}\Omega_{1}^{0}=\\{(X,Y)\mathrel{\mathop{\mathchar
58\relax}}\ 0<X\leq\widehat{X},\ 0<Y\leq X\\},\\\
\Omega_{2}^{0}=\\{(X,Y)\mathrel{\mathop{\mathchar 58\relax}}\ 0\leq
X\leq\widehat{X},\ \phi(X)\leq Y\leq 0\\},\\\
\Omega_{3}^{0}=\\{(X,Y)\mathrel{\mathop{\mathchar 58\relax}}\
\widehat{X}<X\leq\widetilde{X},\ X-\widetilde{X}\leq Y\leq 0\\},\end{array}$

and $\Omega^{n}=\Omega_{1}^{n}\cup\Omega_{2}^{n}\cup\Omega_{3}^{n}$ with

$\displaystyle\begin{array}[]{l}\Omega_{1}^{n}=\\{(X,Y)\mathrel{\mathop{\mathchar
58\relax}}\ \widehat{X}+k\widetilde{X}<X\leq(k+1)\widetilde{X},\
\widehat{X}+k\widetilde{X}<Y\leq X\\},\\\
\Omega_{2}^{n}=\\{(X,Y)\mathrel{\mathop{\mathchar 58\relax}}\
\widehat{X}+k\widetilde{X}<X\leq(k+1)\widetilde{X},\
k\widetilde{X}<Y\leq\widehat{X}+k\widetilde{X}\\},\\\
\Omega_{3}^{n}=\\{(X,Y)\mathrel{\mathop{\mathchar 58\relax}}\
(k+1)\widetilde{X}<X\leq\widehat{X}+(k+1)\widetilde{X},\ X-\widetilde{X}\leq
Y\leq\widehat{X}+k\widetilde{X}\\},\end{array}$

for $n=2k+1(k=0,1,2,\cdots)$, and with

$\displaystyle\begin{array}[]{l}\Omega_{1}^{n}=\\{(X,Y)\mathrel{\mathop{\mathchar
58\relax}}\ k\widetilde{X}<X\leq\widehat{X}+k\widetilde{X},\
k\widetilde{X}<Y\leq X\\},\\\
\Omega_{2}^{n}=\\{(X,Y)\mathrel{\mathop{\mathchar 58\relax}}\
k\widetilde{X}<X\leq\widehat{X}+k\widetilde{X},\
\widehat{X}+(k-1)\widetilde{X}<Y\leq k\widetilde{X}\\},\\\
\Omega_{3}^{n}=\\{(X,Y)\mathrel{\mathop{\mathchar 58\relax}}\
\widehat{X}+k\widetilde{X}<X\leq(k+1)\widetilde{X},\ X-\widetilde{X}\leq Y\leq
k\widetilde{X}\\},\end{array}$

for $n=2k(k=1,2,3,\cdots)$. See Fig. 2.3 for the illustration.

Figure 2.3. The region $\Omega_{i}^{n}$.

According to the local existence result of the Cauchy problem in [8], we know
that there exists a small positive number
$\delta_{2}<\min\\{\widehat{X},-\phi(\widehat{X})\\}$ such that the problem
(2.50) with
$(\theta,w,z,p,q)|_{\Gamma_{0}}=(\bar{\theta},\bar{w},\bar{z},\bar{p},\bar{q})$
admits a local solution $(\theta,w,z,p,q)(X,Y)$ on $\Omega_{2\delta_{2}}^{0}$,
where

$\displaystyle\Omega_{2\delta_{2}}^{0}=\\{(X,Y)\in\Omega_{2}^{0}\mathrel{\mathop{\mathchar
58\relax}}\ {\rm dist}((X,Y),\Gamma_{0})\leq\delta_{2}\\}.$ (2.64)

We next only discuss the local existence of solutions in the region
$\Omega_{1}^{0}$ near point $(0,0)$, the local existence result in the region
$\Omega_{3}^{0}$ near point $(\widehat{X},\phi(\widehat{X}))$ can be obtained
similarly.

Denote $\Omega_{1\delta_{2}}^{0}=\Omega_{1}^{0}\cap\\{X<\delta_{2}\\}$. By
means of the solution on $Y=0(X\in[0,\delta_{2}])$, we can use (2.50), (2.55)
and (2.63) to construct a map for any point
$(X,Y)\in\Omega_{1\delta_{2}}^{0}$,

$\displaystyle(\hat{\theta},\hat{w},\hat{z},\hat{p},\hat{q})=\mathcal{T}_{1}(\theta,w,z,p,q),$
(2.65)

where

$\displaystyle\hat{\theta}(X,Y)=\theta(X,0)+\int_{0}^{Y}\frac{\sin
z}{4c}q(X,Y^{\prime})\ {\rm d}Y^{\prime},$ (2.66) $\displaystyle\hat{w}(X,Y)=$
$\displaystyle
w(X,0)+\int_{0}^{Y}\frac{q}{4c}\bigg{\\{}\frac{c^{\prime}}{c}\bigg{(}\cos^{2}\frac{z}{2}-\cos^{2}\frac{w}{2}\bigg{)}-\sin
w\cos^{2}\frac{z}{2}$ $\displaystyle-\sin
z\cos^{2}\frac{w}{2}-4J(x_{m},t_{m})\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\bigg{\\}}(X,Y^{\prime})\
{\rm d}Y^{\prime},$ (2.67) $\displaystyle\hat{z}(X,Y)=$
$\displaystyle-w(Y,0)-\int_{0}^{Y}\frac{q}{4c}\bigg{\\{}\frac{c^{\prime}}{c}\bigg{(}\cos^{2}\frac{z}{2}-\cos^{2}\frac{w}{2}\bigg{)}-\sin
w\cos^{2}\frac{z}{2}-\sin z\cos^{2}\frac{w}{2}$
$\displaystyle-4J(x_{m},t_{m})\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\bigg{\\}}(Y,Y^{\prime})\
{\rm
d}Y^{\prime}+\int_{Y}^{X}\frac{p}{4c}\bigg{\\{}\frac{c^{\prime}}{c}\bigg{(}\cos^{2}\frac{w}{2}-\cos^{2}\frac{z}{2}\bigg{)}$
$\displaystyle-\sin w\cos^{2}\frac{z}{2}-\sin
z\cos^{2}\frac{w}{2}-4J(x_{n},t_{n})\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\bigg{\\}}(X^{\prime},Y)\
{\rm d}X^{\prime},$ (2.68) $\displaystyle\hat{p}(X,Y)=$ $\displaystyle
p(X,0)+\int_{0}^{Y}\frac{pq}{2c}\bigg{\\{}\frac{c^{\prime}}{4c}(\sin z-\sin
w)-\frac{1}{4}\sin w\sin z-\sin^{2}\frac{w}{2}\cos^{2}\frac{z}{2}$
$\displaystyle-J(x_{m},t_{m})\sin
w\cos^{2}\frac{z}{2}\bigg{\\}}(X,Y^{\prime})\ {\rm d}Y^{\prime},$ (2.69)

and

$\displaystyle\hat{q}(X,Y)=$ $\displaystyle
p(Y,0)+\int_{0}^{Y}\frac{pq}{2c}\bigg{\\{}\frac{c^{\prime}}{4c}(\sin z-\sin
w)-\frac{1}{4}\sin w\sin z-\sin^{2}\frac{w}{2}\cos^{2}\frac{z}{2}$
$\displaystyle-J(x_{m},t_{m})\sin
w\cos^{2}\frac{z}{2}\bigg{\\}}(Y,Y^{\prime})\ {\rm
d}Y^{\prime}+\int_{Y}^{X}\frac{pq}{2c}\bigg{\\{}\frac{c^{\prime}}{4c}(\sin
w-\sin z)$ $\displaystyle-\frac{1}{4}\sin w\sin
z-\sin^{2}\frac{z}{2}\cos^{2}\frac{w}{2}-J(x_{n},t_{n})\sin
z\cos^{2}\frac{w}{2}\bigg{\\}}(X^{\prime},Y)\ {\rm d}X^{\prime}.$ (2.70)

The points $(x_{m},t_{m})$ and $(x_{n},t_{n})$ are defined as follows

$\displaystyle x_{m}(Z,Y^{\prime})=\int_{Y^{\prime}}^{Z}-\frac{1+\cos
w}{4}p(X^{\prime},Y^{\prime})\ {\rm d}X^{\prime},\ \ (Z=X,Y),$ (2.71)
$\displaystyle t_{m}(Z,Y^{\prime})=$ $\displaystyle
t(Y^{\prime},\phi(Y^{\prime}))+\int_{\phi(Y^{\prime})}^{Y^{\prime}}\frac{1+\cos
z}{4c}q(Y^{\prime},Y^{\prime\prime})\ {\rm d}Y^{\prime\prime}$
$\displaystyle+\int_{Y^{\prime}}^{Z}-\frac{1+\cos
w}{4c}p(X^{\prime},Y^{\prime})\ {\rm d}X^{\prime},\ \ (Z=X,Y),$ (2.72)
$\displaystyle
x_{n}(X^{\prime},Y)=x(X^{\prime},\phi(X^{\prime}))+\int_{\phi(X^{\prime})}^{Y}-\frac{1+\cos
z}{4}q(X^{\prime},Y^{\prime})\ {\rm d}Y^{\prime},$ (2.73)

and

$\displaystyle
t_{n}(X^{\prime},Y)=t(X^{\prime},\phi(X^{\prime}))+\int_{\phi(X^{\prime})}^{Y}\frac{1+\cos
z}{4c}q(X^{\prime},Y^{\prime})\ {\rm d}Y^{\prime}.$ (2.74)

It is noticed that $(x_{m},t_{m})(Z,Y^{\prime})$ and
$(x_{n},t_{n})(X^{\prime},Y)$ have finite partial derivatives in $Z$ and $Y$,
respectively.

We denote

$V=(\theta,w,z,p,q),\quad\hat{V}=(\hat{\theta},\hat{w},\hat{z},\hat{p},\hat{q}),$

and

$V_{0}(X)=(\theta,w,z,p,q)(X,0),\ \ \forall\ X\in[0,\delta_{2}].$

Note that the functions $(\theta,w,z,p,q)(X,0)$ are achieved by solving the
problem in $\Omega_{2\delta_{2}}^{0}$. Due to the result in [8], one knows
that the vector function $V_{0}(X)$ is $C^{\alpha}$ continuous. It is obvious
that

$\displaystyle\hat{\theta}(X,0)=\theta(X,0),\ \ \hat{w}(X,0)=w(X,0),\ \
\hat{p}(X,0)=p(X,0),$ (2.75)

and

$\displaystyle\hat{z}(X,0)=$
$\displaystyle-w(0,0)+\int_{0}^{X}\frac{p}{4c}\bigg{\\{}\frac{c^{\prime}}{c}\bigg{(}\cos^{2}\frac{w}{2}-\cos^{2}\frac{z}{2}\bigg{)}$
$\displaystyle-\sin w\cos^{2}\frac{z}{2}-\sin
z\cos^{2}\frac{w}{2}-4J(x_{n},t_{n})\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\bigg{\\}}(X^{\prime},0)\
{\rm d}X^{\prime}$ $\displaystyle=$ $\displaystyle
z(0,0)+\int_{0}^{X}\frac{p}{4c}\bigg{\\{}\frac{c^{\prime}}{c}\bigg{(}\cos^{2}\frac{w}{2}-\cos^{2}\frac{z}{2}\bigg{)}-\sin
w\cos^{2}\frac{z}{2}$ $\displaystyle-\sin
z\cos^{2}\frac{w}{2}-4J(x_{n},t_{n})\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\bigg{\\}}(X^{\prime},0)\
{\rm d}X^{\prime}=z(X,0).$ (2.76)

The relation $\hat{q}(X,0)=q(X,0)$ can be checked similarly by (2.3).

Let $K_{1}$ be a sufficiently large positive constant that only depends on
$\|V_{0}(X)\|_{C^{\alpha}}$. Set

$\displaystyle\mathcal{K}_{1}=\bigg{\\{}V\ \big{|}\ $
$\displaystyle\|V(X,Y)\|_{C^{\alpha}(\Omega_{1\delta_{1}}^{0})}\leq K_{1},\ \
(w+z)(X,X)=0,$ $\displaystyle(p-q)(X,X)=0,\ \ V(X,0)=V_{0}(X)\bigg{\\}},$
(2.77)

where $\delta_{1}\leq\delta_{2}$ is a small positive constant and

$\displaystyle\Omega_{1\delta_{1}}^{0}=\big{\\{}(X,Y)\in\Omega_{1\delta_{2}}^{0}\mathrel{\mathop{\mathchar
58\relax}}\ {\rm dist}((X,Y),(0,0))\leq\delta_{1}\big{\\}}.$ (2.78)

It is easy to see that $\hat{V}(X,0)=V_{0}(X)$ and $\mathcal{K}_{1}$ is a
compact set in $C^{0}(\Omega_{1\delta_{1}}^{0})$ space. To apply the Schauder
fixed point theorem, we only need to show that the map $\mathcal{T}_{1}$ is
continuous under $C^{0}$ norm and maps $\mathcal{K}_{1}$ to itself. By
selecting appropriate constants $K_{1}$ and then $\delta_{1}$, these
properties can be checked based on the facts that $J(x,t)$ is $C^{\alpha}$
continuous in $(x,t)$ and $(x_{m},t_{m})(Z,Y^{\prime})$ and
$(x_{n},t_{n})(X^{\prime},Y)$ have finite partial derivatives with respect to
$Z,Y$. We omit the proof since it is entirely similar to that in [8].

###  2.4. Inverse transformation

We recall (2.50) and (2.55) to calculate

$\displaystyle\bigg{(}\frac{1+\cos w}{4}p\bigg{)}_{Y}=\bigg{(}-\frac{1+\cos
z}{4}q\bigg{)}_{X},\quad\bigg{(}\frac{1+\cos
w}{4c}p\bigg{)}_{Y}=\bigg{(}\frac{1+\cos z}{4c}q\bigg{)}_{X},$ (2.79)

which mean that

$\displaystyle x_{XY}=x_{YX},\quad t_{XY}=t_{YX}.$ (2.80)

Thus two $x$ equations and two $t$ equations in (2.55) are equivalent,
respectively. Therefore, we have by (2.71)-(2.74)

$(x_{m}(X,Y),t_{m}(X,Y))=(x_{n}(X,Y),t_{n}(X,Y))=\mathrel{\mathop{\mathchar
58\relax}}(x,t).$

Moreover, if the solution exists, we can define the functions
$(x(X,Y),t(X,Y))$ based on the region where $(X,Y)$ is located. For example,
if $(X,Y)\in\Omega_{1}^{2}$, then

$\displaystyle x(X,Y)=\int_{Y}^{X}\frac{1+\cos w}{4}p(X^{\prime},Y)\ {\rm
d}X^{\prime}=\pi+\int_{Y+\bar{X}}^{Y}-\frac{1+\cos z}{4}q(X,Y^{\prime})\ {\rm
d}Y^{\prime},$ (2.81)

and

$\displaystyle t(X,Y)=$
$\displaystyle\int_{\phi(Y-\widetilde{X})}^{Y-\widetilde{X}}\frac{1+\cos
z}{4c}q(Y-\bar{X},Y^{\prime})\ {\rm
d}Y^{\prime}+\int_{Y-\widetilde{X}}^{Y}\frac{1+\cos
w}{4c}p(X^{\prime},Y-\widetilde{X})\ {\rm d}X^{\prime}$
$\displaystyle+\int_{Y-\widetilde{X}}^{Y}\frac{1+\cos z}{4c}q(Y,Y^{\prime})\
{\rm d}Y^{\prime}+\int_{Y}^{X}\frac{1+\cos w}{4c}p(X^{\prime},Y)\ {\rm
d}X^{\prime}$ $\displaystyle=$
$\displaystyle\int_{\phi(X-\widetilde{X})}^{X-\widetilde{X}}\frac{1+\cos
z}{4c}q(X-\widetilde{X},Y^{\prime})\ {\rm
d}Y^{\prime}+\int_{X-\widetilde{X}}^{X}\frac{1+\cos
w}{4c}p(X^{\prime},X-\widetilde{X})\ {\rm d}X^{\prime}$
$\displaystyle+\int_{X-\widetilde{X}}^{Y}\frac{1+\cos z}{4c}q(X,Y^{\prime})\
{\rm d}Y^{\prime}.$ (2.82)

We point out that the map from $(X,Y)$ to $(x,t)$ constructed above may not be
one-to-one mapping. But the values of $\theta$ do not depend on the choice of
$(X,Y)$, that is, if $x(X_{1},Y_{1})=x(X_{2},Y_{2})$ and
$t(X_{1},Y_{1})=t(X_{2},Y_{2})$ for two points $(X_{1},Y_{1})$ and
$(X_{2},Y_{2})$ in $\widetilde{\Omega}$, we have

$\theta(x(X_{1},Y_{1}),t(X_{1},Y_{1}))=\theta(x(X_{2},Y_{2}),t(X_{2},Y_{2})).$

To show this assertion, we divide the proof into two cases: Case 1. $X_{1}\leq
X_{2},Y_{1}\leq Y_{2}$ and Case 2. $X_{1}\leq X_{2},Y_{1}\geq Y_{2}$. The
proof for Case 2 is identical to that in Bressan and Zheng [5]. For Case 1, if
$x(X_{1},Y_{1})=x(X_{2},Y_{2})=x^{*}\in(0,\pi)$, as in [5], one considers the
set

$D_{X^{*}}\mathrel{\mathop{\mathchar
58\relax}}=\\{(X,Y)\mathrel{\mathop{\mathchar 58\relax}}\ x(X,Y)\leq x^{*}\\}$

and denote by $\partial D_{X^{*}}$ its boundary. Due to the facts that $x$ is
increasing with $X$ and decreasing with $Y$, this boundary is a Lipschitz
continuous curve in $\Omega$. Hence we can construct a Lipschitz continuous
curve $\gamma_{1}$ connecting points $(X_{1},Y_{1})$ and $(X_{2},Y_{2})$,
which consists a horizontal segment $Y\equiv Y_{1}$, $\partial D_{X^{*}}$ and
a vertical segment $X\equiv X_{2}$. On $\gamma_{1}$, there hold $x(X,Y)\equiv
x(X_{1},Y_{1})$ and $t(X,Y)\equiv t(X_{1},Y_{1})$ by (2.55). Thus we find that

$\frac{1+\cos w}{4}p\ {\rm d}X=\frac{1+\cos z}{4}q\ {\rm d}Y=0,$

along $\gamma_{1}$, which imply by (2.50) that

$\theta(x(X_{2},Y_{2}),t(X_{2},Y_{2}))-\theta(x(X_{1},Y_{1}),t(X_{1},Y_{1}))=\int_{\gamma_{1}}\frac{\sin
w}{4c}p\ {\rm d}X+\frac{\sin z}{4c}q\ {\rm d}Y=0.$

If $x(X_{1},Y_{1})=x(X_{2},Y_{2})=0$ or $\pi$, we then use the line $L_{0}$ or
$L_{\pi}$ to replace the boundary $\partial D_{X^{*}}$ as before. For
instance, if $x(X_{1},Y_{1})=x(X_{2},Y_{2})=\pi$, a Lipschitz continuous curve
$\gamma_{2}$ connecting points $(X_{1},Y_{1})$ and $(X_{2},Y_{2})$ can be
constructed by a horizontal segment $Y\equiv Y_{1}$, $L_{\pi}$ and a vertical
segment $X\equiv X_{2}$. Due to $t(X_{1},Y_{1})=t(X_{2},Y_{2})$, one has by
(2.55) and the boundary condition $(1+\cos w)p=(1+\cos z)q$ on $L_{\pi}$

$\displaystyle 0=$ $\displaystyle
t(X_{2},Y_{2})-t(X_{1},Y_{1})=t(X_{2},X_{2}-\widetilde{X})-t(Y_{1}+\widetilde{X},Y_{1})$
$\displaystyle=$
$\displaystyle\int_{(Y_{1}+\widetilde{X},Y_{1})}^{(X_{2},X_{2}-\widetilde{X})}\frac{(1+\cos
w)p}{4c}\ {\rm d}X+\frac{(1+\cos z)q}{4c}\ {\rm d}Y$ $\displaystyle=$
$\displaystyle\int_{Y_{1}+\widetilde{X}}^{X_{2}}\frac{(1+\cos
w)p}{2c}(X,X-\widetilde{X})\ {\rm d}X,$

which implies that $(1+\cos w)p(X,X-\widetilde{X})=(1+\cos
z)q(X,X-\widetilde{X})=0$ for $X\in[Y_{1}+\widetilde{X},X_{2}]$. Thus we
utilize the equations in (2.50) to get along $\gamma_{2}$

$\displaystyle\theta(x(X_{2},Y_{2}),t(X_{2},Y_{2}))-\theta(x(X_{1},Y_{1}),t(X_{1},Y_{1}))$
$\displaystyle=$
$\displaystyle\theta(\pi,t(X_{2},X_{2}-\widetilde{X}))-\theta(\pi,t(Y_{1}+\widetilde{X},Y_{1}))$
$\displaystyle=$
$\displaystyle\int_{(Y_{1}+\widetilde{X},Y_{1})}^{(X_{2},X_{2}-\widetilde{X})}\frac{\sin
wp}{4c}\ {\rm d}X+\frac{\sin zq}{4c}\ {\rm d}Y=0.$

In addition, one also has

$\displaystyle{\rm d}x{\rm d}t=\frac{pq}{2c(1+R^{2})(1+S^{2})}{\rm d}X{\rm
d}Y=\frac{pq}{2c}\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}{\rm d}X{\rm d}Y.$
(2.83)

We now show that the estimate

$\displaystyle
E(t)=\int_{0}^{\pi}(\theta_{t}^{2}+c^{2}(\theta)\theta_{x}^{2})(x,t)\ {\rm
d}x+2B(\theta(\pi,t))\leq C_{E},$ (2.84)

for $t\in[0,\widetilde{T}]$, where $C_{E}$ is a positive constant depending on
$E(0)$ and $J$, $\widetilde{T}$ is the existence time of solution, and
$B(\theta(\pi,t))$ is the boundary energy defined in (1.22). Let
$\Gamma_{t}\subset\widetilde{\Omega}_{\widetilde{T}}$ be the transformation of
the horizontal segment of $t$ with $x\in[0,\pi]$ in the $(x,t)$ plane, where
$\widetilde{\Omega}_{\widetilde{T}}$ is the corresponding region of
$\Omega_{\widetilde{T}}=[0,\pi]\times[0,\widetilde{T}]$ in the $(X,Y)$ plane.
We use $(X_{0},X_{0})$ and $(X_{\pi},X_{\pi}-\widetilde{X})$ to represent the
coordinates of the intersection points of $\Gamma_{t}$ and the lines $L_{0}$,
$L_{\pi}$, respectively. Denote $\Omega_{t}=[0,\pi]\times[0,t]$ and
$\widetilde{\Omega}_{t}$ the corresponding region of $\Omega_{t}$ in the
$(X,Y)$ plane. Then one has

$\displaystyle\int_{0}^{\pi}(\theta_{t}^{2}+c^{2}(\theta)\theta_{x}^{2})(x,t)\
{\rm d}x=\int_{\Gamma_{t}\cap\\{\cos w\neq-1\\}}\frac{1-\cos w}{4}p\ {\rm
d}X-\int_{\Gamma_{t}\cap\\{\cos z\neq-1\\}}\frac{1-\cos z}{4}q\ {\rm d}Y$
$\displaystyle\leq$ $\displaystyle\int_{\Gamma_{t}}\frac{1-\cos w}{4}p\ {\rm
d}X-\frac{1-\cos z}{4}q\ {\rm d}Y$ $\displaystyle=$
$\displaystyle\bigg{\\{}\int_{\Gamma_{0}}+\int_{(\widehat{X},\phi(\widehat{X}))}^{(X_{\pi},X_{\pi}-\widetilde{X})}-\int_{(0,0)}^{(X_{0},X_{0})}\bigg{\\}}\frac{1-\cos
w}{4}p\ {\rm d}X-\frac{1-\cos z}{4}q\ {\rm d}Y$
$\displaystyle-\iint_{\widetilde{\Omega}_{t}}\frac{pq}{4c}\bigg{(}\sin\frac{w}{2}\cos\frac{z}{2}+\sin\frac{z}{2}\cos\frac{w}{2}\bigg{)}^{2}\
{\rm d}X{\rm d}Y$
$\displaystyle-\iint_{\widetilde{\Omega}_{t}}\frac{pq}{4c}J\bigg{(}\sin
w\cos^{2}\frac{z}{2}+\sin z\cos^{2}\frac{w}{2}\bigg{)}\ {\rm d}X{\rm d}Y$
$\displaystyle=$ $\displaystyle\int_{\Gamma_{0}}\frac{1-\cos w}{4}p\ {\rm
d}X-\frac{1-\cos z}{4}q\ {\rm
d}Y+\int_{(\widehat{X},\phi(\widehat{X}))}^{(X_{\pi},X_{\pi}-\widetilde{X})}\frac{1-\cos
w}{4}p\ {\rm d}X-\frac{1-\cos z}{4}q\ {\rm d}Y$
$\displaystyle-\iint_{\widetilde{\Omega}_{t}}\frac{pq}{4c}\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\bigg{(}\tan\frac{w}{2}+\tan\frac{z}{2}\bigg{)}^{2}\
{\rm d}X{\rm d}Y$
$\displaystyle-\iint_{\widetilde{\Omega}_{t}}\frac{pq}{2c}\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}2J\bigg{(}\tan\frac{w}{2}+\tan\frac{z}{2}\bigg{)}\
{\rm d}X{\rm d}Y.$ (2.85)

Here we have used the boundary conditions $w+z=0,p=q$ on $L_{0}$. Moreover,
recalling the boundary conditions (2.63) on $L_{\pi}$ arrives at

$\displaystyle\int_{(\widehat{X},\phi(\widehat{X}))}^{(X_{\pi},X_{\pi}-\widetilde{X})}\frac{1-\cos
w}{4}p\ {\rm d}X-\frac{1-\cos z}{4}q\ {\rm d}Y$ $\displaystyle=$
$\displaystyle\int_{\phi(\widehat{X})}^{X_{\pi}-\widetilde{X}}\bigg{(}\frac{2\sin^{2}\frac{w}{2}}{4}\frac{1+\tan^{2}\frac{w}{2}}{1+\tan^{2}\frac{z}{2}}-\frac{2\sin^{2}\frac{z}{2}}{4}\bigg{)}q(Y+\widetilde{X},Y)\
{\rm d}Y$ $\displaystyle=$
$\displaystyle\int_{\phi(\widehat{X})}^{X_{\pi}-\widetilde{X}}\bigg{(}\tan^{2}\frac{w}{2}-\tan^{2}\frac{z}{2}\bigg{)}\frac{\cos^{2}\frac{z}{2}}{2}q(Y+\widetilde{X},Y)\
{\rm d}Y.$ (2.86)

Along $L_{\pi}$, we have by (2.55) and the boundary conditions (2.63)

$\displaystyle{\rm d}t=\frac{1+\cos w}{4c}p\ {\rm d}X+\frac{1+\cos z}{4c}q\
{\rm d}Y=\frac{\cos^{2}\frac{z}{2}}{c}q\ {\rm d}Y.$

Putting the above into (2.4) and using the boundary condition
$\theta_{x}=-\iota\theta$ on $x=\pi$ yields

$\displaystyle\int_{(\widehat{X},\phi(\widehat{X}))}^{(X_{\pi},X_{\pi}-\widetilde{X})}\frac{1-\cos
w}{4}p\ {\rm d}X-\frac{1-\cos z}{4}q\ {\rm d}Y$ $\displaystyle=$
$\displaystyle\int_{0}^{t}-2\iota c^{2}(\theta)\theta\theta_{t}(\pi,t)\ {\rm
d}t=-2\int_{\theta(\pi,0)}^{\theta(\pi,t)}\iota c^{2}(s)s\ {\rm
d}s=2B(\theta(\pi,0))-2B(\theta(\pi,t)).$ (2.87)

We now insert (2.4) into (2.4) and utilize (2.83) to obtain

$\displaystyle\int_{0}^{\pi}(\theta_{t}^{2}+c^{2}(\theta)\theta_{x}^{2})(x,t)\
{\rm d}x\leq$
$\displaystyle\int_{0}^{\pi}(\theta_{t}^{2}+c^{2}(\theta)\theta_{x}^{2})(x,0)\
{\rm d}x+2B(\theta(\pi,0))$
$\displaystyle-2B(\theta(\pi,t))-2\iint_{\Omega_{t}}\theta_{t}^{2}\ {\rm
d}x{\rm d}t-2\iint_{\Omega_{t}}J\theta_{t}\ {\rm d}x{\rm d}t,$ (2.88)

from which we get

$\displaystyle E(t)\leq$ $\displaystyle
E(0)-2\iint_{\Omega_{t}}\theta_{t}^{2}\ {\rm d}x{\rm
d}t-\iint_{\Omega_{t}}2|J|\cdot|\theta_{t}|\ {\rm d}x{\rm d}t$
$\displaystyle\leq$ $\displaystyle E(0)+\iint_{\Omega_{t}}|J|^{2}\ {\rm
d}x{\rm d}t\leq E(0)+\bar{J}(T)\pi T,$ (2.89)

which means that

$\displaystyle\max_{0\leq t\leq\widetilde{T}}E(t)\leq E(0)+\bar{J}(T)\pi T.$
(2.90)

Furthermore, the inequality in (2.90) indicates that
$(\theta_{t},\theta_{x})(\cdot,t)$ and then $(R,S)(\cdot,t)$ are square
integrable functions in $x$.

###  2.5. Global existence of the boundary value problem to the semilinear
system

Let $T>0$ be an any fixed time. Denote

$\Omega_{T}=\\{(x,t)\ x\in[0,\pi],\ t\in[0,T]\\}.$

We use $\widetilde{\Omega}_{T}$ to represent the image of $\Omega_{T}$ in the
$(X,Y)$ plane. For any given $J(x,t)$ satisfying

$\displaystyle\|J\|_{C^{\alpha}\cap
L^{\infty}(\Omega_{T})}=\mathrel{\mathop{\mathchar
58\relax}}\bar{J}(T)<\infty,$ (2.91)

we next extend the local solution constructed in Subsection 2.3 to the whole
region $\widetilde{\Omega}_{T}$. It suffices to establish the global a priori
estimates on $p$ and $q$.

######  Lemma 1.

Let $(\theta,w,z,p,q)(X,Y)$ be a solution of the boundary value problem
(2.50), (2.63) on $\widetilde{\Omega}_{T}$. Then there exist two positive
constants $M$ and $N$ depending only on the boundary data on $\Gamma_{0}$ and
$\bar{J}(T)$ such that

$\displaystyle
0<M\leq\max_{(X,Y)\in\widetilde{\Omega}_{T}}\\{p(X,Y),q(X,Y)\\}\leq N.$ (2.92)

###### Proof.

For convenience, we write equations of $p$ and $q$ in (2.50) as

$\displaystyle p_{Y}=Fpq,\quad q_{X}=Gpq.$ (2.93)

We first discuss the region $\Omega^{0}$. From (2.93), we know by $p=q=1$ on
$\Gamma_{0}$ and the boundary conditions on $L_{0},L_{\pi}$ that $p$ and $q$
are positive on $\Omega^{0}$.

For any point $(X,Y)$ in $\Omega_{1}^{0}$, we consider the region
$\Sigma_{1}^{0}$ enclosed by a horizontal segment between $(Y,Y)$ and $(X,Y)$,
a vertical segment between $(X,Y)$ and $(X,\phi(X))$, $\Gamma_{0}$ and
$L_{0}$. By a direct calculate, one obtains

$\displaystyle\int_{\partial\Sigma_{1}^{0}}p\ {\rm d}X^{\prime}-q\ {\rm
d}Y^{\prime}=$ $\displaystyle-\iint_{\Sigma_{1}^{0}}q_{X}+p_{Y}\ {\rm
d}X^{\prime}{\rm d}Y^{\prime}$ $\displaystyle=$
$\displaystyle\iint_{\Sigma_{1}^{0}}\frac{pq}{2c}\bigg{(}\sin\frac{w}{2}\cos\frac{z}{2}+\sin\frac{z}{2}\cos\frac{w}{2}\bigg{)}^{2}\
{\rm d}X^{\prime}{\rm d}Y^{\prime}$
$\displaystyle+\iint_{\Sigma_{1}^{0}}\frac{pq}{2c}J\bigg{(}\sin
z\cos^{2}\frac{w}{2}+\sin w\cos^{2}\frac{z}{2}\bigg{)}\ {\rm d}X^{\prime}{\rm
d}Y^{\prime}.$ (2.94)

In view of the construction of $\Sigma_{1}^{0}$, we apply the facts $p=q=1$ on
$\Gamma_{0}$ and $p=q$ on $L_{0}$ again

$\displaystyle\int_{\partial\Sigma_{1}^{0}}p\ {\rm d}X^{\prime}-q\ {\rm
d}Y^{\prime}=$ $\displaystyle-\int_{Y}^{X}p(X^{\prime},Y)\ {\rm
d}X^{\prime}-\int_{\phi(X)}^{Y}q(X,Y^{\prime})\ {\rm d}Y^{\prime}$
$\displaystyle+\int_{0}^{X}1\ {\rm d}X^{\prime}-1\ {\rm
d}\phi(X^{\prime})-\int_{(0,0)}^{(Y,Y)}p\ {\rm d}X^{\prime}-q\ {\rm
d}X^{\prime}$ $\displaystyle=$ $\displaystyle
X-\phi(X)-\int_{Y}^{X}p(X^{\prime},Y)\ {\rm
d}X^{\prime}-\int_{\phi(X)}^{Y}q(X,Y^{\prime})\ {\rm d}Y^{\prime}.$ (2.95)

Putting (2.5) into (2.5) yields

$\displaystyle\int_{Y}^{X}p(X^{\prime},Y)\ {\rm
d}X^{\prime}+\int_{\phi(X)}^{Y}q(X,Y^{\prime})\ {\rm d}Y^{\prime}$
$\displaystyle=$ $\displaystyle
X-\phi(X)-\iint_{\Sigma_{1}^{0}}\frac{pq}{2c}\bigg{(}\sin\frac{w}{2}\cos\frac{z}{2}+\sin\frac{z}{2}\cos\frac{w}{2}\bigg{)}^{2}\
{\rm d}X^{\prime}{\rm d}Y^{\prime}$
$\displaystyle-\iint_{\Sigma_{1}^{0}}\frac{pq}{2c}J\bigg{(}\sin
z\cos^{2}\frac{w}{2}+\sin w\cos^{2}\frac{z}{2}\bigg{)}\ {\rm d}X^{\prime}{\rm
d}Y^{\prime}$ $\displaystyle=$ $\displaystyle
X-\phi(X)-\iint_{\Sigma_{1}^{0}}\frac{pq}{2c}\bigg{(}\tan\frac{w}{2}+\tan\frac{z}{2}\bigg{)}^{2}\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\
{\rm d}X^{\prime}{\rm d}Y^{\prime}$
$\displaystyle-\iint_{\Sigma_{1}^{0}}\frac{pq}{2c}2J\bigg{(}\tan\frac{w}{2}+\tan\frac{z}{2}\bigg{)}\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\
{\rm d}X^{\prime}{\rm d}Y^{\prime}$ $\displaystyle\leq$ $\displaystyle
X-\phi(X)+\iint_{\Sigma_{1}^{0}}\frac{pq}{2c}|J|^{2}\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\
{\rm d}X^{\prime}{\rm d}Y^{\prime}$ $\displaystyle\leq$
$\displaystyle\widehat{X}+\widetilde{X}+\iint_{\tilde{\Sigma}_{1}^{0}}|J|^{2}\
{\rm d}x{\rm d}t\leq\widehat{X}+\widetilde{X}+\bar{J}(T)\pi T.$ (2.96)

Here $\tilde{\Sigma}_{1}^{0}$ is the region in the $(x,t)$ plane transformed
from $\Sigma_{1}^{0}$. Moreover, if the point $(X,Y)$ on $L_{0}$, that is
$X=Y$, we directly have by (2.5)

$\displaystyle\int_{\phi(Y)}^{Y}q(Y,Y^{\prime})\ {\rm
d}Y^{\prime}\leq\widehat{X}+\widetilde{X}+\bar{J}(T)\pi T.$ (2.97)

Now we integrate (2.93) and employ (2.5), (2.97) to acquire

$\displaystyle p(X,Y)=$ $\displaystyle
p(X,\phi(X))\exp\bigg{(}\int_{\phi(X)}^{Y}Fq(X,Y^{\prime})\ {\rm
d}Y^{\prime}\bigg{)}$ $\displaystyle\leq$
$\displaystyle\exp\bigg{(}\tilde{F}\int_{\phi(X)}^{Y}q(X,Y^{\prime})\ {\rm
d}Y^{\prime}\bigg{)}\leq\exp\bigg{(}\tilde{F}[\widehat{X}+\widetilde{X}+\bar{J}(T)\pi
T]\bigg{)},$ (2.98)

and

$\displaystyle q(X,Y)=$ $\displaystyle
q(Y,Y)\exp\bigg{(}\int_{Y}^{X}Gp(X^{\prime},Y)\ {\rm
d}X^{\prime}\bigg{)}=p(Y,Y)\exp\bigg{(}\int_{Y}^{X}Gp(X^{\prime},Y)\ {\rm
d}X^{\prime}\bigg{)}$ $\displaystyle=$ $\displaystyle
p(Y,\phi(Y))\exp\bigg{(}\int_{\phi(Y)}^{Y}Fq(Y,Y^{\prime})\ {\rm
d}Y^{\prime}\bigg{)}\cdot\exp\bigg{(}\int_{Y}^{X}Gp(X^{\prime},Y)\ {\rm
d}X^{\prime}\bigg{)}$ $\displaystyle\leq$
$\displaystyle\exp\bigg{(}\tilde{F}\int_{\phi(Y)}^{Y}q(Y,Y^{\prime})\ {\rm
d}Y^{\prime}\bigg{)}\cdot\exp\bigg{(}\tilde{F}\int_{Y}^{X}p(X^{\prime},Y)\
{\rm d}X^{\prime}\bigg{)}$ $\displaystyle\leq$
$\displaystyle\exp\bigg{(}2\tilde{F}[\widehat{X}+\widetilde{X}+\bar{J}(T)\pi
T]\bigg{)},$ (2.99)

where

$\displaystyle|F|,|G|\leq\tilde{F}=\frac{1}{2C_{L}}\bigg{(}\frac{C_{1}}{2C_{L}}+2+\bar{J}(T)\bigg{)}.$

Here the assumption (1.4) is applied.

The case $(X,Y)\in\Omega_{2}^{0}$ can be easily handled. We now discuss the
case $(X,Y)\in\Omega_{3}^{0}$. Consider the region $\Sigma_{3}^{0}$ enclosed
by a vertical segment between $(X,Y)$ and $(X,X-\widetilde{X})$, a horizontal
segment between $(X,Y)$ and $(\phi^{-1}(Y),Y)$, $\Gamma_{0}$ and $L_{\pi}$. By
Green’s theorem, one also has

$\displaystyle\int_{\partial\Sigma_{3}^{0}}p\ {\rm d}X^{\prime}-q\ {\rm
d}Y^{\prime}=$ $\displaystyle-\iint_{\Sigma_{3}^{0}}q_{X}+p_{Y}\ {\rm
d}X^{\prime}{\rm d}Y^{\prime}$ $\displaystyle=$
$\displaystyle\iint_{\Sigma_{3}^{0}}\frac{pq}{2c}\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\bigg{(}\tan\frac{w}{2}+\tan\frac{z}{2}\bigg{)}^{2}\
{\rm d}X^{\prime}{\rm d}Y^{\prime}$
$\displaystyle+\iint_{\Sigma_{3}^{0}}\frac{pq}{2c}\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}2J\bigg{(}\tan\frac{w}{2}+\tan\frac{z}{2}\bigg{)}\
{\rm d}X^{\prime}{\rm d}Y^{\prime}.$ (2.100)

Due to the construction of $\Sigma_{3}^{0}$, we employ the boundary conditions
in (2.63) on $\Gamma_{0}$ to get

$\displaystyle\int_{\partial\Sigma_{3}^{0}}p\ {\rm d}X^{\prime}-q\ {\rm
d}Y^{\prime}=$ $\displaystyle\int_{X-\widetilde{X}}^{Y}-q(X,Y^{\prime})\ {\rm
d}Y^{\prime}-\int_{\phi^{-1}(Y)}^{X}p(X^{\prime},Y)\ {\rm d}X^{\prime}$
$\displaystyle\ -\int_{\widehat{X}}^{\phi^{-1}(Y)}1\ {\rm d}X^{\prime}-1\ {\rm
d}\phi(X^{\prime})+\int_{(\widehat{X},\widehat{X}-\widetilde{X})}^{(X,X-\widetilde{X})}p\
{\rm d}X^{\prime}-q\ {\rm d}X^{\prime}$ $\displaystyle=$
$\displaystyle\widehat{X}-\phi^{-1}(Y)+Y-\phi(\widehat{X})-\int_{X-\widetilde{X}}^{Y}q(X,Y^{\prime})\
{\rm d}Y^{\prime}$ $\displaystyle\ -\int_{\phi^{-1}(Y)}^{X}p(X^{\prime},Y)\
{\rm
d}X^{\prime}+\int_{\widehat{X}-\widetilde{X}}^{X-\widetilde{X}}(p-q)(Y^{\prime}+\widetilde{X},Y^{\prime})\
{\rm d}Y^{\prime}.$ (2.101)

We next estimate the last term in (2.5). Making use of the boundary conditions
in (2.63) on $L_{\pi}$ leads to

$\displaystyle\int_{\widehat{X}-\widetilde{X}}^{X-\widetilde{X}}(p-q)(Y^{\prime}+\widetilde{X},Y^{\prime})\
{\rm
d}Y^{\prime}=\int_{\widehat{X}-\widetilde{X}}^{X-\widetilde{X}}q\left[\frac{1+(\tan\frac{z}{2}-2\iota
c\theta)^{2}}{1+\tan^{2}\frac{z}{2}}-1\right](Y^{\prime}+\widetilde{X},Y^{\prime})\
{\rm d}Y^{\prime}$ $\displaystyle=$
$\displaystyle\int_{\widehat{X}-\widetilde{X}}^{X-\widetilde{X}}\left[\frac{1+(\tan\frac{z}{2}-2\iota
c\theta)^{2}}{1+\tan^{2}\frac{z}{2}}-1\right]_{\Gamma_{0}}{\rm
d}Y^{\prime}+\iint_{\Sigma_{3,1}^{0}}\partial_{X}\left[q\left(\frac{(\tan\frac{z}{2}-2\iota
c\theta)^{2}-\tan^{2}\frac{z}{2}}{1+\tan^{2}\frac{z}{2}}\right)\right]\ {\rm
d}X^{\prime}{\rm d}Y^{\prime},$ (2.102)

where $\Sigma_{3,1}^{0}$ is the region enclosed by a horizontal segment
between $(X,X-\widetilde{X})$ and
$(\phi^{-1}(X-\widetilde{X}),X-\widetilde{X})$, $\Gamma_{0}$ and $L_{\pi}$.
Performing direct calculations, one obtains by system (2.50)

$\displaystyle\partial_{X}\left[q\left(\frac{(\tan\frac{z}{2}-2\iota
c\theta)^{2}-\tan^{2}\frac{z}{2}}{1+\tan^{2}\frac{z}{2}}\right)\right]=\partial_{X}\left[q(-2\iota
c\theta\sin z+2\iota^{2}c^{2}\theta^{2}\cos
z+2\iota^{2}c^{2}\theta^{2})\right]$ $\displaystyle=$ $\displaystyle
q_{X}\left[-2\iota c\theta\sin z+2\iota^{2}c^{2}\theta^{2}\cos
z+2\iota^{2}c^{2}\theta^{2}\right]+z_{X}\left[-2\iota qc\theta\cos
z-2\iota^{2}qc^{2}\theta^{2}\sin z\right]$
$\displaystyle+\theta_{X}\left[-2\iota(c^{\prime}\theta+c)q\sin
z+2\iota^{2}q(2cc^{\prime}\theta^{2}+2c^{2}\theta)(\cos z+1)\right]$
$\displaystyle=$
$\displaystyle\frac{pq}{2c}\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}(\iota\Lambda_{1}+\iota^{2}\Lambda_{2}),$
(2.103)

where

$\displaystyle\Lambda_{1}=~{}$ $\displaystyle
c\theta\left[4\tan\frac{w}{2}\sin^{2}\frac{z}{2}+2\tan\frac{z}{2}-2\tan\frac{w}{2}\right]-4c\tan\frac{z}{2}\tan\frac{w}{2}$
$\displaystyle+c^{\prime}\theta\bigg{(}2+\tan^{2}\frac{w}{2}+\tan^{2}\frac{z}{2}-6\tan\frac{z}{2}\tan\frac{w}{2}\bigg{)}-2c\theta\sin
z$ $\displaystyle+8c\theta\sin^{2}\frac{z}{2}J+4c\theta J\cos z+2c\theta\sin
z+2c\theta\sin w-2c^{\prime}\theta,$

and

$\displaystyle\Lambda_{2}=~{}$
$\displaystyle\bigg{[}\frac{c^{\prime}}{2c}\tan\frac{w}{2}-\frac{c^{\prime}}{4c}\sin
z(1+\tan^{2}\frac{w}{2})-\frac{1}{2}\tan\frac{w}{2}\sin
z-\sin^{2}\frac{z}{2}-J\sin z\bigg{]}4c^{2}\theta^{2}$
$\displaystyle+8\tan\frac{w}{2}\left[(cc^{\prime}\theta^{2}+c^{2}\theta)\right]$
$\displaystyle+\bigg{[}\frac{c^{\prime}}{c}\bigg{(}\sin\frac{z}{2}\tan\frac{z}{2}-\cos\frac{z}{2}\tan^{2}\frac{w}{2}\bigg{)}-2\tan\frac{w}{2}-2\sin\frac{z}{2}-4J\cos\frac{z}{2}\bigg{]}\left[-2c^{2}\theta^{2}\sin\frac{z}{2}\right].$

Note that the quadratic terms of $\tan\frac{w}{2}$ and $\tan\frac{z}{2}$
appear at most in $\Lambda_{1},\Lambda_{2}$, and
$\Sigma_{3,1}^{0}\subset\Omega^{0}$ is a bounded region. We use (2.83) and the
estimate (2.90) to acquire

$\displaystyle\left|\iint_{\Sigma_{3,1}^{0}}\partial_{X}\left[q\left(\frac{(\tan\frac{z}{2}-2\iota
c\theta)^{2}-\tan^{2}\frac{z}{2}}{1+\tan^{2}\frac{z}{2}}\right)\right]\ {\rm
d}X^{\prime}{\rm d}Y^{\prime}\right|$ $\displaystyle=$
$\displaystyle\left|\iint_{\Sigma_{3,1}^{0}}\frac{pq}{2c}\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}(\iota\Lambda_{1}+\iota^{2}\Lambda_{2})\
{\rm d}X^{\prime}{\rm
d}Y^{\prime}\right|\lesssim(\iota+\iota^{2})[E(0)+1+\bar{J}(T)]T.$ (2.104)

Here and below, we use $A\lesssim B$ to denote $A\leq CB$ for some uniform
constant $C$. Combining (2.5), (2.5), (2.5) and (2.5), one achieves

$\displaystyle\int_{X-\tilde{X}}^{Y}q(X,Y^{\prime})\ {\rm
d}Y^{\prime}+\int_{\phi^{-1}(Y)}^{X}p(X^{\prime},Y)\ {\rm d}X^{\prime}$
$\displaystyle\leq\ $
$\displaystyle\widehat{X}-\phi^{-1}(Y)+Y-\phi(\widehat{X})+\int_{(\widehat{X},\widehat{X}-\widetilde{X})}^{(X,X-\widetilde{X})}(p-q)(Y^{\prime}+\widetilde{X},Y^{\prime})\
{\rm d}Y^{\prime}$
$\displaystyle-\iint_{\Sigma_{3}^{0}}\frac{pq}{2c}\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\bigg{(}\tan\frac{w}{2}+\tan\frac{z}{2}\bigg{)}^{2}\
{\rm d}X^{\prime}{\rm d}Y^{\prime}$
$\displaystyle-\iint_{\Sigma_{3}^{0}}\frac{pq}{2c}\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}2J\bigg{(}\tan\frac{w}{2}+\tan\frac{z}{2}\bigg{)}\
{\rm d}X^{\prime}{\rm d}Y^{\prime}$ $\displaystyle\lesssim\ $
$\displaystyle\widehat{X}+\widetilde{X}+\bar{J}(T)\pi
T+(\iota+\iota^{2})[E(0)+1+\bar{J}(T)]T.$ (2.105)

With similar arguments as (2.5) and (2.5), we can obtain the upper bounds of
$p(X,Y)$ and $q(X,Y)$ for all $(X,Y)\in\Omega_{3}^{0}$. Thus there exist two
positive constants $M_{0}$ and $N_{0}$ depending only on the boundary data on
$\Gamma_{0}$ and $\bar{J}(T)$ such that

$\displaystyle 0<M_{0}\leq\max_{(X,Y)\in\Omega^{0}}\\{p(X,Y),q(X,Y)\\}\leq
N_{0}.$ (2.106)

Thanks to (2.106), there hold

$\displaystyle 0<M_{0}\leq\max_{(X,Y)\in\Gamma_{1}}\\{p(X,Y),q(X,Y)\\}\leq
N_{0},$ (2.107)

where $\Gamma_{1}=\Gamma_{11}\cup\Gamma_{12}$,

$\Gamma_{11}=\\{(X,Y)\mathrel{\mathop{\mathchar 58\relax}}\ X=\widehat{X},\
0\leq Y\leq\widehat{X}\\},\quad\Gamma_{12}=\\{(X,Y)\mathrel{\mathop{\mathchar
58\relax}}\ \widehat{X}\leq X\leq\widetilde{X},\ Y=0\\},$

which are two boundaries of the region $\Omega^{1}$. For any point $(X,Y)$ in
$\Omega^{1}$, for example, $(X,Y)\in\Omega_{3}^{1}$, we consider the region
$\Sigma_{3}^{1}$ enclosed by a horizontal segment between $(\widehat{X},Y)$
and $(X,Y)$, a vertical segment between $(X,Y)$ and $(X,X-\widetilde{X})$,
$L_{\pi}$, $\Gamma_{12}$ and $\Gamma_{11}$. According to the construction of
$\Sigma_{3}^{1}$, one has as in (2.5)

$\displaystyle\int_{\partial\Sigma_{3}^{1}}p\ {\rm d}X^{\prime}-q\ {\rm
d}Y^{\prime}=-\int_{\widehat{X}}^{X}p(X^{\prime},Y)\ {\rm
d}X^{\prime}-\int_{X-\widetilde{X}}^{Y}q(X,Y^{\prime})\ {\rm d}Y^{\prime}$
$\displaystyle\qquad+\int_{0}^{Y}q(\widehat{X},Y^{\prime})\ {\rm
d}Y^{\prime}+\int_{\widehat{X}}^{\widetilde{X}}p(X^{\prime},0)\ {\rm
d}X^{\prime}+\int_{(\widetilde{X},0)}^{(X,X-\widetilde{X})}p\ {\rm
d}X^{\prime}-q\ {\rm d}(X^{\prime}-\widetilde{X})$ $\displaystyle\leq$
$\displaystyle
N_{0}(Y+\widetilde{X}-\widehat{X})-\int_{\widehat{X}}^{X}p(X^{\prime},Y)\ {\rm
d}X^{\prime}-\int_{X-\widetilde{X}}^{Y}q(X,Y^{\prime})\ {\rm d}Y^{\prime}$
$\displaystyle+\int_{0}^{X-\widetilde{X}}(p-q)(Y^{\prime}+\widetilde{X},Y^{\prime})\
{\rm d}Y^{\prime}.$ (2.108)

The last term in (2.5) can be treated as in (2.5). Hence similar to (2.5), we
can acquire

$\displaystyle\int_{\widehat{X}}^{X}p(X^{\prime},Y)\ {\rm
d}X^{\prime}+\int_{X-\widetilde{X}}^{Y}q(X,Y^{\prime})\ {\rm d}Y^{\prime}$
$\displaystyle\lesssim$ $\displaystyle
N_{0}(\widetilde{X}+\widehat{X})+\bar{J}(T)\pi
T+(\iota+\iota^{2})[E(0)+1+\bar{J}(T)]T=\mathrel{\mathop{\mathchar
58\relax}}C_{0}(\iota,\bar{J},T,N_{0}).$ (2.109)

Integrating (2.93) and utilizing (2.5) and the boundary conditions (2.63) give

$\displaystyle p(X,Y)=$ $\displaystyle
p(X,X-\widetilde{X})\exp\bigg{(}\int_{X-\widetilde{X}}^{Y}Fq(X,Y^{\prime})\
{\rm d}Y^{\prime}\bigg{)}$ $\displaystyle=$
$\displaystyle\frac{1+(\tan\frac{z}{2}-2\iota
c(\theta)\theta)^{2}}{1+\tan^{2}\frac{z}{2}}q(X,X-\widetilde{X})\exp\bigg{(}\int_{X-\widetilde{X}}^{Y}Fq(X,Y^{\prime})\
{\rm d}Y^{\prime}\bigg{)}$ $\displaystyle\lesssim$ $\displaystyle
q(\widehat{X},X-\widetilde{X})\exp\bigg{(}\int_{\widehat{X}}^{X}Gp(X^{\prime},X-\widetilde{X})\
{\rm
d}X^{\prime}\bigg{)}\cdot\exp\bigg{(}\int_{X-\widetilde{X}}^{Y}Fq(X,Y^{\prime})\
{\rm d}Y^{\prime}\bigg{)}$ $\displaystyle\lesssim$ $\displaystyle
N_{0}\exp\bigg{(}2\tilde{F}C_{0}(\iota,\bar{J},T,N_{0})\bigg{)},$ (2.110)

and

$\displaystyle q(X,Y)=$ $\displaystyle
q(\widehat{X},Y)\exp\bigg{(}\int_{\widehat{X}}^{X}Gp(X^{\prime},Y)\ {\rm
d}X^{\prime}\bigg{)}$ $\displaystyle\lesssim$ $\displaystyle
N_{0}\exp\bigg{(}2\tilde{F}C_{0}(\iota,\bar{J},T,N_{0})\bigg{)}.$ (2.111)

The analysis are the same for the point $(X,Y)\in\Omega_{1}^{1}$ and
$(X,Y)\in\Omega_{2}^{1}$. Hence there exist two positive constants $M_{1}$ and
$N_{1}$ depending only on the boundary data on $\Gamma_{0}$ and $\bar{J}(T)$
such that

$\displaystyle 0<M_{1}\leq\max_{(X,Y)\in\Omega^{1}}\\{p(X,Y),q(X,Y)\\}\leq
N_{1}.$ (2.112)

Assume that $l\geq 1$ is the maximum integer such that

$\bigcup_{i=0}^{l-1}\Omega^{i}\subset\widetilde{\Omega}_{T},$

and denote

$\Omega^{\prime}=\widetilde{\Omega}_{T}\setminus\bigg{(}\bigcup_{i=0}^{l-1}\Omega^{i}\bigg{)}.$

We repeat the above process to obtain

$\displaystyle 0<M_{i}\leq\max_{(X,Y)\in\Omega^{i}}\\{p(X,Y),q(X,Y)\\}\leq
N_{i},\ \ i=1,2,\cdots,l-1,$ (2.113)

where $M_{i},N_{i}$ are positive constants depending only on the boundary data
on $\Gamma_{0}$ and $\bar{J}(T)$. Set

$\displaystyle(X_{l},Y_{l})=\left\\{\begin{array}[]{l}(\widehat{X}+k\widetilde{X},k\widetilde{X}),\qquad\quad\
l=2k+1,\\\ (k\widetilde{X},\widehat{X}+(k-1)\widetilde{X}),\ \
l=2k,\end{array}\right.$ (2.116)

Then it follows by (2.113) that

$\displaystyle 0<M_{l-1}\leq\max_{(X,Y)\in\Gamma_{l}}\\{p(X,Y),q(X,Y)\\}\leq
N_{l-1},$ (2.117)

where $\Gamma_{l}=\Gamma_{l1}\cup\Gamma_{l2}$,

$\Gamma_{l1}=\\{(X,Y)\mathrel{\mathop{\mathchar 58\relax}}\ X=X_{l},\
Y_{l}\leq Y\leq X_{l}\\},\quad\Gamma_{l2}=\\{(X,Y)\mathrel{\mathop{\mathchar
58\relax}}\ X_{l}\leq X\leq Y_{l}+\widetilde{X},\ Y=Y_{l}\\},$

which are two boundaries of the region $\Omega^{l}$. Using the same argument
as before, one can show that there exist two positive constants $M_{l}$ and
$N_{l}$ depending only on the boundary data on $\Gamma_{0}$ and $\bar{J}(T)$
such that

$\displaystyle
0<M_{l}\leq\max_{(X,Y)\in\Omega^{\prime}}\\{p(X,Y),q(X,Y)\\}\leq N_{l}.$
(2.118)

By setting

$M=\min\\{M_{0},M_{1},\cdots,M_{l}\\},\quad
N=\max\\{N_{0},N_{1},\cdots,N_{l}\\},$

we complete the proof of the lemma. ∎

###  2.6. Global existence for the wave equation

This subsection is devoted to verifying that $\theta(x,t)$ constructed above
is a weak solution for the wave equation (2.1) on the region $\Omega_{T}$.

We first claim that the function $\theta(x,t)$ is Hölder continuous in both
$x$ and $t$ with exponent $1/2$. To prove this claim, for any point
$(\xi,\tau)\in\Omega_{T}$, we consider the forward characteristic $t\mapsto
x_{+}(t;\xi,\tau)$. By construction, this curve is parameterized by the
function $X\mapsto(t(X,\overline{Y}),x(X,\overline{Y}))$ for a constant
$\overline{Y}$ depending on $(\xi,\tau)$. Integrating along this forward
characteristic from $(\xi_{0},\tau_{0})$ to $(\xi,\tau)$ and using (2.39),
(2.42), (2.50) and (2.55) gives

$\displaystyle\int_{\tau_{0}}^{\tau}[\theta_{t}+c(\theta)\theta_{x}]^{2}(x_{+}(t;\xi,\tau),t)\
{\rm d}t=\int_{X_{0}}^{X_{\tau}}(2cX_{x}u_{X})^{2}(2X_{t})^{-1}\ {\rm d}X$
$\displaystyle=$
$\displaystyle\int_{X_{0}}^{X_{\tau}}(2cX_{x}u_{X})^{2}(2cX_{x})^{-1}\ {\rm
d}X=\int_{X_{0}}^{X_{\tau}}2c\big{(}p\cos^{2}\frac{w}{2}\big{)}^{-1}\cdot\bigg{(}\frac{p}{2c}\sin\frac{w}{2}\cos\frac{w}{2}\bigg{)}^{2}\
{\rm d}X$ $\displaystyle=$
$\displaystyle\int_{X_{0}}^{X_{\tau}}\frac{p}{2c}\sin^{2}\frac{w}{2}\ {\rm
d}X\leq\frac{1}{2C_{L}}\int_{X_{0}}^{X_{\tau}}p(X,\overline{Y})\ {\rm d}X\leq
C,$ (2.119)

where $C$ is positive constant depending only on $\tau$, $(\xi_{0},\tau_{0})$
is a point on $t=0$ or $x=0$ satisfying $\xi_{0}=x_{+}(\tau_{0};\xi,\tau)$,
and $(x(X_{\tau},\overline{Y}),t(X_{\tau},\overline{Y}))=(\xi,\tau)$,
$(x(X_{0},\overline{Y}),t(X_{0},\overline{Y}))=(\xi_{0},\tau_{0})$. Similarly,
one integrates along the backward characteristic $t\mapsto x_{-}(t;\xi,\tau)$
from $(\xi_{\pi},\tau_{\pi})$ to $(\xi,\tau)$ and noting $X=Const.$ on this
kind of characteristics to acquire

$\displaystyle\int_{\tau_{\pi}}^{\tau}[\theta_{t}-c(\theta)\theta_{x}]^{2}(x_{-}(t;\xi,\tau),t)\
{\rm d}t\leq C.$ (2.120)

Combining (2.6) and (2.120) and employing the boundary conditions of $\theta$
on $x=0,\pi$, we can obtain that $\theta(x,t)$ is Hölder continuous with
exponent $1/2$. Moreover, it is concluded that all characteristic curves are
$C^{1}$ with Hölder continuous derivative.

Next we check that

$\displaystyle\int_{0}^{T}\int_{0}^{\pi}\bigg{(}\theta_{t}\varphi_{t}-(c(\theta)\varphi)_{x}c(\theta)\theta_{x}-\theta_{t}\varphi-J\varphi\bigg{)}\
{\rm d}x{\rm d}t+\int_{0}^{T}(c^{2}\varphi\theta_{x})\bigg{|}_{x=0}^{x=\pi}\
{\rm d}t=0,$ (2.121)

for any test function $\varphi\in\mathcal{F}$, where

$\displaystyle\mathcal{F}\mathrel{\mathop{\mathchar
58\relax}}=\bigg{\\{}\varphi\in
C^{\infty}((0,\pi)\times(0,T))\mathrel{\mathop{\mathchar 58\relax}}\
\partial_{t}^{i}\partial_{x}^{j}\varphi\bigg{|}_{t=0,T}=0,\ \ \forall\
i,j=0,1,2\cdots\bigg{\\}}.$ (2.122)

To obtain (2.121), we need to show that

$\displaystyle 0=$
$\displaystyle\int_{0}^{T}\int_{0}^{\pi}\bigg{(}\varphi_{t}[(\theta_{t}+c\theta_{x})+(\theta_{t}-c\theta_{x})]-(c(\theta)\varphi)_{x}[(\theta_{t}+c\theta_{x})-(\theta_{t}-c\theta_{x})]$
$\displaystyle-2\theta_{t}\varphi-2J\varphi\bigg{)}\ {\rm d}x{\rm
d}t+\int_{0}^{T}2(c^{2}\varphi\theta_{x})\bigg{|}_{x=0}^{x=\pi}\ {\rm d}t$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{0}^{\pi}\bigg{(}[\varphi_{t}-(c\varphi)_{x}](\theta_{t}+c\theta_{x})+[\varphi_{t}+(c\varphi)_{x}](\theta_{t}-c\theta_{x})$
$\displaystyle-2\theta_{t}\varphi-2J\varphi\bigg{)}\ {\rm d}x{\rm
d}t+\int_{0}^{T}2(c^{2}\varphi\theta_{x})\bigg{|}_{x=0}^{x=\pi}\ {\rm d}t$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{0}^{\pi}\bigg{(}[\varphi_{t}-(c\varphi)_{x}]R+[\varphi_{t}+(c\varphi)_{x}]S-2\theta_{t}\varphi-2J\varphi\bigg{)}\
{\rm d}x{\rm d}t+\int_{0}^{T}2(c^{2}\varphi\theta_{x})\bigg{|}_{x=0}^{x=\pi}\
{\rm d}t$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{0}^{\pi}\bigg{(}-2cY_{x}\varphi_{Y}R+2cX_{x}\varphi_{X}S+c^{\prime}(\theta_{X}X_{x}+\theta_{Y}Y_{x})\varphi(S-R)$
$\displaystyle-2(\theta_{X}X_{t}+\theta_{Y}Y_{t})\varphi-2J\varphi\bigg{)}\
{\rm d}x{\rm d}t+\int_{0}^{T}2[c^{2}\varphi(\theta_{X}X_{x}+\theta_{Y}Y_{x})]\
{\rm d}t\bigg{|}_{x=0}^{x=\pi}.$ (2.123)

Noting that the identities

$\displaystyle\frac{1}{1+R^{2}}=\cos^{2}\frac{w}{2},\ \
\frac{1}{1+S^{2}}=\cos^{2}\frac{z}{2},\ \ \frac{R}{1+R^{2}}=\frac{\sin w}{2},\
\ \frac{S}{1+S^{2}}=\frac{\sin z}{2},$

and

$\displaystyle{\rm d}x{\rm d}t=\frac{pq}{2c(1+R^{2})(1+S^{2})}{\rm d}X{\rm
d}Y=\frac{pq}{2c}\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}{\rm d}X{\rm d}Y,$

we use (2.39), (2.44), (2.50) and (2.55) to rewrite the integral in (2.6) as

$\displaystyle\iint_{\widetilde{\Omega}_{T}}\bigg{(}\frac{p\sin
w}{2}\varphi_{Y}+\frac{q\sin z}{2}\varphi_{X}\bigg{)}\ {\rm d}X{\rm
d}Y+\iint_{\widetilde{\Omega}_{T}}\bigg{\\{}\frac{c^{\prime}pq}{8c^{2}}[\cos(w+z)-1]$
$\displaystyle\quad-\frac{pq}{4c}\bigg{(}\sin w\cos^{2}\frac{z}{2}+\sin
z\cos^{2}\frac{w}{2}\bigg{)}-\frac{pq}{c}J\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\bigg{\\}}\varphi\
{\rm d}X{\rm d}Y$
$\displaystyle+\int_{(\widehat{X},\phi(\widehat{X}))}^{(X_{\pi},X_{\pi}-\widetilde{X})}2c^{2}\varphi\bigg{(}\frac{\sin
w}{4c}p\cdot\frac{1+R^{2}}{p}-\frac{\sin
z}{4c}q\cdot\frac{1+S^{2}}{q}\bigg{)}\bigg{\\{}\frac{1+\cos w}{4c}p\ {\rm
d}X+\frac{1+\cos z}{4c}q\ {\rm d}Y\bigg{\\}}$
$\displaystyle-\int_{(0,0)}^{(X_{0},X_{0})}2c^{2}\varphi\bigg{(}\frac{\sin
w}{4c}p\cdot\frac{1+R^{2}}{p}-\frac{\sin
z}{4c}q\cdot\frac{1+S^{2}}{q}\bigg{)}\bigg{\\{}\frac{1+\cos w}{4c}p\ {\rm
d}X+\frac{1+\cos z}{4c}q\ {\rm d}Y\bigg{\\}},$ (2.124)

which together with the boundary conditions in (2.63) yields

$\displaystyle\iint_{\widetilde{\Omega}_{T}}\bigg{(}\frac{p\sin
w}{2}\varphi_{Y}+\frac{q\sin z}{2}\varphi_{X}\bigg{)}\ {\rm d}X{\rm
d}Y+\iint_{\widetilde{\Omega}_{T}}\bigg{\\{}\frac{c^{\prime}pq}{8c^{2}}[\cos(w+z)-1]$
$\displaystyle\quad-\frac{pq}{4c}\bigg{(}\sin w\cos^{2}\frac{z}{2}+\sin
z\cos^{2}\frac{w}{2}\bigg{)}-\frac{pq}{c}J\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\bigg{\\}}\varphi\
{\rm d}X{\rm d}Y$ $\displaystyle+\int_{\widehat{X}}^{X_{\pi}}p\sin
w\varphi(X,X-\widetilde{X})\ {\rm d}X-\int_{0}^{X_{0}}p\sin w\varphi(X,X)\
{\rm d}X,$ (2.125)

where $(X_{0},X_{0})$ and $(X_{\pi},X_{\pi}-\widetilde{X})$ are the points in
the $(X,Y)$ plane transformed from $(0,T)$ and $(\pi,T)$, respectively. By
means of Green’s theorem and boundary conditions, one gets

$\displaystyle\iint_{\widetilde{\Omega}_{T}}\bigg{(}\frac{p\sin
w}{2}\varphi_{Y}+\frac{q\sin z}{2}\varphi_{X}\bigg{)}\ {\rm d}X{\rm d}Y$
$\displaystyle=$
$\displaystyle\iint_{\widetilde{\Omega}_{T}}\bigg{\\{}\bigg{(}\frac{p\sin
w}{2}\varphi\bigg{)}_{Y}+\bigg{(}\frac{q\sin
z}{2}\varphi\bigg{)}_{X}\bigg{\\}}\ {\rm d}X{\rm
d}Y-\iint_{\widetilde{\Omega}_{T}}\frac{1}{2}[(p\sin w)_{Y}+(q\sin
z)_{X}]\varphi\ {\rm d}X{\rm d}Y$ $\displaystyle=$
$\displaystyle\int_{\partial\widetilde{\Omega}_{T}}-\frac{p\sin w}{2}\varphi\
{\rm d}X+\frac{q\sin z}{2}\varphi\ {\rm
d}Y-\iint_{\widetilde{\Omega}_{T}}\frac{1}{2}[(p\sin w)_{Y}+(q\sin
z)_{X}]\varphi\ {\rm d}X{\rm d}Y$ $\displaystyle=$
$\displaystyle-\int_{\widehat{X}}^{X_{\pi}}p\sin w\varphi(X,X-\widetilde{X})\
{\rm d}X+\int_{0}^{X_{0}}p\sin w\varphi(X,X)\ {\rm d}X$
$\displaystyle-\iint_{\widetilde{\Omega}_{T}}\frac{1}{2}[(p\sin w)_{Y}+(q\sin
z)_{X}]\varphi\ {\rm d}X{\rm d}Y.$ (2.126)

According to (2.50), we directly compute to achieve

$\displaystyle\frac{1}{2}[(p\sin w)_{Y}+(q\sin z)_{X}]$ $\displaystyle=$
$\displaystyle\frac{c^{\prime}pq}{8c^{2}}[\cos(w+z)-1]-\frac{pq}{4c}\bigg{(}\sin
w\cos^{2}\frac{z}{2}+\sin
z\cos^{2}\frac{w}{2}\bigg{)}-\frac{pq}{c}J\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}.$
(2.127)

Inserting (2.6) and (2.6) into (2.6) leads to

$\displaystyle\iint_{\widetilde{\Omega}_{T}}\bigg{(}\frac{p\sin
w}{2}\varphi_{Y}+\frac{q\sin z}{2}\varphi_{X}\bigg{)}\ {\rm d}X{\rm
d}Y+\iint_{\widetilde{\Omega}_{T}}\bigg{\\{}\frac{c^{\prime}pq}{8c^{2}}[\cos(w+z)-1]$
$\displaystyle\quad-\frac{pq}{4c}\bigg{(}\sin w\cos^{2}\frac{z}{2}+\sin
z\cos^{2}\frac{w}{2}\bigg{)}-\frac{pq}{c}J\cos^{2}\frac{w}{2}\cos^{2}\frac{z}{2}\bigg{\\}}\varphi\
{\rm d}X{\rm d}Y$ $\displaystyle+\int_{\widehat{X}}^{X_{\pi}}p\sin
w\varphi(X,X-\widetilde{X})\ {\rm d}X-\int_{0}^{X_{0}}p\sin w\varphi(X,X)\
{\rm d}X=0,$ (2.128)

which completes the proof of (2.121).

Moreover, we can repeat the proof process of (2.84) to show that for any
$t\in[0,T]$.

$\displaystyle
E(t)=\int_{0}^{\pi}(\theta_{t}^{2}+c^{2}(\theta)\theta_{x}^{2})(x,t)\ {\rm
d}x+2B(\theta(\pi,t))\leq C_{E}.$ (2.129)

In sum, there has

######  Lemma 2.

Let $T>0$ be any fixed number and $J(x,t)$ be any $C^{\alpha}$ function over
$\Omega_{T}$. Assume that the assumptions on initial and boundary conditions
in Theorem 1.3 hold. Then there exists a weak solution of (2.1) over
$\Omega_{T}$ with bounded energy $E(t)\leq C_{E}$ for some $C_{E}$ depending
on $E(0)$ and $J$.

Finally, for this subsection, we give some comments on the map from $J(x,t)$
to $J(X,Y)$. As pointed out in [8], one can use a similar method in Bressan,
Chen and Zhang [3] for variational wave equation to show that the uniqueness
of forward and backward characteristics for (2.1), that is the uniqueness of
the $(X,Y)$ coordinates. This uniqueness leads to the uniqueness of
$(x_{m}(X,Y),t_{m}(X,Y))=(x_{n}(X,Y),t_{n}(X,Y))$. Hence, for any given
$J(x,t)\in C^{\alpha}(\Omega_{T})$, the solution constructed previously
satisfies system (2.50) with a unique source term
$\tilde{J}(X,Y)=J(x(X,Y),t(X,Y))$. Moreover, one can check that
$\tilde{J}(X,Y)$ is $L^{\infty}$. The details can be found in [8].

##  3\. Existence of the coupled system

In this section, we use the Schauder fixed point theorem to show the global
existence of weak solutions to the initial-boundary value problem of the
coupled system (1.3).

###  3.1. Preliminaries for the heat equation

For smooth solutions of (1.3), the variable $J=u_{x}+\theta_{t}$ satisfies

$J_{t}=J_{xx}+\theta_{tt},$

which together with the wave equation in (1.3) gets

$\displaystyle J_{t}-J_{xx}=c(\theta)(c(\theta)\theta_{x})_{x}-\theta_{t}-J.$
(3.1)

According to the initial and boundary conditions (1.20)-(1.8), one obtains

$\displaystyle J(x,0)=J_{0}(x)\mathrel{\mathop{\mathchar
58\relax}}=u_{0}^{\prime}(x)+\theta_{1}(x)\in C^{\alpha}([0,\pi]),\quad
J_{x}(0,t)=J_{x}(\pi,t)=0.$ (3.2)

Let $G_{0}(x,t;\xi,\tau)$ be the fundamental solution of the heat equation

$\displaystyle
G_{0}(x,t;\xi,\tau)=\frac{1}{2\sqrt{\pi(t-\tau)}}\exp\bigg{(}-\frac{(x-\xi)^{2}}{4(t-\tau)}\bigg{)},\
\ (t\geq\tau).$ (3.3)

Then

$\displaystyle G(x,t;\xi,\tau)=\frac{1}{\pi}\sum_{n=-\infty}^{\infty}\bigg{[}$
$\displaystyle
G_{0}\bigg{(}\frac{x}{\pi},\frac{t}{\pi^{2}};2n+\frac{\xi}{\pi},\frac{\tau}{\pi^{2}}\bigg{)}-G_{0}\bigg{(}\frac{x}{\pi},\frac{t}{\pi^{2}};2n-\frac{\xi}{\pi},\frac{\tau}{\pi^{2}}\bigg{)}\bigg{]},$
(3.4)

and

$\displaystyle N(x,t;\xi,\tau)=\frac{1}{\pi}\sum_{n=-\infty}^{\infty}\bigg{[}$
$\displaystyle
G_{0}\bigg{(}\frac{x}{\pi},\frac{t}{\pi^{2}};2n+\frac{\xi}{\pi},\frac{\tau}{\pi^{2}}\bigg{)}+G_{0}\bigg{(}\frac{x}{\pi},\frac{t}{\pi^{2}};2n-\frac{\xi}{\pi},\frac{\tau}{\pi^{2}}\bigg{)}\bigg{]},$
(3.5)

are, respectively, the Green function and the Neumann function for the first
and the second initial-boundary value problem of the heat equation. See Li, Yu
and Shen [29, 30] for details. As a function of $(x,t)$, $G(x,t;\xi,\tau)$
satisfies the equation $G_{t}=G_{xx}$ for $t>\tau$ and $G=0$ for $x=0,\pi$,
while as a function of $(\xi,\tau)$, $G(x,t;\xi,\tau)$ satisfies the adjoint
equation $G_{\tau}=-G_{\xi\xi}$ for $\tau<t$ and $G=0$ for $\xi=0,\pi$.
Moreover, as a function of $(x,t)$, $N(x,t;\xi,\tau)$ satisfies the equation
$N_{t}=N_{xx}$ for $t>\tau$ and $N_{x}=0$ for $x=0,\pi$, while as a function
of $(\xi,\tau)$, $N(x,t;\xi,\tau)$ satisfies the adjoint equation
$N_{\tau}=-N_{\xi\xi}$ for $\tau<t$ and $N_{\xi}=0$ for $\xi=0,\pi$.
Furthermore, we have

$\displaystyle\frac{\partial G}{\partial x}=-\frac{\partial
N}{\partial\xi},\quad\frac{\partial G}{\partial\xi}=-\frac{\partial
N}{\partial x}.$ (3.6)

Set

$\displaystyle
W_{\sigma}(z,t)=t^{-\frac{\sigma}{2}}\exp\bigg{(}-\frac{z^{2}}{16t}\bigg{)}.$
(3.7)

A direct calculation arrives at

$\displaystyle\int_{\mathbb{R}}W_{\sigma}(\xi,t)\ {\rm
d}\xi=4\sqrt{\pi}t^{\frac{1-\sigma}{2}},\ \ (t>0),$ (3.8)

and

$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}W_{\sigma}(\xi,t-\tau)\ {\rm
d}\xi{\rm d}\tau=\frac{8\sqrt{\pi}}{3-\sigma}t^{\frac{3-\sigma}{2}},\ \
(\sigma<3).$ (3.9)

In addition, one also has

$\displaystyle|\eta|^{\beta}\exp\bigg{(}-\frac{\eta}{4}\bigg{)}\lesssim\exp\bigg{(}-\frac{\eta}{16}\bigg{)},$
(3.10)

for $\eta\geq 0$, where $\beta$ is an arbitrary nonnegative real number.

By using the Neumann function $N(x,t;\xi,\tau)$, the solution of the first
initial-boundary value problem (3.1), (3.2) can be expressed as

$\displaystyle J(x,t)=$ $\displaystyle\int_{0}^{\pi}N(x,t;\xi,0)J_{0}(\xi)\
{\rm
d}\xi-\int_{0}^{t}\int_{0}^{\pi}N(x,t;\xi,\tau)(\theta_{\tau}+J)(\xi,\tau)\
{\rm d}\xi{\rm d}\tau$
$\displaystyle+\int_{0}^{t}N(x,t;\xi,\tau)c^{2}(\theta)\theta_{\xi}(\xi,\tau)\bigg{|}_{\xi=0}^{\xi=\pi}\
{\rm
d}\tau-\int_{0}^{t}\int_{0}^{\pi}N(x,t;\xi,\tau)cc^{\prime}\theta_{\xi}^{2}(\xi,\tau)\
{\rm d}\xi{\rm d}\tau$
$\displaystyle-\int_{0}^{t}\int_{0}^{\pi}\partial_{\xi}N(x,t;\xi,\tau)c^{2}\theta_{\xi}(\xi,\tau)\
{\rm d}\xi{\rm d}\tau.$ (3.11)

###  3.2. The existence of fixed point

For any given function $J(x,t)\in C^{\alpha}(\Omega_{T})$, we know by previous
analysis that there exists a weak solution $\theta(x,t)=\theta^{J}(x,t)$ of
the wave equation (2.1). In view of (3.1), we can define a function
$\mathcal{M}(J)$ on $\Omega_{T}$

$\displaystyle\mathcal{M}(J)(x,t)=$
$\displaystyle\int_{0}^{\pi}N(x,t;\xi,0)J_{0}(\xi)\ {\rm
d}\xi-\int_{0}^{t}\int_{0}^{\pi}N(x,t;\xi,\tau)(\theta_{\tau}+J)(\xi,\tau)\
{\rm d}\xi{\rm d}\tau$
$\displaystyle+\int_{0}^{t}N(x,t;\xi,\tau)c^{2}(\theta)\theta_{\xi}(\xi,\tau)\bigg{|}_{\xi=0}^{\xi=\pi}\
{\rm
d}\tau-\int_{0}^{t}\int_{0}^{\pi}N(x,t;\xi,\tau)cc^{\prime}\theta_{\xi}^{2}(\xi,\tau)\
{\rm d}\xi{\rm d}\tau$
$\displaystyle-\int_{0}^{t}\int_{0}^{\pi}\partial_{\xi}N(x,t;\xi,\tau)c^{2}\theta_{\xi}(\xi,\tau)\
{\rm d}\xi{\rm d}\tau.$ (3.12)

It follows by (3.1) that $\mathcal{M}(J)(x,t)$ is a weak solution of

$\displaystyle\mathcal{M}_{t}-\mathcal{M}_{xx}=c(\theta)(c(\theta)\theta_{x})_{x}-\theta_{t}-J.$
(3.13)

Furthermore, we have a map

$\displaystyle\mathcal{T}\mathrel{\mathop{\mathchar 58\relax}}\
J(x,t)\rightarrow\mathcal{M}(J)(x,t),$ (3.14)

on $C^{\alpha}(\Omega_{T})$. Following Chen-Huang-Liu [8], we define a set
$\mathcal{K}$ for some constants $\delta$ and $K$

$\displaystyle\mathcal{K}=\bigg{\\{}$ $\displaystyle J(x,t)\big{|}\
\|J(x,t)-J^{0}(x,t)\|_{C^{\alpha}(\Omega_{\delta})}\leq K,J(x,0)=J_{0}(x),\
J_{x}(0,t)=J_{x}(\pi,t)=0,\text{ a.e. }\bigg{\\}},$ (3.15)

where $\Omega_{\delta}=[0,\pi]\times[0,\delta]$ and

$J^{0}(x,t)=\int_{0}^{\pi}N(x,t;\xi,0)J_{0}(\xi)\ {\rm d}\xi.$

We first show that, for a given large $K$, the map $\mathcal{T}$ has a fixed
point on $\mathcal{K}$ if $\delta$ is sufficiently small. Next we derive the
energy estimate to select a constant $K$ that only depends on the initial
value. Based on this constant $K$, we can fix the small number $\delta>0$ and
then extend the existence on $\Omega_{\delta}$ to $\Omega_{T}$ in finite
steps.

We apply the Schauder fixed point theorem to show the existence of fixed
points on $\mathcal{K}$ for sufficiently small $\delta$. It is clear to see
that $\mathcal{K}$ is a compact set in $L^{\infty}$. To use the Schauder fixed
point theorem, it suffices to check that $\mathcal{T}$ maps from $\mathcal{K}$
to itself and is continuous under the $L^{\infty}$ norm.

######  Lemma 3.

$\mathcal{T}$ maps from $\mathcal{K}$ to itself.

###### Proof.

According to the properties of the Neumann function $N(x,t;\xi,\tau)$, we know
that $\mathcal{M}(J)(x,0)=J_{0}(x)$ and
$\partial_{x}\mathcal{M}(J)(0,t)=\partial_{x}\mathcal{M}(J)(\pi,t)=0$. To
finish the proof of the lemma, it suffices to verify that $L_{0,1,2,3}(x,t)$
are $C^{\alpha}$ functions, where

$\displaystyle L_{0}(x,t)$
$\displaystyle=\int_{0}^{t}N(x,t;\xi,\tau)c^{2}(\theta)\theta_{\xi}(\xi,\tau)\bigg{|}_{\xi=0}^{\xi=\pi}\
{\rm d}\tau,$ $\displaystyle L_{1}(x,t)$
$\displaystyle=\int_{0}^{t}\int_{0}^{\pi}N(x,t;\xi,\tau)(\theta_{\tau}+J)(\xi,\tau)\
{\rm d}\xi{\rm d}\tau,$ $\displaystyle L_{2}(x,t)$
$\displaystyle=\int_{0}^{t}\int_{0}^{\pi}N(x,t;\xi,\tau)cc^{\prime}\theta_{\xi}^{2}(\xi,\tau)\
{\rm d}\xi{\rm d}\tau,$ $\displaystyle L_{3}(x,t)$
$\displaystyle=\int_{0}^{t}\int_{0}^{\pi}\partial_{\xi}N(x,t;\xi,\tau)c^{2}\theta_{\xi}(\xi,\tau)\
{\rm d}\xi{\rm d}\tau.$

We just only consider the function $L_{3}$, and the functions $L_{0,1,2}$ can
be discussed analogously. In view of the expression of $L_{3}$, it is
sufficient to show that $L_{3}$ is $C^{\alpha}$ continuous with respect to
$x$, that is

$\displaystyle\int_{0}^{t}\int_{0}^{\pi}\frac{|\partial_{\xi}N(x_{2},t;\xi,\tau)-\partial_{\xi}N(x_{1},t;\xi,\tau)|}{(x_{2}-x_{1})^{\alpha}}c^{2}|\theta_{\xi}|(\xi,\tau)\
{\rm d}\xi{\rm d}\tau\leq Ct^{\nu},$ (3.16)

for any $x_{1}<x_{2}\in[0,\pi]$, where $C$ is positive constant and
$\nu\in(0,1)$. Recall the expression of $N(x,t;\xi,\tau)$ in (3.4), one has

$\displaystyle\partial_{\xi}N(x,t;\xi,\tau)=$
$\displaystyle\sum_{n=-\infty}^{\infty}\frac{1}{2\sqrt{\pi(t-\tau)}}\bigg{[}\exp\bigg{(}-\frac{(x-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}\cdot\bigg{(}-\frac{x-2n\pi-\xi}{2(t-\tau)}\bigg{)}$
$\displaystyle+\exp\bigg{(}-\frac{(x-2n\pi+\xi)^{2}}{4(t-\tau)}\bigg{)}\cdot\bigg{(}-\frac{x-2n\pi+\xi)}{2(t-\tau)}\bigg{)}\bigg{]},$
(3.17)

from which we obtain

$\displaystyle\partial_{\xi}N(x_{2},t;\xi,\tau)-\partial_{\xi}N(x_{1},t;\xi,\tau)=\sum_{n=-\infty}^{\infty}(L_{31}^{(n)}+L_{32}^{(n)}),$
(3.18)

where

$\displaystyle L_{31}^{(n)}=$
$\displaystyle\frac{1}{4\sqrt{\pi}(t-\tau)^{\frac{3}{2}}}\bigg{[}\exp\bigg{(}-\frac{(x_{1}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}\cdot(x_{1}-2n\pi-\xi)$
$\displaystyle-\exp\bigg{(}-\frac{(x_{2}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}\cdot(x_{2}-2n\pi-\xi)\bigg{]},$

and

$\displaystyle L_{32}^{(n)}=$
$\displaystyle\frac{1}{4\sqrt{\pi}(t-\tau)^{\frac{3}{2}}}\bigg{[}\exp\bigg{(}-\frac{(x_{1}-2n\pi+\xi)^{2}}{4(t-\tau)}\bigg{)}\cdot(x_{1}-2n\pi+\xi)$
$\displaystyle-\exp\bigg{(}-\frac{(x_{2}-2n\pi+\xi)^{2}}{4(t-\tau)}\bigg{)}\cdot(x_{2}-2n\pi+\xi)\bigg{]},$

We first fix $n\in\mathbb{Z}$ and consider the integral

$\displaystyle I^{(n)}=$
$\displaystyle\int_{0}^{t}\int_{0}^{\pi}\frac{|L_{31}^{(n)}|}{(x_{2}-x_{1})^{\alpha}}c^{2}|\theta_{\xi}|(\xi,\tau)\
{\rm d}\xi{\rm d}\tau$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{B_{1}}\frac{|L_{31}^{(n)}|}{(x_{2}-x_{1})^{\alpha}}c^{2}|\theta_{\xi}|(\xi,\tau)\
{\rm d}\xi{\rm
d}\tau+\int_{0}^{t}\int_{B_{2}}\frac{|L_{31}^{(n)}|}{(x_{2}-x_{1})^{\alpha}}c^{2}|\theta_{\xi}|(\xi,\tau)\
{\rm d}\xi{\rm d}\tau$ $\displaystyle=\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle I_{1}^{(n)}+I_{2}^{(n)},$ (3.19)

where

$\displaystyle B_{1}\mathrel{\mathop{\mathchar 58\relax}}\
\big{|}x_{1}-2n\pi-\xi\big{|}<\big{|}x_{2}-2n\pi-\xi\big{|},$ $\displaystyle
B_{2}\mathrel{\mathop{\mathchar 58\relax}}\
\big{|}x_{1}-2n\pi-\xi\big{|}>\big{|}x_{2}-2n\pi-\xi\big{|}.$

By the definitions of $B_{1}$ and $B_{2}$, we find that

$\displaystyle\big{|}x_{2}-x_{1}\big{|}\leq\big{|}x_{2}-2n\pi-\xi\bigg{|}+\big{|}x_{1}-2n\pi-\xi\big{|}\leq
2\bigg{|}x_{2}-2n\pi-\xi\big{|},\quad{\rm on}\ B_{1},$ (3.20)

and

$\displaystyle\big{|}x_{2}-x_{1}\big{|}\leq\big{|}x_{2}-2n\pi-\xi\big{|}+\big{|}x_{1}-2n\pi-\xi\big{|}\leq
2\big{|}x_{1}-2n\pi-\xi\big{|},\quad{\rm on}\ B_{2},$ (3.21)

Then we handle $I_{1}^{(n)}$ as follows

$\displaystyle I_{1}^{(n)}\leq$
$\displaystyle\int_{0}^{t}\int_{B_{1}}\frac{c^{2}}{4\sqrt{\pi}}\cdot\frac{1}{(x_{2}-x_{1})^{\alpha}(t-\tau)^{\frac{3}{2}}}$
$\displaystyle\quad\cdot(x_{2}-x_{1})\exp\bigg{(}-\frac{(x_{2}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}|\theta_{\xi}|(\xi,\tau)\
{\rm d}\xi{\rm d}\tau$
$\displaystyle+\int_{0}^{t}\int_{B_{1}}\frac{c^{2}}{4\sqrt{\pi}}\cdot\frac{1}{(x_{2}-x_{1})^{\alpha}(t-\tau)^{\frac{3}{2}}}$
$\displaystyle\cdot\big{|}x_{1}-2n\pi-\xi\big{|}\cdot\exp\bigg{(}-\frac{(x_{1}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}$
$\displaystyle\cdot\bigg{|}1-\exp\bigg{(}-\frac{[(x_{2}-2n\pi-\xi)^{2}-(x_{1}-2n\pi-\xi)^{2}]}{4(t-\tau)}\bigg{)}\bigg{|}|\theta_{\xi}|(\xi,\tau)\
{\rm d}\xi{\rm d}\tau$ $\displaystyle=\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle I_{11}^{(n)}+I_{12}^{(n)}.$ (3.22)

The term $I_{2}^{(n)}$ can be treated similarly by symmetry. For
$I_{11}^{(n)}$, one gets by (3.20)

$\displaystyle I_{11}^{(n)}\lesssim$
$\displaystyle\int_{0}^{t}\int_{B_{1}}\frac{1}{(t-\tau)^{\frac{3}{2}}}\bigg{|}\frac{x_{2}}{\pi}-2n-\frac{\xi}{\pi}\bigg{|}^{1-\alpha}$
$\displaystyle\cdot\exp\bigg{(}-\frac{(x_{2}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}|\theta_{\xi}|(\xi,\tau)\
{\rm d}\xi{\rm d}\tau$ $\displaystyle\leq$
$\displaystyle\bigg{\\{}\int_{0}^{t}\int_{B_{1}}\frac{1}{(t-\tau)^{3-2r_{1}}}\bigg{|}\frac{x_{2}}{\pi}-2n-\frac{\xi}{\pi}\bigg{|}^{2-2\alpha}$
$\displaystyle\cdot\exp\bigg{(}-\frac{(x_{2}-2n\pi-\xi)^{2}}{2(t-\tau)}\bigg{)}\
{\rm d}\xi{\rm
d}\tau\bigg{\\}}^{\frac{1}{2}}\cdot\bigg{\\{}\int_{0}^{t}\int_{B_{1}}\frac{|\theta_{\xi}|^{2}}{(t-\tau)^{2r_{1}}}\
{\rm d}\xi{\rm d}\tau\bigg{\\}}^{\frac{1}{2}},$ (3.23)

where $r_{1}$ satisfies

$\displaystyle\frac{1}{4}+\frac{\alpha}{2}<r_{1}<\frac{1}{2},\quad{\rm and\
then}\ \sigma_{1}=4-4r_{1}+2\alpha<3.$ (3.24)

Applying the fact

$\displaystyle\bigg{(}\frac{(x_{2}-2n\pi-\xi)^{2}}{t-\tau}\bigg{)}^{1-\alpha}\exp\bigg{(}-\frac{(x_{2}-2n\pi-\xi)^{2}}{2(t-\tau)}\bigg{)}$
$\displaystyle\lesssim$
$\displaystyle\exp\bigg{(}-\frac{(x_{2}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)},$

we have

$\displaystyle I_{11}^{(n)}\lesssim$
$\displaystyle\bigg{\\{}\int_{0}^{t}\int_{B_{1}}\frac{(t-\tau)^{1-\alpha}}{(t-\tau)^{3-2r_{1}}}\cdot\exp\bigg{(}-\frac{(x_{2}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}\
{\rm d}\xi{\rm d}\tau\bigg{\\}}^{\frac{1}{2}}$
$\displaystyle\cdot\bigg{\\{}\int_{0}^{t}\frac{1}{(t-\tau)^{2r_{1}}}\bigg{(}\int_{0}^{\pi}|\theta_{\xi}|^{2}\
{\rm d}\xi\bigg{)}{\rm d}\tau\bigg{\\}}^{\frac{1}{2}}$
$\displaystyle\lesssim\|\theta_{x}\|_{L^{2}([0,\pi])}t^{\frac{1}{2}-r_{1}}\bigg{\\{}\int_{0}^{t}\int_{B_{1}}(t-\tau)^{-\frac{\sigma_{1}}{2}}\exp\bigg{(}-\frac{(x_{2}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}\
{\rm d}\xi{\rm d}\tau\bigg{\\}}^{\frac{1}{2}}.$ (3.25)

Note that

$\displaystyle\bigg{(}\frac{x}{\pi}-2n-\frac{\xi}{\pi}\bigg{)}^{2}\geq\bigg{(}\frac{x-\xi}{\pi}\bigg{)}^{2}+(2|n|-1)^{2}-1,$
$\displaystyle\bigg{(}\frac{x}{\pi}-2n+\frac{\xi}{\pi}\bigg{)}^{2}\geq\bigg{(}\frac{x-\xi}{\pi}\bigg{)}^{2}+(2|n|-2)^{2}-4,$

we see that

$\displaystyle(2|n|-1)^{2}-1\geq 0,\ \ (2|n|-2)^{2}-4\geq 0,\ \ {\rm for}\
|n|\geq 2.$ (3.26)

Then by (3.9) one acquires

$\displaystyle\int_{0}^{t}\int_{B_{1}}(t-\tau)^{-\frac{\sigma_{1}}{2}}\exp\bigg{(}-\frac{(x_{2}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}\
{\rm d}\xi{\rm d}\tau$ $\displaystyle\leq$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}(t-\tau)^{-\frac{\sigma_{1}}{2}}\exp\bigg{(}-\frac{(x_{2}-\xi)^{2}}{4(t-\tau)}\bigg{)}\
{\rm d}\xi{\rm d}\tau\lesssim t^{\frac{3-\sigma_{1}}{2}},$ (3.27)

for $|n|<2$, and

$\displaystyle\int_{0}^{t}\int_{B_{1}}(t-\tau)^{-\frac{\sigma_{1}}{2}}\exp\bigg{(}-\frac{(x_{2}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}\
{\rm d}\xi{\rm d}\tau$ $\displaystyle\leq$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}(t-\tau)^{-\frac{\sigma_{1}}{2}}\exp\bigg{(}-\frac{(x_{2}-\xi)^{2}}{4(t-\tau)}\bigg{)}\cdot\exp\bigg{(}-\frac{(2|n|-1)^{2}-1}{4(t-\tau)}\bigg{)}\
{\rm d}\xi{\rm d}\tau$ $\displaystyle\leq$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}(t-\tau)^{-\frac{\sigma_{1}}{2}}\exp\bigg{(}-\frac{(x_{2}-\xi)^{2}}{4(t-\tau)}\bigg{)}\cdot\exp\bigg{(}-\frac{(2|n|-1)^{2}-1}{4T}\bigg{)}\
{\rm d}\xi{\rm d}\tau$ $\displaystyle\lesssim$ $\displaystyle
t^{\frac{3-\sigma_{1}}{2}}\exp\bigg{(}-\frac{(2|n|-1)^{2}-1}{4T}\bigg{)},$
(3.28)

for $|n|\geq 2$. Set

$\displaystyle
A_{n}=\left\\{\begin{array}[]{l}1,\qquad\qquad\qquad\qquad\qquad\quad|n|<2,\\\
\displaystyle\exp\bigg{(}-\frac{(2|n|-1)^{2}-1}{8T}\bigg{)},\ |n|\geq
2.\end{array}\right.$ (3.31)

Obviously, the positive series $\displaystyle\sum_{n=-\infty}^{\infty}A_{n}$
is convergent. Putting (3.2)-(3.31) into (3.2) yields

$\displaystyle I_{11}^{(n)}$
$\displaystyle\lesssim\|\theta_{x}\|_{L^{2}([0,\pi])}t^{\frac{1}{2}-r_{1}}\cdot
A_{n}t^{\frac{3-\sigma_{1}}{4}}$
$\displaystyle=\|\theta_{x}\|_{L^{2}([0,\pi])}t^{\frac{1}{2}-r_{1}}\cdot
A_{n}t^{\frac{3-(4-4r+2\alpha)}{4}}=\|\theta_{x}\|_{L^{2}([0,\pi])}t^{\frac{1}{4}-\frac{\alpha}{2}}A_{n}.$
(3.32)

To deal with the term $I_{12}^{(n)}$, we recall the following property:

$\displaystyle 1-e^{-x}\leq\frac{1}{\lambda}x^{\lambda},$ (3.33)

for all $x\geq 0$ and $0<\lambda\leq 1$. Since

$\displaystyle\bigg{(}x_{2}-2n\pi-\xi\bigg{)}^{2}-\bigg{(}x_{1}-2n\pi-\xi\bigg{)}^{2}$
$\displaystyle=$ $\displaystyle(x_{2}-x_{1})\cdot
2\bigg{(}\frac{x_{1}+x_{2}}{2}-2n\pi-\xi\bigg{)}\geq 0,$

on the region $B_{1}$, then we use (3.33) to achieve

$\displaystyle 0\leq$ $\displaystyle
1-\exp\bigg{(}-\frac{(x_{2}-2n\pi-\xi)^{2}-(x_{1}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\frac{1}{2}-\mu}\bigg{(}\frac{(x_{2}-x_{1})(\frac{x_{1}+x_{2}}{2}-2n\pi-\xi)}{2(t-\tau)}\bigg{)}^{\frac{1}{2}-\mu},$
(3.34)

for any number $\mu\in(\frac{1}{4},\frac{1-\alpha}{2})$. Thus we have

$\displaystyle I_{12}^{(n)}\lesssim$
$\displaystyle\int_{0}^{t}\int_{B_{1}}\frac{1}{(x_{2}-x_{1})^{\alpha}(t-\tau)^{\frac{3}{2}}}$
$\displaystyle\cdot\bigg{|}x_{1}-2n\pi-\xi\bigg{|}\cdot\exp\bigg{(}-\frac{(x_{1}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}$
$\displaystyle\cdot\bigg{(}\frac{(x_{2}-x_{1})(\frac{x_{1}+x_{2}}{2}-2n\pi-\xi)}{(t-\tau)}\bigg{)}^{\frac{1}{2}-\mu}|\theta_{\xi}|(\xi,\tau)\
{\rm d}\xi{\rm d}\tau.$ (3.35)

Noting that

$\displaystyle(x_{2}-x_{1})\bigg{(}\frac{x_{1}+x_{2}}{2}-2n\pi-\xi\bigg{)}=(x_{2}-x_{1})\bigg{(}\frac{x_{2}-x_{1}}{2}+x_{1}-2n\pi-\xi\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\big{(}x_{2}-x_{1}\big{)}^{2}+(x_{2}-x_{1})\cdot\bigg{(}x_{1}-2n\pi-\xi\bigg{)},$

one has

$\displaystyle\bigg{[}(x_{2}-x_{1})\bigg{(}\frac{x_{1}+x_{2}}{2}-2n\pi-\xi\bigg{)}\bigg{]}^{\frac{1}{2}-\mu}$
$\displaystyle\lesssim$
$\displaystyle\big{(}x_{2}-x_{1}\big{)}^{1-2\mu}+\big{(}x_{2}-x_{1}\big{)}^{\frac{1}{2}-\mu}\cdot\bigg{(}x_{1}-2n\pi-\xi\bigg{)}^{\frac{1}{2}-\mu}.$

Inserting the above into (3.2) arrives at

$\displaystyle I_{12}^{(n)}\lesssim$
$\displaystyle\int_{0}^{t}\int_{B_{1}}\frac{(x_{2}-x_{1})^{1-2\mu-\alpha}}{(t-\tau)^{2-\mu}}\cdot\bigg{|}x_{1}-2n\pi-\xi\bigg{|}$
$\displaystyle\ \
\cdot\exp\bigg{(}-\frac{(x_{1}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}\cdot|\theta_{\xi}|(\xi,\tau)\
{\rm d}\xi{\rm d}\tau$
$\displaystyle+\int_{0}^{t}\int_{B_{1}}\frac{(x_{2}-x_{1})^{\frac{1}{2}-\mu-\alpha}}{(t-\tau)^{2-\mu}}\cdot\bigg{|}x_{1}-2n\pi-\xi\bigg{|}^{\frac{3}{2}-\mu}$
$\displaystyle\ \
\cdot\exp\bigg{(}-\frac{(x_{1}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}\cdot|\theta_{\xi}|(\xi,\tau)\
{\rm d}\xi{\rm d}\tau=\mathrel{\mathop{\mathchar
58\relax}}I_{12,1}^{(n)}+I_{12,2}^{(n)}.$ (3.36)

For $I_{12,1}^{(n)}$, we get by $1-2\mu-\alpha>0$

$\displaystyle I_{12,1}^{(n)}\lesssim$
$\displaystyle\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{1}{(t-\tau)^{4-2\mu-2r_{2}}}\cdot\bigg{|}x_{1}-2n\pi-\xi\bigg{|}^{2}$
$\displaystyle\ \
\cdot\exp\bigg{(}-\frac{(x_{1}-2n\pi-\xi)^{2}}{2(t-\tau)}\bigg{)}\ {\rm
d}\xi{\rm
d}\tau\bigg{)}^{\frac{1}{2}}\cdot\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{|\theta_{\xi}|^{2}}{(t-\tau)^{2r_{2}}}\
{\rm d}\xi{\rm d}\tau\bigg{)}^{\frac{1}{2}}$ $\displaystyle\lesssim$
$\displaystyle\|\theta_{x}\|_{L^{2}([0,\pi])}t^{\frac{1}{2}-r_{2}}\bigg{(}\int_{0}^{t}\int_{B_{1}}t^{-\frac{\sigma_{2}}{2}}\exp\bigg{(}-\frac{(x_{1}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}\
{\rm d}\xi{\rm d}\tau\bigg{)}^{\frac{1}{2}},$ (3.37)

where $r_{2}$ and $\sigma_{2}$ satisfy

$\displaystyle\frac{3}{4}-\mu<r_{2}<\frac{1}{2},\quad\sigma_{2}=6-4r_{2}-4\mu<3.$
(3.38)

Similar to handling (3.2), we can obtain

$\displaystyle
I_{12,1}^{(n)}\lesssim\|\theta_{x}\|_{L^{2}([0,\pi])}t^{\frac{1}{2}-r_{2}}\cdot
t^{\frac{3-\sigma_{2}}{4}}A_{n}=\|\theta_{x}\|_{L^{2}([0,\pi])}t^{\mu-\frac{1}{4}}A_{n}.$
(3.39)

Now for $I_{12,2}^{(n)}$, one acquires by $\frac{1}{2}-\mu-\alpha>0$

$\displaystyle I_{12,2}^{(n)}\lesssim$
$\displaystyle\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{1}{(t-\tau)^{4-2\mu-2r_{3}}}\cdot\bigg{|}x_{1}-2n\pi-\xi\bigg{|}^{3-2\mu}$
$\displaystyle\ \
\cdot\exp\bigg{(}-\frac{(x_{1}-2n\pi-\xi)^{2}}{2(t-\tau)}\bigg{)}\ {\rm
d}\xi{\rm
d}\tau\bigg{)}^{\frac{1}{2}}\cdot\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{|\theta_{\xi}|^{2}}{(t-\tau)^{2r_{3}}}\
{\rm d}\xi{\rm d}\tau\bigg{)}^{\frac{1}{2}}$ $\displaystyle\lesssim$
$\displaystyle\|\theta_{x}\|_{L^{2}([0,\pi])}t^{\frac{1}{2}-r_{3}}\bigg{(}\int_{0}^{t}\int_{B_{1}}t^{-\frac{\sigma_{3}}{2}}\exp\bigg{(}-\frac{(x_{1}-2n\pi-\xi)^{2}}{4(t-\tau)}\bigg{)}\
{\rm d}\xi{\rm d}\tau\bigg{)}^{\frac{1}{2}},$ (3.40)

where $r_{3}$ and $\sigma_{3}$ satisfy

$\displaystyle\frac{1}{2}-\frac{\mu}{2}<r_{3}<\frac{1}{2},\quad\sigma_{3}=5-4r_{3}-2\mu<3.$
(3.41)

Hence we find that

$\displaystyle
I_{12,2}^{(n)}\lesssim\|\theta_{x}\|_{L^{2}([0,\pi])}t^{\frac{1}{2}-r_{3}}\cdot
t^{\frac{3-\sigma_{3}}{4}}A_{n}=\|\theta_{x}\|_{L^{2}([0,\pi])}t^{\frac{\mu}{2}}A_{n}.$
(3.42)

Combining (3.2), (3.39) and (3.42) leads to

$\displaystyle
I_{12}^{(n)}\lesssim\|\theta_{x}\|_{L^{2}([0,\pi])}(t^{\mu-\frac{1}{4}}+t^{\frac{\mu}{2}})A_{n}.$
(3.43)

Thus we sum up (3.2), (3.2) and (3.43) to conclude that

$\displaystyle
I_{1}^{(n)}\lesssim\|\theta_{x}\|_{L^{2}([0,\pi])}(t^{\frac{1}{4}-\frac{\alpha}{2}}+t^{\mu-\frac{1}{4}}+t^{\frac{\mu}{2}})A_{n}.$
(3.44)

The above estimate is also valid for the term $I_{2}^{(n)}$ in (3.2).
Therefore there holds

$\displaystyle
I^{(n)}=\int_{0}^{t}\int_{0}^{\pi}\frac{|L_{31}^{(n)}|}{(x_{2}-x_{1})^{\alpha}}c^{2}|\theta_{\xi}|(\xi,\tau)\
{\rm d}\xi{\rm
d}\tau\lesssim\|\theta_{x}\|_{L^{2}([0,\pi])}(t^{\frac{1}{4}-\frac{\alpha}{2}}+t^{\mu-\frac{1}{4}}+t^{\frac{\mu}{2}})A_{n},$
(3.45)

which together with (3.31) implies that the series
$\displaystyle\sum_{n=-\infty}^{\infty}I$ is uniform convergent. Similarly,
one also has

$\displaystyle\int_{0}^{t}\int_{0}^{\pi}\frac{|L_{32}^{(n)}|}{(x_{2}-x_{1})^{\alpha}}c^{2}|\theta_{\xi}|(\xi,\tau)\
{\rm d}\xi{\rm
d}\tau\lesssim\|\theta_{x}\|_{L^{2}([0,\pi])}(t^{\frac{1}{4}-\frac{\alpha}{2}}+t^{\mu-\frac{1}{4}}+t^{\frac{\mu}{2}})A_{n}.$
(3.46)

Then

$\displaystyle\int_{0}^{t}\int_{0}^{\pi}\frac{|\partial_{\xi}N(x_{2},t;\xi,\tau)-\partial_{\xi}N(x_{1},t;\xi,\tau)|}{(x_{2}-x_{1})^{\alpha}}c^{2}|\theta_{\xi}|(\xi,\tau)\
{\rm d}\xi{\rm d}\tau$ $\displaystyle\leq$
$\displaystyle\int_{0}^{t}\int_{0}^{\pi}\frac{1}{(x_{2}-x_{1})^{\alpha}}\sum_{n=-\infty}^{\infty}(|L_{31}^{(n)}|+|L_{32}^{(n)}|)c^{2}|\theta_{\xi}|(\xi,\tau)\
{\rm d}\xi{\rm d}\tau$ $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{\infty}\int_{0}^{t}\int_{0}^{\pi}\frac{1}{(x_{2}-x_{1})^{\alpha}}(|L_{31}^{(n)}|+|L_{32}^{(n)}|)c^{2}|\theta_{\xi}|(\xi,\tau)\
{\rm d}\xi{\rm d}\tau$ $\displaystyle\lesssim$
$\displaystyle\|\theta_{x}\|_{L^{2}([0,\pi])}(t^{\frac{1}{4}-\frac{\alpha}{2}}+t^{\mu-\frac{1}{4}}+t^{\frac{\mu}{2}})\sum_{n=-\infty}^{\infty}A_{n}\leq
Ct^{\nu},$ (3.47)

for some constant $C>0$, where
$\nu=\min\\{\frac{1}{4}-\frac{\alpha}{2},\mu-\frac{1}{4},\frac{\mu}{2}\\}>0$.
The proof of Lemma 3 is completed by choosing $3C\delta^{\nu}<K$. ∎

######  Lemma 4.

The map $\mathcal{T}$ is continuous under the $L^{\infty}$ norm.

###### Proof.

The proof follows from the fact that $(\theta,w,z,p,q,x,t)(X,Y)$ is Lipschitz
continuously dependent on $\tilde{J}(X,Y)$ in the $L^{\infty}$ distance. The
proof is entirely similar to that in Chen-Huang-Liu [8] and we omit it here. ∎

By means of Lemmas 3 and 4, we know by the Schauder fixed point theorem that
there exists a function $J^{*}(x,t)\in\mathcal{K}$ such that

$\displaystyle\mathcal{M}(J^{*})(x,t)=J^{*}(x,t).$ (3.48)

We fix $J=J^{*}$ and utilize previous results to obtain that

$\displaystyle\theta_{t}(\cdot,t),\ \theta_{x}(\cdot,t)\in
L^{2}([0,\pi]),\quad J\in C^{\alpha}([0,\pi]\times[0,\delta]).$ (3.49)

Furthermore, we consider the initial-boundary value problem for $u$

$\displaystyle\left\\{\begin{array}[]{l}u_{t}-u_{xx}=\theta_{tx},\\\
u(x,0)=u_{0}(x),\\\ u(0,t)=u(\pi,t)=0.\end{array}\right.$ (3.53)

According to the properties of the Green function $G(x,t;\xi,\tau)$ given in
(3.5), the weak solution $u(x,t),(t\in[0,\delta])$ of (3.53) can be expressed
as

$\displaystyle u(x,t)=\int_{0}^{\pi}G(x,t;\xi,0)u_{0}(\xi)\ {\rm
d}\xi-\int_{0}^{t}\int_{0}^{\pi}\partial_{\xi}G(x,t;\xi,\tau)\theta_{\tau}(\xi,\tau)\
{\rm d}\xi{\rm d}\tau.$ (3.54)

Clearly, this weak solution $u(x,t)$ satisfies

$\displaystyle\int_{0}^{\delta}\int_{0}^{\pi}\bigg{(}u\psi_{t}-(u_{x}+\theta_{t})\psi_{x}\bigg{)}\
{\rm d}x{\rm
d}t+\int_{0}^{\delta}(u_{x}+\theta_{t})\psi\bigg{|}_{x=0}^{x=\pi}\ {\rm
d}t=0,$ (3.55)

for any test functions $\psi\in\mathcal{F}$. Letting $\psi=\varphi_{x}$ in
(3.55) gives

$\displaystyle\int_{0}^{\delta}\int_{0}^{\pi}\bigg{(}u_{x}\varphi_{t}+(u_{x}+\theta_{t})\varphi_{xx}\bigg{)}\
{\rm d}x{\rm
d}t-\int_{0}^{\delta}(u_{x}+\theta_{t})\varphi_{x}\bigg{|}_{x=0}^{x=\pi}\ {\rm
d}t=0,$ (3.56)

Finally, we prove $J=u_{x}+\theta_{t}$ in some sense. By (3.13), the function
$J$ satisfies

$\displaystyle\int_{0}^{\delta}\int_{0}^{\pi}\bigg{(}J(\varphi^{0}_{t}+\varphi^{0}_{xx})-[(c\varphi^{0})_{x}c\theta_{x}+(\theta_{t}+J)\varphi^{0}]\bigg{)}\
{\rm d}x{\rm
d}t-\int_{0}^{\delta}[J\varphi^{0}_{x}-c^{2}\varphi^{0}\theta_{x}]\bigg{|}_{x=0}^{x=\pi}\
{\rm d}t=0,$ (3.57)

for any test function $\varphi^{0}(x,t)\in
C^{\infty}_{c}([0,\pi]\times[0,\delta])$, which together with (2.121) yields

$\displaystyle\int_{0}^{\delta}\int_{0}^{\pi}\bigg{(}(J-\theta_{t})\varphi^{0}_{t}+J\varphi^{0}_{xx}\bigg{)}\
{\rm d}x{\rm d}t-\int_{0}^{\delta}J\varphi^{0}_{x}\bigg{|}_{x=0}^{x=\pi}\ {\rm
d}t=0.$ (3.58)

We take the difference of (3.58) and (3.56), with the test function
$\varphi^{0}$ vanishing on the boundaries, to obtain

$\displaystyle\int_{0}^{\delta}\int_{0}^{\pi}[J-(u_{x}+\theta_{t})](\varphi^{0}_{t}+\varphi^{0}_{xx})\
{\rm d}x{\rm
d}t-\int_{0}^{\delta}(J-u_{x}-\theta_{t})\varphi^{0}_{x}\bigg{|}_{x=0}^{x=\pi}\
{\rm d}t=0.$ (3.59)

Hence it is clear that $J=u_{x}+\theta_{t}$, for a.e. any
$(x,t)\in[0,\pi]\times[0,\delta]$.

###  3.3. Energy estimate

In this subsection, we derive the energy estimate in Theorem 1.3, which allows
us to extend the local solution on $\Omega_{\delta}$ to $\Omega_{T}$.

Following [8], we proceed by the inequality in (2.4) and the fact
$J=u_{x}+\theta_{t}$ in $L^{2}(\Omega_{T})$ sense

$\displaystyle\int_{0}^{\pi}(\theta_{t}^{2}+c^{2}(\theta)\theta_{x}^{2})(x,t)\
{\rm d}x\leq$
$\displaystyle\int_{0}^{\pi}(\theta_{t}^{2}+c^{2}(\theta)\theta_{x}^{2})(x,0)\
{\rm d}x+2B(\theta(\pi,0))$
$\displaystyle-2B(\theta(\pi,t))-2\int_{0}^{t}\int_{0}^{\pi}\theta_{t}^{2}\
{\rm d}x{\rm d}t-2\int_{0}^{t}\int_{0}^{\pi}J\theta_{t}\ {\rm d}x{\rm d}t$
$\displaystyle\leq$
$\displaystyle\int_{0}^{\pi}(\theta_{t}^{2}+c^{2}(\theta)\theta_{x}^{2})(x,0)\
{\rm d}x+2B(\theta(\pi,0))$
$\displaystyle-2B(\theta(\pi,t))-2\int_{0}^{t}\int_{0}^{\pi}\theta_{t}^{2}\
{\rm d}x{\rm d}t-2\int_{0}^{t}\int_{0}^{\pi}J(J-u_{x})\ {\rm d}x{\rm d}t.$
(3.60)

Since $u_{t}=J_{x}$ holds in $L^{2}(\Omega_{T})$ sense, one has by integrating
by parts

$\displaystyle\int_{0}^{t}\int_{0}^{\pi}Ju_{x}\ {\rm d}x{\rm d}t=$
$\displaystyle-\int_{0}^{t}\int_{0}^{\pi}u_{t}u\ {\rm d}x{\rm d}t$
$\displaystyle=$ $\displaystyle\frac{1}{2}\int_{0}^{\pi}u_{0}^{2}(x)\ {\rm
d}x-\frac{1}{2}\int_{0}^{\pi}u^{2}(x,t)\ {\rm d}x.$ (3.61)

Putting (3.3) into (3.3) yields

$\displaystyle\int_{0}^{\pi}(\theta_{t}^{2}+c^{2}(\theta)\theta_{x}^{2})(x,t)\
{\rm
d}x+2B(\theta(\pi,t))\leq\int_{0}^{\pi}(\theta_{t}^{2}+c^{2}(\theta)\theta_{x}^{2})(x,0)\
{\rm d}x+2B(\theta(\pi,0))$
$\displaystyle-2\int_{0}^{t}\int_{0}^{\pi}\theta_{t}^{2}\ {\rm d}x{\rm
d}t-2\int_{0}^{t}\int_{0}^{\pi}J^{2}\ {\rm d}x{\rm
d}t+\int_{0}^{\pi}u_{0}^{2}(x)\ {\rm d}x-\int_{0}^{\pi}u^{2}(x,t)\ {\rm d}x,$
(3.62)

which means that

$\displaystyle\frac{1}{2}\int_{0}^{\pi}(\theta_{t}^{2}+c^{2}(\theta)\theta_{x}^{2}+u^{2})(x,t)\
{\rm d}x+B(\theta(\pi,t))$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{0}^{\pi}(\theta_{t}^{2}+c^{2}(\theta)\theta_{x}^{2}+u^{2})(x,0)\
{\rm
d}x+B(\theta(\pi,0))-\int_{0}^{t}\int_{0}^{\pi}(u_{x}+\theta_{t})^{2}+\theta_{t}^{2}\
{\rm d}x{\rm d}t,$ (3.63)

which is the desired inequality (1.23).

##  4\. The problem with Neumann boundary conditions for heat equation

In this section, we provide the proof on the initial-boundary value problem of
(1.3) with the Neumann boundary conditions on $u$.

First, in Section 2, we solve a boundary value problem for $\theta$, so the
result still holds for the case with the Neumann boundary conditions.

For the existence of $(u,\theta)$ by Schaulder fixed point theorem, to avoid
repeat, we only give calculations different from the case with the Dirichlet
boundary condition.

Consider system (1.3) with the initial condition (1.5) and the following the
mixed boundary conditions

$\displaystyle(u_{x}+\theta_{t})(0,t)=(u_{x}+\theta_{t})(\pi,t)$
$\displaystyle=0,$ (4.1)
$\displaystyle\theta(0,t)=\iota\theta(\pi,t)+\theta_{x}(\pi,t)$
$\displaystyle=0.$

Assume that the initial functions $u_{0}(x)$, $\theta_{0}(x)$ and
$\theta_{1}(x)$ satisfy the corresponding compatibility conditions at $0$ and
$\pi$. Based on the results in Section 2, for any given $T>0$ and $J\in
C^{\alpha}$, we obtain the global solution $\theta(x,t)$ for the wave equation

$\theta_{tt}-c(\theta)(c(\theta)\theta_{x})_{x}+\theta_{t}+J=0.$

Corresponding to the Neumann boundary conditions of $u$, the initial-boundary
value problem of $J$ now is

$\displaystyle\left\\{\begin{array}[]{l}J_{t}-J_{xx}=c(\theta)(c(\theta)\theta_{x})_{x}-\theta_{t}-J,\\\
J(x,0)=J_{0}(x)\mathrel{\mathop{\mathchar
58\relax}}=u_{0}^{\prime}(x)+\theta_{1}(x)\in C^{\alpha}([0,\pi]),\\\
J(0,t)=J(\pi,t)=0.\end{array}\right.$ (4.5)

Thanks to the Green function $G(x,t;\xi,\tau)$ given in Section 3, the
solution $J$ of (4.5) can be expressed as

$\displaystyle J(x,t)=\int_{0}^{\pi}G(x,t;\xi,0)J_{0}(\xi)\ {\rm
d}\xi-\int_{0}^{t}\int_{0}^{\pi}G(x,t;\xi,\tau)(\theta_{\tau}+J)(\xi,\tau)\
{\rm d}\xi{\rm d}\tau$
$\displaystyle\qquad-\int_{0}^{t}\int_{0}^{\pi}G(x,t;\xi,\tau)cc^{\prime}\theta_{\xi}^{2}(\xi,\tau)\
{\rm d}\xi{\rm
d}\tau-\int_{0}^{t}\int_{0}^{\pi}\partial_{\xi}G(x,t;\xi,\tau)c^{2}\theta_{\xi}(\xi,\tau)\
{\rm d}\xi{\rm d}\tau.$ (4.6)

The relation (4) defines a map on $C^{\alpha}(\Omega_{T})$

$\displaystyle\widetilde{\mathcal{T}}\mathrel{\mathop{\mathchar 58\relax}}\
J(x,t)\rightarrow\widetilde{\mathcal{M}}(J)(x,t),$ (4.7)

where

$\displaystyle\widetilde{\mathcal{M}}(J)(x,t)=\int_{0}^{\pi}G(x,t;\xi,0)J_{0}(\xi)\
{\rm
d}\xi-\int_{0}^{t}\int_{0}^{\pi}G(x,t;\xi,\tau)(\theta_{\tau}+J)(\xi,\tau)\
{\rm d}\xi{\rm d}\tau$
$\displaystyle\qquad-\int_{0}^{t}\int_{0}^{\pi}G(x,t;\xi,\tau)cc^{\prime}\theta_{\xi}^{2}(\xi,\tau)\
{\rm d}\xi{\rm
d}\tau-\int_{0}^{t}\int_{0}^{\pi}\partial_{\xi}G(x,t;\xi,\tau)c^{2}\theta_{\xi}(\xi,\tau)\
{\rm d}\xi{\rm d}\tau,$ (4.8)

which is a weak solution of the following equation

$\displaystyle\widetilde{\mathcal{M}}_{t}-\widetilde{\mathcal{M}}_{xx}=c(\theta)(c(\theta)\theta_{x})_{x}-\theta_{t}-J.$
(4.9)

Through the parallel procedure as in Subsection 3.2, we can verify that the
map $\widetilde{\mathcal{T}}$ exists a fixed point in the set

$\displaystyle\widetilde{\mathcal{K}}=\bigg{\\{}$ $\displaystyle
J(x,t)\big{|}\
\|J(x,t)-\widetilde{J}^{0}(x,t)\|_{C^{\alpha}(\Omega_{\delta})}\leq
K,J(x,0)=J_{0}(x),\ J(0,t)=J(\pi,t)=0\bigg{\\}},$ (4.10)

for some constants $\delta>0$ and $K>0$, where

$\widetilde{J}^{0}(x,t)=\int_{0}^{\pi}G(x,t;\xi,0)J_{0}(\xi)\ {\rm d}\xi.$

Denote this fixed point still by $J\in\widetilde{\mathcal{K}}$. Then there has

$\displaystyle\widetilde{\mathcal{M}}(J)(x,t)=J(x,t).$ (4.11)

We next show $J=u_{x}+\theta_{t}$. It follows by (4.9) and (4.11) that the
weak solution $J$ of (4.5) satisfies

$\displaystyle\int_{0}^{\delta}\int_{0}^{\pi}\bigg{(}J(\varphi^{0}_{t}+\varphi^{0}_{xx})-[(c\varphi)_{x}c\theta_{x}+(\theta_{t}+J)\varphi^{0}]\bigg{)}\
{\rm d}x{\rm
d}t+\int_{0}^{\delta}(J_{x}+c^{2}\theta_{x})\varphi^{0}\bigg{|}_{x=0}^{x=\pi}\
{\rm d}t=0,$ (4.12)

for any test function $\varphi^{0}(x,t)\in
C^{\infty}_{c}([0,\pi]\times[0,\delta])$. Combining (2.121) and (4.12), one
obtains

$\displaystyle\int_{0}^{\delta}\int_{0}^{\pi}\bigg{(}(J-\theta_{t})\varphi^{0}_{t}+J\varphi^{0}_{xx}\bigg{)}\
{\rm d}x{\rm d}t+\int_{0}^{\delta}J_{x}\varphi^{0}\bigg{|}_{x=0}^{x=\pi}\ {\rm
d}t=0.$ (4.13)

Furthermore, for the variable $u$, we consider the initial-boundary value
problem

$\displaystyle\left\\{\begin{array}[]{l}u_{t}-u_{xx}=\theta_{tx},\\\
u(x,0)=u_{0}(x),\\\ u_{x}(0,t)=0,\ \
u_{x}(\pi,t)=-\theta_{t}(\pi,t).\end{array}\right.$ (4.17)

Let

$\tilde{u}(x,t)=u(x,t)+\frac{1}{\pi}\int_{0}^{x}y\theta_{t}(y,t)\ {\rm d}y.$

Then the initial-boundary value problem for $\tilde{u}$ is

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle\tilde{u}_{t}-\tilde{u}_{xx}=(1-\frac{x}{\pi})\theta_{tx}-\frac{1}{\pi}\theta_{t}+\frac{1}{\pi}xc^{2}\theta_{x}-\frac{1}{\pi}\int_{0}^{x}(c+yc^{\prime}\theta_{y})c\theta_{y}\
{\rm d}y-\frac{1}{\pi}\int_{0}^{x}y(\theta_{t}+J)\ {\rm d}y,\\\\[8.0pt]
\displaystyle\tilde{u}(x,0)=u_{0}(x)+\frac{1}{\pi}\int_{0}^{x}y\theta_{1}(y)\
{\rm d}y,\quad x\in[0,\pi],\\\\[8.0pt]
\displaystyle\tilde{u}_{x}(0,t)=\tilde{u}_{x}(\pi,t)=0.\end{array}\right.$
(4.21)

Here we have used the wave equation for $\theta(x,t)$. According to the
Neumann function $N(x,t;\xi,\tau)$ given in Section 3, the weak solution of
(4.17) can be expressed as

$\displaystyle u(x,t)=-\frac{1}{\pi}\int_{0}^{x}y\theta_{t}(y,t)\ {\rm
d}y+\int_{0}^{\pi}N(x,t;\xi,0)\bigg{(}u_{0}(\xi)+\frac{1}{\pi}\int_{0}^{\xi}y\theta_{1}(y)\
{\rm d}y\bigg{)}\ {\rm d}\xi$ $\displaystyle\ \
+\int_{0}^{t}\int_{0}^{\pi}N(x,t;\xi,\tau)\bigg{(}\frac{1}{\pi}\xi
c^{2}\theta_{\xi}-\frac{1}{\pi}\int_{0}^{\xi}(c+yc^{\prime}\theta_{y})c\theta_{y}\
{\rm d}y-\frac{1}{\pi}\int_{0}^{\xi}y(\theta_{\tau}+J)\ {\rm
d}y\bigg{)}(\xi,\tau)\ {\rm d}\xi{\rm d}\tau$ $\displaystyle\ \
-\int_{0}^{t}\int_{0}^{\pi}\partial_{\xi}N(x,t;\xi,\tau)\left(1-\frac{\xi}{\pi}\right)\theta_{\tau}(\xi,\tau)\
{\rm d}\xi{\rm d}\tau,$ (4.22)

which satisfies

$\displaystyle\int_{0}^{\delta}\int_{0}^{\pi}\bigg{(}u\psi^{0}_{t}-(u_{x}+\theta_{t})\psi^{0}_{x}\bigg{)}\
{\rm d}x{\rm d}t=0,$ (4.23)

for any test functions $\psi^{0}\in C^{\infty}_{c}([0,\pi]\times[0,\delta])$.
Letting $\psi^{0}=\varphi^{0}_{x}$ in (4.23), we get

$\displaystyle\int_{0}^{\delta}\int_{0}^{\pi}\bigg{(}u_{x}\varphi^{0}_{t}+(u_{x}+\theta_{t})\varphi^{0}_{xx}\bigg{)}\
{\rm d}x{\rm d}t-\int_{0}^{\delta}u\varphi^{0}_{t}\bigg{|}_{x=0}^{x=\pi}\ {\rm
d}t=0.$ (4.24)

By taking the difference of (4.13) and (4.24) yields

$\displaystyle\int_{0}^{\delta}\int_{0}^{\pi}[J-(u_{x}+\theta_{t})](\varphi^{0}_{t}+\varphi^{0}_{xx})\
{\rm d}x{\rm
d}t+\int_{0}^{\delta}(u\varphi^{0}_{t}+J_{x}\varphi^{0})\bigg{|}_{x=0}^{x=\pi}\
{\rm d}t=0,$ (4.25)

which implies by the disappearance of the test function $\varphi^{0}$ on the
boundaries that $J=u_{x}+\theta_{t}$, for a.e. any
$(x,t)\in[0,\pi]\times[0,\delta]$.

Finally, the energy estimate in (3.3) still holds for the current case, so we
can extend the local solution on $\Omega_{\delta}$ to $\Omega_{T}$.

## Acknowledgments

The first author is partially supported by NSF with grant DMS-2008504. The
second author is partially supported by NSF of China with grants 12071106 and
12171130.

## References

  * [1] A. Bressan and G. Chen, Lipschitz metric for a class of nonlinear wave equations, Arch. Ration. Mech. Anal., 226 (2017), no. 3, 1303–1343.
  * [2] A. Bressan and G. Chen, Generic regularity of conservative solutions to a nonlinear wave equation, Ann. I. H. Poincaré–AN, 34 (2017), no. 2, 335–354.
  * [3] A. Bressan, G. Chen, and Q. T. Zhang, Unique conservative solutions to a variational wave equation, Arch. Ration. Mech. Anal., 217 (2015), no. 3, 1069–1101.
  * [4] A. Bressan and T. Huang, Representation of dissipative solutions to a nonlinear variational wave equation, Commun. Math. Sci., 14 (2016), no. 1, 31–53.
  * [5] A. Bressan and Y. X. Zheng, Conservative solutions to a nonlinear variational wave equation, Commun. Math. Phys., 266 (2006), no. 2, 471–497.
  * [6] H. Cai, G. Chen, and Y. Du, Uniqueness and regularity of conservative solution to a wave system modeling nematic liquid crystal, J. Math. Pures Appl., 9 (2018), no. 117, 185–220.
  * [7] H. Cai, G. Chen, and Y. Du, Lipschitz optimal transport metric for a wave system modeling nematic liquid crystals, submitted, available at arXiv: 2304.11535.
  * [8] G. Chen, T. Huang, and W. S. Liu, Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model, Arch. Ration. Mech. Anal., 236 (2020), no. 2, 839–891.
  * [9] G. Chen, T. Huang, and X. Xu Singularity formation for full Ericksen-Leslie system of nematic liquid crystal flows in dimension two, submitted, available at arXiv: 2305.03904.
  * [10] G. Chen, W. S. Liu, and M. Sofiani, The Poiseuille flow of the full Ericksen-Leslie model for nematic liquid crystals: The general Case, submitted, available at arXiv: 2302.08616.
  * [11] G. Chen, P. Zhang, and Y. X. Zheng, Energy Conservative Solutions to a Nonlinear Wave System of Nematic Liquid Crystals, Comm. Pure Appl. Anal., 12 (2013), no. 3, 1445–1468.
  * [12] F. Cheng, N. Jiang, and Y. L. Luo, On dissipative solutions to a simplified hyperbolic Ericksen-Leslie system of liquid crystals, Commun. Math. Sci., 19 (2021), no. 1, 175–192.
  * [13] W. H. Duan, Y. B. Hu, and G. D. Wang, Singularity for a multidimensional variational wave equation arising from nematic liquid crystals, J. Math. Anal. Appl., 487 (2020), no. 2, 124026, 13 pp.
  * [14] J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheology, 5 (1961), no. 1, 23–34.
  * [15] J. L. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21  (1987), 381–392.
  * [16] J. L. Ericksen, Liquid crystals with variable degree of orientation, Arch. Ration. Mech. Anal., 113 (1991), no. 2, 97–120.
  * [17] R. T. Glassey, J. K. Hunter, and Y. X. Zheng, Singularities in a nonlinear variational wave equation, J. Differential Equations, 129 (1996), no. 1, 49–78.
  * [18] R. Hardt and D. Kinderlehrer, Mathematical questions of liquid crystal theory, In Theory and applications of liquid crystals (Minneapolis, Minn., 1985), IMA Vol. Math. Appl. Vol 5, pp. 151–184, Springer, New York, 1987.
  * [19] H. Holden and X. Raynaud, Global semigroup of conservative solutions of the nonlinear variational wave equation, Arch. Ration. Mech. Anal., 201 (2011), no. 3, 871–964.
  * [20] M. C. Hong and Z. P. Xin,  Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\mathbb{R}^{2}$,  Adv. Math., 231 (2012), no. 3-4, 1364–1400.
  * [21] Y. B. Hu, Conservative solutions to a one-dimensional nonlinear variational wave equation, J. Differential Equations, 259 (2015), no. 1, 172–200.
  * [22] J. R. Huang, F. H. Lin, and C. Y. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in $\mathbb{R}^{2}$, Commun. Math. Phys., 331 (2014), no. 2, 805–850.
  * [23] J. X. Huang, N. Jiang, Y. L. Luo, and L. F. Zhao, Small data global regularity for the 3-D Ericksen-Leslie hyperbolic liquid crystal model without kinematic transport, SIAM J. Math. Anal., 53 (2021), no. 1, 530–573.
  * [24] N. Jiang and Y. L. Luo, On well-posedness of Ericksen-Leslie’s hyperbolic incompressible liquid crystal model, SIAM J. Math. Anal., 51 (2019), no. 1, 403–434.
  * [25] N. Jiang and Y. L. Luo, The zero inertia limit from hyperbolic to parabolic Ericksen-Leslie system of liquid crystal flow, J. Funct. Anal., 282 (2022), no. 1, 109280, 62 pp.
  * [26] N. Jiang, Y. L. Luo, and S. J. Tang, Zero inertia density limit for the hyperbolic system of Ericksen-Leslie’s liquid crystal flow with a given velocity, Nonlinear Anal. Real World Appl., 45 (2019), 590–608.
  * [27] F. Leslie, Theory of flow phenomena in liquid crystals, Advances in Liquid Crystals, 4, Elsevier, New York, 1979, pp. 1-81.
  * [28] J. K. Li, E. Titi, and Z. P. Xin, On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in $\mathbb{R}^{2}$, Math. Models Methods Appl. Sci., 26 (2016), no. 4, 803–822.
  * [29] T. T. Li, W. C. Yu, and W. X. Shen, Second initial-boundary value problems for quasi-linear hyperbolic-parabolic coupled systems (in chinese), Chinese Ann. of Math., 2 (1981), 65–90.
  * [30] T. T. Li, W. C. Yu, and W. X. Shen, First initial-boundary value problems for quasilinear hyperbolic-parabolic coupled systems, Chinese Ann. of Math., 5B (1984), 77–90.
  * [31] F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and phenomena, Comm. Pure Appl. Math., 42 (1989), no. 6, 789–814.
  * [32] F. H. Lin and C. Liu, Static and dynamic theories of liquid crystals, J. Partial Differential Equations, 14 (2001), no. 4, 289–330.
  * [33] F. H. Lin and C. Y. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), no. 2029, 20130361, 18 pp.
  * [34] F. H. Lin and C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math., 69 (2016), no. 8, 1532–1571.
  * [35] I. W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction, Crc Press, 2019.
  * [36] M. Wang and W. D. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Calc. Var. Partial Differential Equations, 51 (2014), no. 3-4, 915–962.
  * [37] W. Wang, P. W. Zhang, and Z. F. Zhang, Well-posedness of the Ericksen-Leslie system, Arch. Ration. Mech. Anal., 210 (2013), no. 3, 837-855.
  * [38] P. Zhang and Y. X. Zheng, Weak solutions to a nonlinear variational wave equation, Arch. Ration. Mech. Anal., 166 (2003), no. 4, 303–319.
  * [39] P. Zhang and Y. X. Zheng, Conservative solutions to a system of variational wave equations of nematic liquid crystals, Arch. Ration. Mech. Anal., 195 (2010), no. 3, 701–727.
  * [40] P. Zhang and Y. X. Zheng, Energy conservative solutions to a one-dimensional full variational wave system, Comm. Pure Appl. Math., 55 (2012), no. 5, 582–632.