\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 294, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/294\hfil 
Wave equation derivable from a variational principle]
{Global solutions to a one-dimensional nonlinear wave equation
 derivable from a variational principle}

\author[Y. Hu, G. Wang \hfil EJDE-2017/294\hfilneg]
{Yanbo Hu, Guodong Wang}

\address{Yanbo Hu (corresponding author) \newline
Department of Mathematics,
Hangzhou Normal University,
Hangzhou, 310036, China}
\email{yanbo.hu@hotmail.com}

\address{Guodong Wang \newline
School of Mathematics \& Physics,
Anhui Jianzhu University,
 Hefei, 230601, China}
\email{yxgdwang@163.com}

\thanks{Submitted October 13, 2017. Published November 28, 2017.}
\subjclass[2010]{35D05, 35L15, 35L70}
\keywords{Nonlinear wave equation; weak solutions; existence; 
\hfill\break\indent energy-dependent coordinates}

\begin{abstract}
 This article focuses on a one-dimensional nonlinear wave equation which
 is the Euler-Lagrange equation of a variational principle whose Lagrangian
 density involves linear terms and zero term as well as quadratic terms
 in derivatives of the field. We establish the global existence of weak
 solutions to its Cauchy problem by the method of energy-dependent coordinates
 which allows us to rewrite the equation as a semilinear system and resolve
 all singularities by introducing a new set of variables related to the energy.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks


\section{Introduction}\label{S1}

The variational principle whose action is a quadratic
function of the derivatives of the field with coefficients depending on
the field and the independent variables takes the form
\cite{Ali0, Hunter-Saxton, Hunter-Zheng2}
\begin{equation}\label{1.1}
\delta\int A_{\mu\nu}^{ij}(\mathbf{x}, \mathbf{u})
\frac{\partial u^\mu}{\partial x_i}\frac{\partial
u^\nu}{\partial x_j}\,{\rm d}\mathbf{x}=0,
\end{equation}
where the summation convention is employed. Here, $\mathbf{x}\in \mathbb{R}^{d+1}$
 are the space-time independent variables and
$\mathbf{u}: \mathbb{R}^{d+1}\to \mathbb{R}^n$ are the dependent variables.
The coefficients
$A_{\mu\nu}^{ij}: \mathbb{R}^{d+1}\times \mathbb{R}^n\to \mathbb{R}$
are smooth functions.

A particular motivation for studying \eqref{1.1} comes from the theory of
nematic liquid crystals. In the regime in which inertia effects dominate
viscosity, Saxton \cite{Saxton} modelled the
propagation of the orientation waves in the director field of a nematic
liquid crystal by the least action principle
\begin{equation}\label{1.2}
\delta\int \Big(\frac{1}{2}\partial_t\mathbf{n}\cdot\partial_t\mathbf{n}-W_n(\mathbf{n},
\nabla \mathbf{n})\Big){\rm d}\mathbf{x}{\rm d}t=0,\quad \mathbf{n}\cdot\mathbf{n}=1,
\end{equation}
where $\mathbf{n}(\mathbf{x}, t)$ is the director field and
$W_n(\mathbf{n},\nabla \mathbf{n})$ is the well-known Oseen-Franck potential
energy density,
$$
W_n(\mathbf{n},\nabla \mathbf{n})=\frac{1}{2}k_1(\nabla\cdot\mathbf{n})^2
+\frac{1}{2}k_2(\mathbf{n}\cdot\nabla\times\mathbf{n})^2
+\frac{1}{2}k_3|\mathbf{n}\times(\nabla\times\mathbf{n})|^2,
$$
which is a quadratic function of $\nabla\mathbf{n}$ with coefficients
depending on $\mathbf{n}$. Here $k_1, k_2$ and $k_3$ are the splay,
twist and bend elastic
constants of the liquid crystal, respectively.

The hyperbolic systems of nonlinear wave equations derivable from the variational
principle \eqref{1.2} have been widely explored since they were introduced.
Hunter and Saxton \cite{Hunter-Saxton} considered the planar deformations
of the director field $\mathbf{n}$ that depend only on a single space
variable $x$ with $\mathbf{n}=(\cos u(t,x), \sin u(t,x), 0)$ and derived
the Euler-Lagrange equation of \eqref{1.2} given by
\begin{equation}\label{1.3}
u_{tt}-c(u)[c(u)u_x]_x=0
\end{equation}
with $c^2(u)=k_1\sin^2u+k_3\cos^2u$. For another important application,
 equation \eqref{1.3} describes the motion of long waves on a neutral
dipole chain in the continuum limit \cite{Zorski}. A basic feature of
\eqref{1.3} is that, even for smooth initial data, cusp-type singularities
can form in finite time, see \cite{Glassey1, Glassey2}.
In a series of papers \cite{Zha-Zhe3, Zha-Zhe4, Zha-Zhe7, Zha-Zhe8},
Zhang and Zheng have studied carefully the global
existence of dissipative weak solutions to the initial value problems
for \eqref{1.3} and its asymptotic models. Bressan and Huang \cite{Bres-Huang}
proposed another way to construct a global dissipative solution to \eqref{1.3}.
The global existence and uniqueness of conservative weak solutions to its
Cauchy problem for initial data of finite energy were established, respectively,
by Bressan and Zheng \cite{Bres-Zheng} and Bressan, Chen and Zhang \cite{BCZ}.
Holden and Raynaud \cite{Holden} carried out
a detailed construction of a global semigroup for its conservative weak solutions.
A Lipschitz continuous metric of conservative weak solutions to \eqref{1.3}
 has been constructed recently by Bressan and Chen \cite{BC2}.
The generic properties of conservative solutions was studied in \cite{BC1}.
 In \cite{Hu4}, the first author investigated a more general nonlinear wave
equation than \eqref{1.3} arising from the variational principle \eqref{1.1}
and constructed a global energy-conservative weak solution to its initial
value problem.

For the three-dimensional deformations depending on a single space variable $x$
and the director field $\mathbf{n}$ taking the form
\[
\mathbf{n}=(\cos u(x,t), \sin u(x,t)\cos v(x,t), \sin u(x,t)\sin v(x,t)),
\]
the Euler-Lagrange equations of \eqref{1.2} are
\begin{equation}\label{1.5}
\begin{gathered}
\begin{aligned}
 u_{tt}-(c_{1}^2(u)u_x)_x
&=-c_1(u)\partial_uc_{1}(u)u_{x}^2 +a(u)\partial_ua(u)[v_{t}^2-c_{2}^2(u)v_{x}^2]\\
&\quad -a^2(u)c_{2}(u,x)\partial_uc_{2}(u)v_{x}^2, 
\end{aligned}\\
 (a^2(u)v_t)_t-[a^2(u)c_{2}^2(u)v_x]_x=0
 \end{gathered}
\end{equation}
where $c_{1}^2(u)=k_1\sin^2u+k_3\cos^2u, c_{2}^2(u)=k_2\sin^2u+k_3\cos^2u$
and $a^2(u)=\sin^2u$. System \eqref{1.5} was first derived by Ali and
Hunter \cite{Ali} to describe the propagation of splay and twist waves
in nematic liquid crystals. They also analyzed some properties for the
asymptotic equations of \eqref{1.5} in \cite{Ali}.
Recently, Zhang and Zheng \cite{Zha-Zhe9, Zha-Zhe10} demonstrated the
global existence of conservative weak solutions to the Cauchy problem
of \eqref{1.5} under some assumptions. The relevant results about the
related nonlinear wave system derived from the variational principle
of \eqref{1.1} was provided by the author \cite{Hu1}. We also refer the
reader to \cite{Chen1} by Chen, Zhang and Zheng for the discussion of the
nonlinear wave system obtained from \eqref{1.2} by considering the director
field $\mathbf{n}$ in its natural three-component form, also see
Chen and Zheng \cite{Chen-Zheng} for the analysis of the corresponding
viscous system.

In a recent paper \cite{Hu5}, we studied a nonlinear wave system coming
from the theory of cholesteric liquid crystals, in which the Oseen-Franck
potential energy density is expressed as the sum of an elastic and a chiral
contribution (neglecting a constant factor)
\begin{equation}\label{1.6}
W_c(\mathbf{n},\nabla \mathbf{n})
=\frac{1}{2}k_1(\nabla\cdot\mathbf{n})^2
+\frac{1}{2}k_2(\mathbf{n}\cdot\nabla\times\mathbf{n})^2
+\frac{1}{2}k_3|\mathbf{n}\times(\nabla\times\mathbf{n})|^2
+\lambda\mathbf{n}\cdot\nabla\times\mathbf{n},
\end{equation}
where $\lambda$ is a pseudoscalar material parameter that represents molecular
chirality, see e.g. \cite{Frank, Gennes}. Compared with the nematic, linear
terms in derivatives of the director field $\mathbf{n}$ must be considered
in cholesteric liquid crystals. For detailed information regarding cholesteric
liquid crystals, see, for example, \cite{Gennes, Leslie, Oswald, Stephen}.
Considering the three-dimensional deformations and replacing
$W_n(\mathbf{n},\nabla \mathbf{n})$ by $W_c(\mathbf{n},\nabla \mathbf{n})$,
the variational principle \eqref{1.2} gives rise to the associated
 Euler-Lagrange equations
\begin{equation}\label{1.7}
 \begin{gathered}
\begin{aligned}
 u_{tt}-(c_{1}^2(u)u_x)_x
&=-c_1(u)\partial_uc_{1}(u)u_{x}^2 +a(u)\partial_ua(u)[v_{t}^2-c_{2}^2(u)v_{x}^2] \\
&\quad -a^2(u)c_{2}(u,x)\partial_uc_{2}(u)v_{x}^2 +2\lambda a(u)\partial_ua(u)v_x,
\end{aligned} \\
 (a^2(u)v_t)_t-[a^2(u)c_{2}^2(u)v_x-\lambda a^2(u)]_x=0.
 \end{gathered}
\end{equation}
In contrast to equations \eqref{1.5}, the linear terms in \eqref{1.6}
cause the total energy of solution for system \eqref{1.7} to be not
conservative. The global existence of weak solutions to its Cauchy
problem was established in \cite{Hu5} by using the method of energy-dependent
coordinates and the Young measure theory.

Motivated by \eqref{1.6}, we are interested in the variational principle
whose Lagrangian density includes linear terms and zero term as well as
quadratic terms in derivatives of the field
\begin{equation}\label{1.8}
\delta\int \Big\{A_{\mu\nu}^{ij}(\mathbf{x}, \mathbf{u})
\frac{\partial u^\mu}{\partial x_i}\frac{\partial
u^\nu}{\partial x_j}+B_{\mu}^{i}(\mathbf{x}, \mathbf{u})\frac{\partial u^\mu}{\partial x_i}
+F(\mathbf{x}, \mathbf{u})\Big\}\,{\rm d}\mathbf{x}=0,
\end{equation}
where $\mathbf{x}$, $\mathbf{u}$ and $A_{\mu\nu}^{ij}$ are as in \eqref{1.1},
$B_{\mu}^{i}$ and $F$ are smooth functions. The term $F$ can be regarded as
the contribution from the external electrical or magnetic field.
Consider $n=1$ and $d=1$, then the Euler-Lagrange equation for \eqref{1.8}
reads that
\begin{equation}\label{1.9}
\begin{aligned}
&(2A^{11}u_t+(A^{12}+A^{21})u_x+B^1)_t+((A^{12}+A^{21})u_t+2A^{22}u_x+B^2)_x
 \\
&=\frac{\partial A^{11}}{\partial u}u_{t}^2+ \frac{\partial (A^{12}+A^{21})}{\partial u}u_{t}u_x
+ \frac{\partial A^{22}}{\partial u}u_{x}^2 + \frac{\partial B^1}{\partial u}u_t
+\frac{\partial B^2}{\partial u}u_x +\frac{\partial F}{\partial u}.
\end{aligned}
\end{equation}

In this article, we consider the initial value problem for equation \eqref{1.9} with
\begin{equation}\label{1.10}
(A^{ij})_{2\times2}=
\frac{1}{2}\begin{pmatrix}
\alpha^2 & \beta_1 \\
\beta_2 & -\gamma^2
\end{pmatrix}(x,u),
\end{equation}
and $B^1=\kappa(x,u)$, $B^2=\lambda(x,u)$, $F=F(x,u)$, where
$\alpha, \gamma, \kappa, \lambda, F$ and $\beta_1+\beta_2:=2\beta$ are
smooth functions, independent of $t$, satisfying
\begin{equation}\label{1.11}
\begin{gathered}
0<\alpha_1\leq\alpha(z)\leq\alpha_2, \quad
|\beta(z)|+|\kappa(z)|+|\lambda(z)|+|F(z)|\leq \Lambda,\\
 0<\gamma_1\leq\gamma(z)\leq\gamma_2, \\
\sup_{z}\{|\nabla\alpha(z)|, |\nabla\beta(z)|, |\nabla\gamma(z)|,
|\nabla\kappa(z)|, |\nabla\lambda(z)|, |\nabla F(z)|\}\leq\Lambda,
 \quad \forall z\in\mathbb{R}^2
\end{gathered}
\end{equation}
for positive numbers $\alpha_1, \alpha_2, \gamma_1, \gamma_2$ and $\Lambda$.
Under the above assumptions, \eqref{1.9} reduces to
\begin{equation}\label{1.12}
\begin{aligned}
&(\alpha^2u_t+\beta u_x+\kappa)_t+(\beta u_t-\gamma^2u_x+\lambda)_x \\
&=\alpha\alpha_uu_{t}^2+\beta_uu_tu_x-\gamma\gamma_uu_{x}^2+\kappa_uu_t
 +\lambda_uu_x+F_u,
\end{aligned}
\end{equation}
which is strictly hyperbolic with two eigenvalues
\begin{equation}\label{1.13}
\lambda_+=\frac{\sqrt{\beta^2+\alpha^2\gamma^2}+\beta}{\alpha^2}>0, \quad  
\lambda_- = - \frac{\sqrt{\beta^2+\alpha^2\gamma^2}-\beta}{\alpha^2}<0.
\end{equation}
Equation \eqref{1.12} is a second order quasilinear hyperbolic equation
whose main difficulty arises from the possible cusp-type singularities
of solutions in finite time.

The purpose of this article is to establish the global existence of weak
solutions to the initial value problem \eqref{1.12} with the initial data
\begin{equation}\label{1.14}
u(0,x)=u_0(x)\in H^1,\quad u_t(0,x)=u_1(x)\in L^2
\end{equation}
under the assumption \eqref{1.11}. The approach we used here follows the
method of energy-dependent coordinates proposed by Bressan, Zhang and
Zheng \cite{Bres-Zha-Zhe, Bres-Zheng} to deal with \eqref{1.3}
 and its related asymptotic model. However, in contrast to equation \eqref{1.3},
the energy of solution for \eqref{1.12}, denoted by
$$
\mathcal{E}(t):=\frac{1}{2}\int\Big\{\alpha^2(x,u)u_{t}^2
+\gamma^2(x, u)u_{x}^2\Big\}{\rm d}x,
$$
is not conservative. In spite of this, we can still establish a priori estimates
of solutions for the equivalent semilinear system in the energy-dependent
coordinates. By returning the solution in terms of the original variables,
 we thus recover a global weak solution to \eqref{1.12}.

Before stating the main results, let us first give the definition of weak
solutions to problem \eqref{1.12} \eqref{1.14}.

\begin{definition}[Weak solution]\rm
A function $u(t,x)$ with $(t,x)\in\mathbb{R}^+\times\mathbb{R}$ is a
\emph{weak solution} to the Cauchy problem \eqref{1.12} \eqref{1.14}
if the following hold:

\noindent(i) In the $t$-$x$ plane, the function $u$ is locally
H\"{o}lder continuous with exponent $1/2$. The function
$t\mapsto u(t,\cdot)$ is continuously differentiable as a map with values in
 $ L_{\rm loc}^\theta$ for all $1\leq \theta<2$. Moreover, for any $T>0$,
it satisfies the Lipschitz continuity property
\begin{equation}\label{1.16}
\|u(t,\cdot)-u(s,\cdot)\|_{L_{\rm loc}^2}\leq L|t-s|, \quad \forall t,s\in(0,T]
\end{equation}
for some constant $L$ depending on $T$ with $L=O(\sqrt{T})$.

\noindent(ii) The function $u(t,x)$ takes on the initial
condition in \eqref{1.14} pointwise, while its temporal derivative holds in
$ L_{\rm loc}^\theta$ for $\theta\in[1,2)$.

\noindent(iii) Equation \eqref{1.12} is satisfied in the
distributional sense, that is
\begin{equation}\label{1.17}
\begin{aligned}
&\iint_{\mathbb{R}^+\times\mathbb{R}}\Big\{\varphi_t(\alpha^2u_t+\beta u_x+\kappa)
+\varphi_x(\beta u_t-\gamma^2u_x+\lambda) \\
 &\ +\varphi\Big(\alpha\alpha_uu_{t}^2+\beta_uu_tu_x-\gamma\gamma_uu_{x}^2
+\kappa_uu_t+\lambda_uu_x+F_u\Big)\Big\}{\rm d}x{\rm d}t=0
\end{aligned}
\end{equation}
for all test functions $\varphi\in C_{c}^1(\mathbb{R}^+\times\mathbb{R})$.
\end{definition}

The conclusions of this paper are as follows.

\begin{theorem}[Existence]\label{ex}
Let condition \eqref{1.11} be satisfied. Then the Cauchy problem
\eqref{1.12} \eqref{1.14} admits a global weak solution defined for
all $(t,x)\in\mathbb{R}^+\times\mathbb{R}$.
\end{theorem}

The continuous dependence of the solution upon the initial data follows
 directly from the constructive procedure.

\begin{theorem}[Continuous dependence]\label{C-D}
Suppose the assumptions in Theorem \ref{ex} hold. For the Cauchy
 problem \eqref{1.12} \eqref{1.14}, let a sequence of initial conditions satisfy
\begin{align*}
\|(u_{0}^\nu)_x-(u_{0})_x\|_{L^2}\to 0,\quad \|u_{1}^\nu-u_{1}\|_{L^2}\to 0,
\end{align*}
and $u_{0}^\nu\to u_{0}$ uniformly on
compact sets, as $\nu\to \infty$. Then one has the
convergence of the corresponding solutions $u^\nu\to u$,
uniformly on bounded subsets of the $(t,x)$-plane with $t>0$.
\end{theorem}

The article is organized as follows. In Section \ref{S2}, we
introduce a new set of dependent and independent variables,
and derive an equivalent semilinear system of \eqref{1.12} for smooth solutions.
Section \ref{S3} presents the existence and continuous dependence results
for solutions to the equivalent semilinear system. We show the H\"{o}lder
continuity of solutions $(u,v)$ in terms of the original independent variables
$t,x$, and verify that the integral equation \eqref{1.17} is satisfied in
Section \ref{S4}.
Finally, in Section \ref{S5}, we establish the Lipschitz continuity
property \eqref{1.16} and the continuity of the maps $t\mapsto u_t(t,\cdot)$,
$t\mapsto u_x(t,\cdot)$ as functions with values in
$L_{\rm loc}^\theta (1\leq \theta<2)$, which complete the proof of
Theorems \ref{ex}.


\section{New formulation in energy-dependent coordinates}\label{S2}

In this section, we derive an equivalent system of
\eqref{1.12} for smooth solutions by introducing a new set of
variables to replace the original variables $u, t, x$.


\subsection{Energy-dependent coordinates}

Denote $c_1:=\alpha\lambda_-<0$ and $c_2:=\alpha\lambda_+>0$, and
\begin{equation}\label{2.1}
R:=\alpha u_t+c_2u_x,\quad S:=\alpha u_t+c_1u_x.
\end{equation}
Then \eqref{1.12} can be rewritten as
\begin{equation}\label{2.3}
 \begin{gathered}
 \alpha(x,u) R_{t}+c_1(x,u)R_{x}
=a_1R^2-(a_1+a_2)RS+a_2S^2+c_2bS-d_1R-\lambda_x+F_u,\\
 \alpha(x,u) S_{t}+c_2(x,u)S_{x}
=-a_1R^2+(a_1+a_2)RS-a_2S^2+c_1bR-d_2S-\lambda_x+F_u,\\
 \alpha(x,u) u_{t}+c_1(x,u)u_{x}=S,
 \end{gathered}
\end{equation}
where
\begin{equation}\label{2.4}
\begin{gathered}
a_i=\frac{c_i\partial_u\alpha-\alpha\partial_uc_i}{2\alpha(c_2-c_1)}, \quad
b=\frac{\alpha\partial_x(c_1-c_2)+(c_1-c_2)\partial_x\alpha}{2\alpha(c_2-c_1)} \\
d_i=\frac{c_2\partial_xc_1-c_1\partial_xc_2}{2(c_2-c_1)}
+\frac{\alpha\partial_xc_i-c_i\partial_x\alpha}{2\alpha},\quad (i=1,2)
\end{gathered}
\end{equation}
and $\partial_x$ and $\partial_u$ denote, respectively, partial derivatives with
respect to the arguments $x$ and $u$.
System \eqref{2.3} is equivalent to equation \eqref{1.7} for smooth
solutions if we supplement it with initial restriction at $t=0$,
\begin{equation}\label{2.5}
 u_x=\frac{R-S}{c_2(x,u)-c_1(x,u)}.
\end{equation}

For convenience to deal with possibly unbounded values of $R$ and
$S$, we introduce a new set of dependent variables
\begin{equation}\label{2.6}
\ell:=\frac{R}{1+R^2},\quad
h:=\frac{1}{1+R^2}, \quad
m:=\frac{S}{1+S^2},\quad g:=\frac{1}{1+S^2},
\end{equation}
from which one easily checks that
\begin{equation}\label{2.7}
\ell^2+h^2=h,\quad m^2+g^{2}=g.
\end{equation}
Define the forward and backward characteristics as follows:
\begin{equation}\label{2.12}
 \begin{gathered}
 \frac{{\rm d}}{{\rm d}s}x^{\pm}(s;t,x)
=\lambda_\pm(x^{\pm}(s;t,x),u(s;x^{\pm}(s;t,x))),\\
 x^{\pm}|_{s=t}=x,
 \end{gathered}
\end{equation}
where $\lambda_\pm$ are given in \eqref{1.13}.
Now we define the coordinate transformation $(t,x)\to (X,Y)$ where
\begin{equation}\label{2.13}
 X:=\int_0^{x^{-}(0;t,x)}\Big(1+ R^2(0,\xi)\Big){\rm d}\xi,\quad
 Y:=\int_{x^{+}(0;t,x)}^0\ \Big(1+ S^2(0,\xi)\Big){\rm d}\xi,
\end{equation}
which indicates that
\begin{equation}\label{2.14}
\alpha(x,u) X_t+c_1(x,u)X_x=0,\quad \alpha(x,u) Y_t+c_2(x,u)Y_x=0,
\end{equation}
from which, it turns out that
\begin{equation}\label{2.15}
 f_t=-\frac{c_1}{\alpha}X_xf_X-\frac{c_2}{\alpha}Y_xf_Y,\quad
 f_x=X_xf_X+Y_xf_Y
\end{equation}
for any smooth function $f(t,x)$. Moreover, we introduce the new variables
\begin{equation}\label{2.16}
p:=\frac{1+R^2}{X_x},\quad q:=\frac{1+S^2}{-Y_x}.
\end{equation}

Then we can obtain a semilinear hyperbolic system
with smooth coefficients for the variables $g, h, \ell, m, p, q, u, x$
in $(X,Y)$ coordinates as follows:
\begin{equation}\label{2.20}
\begin{gathered}
\begin{aligned}
 \ell_Y&=\frac{q(2h-1)}{c_2-c_1}\Big\{a_1g+a_2h-(a_1+a_2)(gh+m\ell)
+c_2bhm \\
&\quad -d_1g\ell +(F_u-\lambda_x)gh\Big\},
\end{aligned} \\
\begin{aligned}
 m_X&=\frac{p(2g-1)}{c_2-c_1}\Big\{-a_1g-a_2h+(a_1+a_2)(gh+m\ell)
+c_1bg\ell \\
&\quad -d_2hm +(F_u-\lambda_x)gh\Big\}, 
\end{aligned}\\
 u_X=\frac{1}{c_2-c_1}p\ell \quad
\Big({\rm or}\ u_Y=\frac{1}{c_2-c_1}qm\Big), \\
 x_X=\frac{c_2}{c_2-c_1}ph \quad
\Big({\rm or}\ x_Y=\frac{c_1}{c_2-c_1}qg\Big),
\end{gathered}
\end{equation}

\begin{equation}\label{2.21}
 \begin{gathered}
\begin{aligned}
 h_Y&=-\frac{2q\ell}{c_2-c_1}\Big\{a_1g+a_2h-(a_1+a_2)(gh+m\ell)
 +c_2bhm \\
&\quad -d_1g\ell +(F_u-\lambda_x)gh\Big\},
\end{aligned} \\
\begin{aligned}
g_X&=-\frac{2pm}{c_2-c_1}\Big\{-a_1g-a_2h+(a_1+a_2)(gh+m\ell)
 +c_1bg\ell \\
&\quad -d_2hm +(F_u-\lambda_x)gh\Big\},
\end{aligned}
\end{gathered}
\end{equation}
and
\begin{equation}\label{2.22}
\begin{aligned}
p_Y&=\frac{2pq}{c_2-c_1}\Big\{a_2(\ell-m)+(a_1+a_2)(hm-g\ell)+c_2bm\ell+d_1gh \\
&\quad +\frac{c_1\partial_xc_2-c_2\partial_xc_1}{2(c_2-c_1)}g +(F_u-\lambda_x)g\ell\Big\},
\\
q_X&=\frac{2pq}{c_2-c_1}\Big\{a_1(\ell-m)+(a_1+a_2)(hm-g\ell)+c_1bm\ell+d_2gh \\
&\quad +\frac{c_1\partial_xc_2-c_2\partial_xc_1}{2(c_2-c_1)}h +(F_u-\lambda_x)mh\Big\}.
\end{aligned}
\end{equation}
We here point out that
\begin{equation}\label{2.22a}
\Big(\frac{qm}{c_2-c_1}\Big)_X=\Big(\frac{p\ell}{c_2-c_1}\Big)_Y,\quad
\Big(\frac{c_1qg}{c_2-c_1}\Big)_X=\Big(\frac{c_2ph}{c_2-c_1}\Big)_Y,
\end{equation}
which imply that $u_{XY}=u_{YX}$ and $x_{XY}=x_{YX}$,
so we may use either $u_{X}, x_{X}$ or $u_{Y}, x_{Y}$ in \eqref{2.20}.


\subsection{Initial data conversion}

We now consider the boundary conditions of system
\eqref{2.20}-\eqref{2.22} in the energy-dependent coordinates $(X, Y)$,
corresponding to \eqref{1.14} in the original coordinates $(t, x)$.

It is easily known by \eqref{1.11} and \eqref{1.14} that
\begin{gather*}
R(0,x)=\alpha(x, u_0(x))u_{1}(x)+c_2(x, u_0(x))u'_{0}(x)\in L^2,\\
 S(0,x)=\alpha(x, u_0(x))u_{1}(x)+c_1(x, u_0(x))u'_{0}(x)\in L^2.
\end{gather*}
The initial line $t=0$ in the $(t,x)$ plane is transformed to a curve $\Gamma_0:
Y=\phi(X)$ defined through a parametric $x\in\mathbb{R}$
\begin{equation}\label{2.24}
 X=\int_{0}^{x}[1+R^2(0,\xi)]{\rm d}\xi,\quad
 Y=\int_{x}^{0}[1+S^2(0,\xi)]{\rm d}\xi,
\end{equation}
which, clearly, is non-characteristic. We see by \eqref{1.11} and \eqref{1.13}
that the two functions $X=X(x)$, $Y=Y(x)$ are well defined and absolutely continuous.
Moreover, $X$ is strictly increasing while $Y$ is strictly decreasing.
Hence the map $X\mapsto\phi(X)$ is continuous and strictly decreasing.
In addition, applying \eqref{1.11} and \eqref{1.13} again arrives at
\begin{equation}\label{2.25}
\begin{aligned}
\frac{1}{\underline{M}}
&:=\frac{\alpha_{1}^2\gamma_{1}^2}
{2(\Lambda^2+\alpha_{2}^2\gamma_{2}^2 +\Lambda\sqrt{\Lambda^2
+\alpha_{2}^2\gamma_{2}^2})} \\
&\leq \big|\frac{c_i}{c_2-c_1}\big| \\
&\leq \frac{\Lambda+\sqrt{\Lambda^2+\alpha_{2}^2\gamma_{2}^2}}
 {2\alpha_1\gamma_1}:=\overline{M}
\end{aligned}
\end{equation}
for $i=1,2$. Therefore, the map $X\mapsto\phi(X)$ satisfies
\begin{equation}\label{2.26}
\begin{aligned}
|X+\phi(X)|
&\leq \int_{\mathbb{R}}\Big(R^2(0,\xi) +S^2(0,\xi)\Big){\rm d}\xi \\
&= \underline{M}\int_{\mathbb{R}}\Big(\alpha^2(\xi, u_0(\xi))u_{1}^2(\xi)
+\gamma^2(\xi, u_0(\xi))(u_{0}'(\xi))^2\Big){\rm d}\xi \\
&=:2\underline{M}\mathcal{{E}}_0,
\end{aligned}
\end{equation}
which is a finite number. The coordinate
transformation maps the domain $[0,\infty)\times \mathbb{R}$ in the
$(t,x)$ plane into the set
\begin{equation}\label{2.27}
\Omega^+:=\{(X,Y); Y\geq\phi(X)\}
\end{equation}
in the $(X,Y)$ plane. Along the curve $\Gamma_0$ parameterized by
$x\mapsto(X(x),Y(x))$, we can thus assign the boundary data
$(\bar{h}, \bar{g}, \bar{\ell}, \bar{m}, \bar{p},\bar{q},
\bar{u})\in L^\infty$ defined by
\begin{equation}\label{2.28}
\begin{gathered}
 \bar{h}=\frac{1}{1+R^2(0,x)},\quad
 \bar{g}=\frac{1}{1+S^2(0,x)}, \\
 \bar{\ell}=R(0,x)\bar{h},\quad
 \bar{m}=S(0,x)\bar{g}, \\
 \bar{p}=1,\quad
 \bar{q}=1, \\
 \bar{u}=u_{0}(x),\quad \bar{x}=x.
\end{gathered}
\end{equation}
It is easily checked that $h^2+\ell^2-h=0$ and
 $g^2+m^2-g=0$ on $\Gamma_0$.

\section{Solutions to the equivalent system}\label{S3}

This section is devoted to establishing the existence of a unique global
solution for system \eqref{2.20}-\eqref{2.22} with boundary data \eqref{2.28}
in the energy coordinates $(X,Y)$. The method follows from Bressan
and Zheng \cite{Bres-Zheng}, and also see Zhang and Zheng \cite{Zha-Zhe9}
and Hu \cite{Hu4}.

We first derive several identities from system \eqref{2.20}-\eqref{2.22},
which are useful to derive the
desired a priori estimates for the solutions.

(i) Consistency:
\begin{equation}\label{3.1}
\partial_Y(h^2+\ell^2-h)=0,\quad \partial_X(g^2+m^2-g)=0.
\end{equation}

\begin{proof}
The proof is directly from \eqref{2.20} and \eqref{2.21}.
\end{proof}

Thanks to \eqref{3.1} and the boundary conditions \eqref{2.28}, we find that
\begin{equation}\label{3.2}
h^2+\ell^2=h,\quad g^2+m^2=g,\quad \forall\ (X,Y)\in\Omega^+.
\end{equation}

(ii) An identity:
\begin{equation}\label{3.3}
\Big(\frac{c_2q(1-g)}{2(c_2-c_1)} +\frac{\lambda qm}{c_2-c_1}
+\frac{c_1Fqg}{c_2-c_1}\Big)_X-\Big(\frac{c_1p(1-h)}{2(c_2-c_1)}
+\frac{\lambda p\ell}{c_2-c_1} +\frac{c_2Fph}{c_2-c_1}\Big)_Y=0.
\end{equation}

\begin{proof}
It first follows from \eqref{2.22a} that
\begin{gather*}
\Big(\frac{\lambda qm}{c_2-c_1}\Big)_X-\Big( \frac{\lambda p\ell}{c_2-c_1}\Big)_Y
 =\frac{-\lambda_xpq}{(c_2-c_1)^2}(c_1g\ell-c_2mh),\\
\Big(\frac{c_1Fqg}{c_2-c_1}\Big)_X-\Big( \frac{c_2Fph}{c_2-c_1}\Big)_Y
 =\frac{F_upq}{(c_2-c_1)^2}(c_1g\ell-c_2mh),
\end{gather*}
from which one deduces
\begin{equation}\label{3.4}
\begin{aligned}
&\Big(\frac{\lambda qm}{c_2-c_1} +\frac{c_1Fqg}{c_2-c_1}\Big)_X
-\Big(\frac{\lambda p\ell}{c_2-c_1} +\frac{c_2Fph}{c_2-c_1}\Big)_Y\\
&=\frac{pq}{(c_2-c_1)^2}(F_u-\lambda_x)(c_1g\ell-c_2mh).
\end{aligned}
\end{equation}
On the other hand,  by \eqref{2.15} and \eqref{2.6} we compute
\begin{equation}
\Big(\frac{c_2}{c_2-c_1}\Big)_X
=\frac{p}{(c_2-c_1)^2}\Big\{
\frac{c_2\partial_uc_1-c_1\partial_uc_2}{c_2-c_1}\ell+\frac{c_2(c_2\partial_xc_1-c_1\partial_xc_2)}{c_2-c_1}h
\Big\}.\label{3.5}
\end{equation}
According to \eqref{2.21} and \eqref{2.22} and employing \eqref{3.2} yields
\begin{equation}\label{3.7}
\begin{aligned}
\Big(q(1-g)\Big)_X
&= \frac{2pq}{c_2-c_1}\Big\{a_1(\ell-m-g\ell+hm)+c_1bm\ell \\
&\quad  +\frac{c_1\partial_xc_2-c_2\partial_xc_1}{2(c_2-c_1)}(1-g)h +(F_u-\lambda_x)mh\Big\}.
\end{aligned}
\end{equation}
We combine \eqref{3.5} and \eqref{3.7} and use \eqref{2.4} to deduce
\begin{equation}\label{3.9}
\begin{aligned}
\Big(\frac{c_2q(1-g)}{2(c_2-c_1)}\Big)_X
& = \frac{pq}{(c_2-c_1)^2}\Big\{
c_1a_2(1-g)\ell-c_2a_1(1-h)m \\
&\quad +c_1c_2bm\ell+(F_u-\lambda_x)c_2mh\Big\}.
\end{aligned}
\end{equation}
Similarly, one finds
\begin{equation}\label{3.10}
\begin{aligned}
\Big(\frac{c_1p(1-h)}{2(c_2-c_1)}\Big)_Y
&=\frac{pq}{(c_2-c_1)^2}\Big\{
c_1a_2(1-g)\ell-c_2a_1(1-h)m \\
&\quad +c_1c_2bm\ell+(F_u-\lambda_x)c_1g\ell\Big\}.
\end{aligned}
\end{equation}
Combining \eqref{3.9} and \eqref{3.10} leads to
\begin{align*}
\Big(\frac{c_2q(1-g)}{2(c_2-c_1)}\Big)_X -\Big(\frac{c_1p(1-h)}{2(c_2-c_1)}\Big)_Y
=\frac{pq}{(c_2-c_1)^2}(F_u-\lambda_x)(c_2mh-c_1g\ell),
\end{align*}
which together with \eqref{3.4} gives \eqref{3.3}.
\end{proof}

We now establish a priori estimates for solutions to the semilinear hyperbolic
system \eqref{2.20}-\eqref{2.22} in $\Omega^+$. Obviously, it turns out by
\eqref{3.2} that
\begin{equation}\label{3.11}
0\leq h\leq 1,\quad 0\leq g\leq 1, \quad
|\ell|\leq\frac{1}{2}, \quad |m|\leq\frac{1}{2}.
\end{equation}
Based on \eqref{3.3} and \eqref{2.22}, We next estimate the functions $p$ and $q$.
It is easy to see from \eqref{2.22} and the initial condition $\bar{p}=\bar{q}=1$
that $p$ and $q$ are positive in $\Omega^+$.
In view of \eqref{2.22a} the differential form
\begin{equation}\label{3.12}
\frac{c_2ph}{c_2-c_1}{\rm d}X+\frac{c_1qg}{c_2-c_1}{\rm d}Y
\end{equation}
has zero integral along every closed curve contained in $\Omega^+$.
Then, for every $(X,Y)\in\Omega^+$, we construct the closed curve $S$
composed of the following three parts: the vertical segment with the
endpoints $(X,\phi(X))$ and $(X,Y)$, the horizontal segment with the
endpoints $(X,Y)$ and $(\phi^{-1}(Y),Y)$, and the boundary curve $\Gamma$
with the endpoints $(\phi^{-1}(Y),Y)$ and $(X,\phi(X))$.
Here $\phi^{-1}$ denotes the inverse of $\phi$. We integrate \eqref{3.12}
along the closed curve $S$ and use \eqref{2.25} and the boundary data \eqref{2.28}
to obtain
\begin{equation}\label{3.13}
\begin{aligned}
&\int_{\phi^{-1}(Y)}^{X}\frac{c_2}{c_2-c_1}ph(X',Y){\rm d}X'
 +\int_{\phi(X)}^{Y}\frac{-c_1}{c_2-c_1}qg(X,Y'){\rm d}Y' \\
&= \int_{\phi^{-1}(Y)}^{X}\frac{c_2\bar{h}}{c_2-c_1}(X',\phi(X')){\rm d}X'
 +\int_{\phi(X)}^{Y}\frac{-c_1\bar{g}}{c_2-c_1}(\phi^{-1}(Y'),Y'){\rm d}Y' \\
&\leq \overline{M}\Big(X-\phi^{-1}(Y)+Y-\phi(X)\Big) \\
&\leq 2\overline{M}(|X|+|Y|+2\underline{M}\mathcal{{E}}_0),
\end{aligned}
\end{equation}
from which, we employ \eqref{2.25} again to arrive at
\begin{equation}\label{3.14}
\int_{\phi^{-1}(Y)}^{X}ph(X',Y){\rm
d}X'+\int_{\phi(X)}^{Y}qg(X,Y'){\rm d}Y'
\leq 2\underline{M}\overline{M}(|X|+|Y|+2\underline{M}\mathcal{{E}}_0).
\end{equation}
On the other hand, making use of \eqref{1.11}, \eqref{2.25} and the
boundary data \eqref{2.28} yields
\begin{align*}
&\int_{\phi^{-1}(Y)}^{X}\Big[\frac{-c_1}{2(c_2-c_1)}p(1-h)
 -\frac{\lambda}{c_2-c_1}p\ell -\frac{c_2F}{c_2-c_1}ph\Big](X',\phi(X')){\rm
d}X' \\
& +\int_{\phi(X)}^{Y}\Big[\frac{c_2}{2(c_2-c_1)}q(1-g)
 +\frac{\lambda}{c_2-c_1}qm +\frac{c_1F}{c_2-c_1}qg\Big](\phi^{-1}(Y'),Y'){\rm d}Y' \\
&\leq \Big(\frac{1}{2}\overline{M}+\frac{\Lambda}{4\gamma_1}
 +\Lambda\overline{M}\Big)\cdot\Big(X-\phi^{-1}(Y)+Y-\phi(X)\Big) \\
&\leq \Big(\overline{M}+\frac{\Lambda}{2\gamma_1}
 +2\Lambda\overline{M}\Big) (|X|+|Y|+2\underline{M}\mathcal{{E}}_0).
\end{align*}
Then, integrating \eqref{3.3} along the closed curve $S$ obtains
\begin{align*}
&\int_{\phi^{-1}(Y)}^{X}\frac{-c_1}{2(c_2-c_1)}p(X',Y){\rm d}X'
+\int_{\phi(X)}^{Y}\frac{c_2}{2(c_2-c_1)}q(X,Y'){\rm d}Y' \\
&\leq \int_{\phi^{-1}(Y)}^{X}\frac{c_1+2c_2F}{2(c_2-c_1)}ph(X',Y){\rm d}X'
+\int_{\phi(X)}^{Y}\frac{-c_2-2c_1F}{2(c_2-c_1)}qg(X,Y'){\rm d}Y' \\
&\quad +\int_{\phi^{-1}(Y)}^{X}\frac{\lambda}{c_2-c_1}p\ell(X',Y){\rm d}X'
+\int_{\phi(X)}^{Y}\frac{-\lambda}{2(c_2-c_1)}qm(X,Y'){\rm d}Y' \\
&\quad +\Big(\overline{M}+\frac{\Lambda}{2\gamma_1}
+2\Lambda\overline{M}\Big) (|X|+|Y|+2\underline{M}\mathcal{{E}}_0),
\end{align*}
from which and the following inequalities
\begin{align*}
\big|\frac{\lambda p\ell}{c_2-c_1}\big|
\leq\frac{-c_1}{4(c_2-c_1)}p+\frac{\overline{M}\Lambda^2}{\gamma_{1}^2}ph,\quad
\big|\frac{\lambda qm}{c_2-c_1}\big|
\leq\frac{c_2}{4(c_2-c_1)}q+\frac{\overline{M}\Lambda^2}{\gamma_{1}^2}qg,
\end{align*}
one has
\begin{align*}
&\int_{\phi^{-1}(Y)}^{X}\frac{-c_1}{4(c_2-c_1)}p(X',Y){\rm d}X'
+\int_{\phi(X)}^{Y}\frac{c_2}{4(c_2-c_1)}q(X,Y'){\rm d}Y' \\
&\leq \overline{M}\Big(\frac{1}{2}+\Lambda+\frac{\Lambda^2}{\gamma_{1}^2}\Big)
\Big\{\int_{\phi^{-1}(Y)}^{X}ph(X',Y){\rm d}X'
 +\int_{\phi(X)}^{Y}qg(X,Y'){\rm d}Y'\Big\} \\
&+\Big(\overline{M}+\frac{\Lambda}{2\gamma_1}
+2\Lambda\overline{M}\Big) (|X|+|Y|+2\underline{M}\mathcal{{E}}_0),
\end{align*}
which combined with \eqref{3.14} and \eqref{2.25} concludes
\begin{equation}\label{3.15}
\begin{aligned}
&\int_{\phi^{-1}(Y)}^{X}p(X',Y){\rm d}X'
 +\int_{\phi(X)}^{Y}q(X,Y'){\rm d}Y' \\
&\leq 4\underline{M}\Big[\underline{M}\overline{M}^2
 \Big(1+2\Lambda+\frac{2\Lambda^2}{\gamma_{1}^2}\Big)
 + \overline{M}+\frac{\Lambda}{2\gamma_1}
 +2\Lambda\overline{M}\Big](|X|+|Y|+2\underline{M}\mathcal{{E}}_0).
\end{aligned}
\end{equation}
For any $(X,Y)\in\Omega^+$, we now integrate the first equation of \eqref{2.22}
 vertically and apply \eqref{3.15} to have
\begin{equation}\label{3.16}
\exp\Big\{-\widetilde{C}(|X|+|Y|+2\underline{M}\mathcal{{E}}_0)\Big\}
\leq p(X,Y)\leq \exp\Big\{\widetilde{C}(|X|+|Y|+2\underline{M}\mathcal{{E}}_0)
\Big\}
\end{equation}
for some constant $\widetilde{C}$ depending only on
 $\Lambda, \alpha_1, \alpha_2, \gamma_1$ and $\gamma_2$.
A similar inequality also holds for $q$. To estimate the function $u$,
we integrate the equation for $u$ in \eqref{2.20} horizontally and
use \eqref{3.15} to obtain
\begin{equation}\label{3.18}
\begin{aligned}
|u(X,Y)|
&\leq |u_0|+\frac{1}{4\gamma_1}\int_{\varphi^{- 1}(Y)}^{X}p(X',Y){\rm d}X' \\
&\leq |u_0|+\frac{\underline{M}}{\gamma_1}
\Big[\underline{M}\overline{M}^2\Big(1+2\Lambda+\frac{2\Lambda^2}{\gamma_{1}^2}\Big)
 + \overline{M} \\
&\quad +\frac{\Lambda}{2\gamma_1}+2\Lambda\overline{M}\Big]
 (|X|+|Y|+2\underline{M}\mathcal{{E}}_0).
\end{aligned}
\end{equation}
Integrating the equation for $x$ in \eqref{2.20} horizontally,
it suggests by \eqref{3.13} that
\begin{equation}\label{3.20}
\begin{aligned}
|x(X,Y)|
&\leq |x(\varphi^{- 1}(Y), Y)|+\int_{\varphi^{- 1}(Y)}^{X}
 \frac{c_2}{c_2-c_1}ph(X',Y){\rm d}X' \\
&\leq |x(\varphi^{- 1}(Y), Y)|+ 2\overline{M}(|X|+|Y|
 +2\underline{M}\mathcal{{E}}_0) \\
&\leq (2\overline{M}+1)(|X|+|Y|+2\underline{M}\mathcal{{E}}_0).
\end{aligned}
\end{equation}

Since all right-hand side functions in system \eqref{2.20}-\eqref{2.22}
are locally Lipschitz continuous, then the local existence of solutions
follows straightforward from the fixed point method.
from the  a priori estimates \eqref{3.11}, \eqref{3.16}-\eqref{3.18}
and \eqref{3.20}, we can extend this local solution to the entire domain
$\Omega^+$ by using the technique in Bressan and Zheng \cite{Bres-Zheng}.
Thus we have the global existence theorem.

\begin{theorem}
Let \eqref{1.11} and \eqref{1.14} be satisfied. Then problem
\eqref{2.20}-\eqref{2.22} with boundary data \eqref{2.28} has a
unique global solution defined for all $(X,Y)\in\Omega^+$.
\end{theorem}

The above construction leads directly to a useful consequence.

\begin{corollary}\label{conv}
Suppose that \eqref{1.11} holds. If the initial data $(u_{0},u_{1})$ are smooth,
then the solution of \eqref{2.20}-\eqref{2.22} \eqref{2.28} is a smooth
function of the variables $(X,Y)$. Moreover, assume that a sequence of smooth
functions $(u_{0}^{\nu},u_{1}^{\nu})_{\nu\geq1}$
satisfies
$$
u_{0}^{\nu}\to u_{0},\quad
(u_{0}^{\nu})_x\to (u_{0})_x,\quad
u_{1}^{\nu}\to u_{1},
$$
uniformly on compact subsets of $\mathbb{R}$. Then it has the following
convergence properties:
$$
(u^\nu, h^{\nu}, g^{\nu}, \ell^{\nu}, m^{\nu}, p^\nu, q^\nu)
\to (u, h, g, \ell, m, p, q),
$$
uniformly on bounded subsets of the $X$-$Y$ plane.
\end{corollary}


\section{Solutions in the original variables}\label{S4}

This section is devoted to returning the solution in the $X$-$Y$
plane to the original variables $(t,x)$. The H\"older continuous of solution
and the integral equation \eqref{1.17} are also verified in this section.

We first examine the regularity of the solution constructed in the previous section.
 Since the initial data $(u_{0})_x$ and $u_{1}$ are assumed only to be in $L^2$,
we see that, on bounded subsets of the $X$-$Y$ plane,

\noindent-- The functions $h$, $\ell$ and $p$ are
Lipschitz continuous with respect to $Y$, measurable with respect to $X$,

\noindent-- The functions $g, m$ and $q$ are
Lipschitz continuous with respect to $X$, measurable with respect to $Y$,

\noindent-- The functions $u$ and $x$ are Lipschitz
continuous with respect to both $X$ and $Y$.

To return the solution $u$ to the original variables $(t,x)$, we need the
inverse functions $X=X(t,x)$, $Y=Y(t,x)$. The function $x=x(X,Y)$ can be
obtained by solving problem \eqref{2.20}-\eqref{2.22} \eqref{2.28},
so it suffices to construct the function $t=t(X,Y)$.
Owning to \eqref{2.15}, it gives
\begin{equation}\label{4.1}
t_X=\frac{\alpha ph}{c_2-c_1},\quad t_Y=\frac{\alpha qg}{c_2-c_1}.
\end{equation}
It is not difficult to show that $t_{XY}=t_{YX}$, which indicates that we
may integrate one of the equations in \eqref{4.1} to obtain the function $t=t(X,Y)$.
Note that the map $(X,Y)\mapsto(t,x)$ may not be
one-to-one mapping, which, however, does not cause any real difficulty due to
the following assertion: for any fixed $(t,x)$, the values of $u$ do not depend
on the choice of $(X,Y)$. We omit the proof of this assertion since it is
completely analogous to Bressan and Zheng\cite{Bres-Zheng}. Then,
for each given point $(t^*,x^*)$, we can choose an arbitrary point $(X^*,Y^*)$
satisfying $t(X^*,Y^*)=t^*$, $x(X^*,Y^*)=x^*$, and define $u(t^*,x^*):=u(X^*,Y^*)$.

We now show that the function $u(t,x)$, obtained as above, is H\"{o}lder
continuous on bounded sets.
In fact, integrating along any forward characteristic $t\mapsto x^+(t)$
and noting $Y=const.$ on this kind of characteristics achieves
\begin{equation}\label{4.2}
\begin{aligned}
\int_{0}^{\tau}[\alpha(x,u)u_{t}+c_2(x,u)u_{x}]^2{\rm d}t
&=\int_{X_0}^{X_\tau}[(c_2-c_1)u_XX_x]^2t_X{\rm d}X \\
&=\int_{X_0}^{X_\tau}\frac{\alpha(1-h)p}{c_2-c_1}{\rm d}X
\leq C_{\tau}
\end{aligned}
\end{equation}
for some constant $C_\tau$ depending only on $\tau$. Analogously, one has
\begin{equation}\label{4.3}
\int_{0}^{\tau}[\alpha(x,u)u_{t}+c_1(x,u)u_{x}]^2{\rm d}t\leq C_\tau,
\end{equation}
which together with \eqref{4.2} and \eqref{1.11} means that the function
$u=u(t,x)$ is H\"{o}lder continuous with exponent 1/2.
Moreover, it leads by \eqref{2.12} to
the fact that all characteristic curves are $C^1$ with H\"{o}lder
continuous derivative.
In addition, by \eqref{4.2} and \eqref{4.3}, the functions $R$ and $S$
at \eqref{2.1} are square integrable on bounded subsets of the $t$-$x$ plane.
From the  identity
\begin{align*}
\alpha(x,u)u_t+c_2(x,u)u_x=(c_2-c_1)u_XX_x=\frac{\ell}{h}=R,
\end{align*}
the function $R$ is indeed the same as recovered from \eqref{2.6}.
It is also true for $S$.

We next demonstrate that the function $u=u(t,x)$ satisfies \eqref{1.12}
in the distributional sense.
For any test function $\varphi\in C^{1}_c(\mathbb{R}^+\times \mathbb{R})$,
it suggests by \eqref{2.15} that
\begin{equation}\label{4.4}
\begin{aligned}
&\varphi_t(\alpha^2u_t+\beta u_x+\kappa)+\varphi_x(\beta u_t-\gamma^2u_x+\lambda) \\
&=\Big(\frac{c_2}{\alpha qg}\varphi_Y-\frac{c_1}{\alpha ph}\varphi_X\Big)\cdot
 \Big[\frac{\alpha(c_2S-c_1R)}{c_2-c_1}+\frac{\beta(R-S)}{c_2-c_1}+\kappa\Big] \\
&\quad +\Big(-\frac{1}{qg}\varphi_Y+\frac{1}{ph}\varphi_X\Big)\cdot
 \Big[\frac{\beta(c_2S-c_1R)}{\alpha(c_2-c_1)}
 -\frac{\gamma^2(R-S)}{c_2-c_1}+\lambda\Big] \\
&= \frac{1}{qg}\Big\{\frac{c_2-c_1}{2}R+\frac{c_2}{\alpha}\kappa-\lambda\Big\}\varphi_Y
 +\frac{1}{ph}\Big\{\frac{c_2-c_1}{2}S-\frac{c_1}{\alpha}\kappa+\lambda\Big\}\varphi_X \\
&= \frac{1}{qgh}\Big\{\frac{c_2-c_1}{2}\ell
 +\Big(\frac{c_2}{\alpha}\kappa-\lambda\Big)h\Big\}\varphi_Y
 +\frac{1}{pgh}\Big\{\frac{c_2-c_1}{2}m
 -\Big(\frac{c_1}{\alpha}\kappa-\lambda\Big)g\Big\}\varphi_X,
\end{aligned}
\end{equation}
which combined with the Jacobian
\begin{equation}\label{4.5}
\frac{\partial(x,t)}{\partial(X,Y)}=\frac{\alpha pqgh}{c_2-c_1},
\end{equation}
ensures
\begin{equation}\label{4.6}
\begin{aligned}
&\iint_{\mathbb{R}^+\times\mathbb{R}}\{\varphi_t(\alpha^2u_t+\beta u_x
 +\kappa)+\varphi_x(\beta u_t-\gamma^2u_x+\lambda)\}{\rm d}x{\rm d}t \\
&=\iint_{(X,Y)\in\Omega^+}\Big\{\Big[\frac{\alpha}{2}qm
 +\frac{\alpha\lambda-c_1\kappa}{c_2-c_1}qg\Big]\varphi_X
 +\Big[\frac{\alpha}{2}p\ell-\frac{\alpha\lambda-c_2\kappa}{c_2-c_1}ph\Big]
 \varphi_Y\Big\}{\rm d}X{\rm d}Y
 \\
&=-\iint_{(X,Y)\in\Omega^+}\varphi\Big\{\Big[\frac{\alpha}{2}qm
 +\frac{\alpha\lambda-c_1\kappa}{c_2-c_1}qg\Big]_X
 +\Big[\frac{\alpha}{2}p\ell-\frac{\alpha\lambda-c_2\kappa}{c_2-c_1}ph
 \Big]_Y\Big\}{\rm d}X{\rm d}Y.
\end{aligned}
\end{equation}
A straightforward computation yields
\begin{equation}\label{4.7}
\begin{aligned}
&\Big(\frac{\alpha}{2}qm\Big)_X+\Big(\frac{\alpha}{2}p\ell\Big)_Y \\
&=\frac{\alpha pqgh}{c_2-c_1}\Big\{a_2\frac{1-g}{g}-a_1\frac{1-h}{h}
+\Big(a_1-a_2+\frac{\alpha_u}{\alpha}\Big)\frac{m\ell}{gh} +(F_u-\lambda_x)\Big\},
\end{aligned}
\end{equation}
and
\begin{equation}\label{4.8}
\begin{aligned}
&\Big\{\frac{\alpha\lambda-c_1\kappa}{c_2-c_1}qg\Big\}_X
 -\Big\{\frac{\alpha\lambda-c_2\kappa}{c_2-c_1}ph\Big\}_Y \\
&=\frac{\alpha pqgh}{c_2-c_1}\Big\{\frac{\lambda_u}{c_2-c_1}\cdot\frac{g\ell-mh}{gh}
 +\frac{\kappa_u}{\alpha(c_2-c_1)}\cdot\frac{c_2mh-c_1g\ell}{gh}+\lambda_x\Big\}.
\end{aligned}
\end{equation}
Inserting \eqref{4.7} and \eqref{4.8} into \eqref{4.6} and using the
Jacobian \eqref{4.5} again leads to
\begin{align*}
&\iint_{\mathbb{R}^+\times\mathbb{R}}\{\varphi_t(\alpha^2u_t+\beta u_x+\kappa)
 +\varphi_x(\beta u_t-\gamma^2u_x+\lambda)\}{\rm d}x{\rm d}t \\
&=-\iint_{\mathbb{R}^+\times\mathbb{R}}\varphi\Big\{a_2\frac{1-g}{g}
 -a_1\frac{1-h}{h} +\Big(a_1-a_2+\frac{\alpha_u}{\alpha}\Big)\frac{m\ell}{gh}+F_u \\
&\quad +\frac{\lambda_u}{c_2-c_1}\frac{g\ell-mh}{gh}
 +\frac{\kappa_u}{\alpha(c_2-c_1)}\frac{c_2mh-c_1g\ell}{gh}\Big\}{\rm d}x{\rm d}t
 \\
&=-\iint_{\mathbb{R}^+\times\mathbb{R}}\varphi\Big\{a_2S^2-a_1R^2
 +\Big(a_1-a_2+\frac{\alpha_u}{\alpha}\Big)RS+F_u \\
&\quad +\frac{\lambda_u}{c_2-c_1}(R-S)
 +\frac{\kappa_u}{\alpha(c_2-c_1)}(c_2S-c_1R)\Big\}{\rm d}x{\rm d}t
 \\
&=-\iint_{\mathbb{R}^+\times\mathbb{R}}\varphi\Big\{\alpha\alpha_uu_{t}^2
 +\beta_uu_tu_x -\gamma\gamma_uu_{x}^2+\kappa_uu_t
 +\lambda_uu_x+F_u\Big\}{\rm d}x{\rm d}t,
\end{align*}
which finishes the proof of \eqref{1.17}.


\section{Regularity of trajectories}\label{S5}

In this section, we complete the proof of Theorem \ref{ex}. For $M>0$, denote
$$
\mathcal{{E}}_M(t)
=\frac{1}{2}\int_{-M}^M\Big\{\alpha^2(x,u)u_{t}^2
+\gamma^2(x,u)u_{x}^2\Big\}{\rm d}x.
$$
We first show the following lemma

\begin{lemma}\label{lemma}
For any $M>0$ and $t>0$, the solution constructed in previous section satisfies
\begin{equation}\label{5.1}
\mathcal{{E}}_M(t)\leq \widehat{C}(\mathcal{{E}}_0+M+t)
\end{equation}
for some positive constant $\widehat{C}$ independent of $M$ and $t$.
Here $\mathcal{{E}}_0$ is defined as in \eqref{2.26}.
\end{lemma}

\begin{proof}
Fix any $\tau>0$, we denote $\Gamma_\tau:=\{(X,Y):\ t(X,Y)=\tau\}$.
Let $A_1$ and $A_2$ on $\Gamma_\tau$ be any two corresponding points of
the points $(\tau,-M)$ and $(\tau,M)$ in $t$-$x$ plane, respectively.
Then we draw the horizontal and vertical lines from $A_1$ and $A_2$ up to
$\Gamma_0$ at points $A_4$ and $A_3$, respectively.
Consider the region $D$ bounded by $\Gamma_0$, $\Gamma_\tau$, $A_1A_4$
and $A_2A_3$, see Figure \ref{fig1}.

\begin{figure}[htbp]
\begin{center}
 \includegraphics[width=0.4\textwidth]{fig1} 
\end{center}
\caption{Region $D$.} \label{fig1}
\end{figure}

We integrate \eqref{3.3} along the boundary of $D$ to obtain
\begin{equation}\label{5.2}
\begin{aligned}
&\int_{A_1A_2}\frac{-c_1p(1-h)}{2(c_2-c_1)}{\rm d}X
 -\frac{c_2q(1-g)}{2(c_2-c_1)}{\rm d}Y \\
&=\int_{A_4A_3}\frac{-c_1p(1-h)}{2(c_2-c_1)}{\rm d}X
 -\frac{c_2q(1-g)}{2(c_2-c_1)}{\rm d}Y +I_1+I_2+I_3+I_4,
\end{aligned}
\end{equation}
where
\begin{gather*}
I_1=\int_{A_4A_3}\Big(\frac{-\lambda p\ell}{c_2-c_1}+\frac{-c_2Fph}{c_2-c_1}\Big){\rm d}X
  -\Big(\frac{\lambda qm}{c_2-c_1}+\frac{c_1Fqg}{c_2-c_1}\Big){\rm d}Y,\\
I_2=\int_{A_1A_2}\Big(\frac{\lambda p\ell}{c_2-c_1}+\frac{c_2Fph}{c_2-c_1}\Big){\rm d}X
 +\Big(\frac{\lambda qm}{c_2-c_1}+\frac{c_1Fqg}{c_2-c_1}\Big){\rm d}Y, \\
I_3=-\int_{A_3A_2}\Big\{\frac{c_2q(1-g)}{2(c_2-c_1)}
 +\frac{\lambda qm}{c_2-c_1}+\frac{c_1Fqg}{c_2-c_1}\Big\}{\rm d}Y, \\
I_4=-\int_{A_4A_1}\Big\{\frac{-c_1p(1-h)}{2(c_2-c_1)}-\frac{\lambda p\ell}{c_2-c_1}
 -\frac{c_2Fph}{c_2-c_1}\Big\}{\rm d}X.
\end{gather*}
Notice that the length of the segment corresponding to $A_4A_3$ in the
initial line $t=0$ is less or equal to $2(M+\max\{|\lambda_\pm|\}\tau)$,
then we have
\begin{equation}\label{5.3}
\begin{aligned}
|I_1|
&\leq \int_{A_4A_3}\Big(\frac{\Lambda}{2(c_2-c_1)}
 +\frac{c_2\Lambda}{c_2-c_1}\Big){\rm d}X
 -\Big(\frac{\Lambda}{2(c_2-c_1)}+\frac{-c_1\Lambda}{c_2-c_1}\Big){\rm d}Y \\
&\leq \Lambda\Big(\frac{1}{\gamma_1}+4\overline{M}\Big)
 \Big(M+\max\{|\lambda_\pm|\}\tau\Big) \\
& \leq \Lambda\Big(\frac{1}{\gamma_1}+4\overline{M}\Big)
 \Big(M+\frac{2\gamma_1\overline{M}}{\alpha_1}\tau\Big).
\end{aligned}
\end{equation}
From the inequalities
\begin{gather*}
\big|\frac{\lambda p\ell}{c_2-c_1}\big|
 \leq \frac{\Lambda^2 ph}{-c_1(c_2-c_1)}+\frac{-c_1 p(1-h)}{4(c_2-c_1)}, \quad
\big|\frac{\lambda qm}{c_2-c_1}\big|
\leq \frac{\Lambda^2 qg}{c_2(c_2-c_1)}+\frac{c_2 q(1-g)}{4(c_2-c_1)},
\end{gather*}
one gets
\begin{equation}\label{5.4}
\begin{aligned}
|I_2|&\leq\frac{1}{2}\int_{A_1A_2}\frac{-c_1p(1-h)}{2(c_2-c_1)}{\rm d}X
 +\frac{c_2q(1-g)}{2(c_2-c_1)}{\rm d}Y \\
&\quad +\Big(\frac{4\Lambda^2\overline{M}}{\gamma_{1}^2}+\Lambda\overline{M}\Big)
\int_{A_1A_2}ph{\rm d}X-qg{\rm d}Y.
\end{aligned}
\end{equation}
Using the above inequalities  yields
\begin{equation}\label{5.5}
\begin{aligned}
|I_3|
&\leq\int_{A_3A_2}\Big\{\frac{-c_2q(1-g)}{4(c_2-c_1)}
 +\Big(\frac{\Lambda^2}{c_2(c_2-c_1)}+\Lambda\overline{M}\Big)qg\Big\}{\rm d}Y, \\
&\leq \Big(\frac{4\Lambda^2\overline{M}}{\gamma_{1}^2}+\Lambda\overline{M}\Big)
 \int_{A_3A_2}qg{\rm d}Y,
\end{aligned}
\end{equation}
and
\begin{equation}\label{5.6}
\begin{aligned}
|I_4|
&\leq\int_{A_4A_1}\Big\{\frac{c_1q(1-g)}{4(c_2-c_1)}
 +\Big(\frac{\Lambda^2}{-c_1(c_2-c_1)}+\Lambda\overline{M}\Big)ph\Big\}{\rm d}X, \\
&\leq \Big(\frac{4\Lambda^2\overline{M}}{\gamma_{1}^2}+\Lambda\overline{M}\Big)
 \int_{A_4A_1}ph{\rm d}X.
\end{aligned}
\end{equation}
Putting \eqref{5.3}--\eqref{5.6} in \eqref{5.2} gives
\begin{equation}\label{5.7}
\begin{aligned}
&\int_{A_1A_2}\frac{-c_1p(1-h)}{2(c_2-c_1)}{\rm d}X
 -\frac{c_2q(1-g)}{2(c_2-c_1)}{\rm d}Y \\
&\leq 2\mathcal{{E}}_0+2\Lambda\Big(\frac{1}{\gamma_1}+4\overline{M}\Big)
 \Big(M+\frac{2\gamma_1\overline{M}}{\alpha_1}\tau\Big) 
 +\Big(\frac{8\Lambda^2\overline{M}}{\gamma_{1}^2}+2\Lambda\overline{M}\Big)\\
&\quad\times \Big\{\int_{A_1A_2}ph{\rm d}X-qg{\rm d}Y+\int_{A_4A_1}ph{\rm d}X
 +\int_{A_3A_2}qg{\rm d}Y\Big\},
\end{aligned}
\end{equation}
where we used the fact that the variables $h, g$ never assume the value
zero at the initial time.
Moreover, integrating \eqref{3.12} along the boundary of $D$ arrives at
\begin{align*}
&\int_{A_1A_2}\frac{c_2ph}{c_2-c_1}{\rm d}X-\frac{-c_1qg}{c_2-c_1}{\rm d}Y \\
&=\int_{A_4A_3}\frac{c_2ph}{c_2-c_1}{\rm d}X-\frac{-c_1qg}{c_2-c_1}{\rm d}Y
 -\int_{A_4A_1}\frac{c_2ph}{c_2-c_1}{\rm d}X
 -\int_{A_3A_2}\frac{-c_1qg}{c_2-c_1}{\rm d}Y \\
&\leq 4\overline{M}\Big(M+\frac{2\gamma_1\overline{M}}{\alpha_1}\tau\Big),
\end{align*}
which along with \eqref{2.25} leads to
\begin{equation}\label{5.8}
\int_{A_1A_2}ph{\rm d}X-qg{\rm d}Y
\leq 4\underline{M}\overline{M}\Big(M+\frac{2\gamma_1\overline{M}}{\alpha_1}\tau\Big).
\end{equation}
To estimate the last two terms in \eqref{5.7}, we extend the segments
$A_4A_1$ and $A_3A_2$ such that they intersect a point $A_5$ and
integrate \eqref{3.12} along the closed curve composed of $A_4A_5, A_5A_3$
and $A_3A_4$ to find
\begin{align*}
\int_{A_4A_5}\frac{c_2ph}{c_2-c_1}{\rm d}X +\int_{A_3A_5}\frac{-c_1qg}{c_2-c_1}{\rm d}Y
&=\int_{A_4A_3}\frac{c_2ph}{c_2-c_1}{\rm d}X-\frac{-c_1qg}{c_2-c_1}{\rm d}Y \\
& \leq 4\overline{M}\Big(M+\frac{2\gamma_1\overline{M}}{\alpha_1}\tau\Big),
\end{align*}
from which, one has
\begin{equation}\label{5.9}
\begin{aligned}
\int_{A_4A_1}ph{\rm d}X +\int_{A_3A_2}qg{\rm d}Y
&\leq\underline{M}\Big\{\int_{A_4A_5}\frac{c_2ph}{c_2-c_1}{\rm d}X
+\int_{A_3A_5}\frac{-c_1qg}{c_2-c_1}{\rm d}Y\Big\} \\
& \leq 4\underline{M}\overline{M}
\Big(M+\frac{2\gamma_1\overline{M}}{\alpha_1}\tau\Big).
\end{aligned}
\end{equation}
Inserting \eqref{5.8} and \eqref{5.9} into \eqref{5.7}, we conclude that
\begin{equation}\label{5.10}
\begin{aligned}
&\int_{A_1A_2}\frac{-c_1p(1-h)}{2(c_2-c_1)}{\rm d}X
 -\frac{c_2q(1-g)}{2(c_2-c_1)}{\rm d}Y \\
&\leq 2\mathcal{{E}}_0+2\Lambda\Big\{\frac{1}{\gamma_1}+4\overline{M}
 +8\underline{M}\overline{M}^2\Big(\frac{4\Lambda}{\gamma_{1}^2}+1\Big)\Big\}
\Big(M+\frac{2\gamma_1\overline{M}}{\alpha_1}\tau\Big) \\
&\leq\widehat{C}(\mathcal{{E}}_0+M+\tau)
\end{aligned}
\end{equation}
for some constant $\widehat{C}$ independent of $M$ and $\tau$.
On the other hand, it follows that
\begin{align*}
&\int_{-M}^M\frac{1}{2}\Big\{\alpha^2(x,u(\tau,x))u_{t}^2(\tau,x)
 +\gamma^2(x,u(\tau,x))u_{x}^2(\tau,x)\Big\}{\rm d}x \\
=&\int_{A_1A_2\cap \{h\neq0\}}\frac{-c_1p(1-h)}{2(c_2-c_1)}{\rm d}X
 -\int_{A_1A_2\cap \{g\neq0\}}\frac{c_2q(1-g)}{2(c_2-c_1)}{\rm d}Y,
\end{align*}
which together with \eqref{5.10} concludes \eqref{5.1}.
\end{proof}

We now use \eqref{5.1} to prove \eqref{1.16}.
For any $t,s\in\mathbb{R}^+$, we see that
\begin{equation}\label{5.11}
\begin{aligned}
\|u(t,x)-u(s,x)\|_{L^2([-M,M])}
&\leq |t-s|\int_{0}^{1}\|u_{t}(s+\xi(t-s),\cdot)\|_{L^2([-M,M])}{\rm d}\xi \\
&\leq \sqrt{\widehat{C}(\mathcal{{E}}_0+M+t+s)}|t-s|
\end{aligned}
\end{equation}
for any $M>0$, where the constant $\widehat{C}$ is independent of $t, s$
and $M$. This proves \eqref{1.16}.

We next prove that, for any $M>0$, the functions $t\mapsto u_{t}(t,\cdot)$ and
$t\mapsto u_{x}(t,\cdot)$ are continuous with values in $L^\theta([-M,M])$
($1\leq \theta<2$), which will complete the proof of Theorem \ref{ex}.
Let us first consider the arguments for smooth initial data with compact support,
in which, the solution $u=u(X,Y)$ remains smooth on the
entire region $\Omega^+$. For a fixed time $\tau$ and any fixed $M>0$,
we assert that,
\begin{equation}\label{5.12}
\frac{{\rm d}}{{\rm d}t}u(t,\cdot)|_{t=\tau}=u_{t}(\tau,\cdot)
\end{equation}
in interval $[-M,M]$, where
\begin{equation}\label{5.13}
u_{t}(\tau, x):=u_{X}X_t+u_{Y}Y_t
=\frac{-c_1}{\alpha(c_2-c_1)}\cdot\frac{\ell}{h}+\frac{c_2}{\alpha(c_2-c_1)}\cdot
\frac{m}{g},
\end{equation}
which defines the value of $u_t(\tau,\cdot)$ at almost every point
$x\in[-M,M]$ by \eqref{5.1}.

To verify the assertion \eqref{5.12}, we consider the curve segment
$A_1A_2$ as before. For any $\theta\in[1,2)$, let $\sigma:=2/(2-\theta)$
be the conjugate exponent of $2/\theta$ and denote
$\widetilde{M}:=\widehat{C}(\mathcal{{E}}_0+M+2\tau)$.
Given any $\varepsilon>0$, it is clear that there exist finitely
many disjoint intervals $[a_i,b_i]\subset[-M,M]$,
$i=1,2,\dots,N$, such that
\begin{equation}\label{5.14}
\min\{h(P), g(P)\}<\frac{2\varepsilon}{(\widetilde{M}+1)^{\sigma}}
\end{equation}
for every point $P=(X(x_P,\tau),Y(x_P,\tau))$ and
\begin{equation}\label{5.15}
h(Q)>\frac{\varepsilon}{(\widetilde{M}+1)^{\sigma}},\quad
g(Q)>\frac{\varepsilon}{(\widetilde{M}+1)^{\sigma}}
\end{equation}
for every point $Q=(X(x_Q,\tau),Y(x_Q,\tau))$, where
$x_P\in J:= \cup_{i=1}^{N}[a_i,b_i]$ and
$x_Q\in J':=[-M,M]\setminus J$. Obviously, the function $u=u(t,x)$ is
smooth in a neighborhood of the set $\{\tau\}\times J'$ by the
construction of $J'$. By employing Minkowski's inequality, we find that
\begin{equation}\label{5.16}
\begin{aligned}
&\lim_{\rho\to 0}\frac{1}{\rho}\Big[\int_{-M}^M|u(\tau+
\rho,x)-u(\tau,x)-\rho u_{t}(\tau,x)|^\theta{\rm d}x\Big]^{1/\theta}
 \\
&\leq \lim_{\rho\to 0}\frac{1}{\rho}\Big[\int_{J}|u(\tau+
\rho,x)-u(\tau,x)|^\theta{\rm
d}x\Big]^{1/\theta}+\Big[\int_{J}|u_{t}(\tau,x)|^\theta{\rm
d}x\Big]^{1/\theta}.
\end{aligned}
\end{equation}
We use \eqref{5.14} and \eqref{5.10} to estimate the measure of the "bad"
set $J$,
\begin{equation}\label{5.17}
\begin{aligned}
\operatorname{meas}(J)
&=\int_{J}{\rm d}x
 =\sum_{i=1}^{N}\int_{(X_{a_i},Y_{a_i})}^{(X_{b_i},Y_{b_i})}
\frac{c_2 ph}{c_2-c_1}{\rm d}X+\frac{c_1 qg}{c_2-c_1}{\rm d}Y \\
&\leq \underline{M}\overline{M}\sum_{i=1}^{N}\int_{(X_{a_i},Y_{a_i})}^{(X_{b_i},Y_{b_i})}
\frac{-c_1 ph}{c_2-c_1}{\rm d}X-\frac{c_2 qg}{c_2-c_1}{\rm d}Y \\
&\leq \frac{\underline{M}\overline{M}\frac{4\varepsilon}{(\widetilde{M}+1)^\sigma}}{1 -\frac{2\varepsilon}{(\widetilde{M}+1)^\sigma}}\sum_{i=1}^{N}
\int_{(X_{a_i},Y_{a_i})}^{(X_{b_i},Y_{b_i})} \frac{-c_1 p(1-h)}{2(c_2-c_1)}{\rm
d}X-\frac{c_2 q(1-g)}{2(c_2-c_1)}{\rm d}Y \\
&\leq \frac{4\underline{M}\overline{M}\varepsilon}{(1-2\varepsilon)(\widetilde{M}+1)^\sigma}
\int_{A_1A_2}\frac{-c_1p(1-h)}{2(c_2-c_1)}{\rm
d}X-\frac{c_2q(1-g)}{2(c_2-c_1)}{\rm d}Y \\
&\leq \frac{4\underline{M}\overline{M}\widetilde{M}\varepsilon}
{(1-2\varepsilon)(\widetilde{M}+1)^\sigma},
\end{aligned}
\end{equation}
where $(X_{a_i},Y_{a_i})=(X(a_i,\tau),Y(a_i,\tau))$ and
$(X_{b_i},Y_{b_i})=(X(b_i,\tau),Y(b_i,\tau))$. Applying H\"{o}lder's
inequality and recalling \eqref{5.11} yields
\begin{align*}
&\int_{J}|u(\tau+\rho,x)-u(\tau,x)|^\theta{\rm d}x \\
&\leq \operatorname{meas}(J)^{1/\sigma}
\Big(\int_{J}|u(\tau+\rho,x)-u(\tau,x)|^2{\rm
d}x\Big)^{\theta/2} \\
&\leq \Big(\frac{4\underline{M}\overline{M}\widetilde{M}\varepsilon}{(1-2\varepsilon)
(\widetilde{M}+1)^\sigma} \Big)^{1/\sigma}\|u(\tau+\rho,\cdot)
-u(\tau,\cdot)\|_{L^2([-M,M])}^{\theta}
 \\
&\leq \Big(\frac{4\underline{M}\overline{M}\widetilde{M}\varepsilon}{(1-2\varepsilon)
(\widetilde{M}+1)^\sigma} \Big)^{1/\sigma}
(\widetilde{M}+\widehat{C}\rho)^{\theta/2}\rho^\theta,
\end{align*}
from which, we have
\begin{equation}\label{5.18}
\begin{aligned}
\limsup_{\rho\to 0}\frac{1}{\rho}\Big(\int_{J}|u(\tau+\rho,x)-
u(\tau,x)|^\theta{\rm d}x\Big)^{1/\theta}
&\leq \sqrt{\widetilde{M}}
\Big(\frac{4\underline{M}\overline{M}\widetilde{M}\varepsilon}{(1-2\varepsilon)
(\widetilde{M}+1)^\sigma} \Big)^{\frac{1}{\sigma\theta}} \\
&\leq \Big(\frac{4\underline{M}\overline{M}\varepsilon}{1-2\varepsilon}
\Big)^{\frac{1}{\sigma\theta}}.
\end{aligned}
\end{equation}
Analogously, one has
\begin{equation}\label{5.19}
\begin{aligned}
\Big(\int_{J}|u_{t}(\tau,x)|^\theta{\rm d}x\Big)^{1/\theta}
&\leq \operatorname{meas}(J)^{\frac{1}{\sigma\theta}}
\Big(\int_{J}|u_{t}(\tau,x)|^2{\rm d}x\Big)^{1/2} \\
&\leq \Big( \frac{4\underline{M}\overline{M}\widetilde{M}\varepsilon}
{(1-2\varepsilon)(\widetilde{M}+1)^\sigma} \Big)^{\frac{1}{\sigma\theta}}
\|u_t(\tau,\cdot)\|_{L^2([-M,M])} \\
&\leq \sqrt{2\widetilde{M}}\Big(
\frac{4\underline{M}\overline{M}\widetilde{M}\varepsilon}{(1-2\varepsilon)(\widetilde{M}+1)^\sigma}
\Big)^{\frac{1}{\sigma\theta}} \\
&\leq\sqrt{2} \Big(\frac{4\underline{M}\overline{M}\varepsilon}{1-2\varepsilon}
\Big)^{\frac{1}{\sigma\theta}}.
\end{aligned}
\end{equation}
Combining with \eqref{5.16}, \eqref{5.18} and \eqref{5.19},
it follows by the arbitrariness of $\varepsilon>0$ that
\begin{equation}\label{5.20}
\lim_{\rho\to 0}\frac{1}{\rho}\Big(\int_{-M}^M|u(\tau+\rho,x)-
u(\tau,x)-\rho u_{t}(\tau,x)|^\theta{\rm d}x\Big)^{1/\sigma}=0.
\end{equation}
Based on the same method, we can establish the continuity of the function
$t\mapsto u_{t}(t,\cdot)$.

To extend the result to general initial data $(u_{0}),u_{1}\in L^2$,
we let $\{(u_{0}^{\nu})_x\},\{u_{1}^{\nu}\}\in C_{c}^{\infty}$ be a
sequence of smooth initial data such that
$u_{0}^{\nu}\to u_{0}$ uniformly,
$(u_{0}^{\nu})_x\to (u_{0})_x$ almost
everywhere and in $L^2$, $u_{1}^\nu\to u_{1}$ almost everywhere and in $L^2$.
The proof is concluded
by Corollary \ref{conv}. The continuity of the function $t\to u_{x}(t,\cdot)$
as a map with values in $L^\theta([-M,M]), 1\leq \theta<2$ can be verified
in an entirely similar way, so we omit it here.


\subsection*{Acknowledgements}
Y. Hu was supported by the NSF of Zhejiang Province of China (LY17A010019)
and NSFC (11301128, 11571088). G. Wang was supported by the NSF of Anhui
 Province of China (1508085MA08).

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