# Local solvability for a quasilinear wave equation with the far field
degeneracy: 1D case

Yuusuke<EMAIL_ADDRESS>The University of shiga
prefecture

###### Abstract

We study the Cauchy problem for the quasilinear wave equation
$\partial^{2}_{t}u=u^{2a}\partial^{2}_{x}u+F(u)u_{x}$ with $a\geq 0$ and show
a result for the local in time existence under new conditions. In the previous
results, it is assumed that $u(0,x)\geq c_{0}>0$ for some constant $c_{0}$ to
prove the existence and the uniqueness. This assumption ensures that the
equation does not degenerate. In this paper, we allow the equation to
degenerate at spacial infinity. Namely we consider the local well-posedness
under the assumption that $u(0,x)>0$ and $u(0,x)\rightarrow 0$ as
$|x|\rightarrow\infty$. Furthermore, to prove the local well-posedness, we
find that the so-called Levi condition appears. Our proof is based on the
method of characteristic and the contraction mapping principle via weighted
$L^{\infty}$ estimates.

##  1 Introduction

In this paper, we consider the following Cauchy problem of the model
quasilinear wave equation in $\mathbb{R}$:

$\displaystyle\left\\{\begin{array}[]{ll}\partial^{2}_{t}u=\partial_{x}(u^{2a}\partial_{x}u)+F(u)u_{x},\
\ (t,x)\in(0,T]\times\mathbb{R},\\\ u(0,x)=u_{0}(x),\ \ x\in\mathbb{R},\\\
\partial_{t}u(0,x)=u_{1}(x),\ \ x\in\mathbb{R},\end{array}\right.$ (1.4)

where $F$ is a given function and $a\geq 0$ . The purpose of this paper is to
show the local existence and the uniqueness under new conditions. The
existence of solutions to the more general quasi-linear wave equations has
been widely known since the 1970s. Kato [12] and Hughes, Kato and Marsden [9]
have shown an abstract theorem about the well-posedness of the system of
general quasi-linear wave equations in $L^{2}$ Sobolev space. In $1$
dimensional case, the well-posedness in $C^{1}_{b}$ class for first order
hyperbolic equations has been studied by Douglis [5] and Hartman and Winter
[7] (see also Majda [14] and Courant and Lax [3]), where $C^{1}_{b}$ is a set
of continuous and bounded functions whose derivatives are also bounded. In
order to apply these results to the existence problem of (1.4), the following
assumption is required:

$\displaystyle u_{0}(x)\geq c_{0}>0$ (1.5)

for a constant $c_{0}$. This condition ensures that the equation in (1.4) is
the strictly hyperbolic type near $t=0$. This paper relaxes this condition. We
show the the local existence and the uniqueness of solutions of (1.4) under
the assumption that the equation degenerates at spacial infinity. Namely we
weaken (1.5) by $u(0,x)>0$ and allow that $u(0,x)$ can decay to $0$ as
$|x|\rightarrow\infty$ (more precise assumptions are given later). To the best
of my knowledge, the well-possessedness has never been studied under these
types of assumptions.

###  1.1 Known results

Let us review some results on the solvability for degenerate wave equations
(weakly hyperbolic equations). The existence, nonexistence and regularity of
solutions to the following type of linear weakly hyperbolic equations have
been studied by many authors (e.g. Oleinik [18], Colombini and Spagnolo [2],
Ivrii and Petkov [11] and Taniguchi and Tozaki [22]),

$\displaystyle\partial^{2}_{t}u-\sum_{i,j=1}^{n}a_{i,j}(t,x)u_{x_{i}x_{j}}+\sum_{j=1}^{n}b_{j}(t,x)u_{x_{j}}=0,$
(1.6)

where $a_{i,j}$ and $b_{j}$ are smooth functions and
$\sum_{i,j=1}^{n}a_{i,j}(t,x)\xi_{i}\xi_{j}\geq 0$ is assumed for
$(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$. We note that
$\sum_{i,j=1}^{n}a_{i,j}(t,x)\xi_{i}\xi_{j}=0$ corresponds to the degeneracy.
In Oleinik [18], (1.6) have been solved under the so-called Levi condition:

$\displaystyle C_{1}\left(\sum_{j=1}^{n}b_{j}\xi_{j}\right)^{2}\leq
C_{2}\left(\sum_{i,j=1}^{n}a_{i,j}\xi_{i}\xi_{j}+\partial_{t}a_{i,j}\xi_{i}\xi_{j}\right).$
(1.7)

Even if the Levi condition is assumed, we can only obtain the following energy
estimate with the regularity loss for weakly hyperbolic equations:

$\displaystyle\|u\|_{H^{s}}+\|u_{t}\|_{H^{s-1}}\leq
C(\|u_{0}\|_{H^{s+r_{1}}}+\|u_{1}\|_{H^{s-1+r_{2}}}),$ (1.8)

where $s$ is an arbitrary real number and $r_{1}$ and $r_{2}$ are non-negative
numbers. It is known that this estimate is optimal in the sense of the
regularity by observing some explicit solution to some special linear weakly
hyperbolic equations. Ivrii and Petkov in [11] have treated the the following
model of the $1D$ weakly hyperbolic equation:

$\displaystyle u_{tt}-t^{2l}u_{xx}+t^{k}u_{x}=0.$

They have shown that the Levi condition ($k\geq l-1$) is necessary for the
Cauchy problem of this equations to be $C^{\infty}$ well-posed. Colombini and
Spagnolo in [2] have given an example of a $C^{\infty}$ function $a(t)\geq 0$
such that

$u_{tt}-a(t)u_{xx}=0$

is not well-posed in $C^{\infty}$. Roughly speaking, highly oscillatory
behaviors of $a(t)$ near the point that $a(t)=0$ causes the ill-posedness. In
[10], Han has derived an energy inequality with a regularity loss for the
linear weakly hyperbolic equation:

$\displaystyle\partial^{2}_{t}u-a(t,x)u_{xx}=0,$

where
$a(t,x)=t^{m}+a_{1}(x)t^{m-1}+a_{2}(x)t^{m-2}+\cdots+a_{m-1}(x)t+a_{m}(x)$.

Manfrin in [17] have established the local existence and the uniqueness for
following $1D$ degenerate quasilinear wave equations with $u_{0},u_{1}\in
C^{\infty}_{0}(\mathbb{R}^{n})$:

$\displaystyle u_{tt}=a(u)\Delta u,$

where $a(u)$ is a analytic function satisfying $a(0)=0$. This result can be
extended to more general degenerate wave equations (see also Manfrin [15,
16]). In Dreher’s paper [6], he also has shown the local solvability for
$\partial^{2}_{t}u=\partial_{x}(|\partial_{x}u|^{p-2}\partial_{x})$ with $p>5$
and under the initial condition that $u_{0},u_{1}\in
C^{k}_{0}(\mathbb{R}^{n})$ for a large natural number $k$. Since their proof
is based on the Nash-Moser implicit function theorem and the argument in the
Oleinik’s paper [18], the compactness of the support of initial data is
essentially used. Hence it does not seem difficult to extend Manfrin’s method
to the case that initial data are not compactly supported.

In [8], Hu and Wang have shown the local existence and uniqueness of solutions
to the following variational wave equation:

$\displaystyle\partial^{2}_{t}u=c(u,x)\partial_{x}(c(u,x)\partial_{x}u)$ (1.9)

with initial data and the function $c(u,x)$ satisfying

$\displaystyle c(u(0,x),x)=0,$ $\displaystyle\partial_{t}u(0,x)\geq c_{0}>0,$
$\displaystyle c_{u}(u,x)\geq c_{1}>0$

for some constants $c_{0}$ and $c_{1}$. The choice of initial data implies
that the equation degenerates at $t=0$ and that $c(u,x)$ becomes positive
uniformly and immediately after $t=0$. The method in [8] is inspired by Zhang
and Zheng’s paper [24] which studies the existence of solutions to Euler type
equation in gas dynamics. In [24] and [8], they use method of characteristic
for a new dependent variable and the fixed point theorem in a special metric
space.

###  1.2 Assumptions and main theorem

Before stating main theorem of this paper, we introduce assumptions on initial
data and the function $F$. We set $\gamma=\gamma(a,\alpha)$ as below:

$\displaystyle\gamma=\left\\{\begin{array}[]{ll}0,&a\geq 1,\\\
(1-a)\alpha,&\mbox{otherwise}.\end{array}\right.$

For initial data $u_{0}\in C^{2}(\mathbb{R})$ and $u_{1}\in
C^{1}_{b}(\mathbb{R})$, we assume that

$\displaystyle c_{1}\braket{x}^{-\alpha}\leq u_{0}(x)\leq c_{2},$ (1.11)
$\displaystyle|u_{1}(x)\pm u_{0}^{a}u^{\prime}_{0}(x)|\leq
c_{3}\braket{x}^{-\beta},$ (1.12) $\displaystyle\left|\frac{d}{dx}(u_{1}(x)\pm
u_{0}^{a}u^{\prime}_{0}(x))\right|\leq c_{4}\braket{x}^{-\gamma}$ (1.13)

with conditions on $\alpha\geq 0$ and $\beta\geq 0$ that

$\displaystyle\alpha\leq\beta,$ (1.14) $\displaystyle a\alpha\leq\beta,$
(1.15)

where $u^{\prime}_{0}=du_{0}/dx$ and $\braket{x}$ is defined by
$\braket{x}=(1+x^{2})^{1/2}$ and $c_{1},c_{2},c_{3},c_{4}$ are positive
constant. The assumption (1.11) indicates that the equation in (1.4) can
degenerate at spatial infinity. For the function $F\in C([0,\infty))\cap
C^{1}((0,\infty))$, we assume that

$\displaystyle|F(\theta)|\leq C_{K}\theta^{a},$ (1.16)
$\displaystyle|F^{\prime}(\theta)|\leq C_{K}\theta^{a-1}$ (1.17)

for $\theta\in(0,K]$ and $C_{K}$ is a positive constant. Typical example of
$F$ is $F(\theta)=\theta^{b}$ with $b\geq a$. This condition appears in the
study of weakly hyperbolic equations called a sufficient Levi condition (e.g.
Manfrin’s paper [17]) . Main theorem of this paper is as follows.

######  Theorem 1.1.

Let $u_{0}\in C^{2}(\mathbb{R})$ and $u_{1}\in C^{1}_{b}(\mathbb{R})$. Suppose
that the conditions (1.11)-(1.17) hold. Then there exists a number $T>0$
depending on the constants in (1.11)-(1.17) such that the Cauchy problem (1.4)
has a unique local solution $u\in C^{2}([0,T]\times\mathbb{R})$ satisfying
that for all $(t,x)\in[0,T]\times\mathbb{R}$

$\displaystyle C_{1}\braket{x}^{-\alpha}\leq u(t,x)\leq C_{2},$ (1.18)
$\displaystyle|(u_{t}\pm u^{a}u_{x})(t,x)|\leq C_{3}\braket{x}^{-\beta},$
(1.19) $\displaystyle|\partial_{t}\left((u_{t}\pm
u^{a}u_{x})(t,x)\right)|+|\partial_{t}\left((u_{t}\pm
u^{a}u_{x})(t,x)\right)|\leq C_{4}\braket{x}^{-\gamma},$ (1.20)

where $C_{1},C_{2},C_{3},C_{4}$ are positive constants.

Theorem 1.1 asserts the local existence and uniqueness of solutions of (1.4)
under the Levi type condition without the regularity loss. Our proof is based
on the method of characteristic and the contraction mapping principle via
weighted $L^{\infty}$ estimates. In contrast to previous results on the
existence for strictly hyperbolic equations, $1/u$ is not bounded . To avoid
this crux, we use the spatial decay of $u_{t}\pm u^{a}u_{x}$. In particularly,
this property helps to show the boundedness of the derivative of
characteristic curves $x_{\pm}(t)$ with initial position (see Lemma 2.2). We
also remark that our approach is applicable to various type of 1D quasilinear
wave equations (e.g. the equation as (1.9) under suitable condition on
$c(u,x)$).

######  Remark 1.2.

Suppose that initial data is
$(u_{0}(x),u_{1}(x))=(\braket{x}^{-\alpha_{1}},\braket{x}^{-\alpha_{2}})$ with
$\alpha_{1},\alpha_{2}\geq 0$. If $\alpha_{2}\geq\alpha_{1}$ and
$a\alpha_{1}\leq\alpha_{2}$ are satisfied, then all assumptions (1.11)-(1.15)
on initial data are satisfied. We also remark that the condition (1.15) is not
necessary in the case that $a\leq 1$, since (1.14) implies (1.15). While if
$a\geq 1$, (1.14) is not necessary.

###  1.3 Notation and plan of the paper

For a domain $\Omega\subset\mathbb{R}^{n}$, we define $C^{m}_{b}(\Omega)$ with
$m\in\mathbb{N}$ as follows

$\displaystyle C^{m}_{b}(\Omega)=\\{f\in C^{m}(\Omega)\ |\ \sum_{|\alpha|\leq
m}\sup_{x\in\Omega}|\partial_{x}^{\alpha}f(x)|<\infty\\}.$

We write $C_{b}(\Omega)=C^{0}_{b}(\Omega)$ and denote the Lebesgue space for
$1\leq p\leq\infty$ on $\mathbb{R}^{n}$ by $L^{p}$ with the norm
$\|\cdot\|_{L^{p}}$. For a Banach space $X$, $1\leq p\leq\infty$ and $T>0$, we
denote the set of all $X$-valued $L^{p}$ functions with $t\in[0,T]$ by
$L^{p}([0,T];X)$. For convenience, we denote $L^{p}([0,T];X)$ by $L^{p}_{T}X$.
The norm of $L^{p}_{T}X$ is denoted by $\|f\|_{L^{p}_{T}X}$. Various constants
are simply denoted by $C$ or $C_{j}$ for $j\in\mathbb{N}$. We denote that
$\braket{x}=(1+x^{2})^{1/2}$.

The remainder of the present paper is organized as follows. In Section 2, we
review several formulas for the unknown valuable $R=u_{t}+u^{a}u_{x}$ and
$S=u_{t}-u^{a}u_{x}$, which are called Riemann invariant in the study of the
1D hyperbolic conservation law, and give some estimates for characteristic
curves. In Section 3, we show Theorem 1.1 by using the method of
characteristic, weighted $L^{\infty}$ estimates and the contraction mapping
principle. Concluding remarks are given in Section 5.

##  2 Preliminaries

###  2.1 Basic formulation for unknown variables $R$ and $S$

We set $R(t,x)$ and $S(t,x)$ as follows

$\displaystyle\left\\{\begin{array}[]{ll}R=\partial_{t}u+u^{a}\partial_{x}u,\\\
S=\partial_{t}u-u^{a}\partial_{x}u.\end{array}\right.$ (2.3)

By (1.4), $R$ and $S$ are solutions to the system of the following first order
equations:

$\displaystyle\left\\{\begin{array}[]{ll}\partial_{t}R-u^{a}\partial_{x}R=N_{1}(u,R,S)+L(u,R,S),\\\
\partial_{x}u=\dfrac{1}{2u^{a}}(R-S),\\\
\partial_{t}S+u^{a}\partial_{x}S=N_{2}(u,R,S)+L(u,R,S),\end{array}\right.$
(2.7)

where we set

$L(u,R,S)=\dfrac{F(u)(R-S)}{2u^{a}},$ $N_{1}(u,R,S)=\dfrac{a}{2u}(R^{2}-RS)$

and

$N_{2}(u,R,S)=\dfrac{a}{2u}(S^{2}-RS).$

Let $x_{\pm}(t)$ be characteristic curves on the first and third equations of
(2.7) respectively. That is, $x_{+}(t)$ and $x_{-}(t)$ are solutions to the
following differential equations respectively:

$\displaystyle\dfrac{d}{dt}x_{\pm}(t)=\pm u^{a}(t,x_{\pm}(t)).$ (2.8)

When we emphasize the characteristic curves go through $(s,y)$, we denote
$x_{\pm}(t)$ by $x_{\pm}(t;s,y)$. That is, $x_{\pm}(t;s,y)$ satisfies that

$\displaystyle
x_{\pm}(t;s,y)=y\pm\int_{s}^{t}u^{a}(\tau,x_{\pm}(\tau;s,y))d\tau.$ (2.9)

On the characteristic curves, $R$ and $S$ satisfy that

$\displaystyle\left\\{\begin{array}[]{ll}\dfrac{d}{dt}R(t,x_{-}(t))=N_{1}(u,R,S)(t,x_{-}(t))+L(u,R,S)(t,x_{-}(t)),\\\
\dfrac{d}{dt}S(t,x_{+}(t))=N_{2}(u,R,S)(t,x_{+}(t))+L(u,R,S)(t,x_{+}(t)).\end{array}\right.$
(2.12)

###  2.2 Some estimates of characteristic curves

We prepare some estimates for characteristic curves for $u\in
C^{1}([0,T]\times\mathbb{R})$ satisfying for $\alpha\geq 0$

$\displaystyle\braket{x}^{-\alpha}A_{0}\leq u(t,x)\leq A_{1},$ (2.13)

where $A_{0}$ and $A_{1}$ are positive constants. In addition, we assume that

$\displaystyle 0\leq u^{a}|u_{x}(t,x)|\leq A_{2}\braket{x}^{-\alpha}$ (2.14)

for a constant $B_{1}$. The boundedness of $u$ and (2.9) implies the following
estimate with $s,t\in[0,T]$:

$\displaystyle x-A_{1}^{a}|t-s|\leq x_{\pm}(s;t,x)\leq x+A_{1}^{a}|t-s|.$
(2.15)

Next we show a lemma ensures a uniform Lipschitz continuity of
$x_{\pm}(t;s,y)$. This lemma helps to show that a sequence of characteristic
curves satisfies an assumption of the Arzelá-Ascoli theorem.

######  Lemma 2.1.

Let $u\in C^{1}([0,T]\times\mathbb{R})$. Suppose that (2.13) and (2.14) hold.
Then the characteristic curves fulfill that for $x_{1},x_{2}\in\mathbb{R}$ and
$t_{1},t_{2},t_{3},t_{4}\in[0,T]$

$\displaystyle|x_{\pm}(t_{3};t_{1},x_{1})-x_{\pm}(t_{4};t_{2},x_{2})|\leq
3(1+A_{1}^{a})(|x_{1}-x_{2}|+|t_{1}-t_{2}|+|t_{3}-t_{4}|),$ (2.16)

if $T>0$ is sufficiently small.

###### Proof.

First we show the case that $t_{3}=t_{4}=t\in[0,T]$ and $t\geq t_{1},t_{2}$.
From (2.9), we can easily compute that

$\displaystyle|x_{\pm}(t;t_{1},x_{1})-x_{\pm}(t;t_{2},x_{2})|\leq$
$\displaystyle|x_{1}-x_{2}|$ $\displaystyle+$
$\displaystyle\left|\int_{t_{1}}^{t}u^{a}(\tau,x_{\pm}(\tau;t_{1},x_{1}))d\tau-\int_{t_{2}}^{t}u^{a}(\tau,x_{\pm}(\tau;t_{1},x_{2}))d\tau\right|$
$\displaystyle\leq$
$\displaystyle|x_{1}-x_{2}|+\left|\int^{t_{1}}_{t_{2}}u^{a}(\tau,x_{\pm}(\tau;t_{2},x_{2}))d\tau\right|$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\left|u^{a}(\tau,x_{\pm}(\tau;t_{1},x_{1}))-u^{a}(\tau,x_{\pm}(\tau;t_{2},x_{2}))\right|d\tau.$
(2.17)

From (2.13) and (2.14), we have that $u^{a-1}u_{x}$ is bounded, from which, we
have for the third term of the right hand side in (2.17) that

$\displaystyle\left|u^{a}(\tau,x_{\pm}(\tau;t_{1},x_{1}))-u^{a}(\tau,x_{\pm}(\tau;t_{2},x_{2}))\right|\leq$
$\displaystyle\left|\int_{x_{\pm}(\tau;t_{2},x_{2})}^{x_{\pm}(\tau;t_{1},x_{1})}u^{a-1}u_{x}(\tau,y)dy\right|$
$\displaystyle\leq$ $\displaystyle
C\left|x_{\pm}(\tau;t_{1},x_{1})-x_{\pm}(\tau;t_{2},x_{2})\right|.$

Hence we have

$\displaystyle|x_{\pm}(t;t_{1},x_{1})-x_{\pm}(t;t_{2},x_{2})|\leq$
$\displaystyle|x_{1}-x_{2}|+A_{1}^{\alpha}|t_{1}-t_{2}|$ $\displaystyle+$
$\displaystyle
C\int_{0}^{t}\left|x_{\pm}(\tau;t_{1},x_{1})-x_{\pm}(\tau;t_{2},x_{2})\right|d\tau.$
$\displaystyle\leq$ $\displaystyle(1+A_{1}^{a})(|x_{1}-x_{2}|+|t_{1}-t_{2}|)$
$\displaystyle+$ $\displaystyle
C\int_{0}^{t}\left|x_{\pm}(\tau;t_{1},x_{1})-x_{\pm}(\tau;t_{2},x_{2})\right|d\tau.$

Thus we have (2.16) from the Grownwall inequality

$\displaystyle|x_{\pm}(t;t_{1},x_{1})-x_{\pm}(t;t_{2},x_{2})|\leq(1+A_{1}^{a})(|x_{1}-x_{2}|+|t_{1}-t_{2}|)e^{Ct}.$

Hence if $T$ is small, then we have that with $t_{1}\geq t_{2}$

$\displaystyle|x_{\pm}(t;t_{1},x_{1})-x_{\pm}(t;t_{2},x_{2})|\leq
2(1+A_{1}^{a})(|x_{1}-x_{2}|+|t_{1}-t_{2}|).$ (2.18)

In the same way as above, we can show (2.18) with the case that $t<t_{1}$ or
$t<t_{2}$. We omit the proof of this case. Next we show (2.16). The left hand
side of (2.16) is written by

$\displaystyle|x_{\pm}(t_{3};t_{1},x_{1})-x_{\pm}(t_{4};t_{2},x_{2})|\leq$
$\displaystyle|x_{\pm}(t_{3};t_{1},x_{1})-x_{\pm}(t_{3};t_{2},x_{2})|$
$\displaystyle+|x_{\pm}(t_{3};t_{1},x_{1})-x_{\pm}(t_{4};t_{1},x_{x})|$

From (2.18), the first term of the right hand side is estimated by
$2(1+A_{1}^{a})(|x_{1}-x_{2}|+|t_{1}-t_{2}|)$. From (2.9) and (2.13), the
second term is estimated by $A_{1}^{a}|t_{3}-t_{4}|$. Therefore, we have the
desired inequality. ∎

Following lemma is used to show the boundedness of the derivatives of $R$ and
$S$.

######  Lemma 2.2.

Let $u\in C^{1}([0,T]\times\mathbb{R})$. Suppose that (2.13) and (2.14) hold.
Then the characteristic curves $x_{\pm}(t;s,x)$ are differentiable with $x$
and $\partial_{x}x_{\pm}(t;s,x)$ satisfies that with
$(t,x)\in[0,T]\times\mathbb{R}$ and $s\in[0,T]$ for small $T>0$

$\displaystyle\left\\{\begin{array}[]{ll}\dfrac{d}{ds}\partial_{x}x_{\pm}(s;t,x)=\pm
au^{a-1}u_{x}(t,x_{\pm}(s;t,x))\partial_{x}x_{\pm}(s;t,x),\\\
\partial_{x}x_{\pm}(t;t,x)=1\end{array}\right.$ (2.21)

and

$\displaystyle|\partial_{x}x_{\pm}(s;t,x)|\leq e^{C|t-s|},$ (2.22)

where the positive constant $C$ is depending on $A_{0},A_{1}$ and $A_{2}$.

###### Proof.

The differentiability of $x_{\pm}(s;t,x)$ and (2.21) are well-known as a basic
fact(e.g. textbook of Sideris [19]). We estimate $\partial_{x}x_{-}(s;t,x)$.
We only show (2.22) with the case that $t\geq s$. From (2.13) and the
boundedness of $\braket{x}^{\alpha}u^{a}u_{x}$, we obtain that

$\displaystyle|\partial_{x}x_{\pm}(s;t,x)|\leq$ $\displaystyle
1+a\int_{s}^{t}|u^{a-1}u_{x}||\partial_{x}x_{\pm}(\tau;t,x)|d\tau$
$\displaystyle\leq$ $\displaystyle
1+C\int_{s}^{t}|\partial_{x}x_{\pm}(\tau;t,x)|d\tau.$

Hence, from the Gronwall inequality, we obtain (2.22) for small $T$. ∎

##  3 Proof of the main theorem

As in Introduction, we set

$\displaystyle\gamma=\left\\{\begin{array}[]{ll}0,&a\geq 1,\\\
(1-a)\alpha,&\mbox{otherwise}.\end{array}\right.$

We treat functions satisfying the following conditions for $\alpha,\beta\geq
0$ with $\alpha\leq\beta$ such that

$\displaystyle A_{0}\braket{x}^{-\alpha}\leq f(t,x)\leq A_{1}$ (3.2)

and

$\displaystyle f^{a}(t,x)|f_{x}(t,x)|\leq A_{2}\braket{x}^{-\beta},$ (3.3)
$\displaystyle|f_{t}(t,x)|\leq A_{3}\braket{x}^{-\beta}$ (3.4)

or

$\displaystyle|f(t,x)|\leq A_{3}\braket{x}^{-\beta},$ (3.5)

and

$\displaystyle|f_{x}(t,x)|\leq A_{4}\braket{x}^{-\gamma},$ (3.6)
$\displaystyle|f_{t}(t,x)|\leq A_{5}\braket{x}^{-\gamma},$ (3.7)

where $A_{j}$ are positive constants with $j=1,\cdots,5$. We define sets of
$C^{1}$ functions $X_{\alpha}$, $Y_{\beta,1}$, $Y_{\beta,2}$ as follows:

$\displaystyle X_{\alpha}=\\{f\in C^{1}\cap C_{b}\ |\ f(0,x)=u_{0}(x)\
\mbox{and \ \eqref{c1}, \eqref{c1-2} and \eqref{c1-3} hold.}\\},$
$\displaystyle Y_{\beta,1}=\\{f\in C^{1}_{b}\ |\ f(0,x)=R_{0}(x)\ \mbox{and \
\eqref{c2}, \eqref{c3} and \eqref{c4} hold.}\\},$ $\displaystyle
Y_{\beta,2}=\\{f\in C^{1}_{b}\ |\ f(0,x)=S_{0}(x)\ \mbox{and \ \eqref{c2},
\eqref{c3} and \eqref{c4} hold.}\\},$

where given functions $(u_{0},R_{0},S_{0})$ belongs to $C^{1}\cap C_{b}\times
C^{1}_{b}\times C^{1}_{b}$. For given functions $(v,\bar{R},\bar{S})\in
X_{\alpha}\times Y_{\beta,1}\times Y_{\beta,2}$, we consider the first order
linear hyperbolic equation:

$\displaystyle\left\\{\begin{array}[]{ll}R_{t}-v^{a}R_{x}=N_{1}(v,\bar{R},\bar{S})+L(v,\bar{R},\bar{S}),\\\
S_{t}+v^{a}S_{x}=N_{2}(v,\bar{R},\bar{S})+L(v,\bar{R},\bar{S})\end{array}\right.$
(3.10)

with initial condition $(R(0,x),S(0,x))=(R_{0},S_{0})\in C^{1}_{b}\times
C^{1}_{b}$. We set

$\displaystyle u=u_{0}(x)+\int_{0}^{t}\frac{R+S}{2}(s,x)ds.$ (3.11)

We find that (3.10) with $C^{1}$ initial data has unique and time-global
solutions such that $R,S\in C^{1}([0,T]\times\mathbb{R})\cap
C_{b}([0,T]\times\mathbb{R})$ with arbitrary fixed $T>0$ from the method of
characteristic. From (3.11), it holds that $u\in C^{1}\cap C_{b}$. Namely we
can define the map

$\Phi:X_{\alpha}\times Y_{\beta,1}\times Y_{\beta,2}\rightarrow C^{1}\times
C^{1}\times C^{1}$

such that $\Phi(v,\bar{R},\bar{S})=(u,R,S)$. We take four positive numbers
$A_{0},A_{1},A_{3},A_{4}$ satisfying that

$\displaystyle 2A_{0}\braket{x}^{-\alpha}\leq u_{0}(x)\leq\frac{A_{1}}{2},$
(3.12)
$\displaystyle\|\braket{x}^{\beta}R_{0}\|_{L^{\infty}}+\|\braket{x}^{\beta}S_{0}\|_{L^{\infty}}\leq\frac{A_{3}}{4},$
(3.13)
$\displaystyle\|\braket{x}^{\gamma}R^{\prime}_{0}\|_{L^{\infty}}+\|\braket{x}^{\gamma}S^{\prime}_{0}\|_{L^{\infty}}\leq\frac{A_{4}}{8}.$
(3.14)

The constants $A_{2}$ and $A_{5}$ in (3.7) will be taken later. Moreover, we
assume that

$\displaystyle\|\braket{x}^{\beta}u_{0}^{a}u^{\prime}_{0}\|_{L^{\infty}}\leq
B_{1}$ (3.15)

for a positive constant $B_{1}$. In the following, we show that $(u,R,S)\in
X_{\alpha}\times Y_{\beta,1}\times Y_{\beta,2}$ and $\Phi$ is a contraction
mapping in the topology of $L^{\infty}$ for sufficient small $T$. $X_{\alpha}$
and $Y_{\beta,j}$ with $j=1,2$ are not closed set of $L^{\infty}$ space.
Nevertheless it is possible to show that the fixed point belongs to
$X_{\alpha}\times Y_{\beta,1}\times Y_{\beta,2}$. Furthermore we will show
that the regularity is improved as $u\in C^{2}$. First we show the following
proposition.

######  Proposition 3.1.

Let $(u_{0},R_{0},S_{0})\in\ C^{1}\times C^{1}\times C^{1}$ satisfying
(3.12)-(3.14) and (3.15). Suppose that $v,\bar{R},\bar{S}\in X_{\alpha}\times
Y_{\beta,1}\times Y_{\beta,2}$. Then $\Phi(v,\bar{R},\bar{S})=(u,R,S)\in
X_{\alpha}\times Y_{\beta,1}\times Y_{\beta,2}$ for sufficiently small $T>0$.

###### Proof.

From the method of characteristic, we can see that the solution of (3.10) can
be written by

$\displaystyle\left\\{\begin{array}[]{ll}R(t,x)=R(0,x_{-}(0))+\int_{0}^{t}N_{1}(v,\bar{R},\bar{S})(s,x_{-}(s))+L(v,\bar{R},\bar{S})(s,x_{-}(s))ds,\\\
S(t,x)=S(0,x_{+}(0))+\int_{0}^{t}N_{2}(v,\bar{R},\bar{S})(s,x_{+}(s))+L(v,\bar{R},\bar{S})(s,x_{+}(s))ds,\end{array}\right.$
(3.18)

where the characteristic curves for the linear equation (3.10) are defined as
follows:

$\displaystyle\dfrac{d}{dt}x_{\pm}(t)=\pm v^{a}(t,x_{\pm}(t))$

with initial data $x_{\pm}(t)=x$. From this expression, we have that
$(u,R,S)\in C^{1}\times C^{1}\times C^{1}$. Now we estimate
$\|\braket{x}^{\beta}R\|_{L^{\infty}_{T}L^{\infty}}$. From (2.15), if $T$ is
small, we have

$\displaystyle\frac{\braket{x}^{\beta}}{2}\leq\braket{x_{-}(s;t,x)}^{\beta}\leq
2\braket{x}^{\beta}.$

Using these inequalities, from (3.18), we have that if $T$ is small

$\displaystyle|\braket{x}^{\beta}R(t,x)|\leq$ $\displaystyle
2\|\braket{x}^{\beta}R(0,\cdot)\|_{L^{\infty}}$
$\displaystyle+\int_{0}^{t}\dfrac{\braket{x_{-}(s)}^{\beta}}{v}|\bar{R}^{2}-\bar{R}\bar{S}|+\braket{x_{-}(s)}^{\beta}\frac{F(v)}{v^{a}}|\bar{R}-\bar{S}|ds$
$\displaystyle\leq$
$\displaystyle\frac{A_{3}}{2}+\int_{0}^{t}\dfrac{\braket{x_{-}(s)}^{\alpha+\beta}}{\braket{x_{-}(s)}^{\alpha}v}|\bar{R}^{2}-\bar{R}\bar{S}|+C\braket{x_{-}(s)}^{\beta}|\bar{R}-\bar{S}|ds$
$\displaystyle\leq$
$\displaystyle\frac{A_{3}}{2}+CT\left(\|\braket{x}^{\beta}\bar{R}\|_{L^{\infty}_{T}L^{\infty}}\|\braket{x}^{\beta}\bar{S}(s)\|_{L^{\infty}_{T}L^{\infty}}+\|\braket{x}^{\beta}\bar{S}\|^{2}_{L^{\infty}_{T}L^{\infty}}\right)$
$\displaystyle+CT(\|\braket{x}^{\beta}\bar{R}\|_{L^{\infty}_{T}L^{\infty}}+\|\braket{x}^{\beta}\bar{S}\|_{L^{\infty}_{T}L^{\infty}})$
$\displaystyle\leq\frac{A_{3}}{2}+CT,$

where $C$ is a positive constant depending on $A_{0},A_{1},A_{3}$ and the
assumptions that $\beta\geq\alpha$ and (1.16) are used. Hence we obtain that
for sufficiently small $T$

$\displaystyle\|\braket{x}^{\beta}R\|_{L^{\infty}_{T}L^{\infty}}\leq A_{3}.$
(3.19)

Similarly for $S$, we have that

$\displaystyle\|\braket{x}^{\beta}S\|_{L^{\infty}_{T}L^{\infty}}\leq A_{3}.$
(3.20)

Next we estimate $\|\braket{x}^{\gamma}R_{x}\|_{L^{\infty}_{T}L^{\infty}}$ and
$\|\braket{x}^{\gamma}S_{x}\|_{L^{\infty}_{T}L^{\infty}}$. Differentiating the
both side of the equations (3.10) with $x$, we can obtain integral equations
for $R_{x}$ and $S_{x}$ as follows:

$\displaystyle V(t,x)$
$\displaystyle=V_{0}(x_{-}(0;t,x))\partial_{x}x_{-}(0;t,x)$ $\displaystyle+$
$\displaystyle\int_{0}^{t}\partial_{x}x_{-}(s;t,x)\left(N_{1u}v_{x}+N_{1R}\bar{V}+N_{1S}\bar{W}\right)(t,x_{-}(s;t,x))ds$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\partial_{x}x_{-}(s;t,x)\left(L_{u}v_{x}+L_{R}\bar{V}+L_{S}\bar{W}\right)(t,x_{-}(s;t,x))ds$
(3.21)

and

$\displaystyle W(t,x)$
$\displaystyle=W_{0}(x_{-}(0;t,x))\partial_{x}x_{+}(0;t,x)$ $\displaystyle+$
$\displaystyle\int_{0}^{t}\partial_{x}x_{+}(s;t,x)\left(N_{2u}v_{x}+N_{2R}W+N_{2S}V\right)(t,x_{+}(s;t,x))ds$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\partial_{x}x_{+}(s;t,x)\left(L_{u}v_{x}+L_{R}\bar{V}+L_{S}\bar{W}\right)(t,x_{+}(s;t,x))ds,$
(3.22)

where we denote $\bar{V}=\bar{R}_{x}$ and $\bar{W}=\bar{S}_{x}$ and
$(V_{0},W_{0})=(R^{\prime}_{0}(\cdot),S^{\prime}_{0}(\cdot))$ and
$N_{ju},N_{jS},N_{jR}$ ($j=1,2$) are partial derivatives of
$N_{j}=N_{j}(u,R,S)$ with $u,S,R$ respectively (the same manners are also used
for $L$). From Lemma 2.2, we obtain that $|\partial_{x}x_{+}(0;t,x)|$ is
bounded by $2$, if $T$ is small (note that smallness of $T$ depends on
$A_{0},A_{1}$ and $A_{2}$). Hence we have that

$\displaystyle|W(t,x)|\leq$
$\displaystyle\frac{A_{4}\braket{x}^{-\gamma}}{2}+2\int_{0}^{t}\left|N_{1u}v_{x}+N_{1R}\bar{V}+N_{1S}\bar{W}\right|(t,x_{-}(s;t,x))ds$
$\displaystyle+2\int_{0}^{t}|L_{u}v_{x}+L_{R}\bar{V}+L_{S}\bar{W}|(t,x_{-}(s;t,x))ds.$
(3.23)

From (3.12)-(3.14), $|N_{1R}\bar{V}|$ and $|N_{1S}\bar{W}|$ are trivially
estimated as

$\displaystyle|N_{1R}\bar{V}|+|N_{1S}\bar{W}|\leq$
$\displaystyle\frac{C}{v}(|\bar{R}|+|\bar{S}|)(|\bar{V}|+|\bar{W}|)$
$\displaystyle\leq$ $\displaystyle C\braket{x}^{\alpha-\beta-\gamma}$
$\displaystyle\leq$ $\displaystyle C\braket{x}^{-\gamma}.$

From (3.12)-(3.14) and (1.14) and (1.15), we have that for $|N_{1u}v_{x}|$

$\displaystyle|N_{1u}v_{x}|\leq$
$\displaystyle\frac{Cv^{a}v_{x}}{u^{2+a}}(|\bar{R}|^{2}+|\bar{S}|^{2})$
$\displaystyle\leq$
$\displaystyle\frac{C}{u^{2+a}}(|\bar{R}|^{3}+|\bar{S}|^{3})$
$\displaystyle\leq$ $\displaystyle C\braket{x}^{(2+a)\alpha-3\beta}$
$\displaystyle\leq$ $\displaystyle C\braket{x}^{-\gamma}.$

From (1.14), (1.15), (1.16) and (1.17), we estimate $L_{u}v_{x}$ as

$\displaystyle|L_{u}v_{x}|\leq$ $\displaystyle
C\left(\frac{|F(v)v_{x}|}{v^{a+1}}+\frac{|F^{\prime}(v)v_{x}|}{v^{a}}\right)$
$\displaystyle\leq$ $\displaystyle\frac{C(|R|+|S|)}{v^{a}}$
$\displaystyle\leq$ $\displaystyle C\braket{x}^{\alpha a-\beta}$
$\displaystyle\leq$ $\displaystyle C\braket{x}^{-\gamma}.$

(1.16) and (3.14) directly implies that

$\displaystyle|L_{R}\bar{V}|+|L_{S}\bar{W}|\leq C\braket{x}^{-\gamma}.$

Applying these estimates to (3.23), we obtain that with small $T$

$\displaystyle\|\braket{x}^{\gamma}W\|_{L^{\infty}_{T}L^{\infty}}\leq\frac{A_{4}}{2}.$
(3.24)

Similarly

$\displaystyle\|\braket{x}^{\gamma}V\|_{L^{\infty}_{T}L^{\infty}}\leq\frac{A_{4}}{2}.$
(3.25)

Estimates of $R_{t}$ and $S_{t}$ are obtained from (3.10). In fact, we have
that

$\displaystyle|R_{t}(t,x)|+|S_{t}(t,x)|\leq$
$\displaystyle|v^{a}R_{x}|+|v^{a}S_{x}|+|N_{1}(v\bar{R},\bar{S})|$
$\displaystyle+|N_{2}(v\bar{R},\bar{S})|+2|L(v\bar{R},\bar{S})|$
$\displaystyle\leq$ $\displaystyle C\braket{x}^{-\gamma}+C\braket{x}^{-\beta}$
$\displaystyle\leq$ $\displaystyle C_{A}\braket{x}^{-\gamma},$ (3.26)

where $C_{A}$ is a positive constant depending on $A_{1}$, $A_{3}$ and $A_{4}$
(independent of $A_{5}$). Here we choose $A_{5}$ as $A_{5}=C_{A}$. From the
above estimates, we have that $(R,S)\in Y_{\beta,1}\times Y_{\beta,2}$. Next
we show that $u\in X_{\alpha}.$ From (3.11), (3.19) and (3.20), it follows for
sufficiently small $T$ that

$\displaystyle\braket{x}^{\alpha}u(t)\geq$
$\displaystyle\braket{x}^{\alpha}u_{0}(x)-\int_{0}^{t}\dfrac{\braket{x}^{\alpha}(|R|+|S|)}{2}ds$
$\displaystyle\geq$ $\displaystyle
2A_{0}-\int_{0}^{t}\dfrac{\braket{x}^{\beta}(|R|+|S|)}{2}ds$
$\displaystyle\geq$ $\displaystyle 2A_{0}-TA_{3}$ $\displaystyle\geq$
$\displaystyle A_{0}.$ (3.27)

Similarly we can easily check that $\|u\|_{L^{\infty}_{T}L^{\infty}}\leq
A_{1}$, if $T$ is small. Next we show that

$\displaystyle\|\braket{x}^{\beta}u^{a}u_{x}\|_{L^{\infty}_{T}L^{\infty}}\leq
A_{2},$ (3.28)

(3.11) directly implies that

$\displaystyle u^{a}u_{x}=$
$\displaystyle\left(u_{0}+\int_{0}^{t}\frac{R+S}{2}ds\right)^{a}u^{\prime}_{0}$
$\displaystyle+$
$\displaystyle\left(u_{0}+\int_{0}^{t}\frac{R+S}{2}ds\right)^{a}\int_{0}^{t}\frac{R_{x}+S_{x}}{2}ds.$
(3.29)

From (3.12) and the boundedness of
$\braket{x}^{\beta}u^{a}_{0}|u^{\prime}_{0}|$ the first term of (3.29) is
estimated as

$\displaystyle\left(u_{0}+\int_{0}^{t}\frac{R+S}{2}ds\right)^{a}|u^{\prime}_{0}|\leq$
$\displaystyle 2^{a}(u^{a}_{0}+CT^{a}\braket{x}^{-a\beta})|u^{\prime}_{0}|$
$\displaystyle\leq$ $\displaystyle(2^{2}+CT^{a})u^{a}_{0}|u^{\prime}_{0}|$
$\displaystyle\leq$ $\displaystyle C_{1,A}\braket{x}^{-\beta},$

where we note that the positive constant $C_{1,A}$ does not depend on $A_{2}$.
Deducing $R_{x}+S_{x}$ via (3.10), we also obtain

$\displaystyle\left(u_{0}+\int_{0}^{t}\frac{R+S}{2}ds\right)^{a}\left|\int_{0}^{t}\frac{R_{x}+S_{x}}{2}ds\right|$
$\displaystyle=\left(u_{0}+\int_{0}^{t}\frac{R+S}{2}ds\right)^{a}\left|\int_{0}^{t}\frac{1}{2v^{a}}(R_{t}-S_{t}+N_{2}-N_{1})ds\right|$
$\displaystyle\leq
2^{a}(u^{a}_{0}+T^{a}\braket{x}^{-a\beta})\left|\int_{0}^{t}\frac{1}{2v^{a}}(R_{t}-S_{t}+N_{2}-N_{1})ds\right|.$

Since the spatial decay of $R_{t}$ and $S_{t}$ is not enough to show (3.28),
we need to change this term. From the integration by parts and the property of
$X_{\alpha}$ that $v_{0}=u_{0}$, we obtain that

$\displaystyle\int_{0}^{t}\frac{R_{t}-S_{t}}{2v^{a}}ds=\frac{R-S}{2v^{a}}-\frac{R_{0}-S_{0}}{2u^{a}_{0}}+\int_{0}^{t}\frac{a(R-S)v_{t}}{2v^{a+1}}ds.$
(3.30)

While, from (3.4), we have that

$\displaystyle u_{0}-A_{3}T\braket{x}^{-\beta}\leq v\leq
u_{0}+A_{3}T\braket{x}^{-\beta}.$

Hence, with the help of (3.2), it holds that

$\displaystyle\frac{1}{2}\leq 1-A_{3}A_{0}T\leq\frac{v}{u_{0}}\leq
1+A_{3}A_{0}T\leq 2$ (3.31)

for small $T$. The third term of the right hand side in (3.30) is estimated as

$\displaystyle\int_{0}^{t}\frac{a(R-S)|v_{t}|}{2v^{a+1}}ds\leq$ $\displaystyle
C\int_{0}^{t}\frac{\braket{x}^{\alpha}(R-S)|v_{t}|}{v^{a}}ds$
$\displaystyle\leq C$ $\displaystyle\braket{x}^{-\beta}\int_{0}^{t}v^{-a}ds.$
(3.32)

Applying (3.31) and (3.32) to (3.30), we have

$\displaystyle
2^{a}(u^{a}_{0}+T^{a}\braket{x}^{-a\beta})\left|\int_{0}^{t}\frac{1}{2v^{a}}(R_{t}-S_{t})ds\right|\leq
C_{2,A}\braket{x}^{-\beta}$

and similarly

$\displaystyle
2^{a}(u^{a}_{0}+T^{a}\braket{x}^{-a\beta})\left|\int_{0}^{t}\frac{1}{2v^{a}}(N_{2}-N_{1})ds\right|\leq
C_{3,A}\braket{x}^{-\beta},$

where positive constants $C_{2,A}$ and $C_{3,A}$ are independent of $A_{2}$.

$\displaystyle\|\braket{x}^{-\beta}u^{a}u_{x}\|_{L^{\infty}_{T}L^{\infty}}\leq
C_{1,A}+C_{2,A}+C_{3,A}.$

Taking $A_{5}=C_{1,A}+C_{2,A}+C_{3,A}$, we obtain (3.28). (3.11) and (3.5)
directly yield that

$\displaystyle\|\braket{x}^{-\beta}u_{t}\|_{L^{\infty}_{T}L^{\infty}}\leq$
$\displaystyle\left\|\frac{\braket{x}^{-\beta}(R+S)}{2}\right\|_{L^{\infty}_{T}L^{\infty}}$
$\displaystyle\leq$ $\displaystyle A_{3}.$

Therefore we have that $(u,R,S)\in X_{\alpha}\times Y_{\beta,1}\times
Y_{\beta,2}$. In the end of the proof, we show that $(u,R,S)$ is Lipschitz
continuous. From (3.24), (3.25) and (3.26), we can obviously check that $R$
and $S$ satisfies the following uniform Lipschitz estimate:

$\displaystyle|R(t,x)-R(s,y)|+|S(t,x)-S(s,y)|\leq
2(A_{4}+A_{5})(|t-s|+|x-y|).$ (3.33)

Next we check that $u$ is Lipschitz continuous. From (3.28) and the condition
that $\beta\geq a\alpha$, we have that

$\displaystyle|u(t,x)-u(t,y)|\leq$
$\displaystyle\left|\int^{x}_{y}|u_{x}(t,z)|dz\right|$ $\displaystyle\leq$
$\displaystyle
C\left|\int^{x}_{y}\braket{z}^{a\alpha}u^{a}|u_{x}|(t,z)dz\right|$
$\displaystyle\leq$ $\displaystyle C|x-y|.$

Combining this estimate with the boundedness of $R$ and $S$, we have from
(3.11) that $u$ is Lipschitz continuous such that for any
$t_{1},t_{2}\in[0,T]$, $x_{1},x_{2}\in\mathbb{R}$ with

$\displaystyle|u(t_{1},x_{1})-u(t_{2},x_{2})|\leq
C(|t_{1}-t_{2}|+|x_{1}-x_{2}|),$ (3.34)

where C is a positive constant depending on $A_{0},A_{3},A_{4}$ and $A_{5}$.
These additional properties are used to show that the fixed point of $\Phi$
satisfies integral equations. ∎

######  Proposition 3.2.

Under the same assumptions on (3.1), $\Phi$ is a contraction mapping in the
topology of $L^{\infty}$-norm with small $T>0$. Namely, If $T>0$ is small,
then there exists a constant $c\in(0,1)$ such that $\Phi$ satisfies that

$\displaystyle\|u_{1}-u_{2}\|_{L^{\infty}_{T}L^{\infty}}+\|R_{1}-R_{2}\|_{L^{\infty}_{T}L^{\infty}}+\|S_{1}-S_{2}\|_{L^{\infty}_{T}L^{\infty}}$
$\displaystyle\leq
c\left(\|v_{1}-v_{2}\|_{L^{\infty}_{T}L^{\infty}}+\|\bar{R}_{1}-\bar{R}_{2}\|_{L^{\infty}_{T}L^{\infty}}+\|\bar{S}_{1}-\bar{S}_{2}\|_{L^{\infty}_{T}L^{\infty}}\right),$

where $(u_{j},R_{j},S_{j})=\Phi(v_{j},\bar{R}_{j},\bar{S}_{j})$ with $j=1,2$.

###### Proof.

Put $\tilde{u}=u_{1}-u_{2}$, $\tilde{R}=R_{1}-R_{2}$, $\tilde{S}=S_{1}-S_{2}$.
From (3.10), we have

$\displaystyle\tilde{R}_{t}-v^{a}_{1}\tilde{R}_{x}=$ $\displaystyle
N_{1}(v_{1},\bar{R}_{1},\bar{S}_{1})-N_{1}(v_{2},\bar{R}_{2},\bar{S}_{2})$
$\displaystyle+L(v_{1},\bar{R}_{1},\bar{S}_{1})-L(v_{2},\bar{R}_{2},\bar{S}_{2})$
$\displaystyle+(v^{a}_{1}-v^{a}_{2})R_{2x}.$

From the method of characteristic, we have that

$\displaystyle\tilde{R}(t,x)=$
$\displaystyle\int_{0}^{t}(N_{1}(v_{1},\bar{R}_{1},\bar{S}_{1})-N_{1}(v_{2},\bar{R}_{2},\bar{S}_{2}))ds$
$\displaystyle+\int_{0}^{t}(L(v_{1},\bar{R}_{1},\bar{S}_{1})-L(v_{2},\bar{R}_{2},\bar{S}_{2}))ds$
$\displaystyle+\int_{0}^{t}(v^{a}_{1}-v^{a}_{2})R_{2x}ds.$ (3.35)

The second term of the right hand side in (3.35) can be written as

$\displaystyle\int_{0}^{t}(L(v_{1},\bar{R}_{1},\bar{S}_{1})-L(v_{2},\bar{R}_{2},\bar{S}_{2}))ds=$
$\displaystyle\int_{0}^{t}(G(v_{1})-G(v_{2}))(\bar{R}_{1}-\bar{S}_{1})ds$
$\displaystyle+\int_{0}^{t}G(v_{2})(\bar{R}_{1}-\bar{S}_{1}-\bar{R}_{2}+\bar{S}_{2})ds,$
(3.36)

where we set $G(\theta)=F(\theta)/2\theta^{a}$. Using (1.16), (1.17) and (3.2)
, we obtain that

$\displaystyle|G(v_{1})-G(v_{2})|\leq\int_{0}^{1}|G^{\prime}(\theta
v_{1}+(1-\theta)v_{2})|d\theta|v_{1}-v_{2}|$
$\displaystyle\leq\int_{0}^{1}\frac{Cd\theta}{\theta
v_{1}+(1-\theta)v_{2}}|v_{1}-v_{2}|$ $\displaystyle\leq
C\braket{x}^{\alpha}|v_{1}-v_{2}|,$

which implies that the first term of the right hand side in (3.36) is
estimated as

$\displaystyle\int_{0}^{t}|G(v_{1})-G(v_{2})||\bar{R}_{1}-\bar{S}_{1}|ds\leq
CTA_{3}\|v_{1}-v_{2}\|_{L^{\infty}_{T}L^{\infty}}.$

From (1.16), the second term is estimated by

$\displaystyle\int_{0}^{t}|G(v_{2})||\bar{R}_{1}-\bar{S}_{1}-\bar{R}_{2}+\bar{S}_{2}|ds\leq
CT(\|\bar{R}_{1}-\bar{R}_{2}\|_{L^{\infty}_{T}L^{\infty}}+\|\bar{S}_{1}-\bar{S}_{2}\|_{L^{\infty}_{T}L^{\infty}}).$

Setting $N_{1}(v,\bar{R},\bar{S})=\frac{a}{2v}Q(\bar{R},\bar{S})$ and
$Q(\bar{R},\bar{S})=(\bar{R}^{2}-\bar{R}\bar{S})$, we change the first term of
the right hand side in (3.36) to

$\displaystyle\int_{0}^{t}(N_{1}(v_{1},\bar{R}_{1},\bar{S}_{1})-N_{1}(v_{2},\bar{R}_{2},\bar{S}_{2}))ds=\int_{0}^{t}a\left(\dfrac{v_{2}-v_{1}}{2v_{1}v_{2}}\right)Q(\bar{R}_{1},\bar{S}_{1})ds$
$\displaystyle+\int_{0}^{t}\dfrac{a}{2v_{2}}\left(Q(\bar{R}_{1},\bar{S}_{1})-Q(\bar{R}_{2},\bar{S}_{2})\right)ds.$
(3.37)

The first term of the right hand side in (3.37) is estimated as

$\displaystyle\left|\int_{0}^{t}a\left(\dfrac{v_{2}-v_{1}}{2v_{1}v_{2}}\right)Q(\bar{R}_{1},\bar{S}_{1})ds\right|\leq\int_{0}^{t}a\dfrac{|v_{1}-v_{2}|}{2\braket{x}^{2\alpha}|v_{1}v_{2}|}\braket{x}^{2\alpha}|Q(\bar{R}_{1},\bar{S}_{1})|ds$
$\displaystyle\leq\int_{0}^{t}a\dfrac{2|v_{1}-v_{2}|}{A^{2}_{0}}\braket{x}^{2\beta}|Q(\bar{R}_{1},\bar{S}_{1})|ds\leq
CT\|v_{1}-v_{2}\|_{L^{\infty}_{T}L^{\infty}}.$

The second term can be estimated as

$\displaystyle\left|\int_{0}^{t}\left(\dfrac{a}{2v_{2}}\right)\left(Q(\bar{R}_{1},\bar{S}_{1})-Q(\bar{R}_{2},\bar{S}_{2})\right)ds\right|\leq\int_{0}^{t}\dfrac{a}{2A_{0}}\braket{x}^{\alpha}\left|Q(\bar{R}_{1},\bar{S}_{1})-Q(\bar{R}_{2},\bar{S}_{2})\right|ds$
$\displaystyle\leq\int_{0}^{t}\dfrac{a}{A_{0}}\braket{x}^{\beta}\left|Q(\bar{R}_{1},\bar{S}_{1})-Q(\bar{R}_{2},\bar{S}_{2})\right|ds$
$\displaystyle\leq
CT\left(\|\bar{R}_{1}-\bar{R}_{2}\|_{L^{\infty}_{T}L^{\infty}}+\|\bar{S}_{1}-\bar{S}_{2}\|_{L^{\infty}_{T}L^{\infty}}\right).$

Next we estimate the third term of the right hand side in (3.37). When $a\geq
1$, from the mean-value theorem for $|v^{a}_{1}-v^{a}_{2}|$ and the
boundedness of $R_{x}$, we obtain that

$\displaystyle|(v^{a}_{1}-v^{a}_{2})R_{2x}|\leq
C\|v_{1}-v_{2}\|_{L^{\infty}_{T}L^{\infty}}.$

While, $a<1$, by using the boundedness of $\braket{x}^{\gamma}R_{x}$, we have
that

$\displaystyle|(v^{a}_{1}-v^{a}_{2})R_{2x}|\leq$ $\displaystyle
a\int_{0}^{1}(\theta v_{1}+(1-\theta)v_{2})^{a-1}|v_{1}-v_{2}||R_{2x}|d\theta$
$\displaystyle\leq C\braket{x}^{\gamma}|v_{1}-v_{2}||R_{2x}|$
$\displaystyle\leq C\|v_{1}-v_{2}\|_{L^{\infty}_{T}L^{\infty}}.$

Therefore, we obtain that for sufficiently small $T$

$\displaystyle\|\tilde{R}\|_{L^{\infty}_{T}L^{\infty}}\leq\dfrac{1}{6}\left(\|v_{1}-v_{2}\|_{L^{\infty}_{T}L^{\infty}}+\|\bar{R}_{1}-\bar{R}_{2}\|_{L^{\infty}_{T}L^{\infty}}+\|\bar{S}_{1}-\bar{S}_{2}\|_{L^{\infty}_{T}L^{\infty}}\right).$

In the same way as in the estimate of $\tilde{R}$, we have that

$\displaystyle\|\tilde{S}\|_{L^{\infty}_{T}L^{\infty}}\leq\dfrac{1}{6}\left(\|v_{1}-v_{2}\|_{L^{\infty}_{T}L^{\infty}}+\|\bar{R}_{1}-\bar{R}_{2}\|_{L^{\infty}_{T}L^{\infty}}+\|\bar{S}_{1}-\bar{S}_{2}\|_{L^{\infty}_{T}L^{\infty}}\right).$

From (3.11), the above two estimates on $\tilde{R}$ and $\tilde{S}$ imply that
for sufficiently small $T$

$\displaystyle\|\tilde{u}\|_{L^{\infty}_{T}L^{\infty}}\leq$
$\displaystyle\dfrac{T}{2}\left(\|\tilde{R}\|_{L^{\infty}_{T}L^{\infty}}+\|\tilde{S}\|_{L^{\infty}_{T}L^{\infty}}\right)$
$\displaystyle\leq$
$\displaystyle\dfrac{1}{6}\left(\|v_{1}-v_{2}\|_{L^{\infty}_{T}L^{\infty}}+\|\bar{R}_{1}-\bar{R}_{2}\|_{L^{\infty}_{T}L^{\infty}}+\|\bar{S}_{1}-\bar{S}_{2}\|_{L^{\infty}_{T}L^{\infty}}\right).$

Therefore, we find that $\Phi$ is a contraction mapping for sufficiently small
$T>0$. ∎

Next we construct a unique solution $(u,R,S)$ of the nonlinear problem and the
characteristic curves $x_{\pm}(\cdot;t,x)$.

######  Proposition 3.3.

Under the same assumptions as in Proposition 3.2, if $T$ is small, then there
uniquely exist $(u,R,S)\in X_{\alpha}\times Y_{\beta,1}\times Y_{\beta,2}$ and
$x_{\pm}(s)=x_{\pm}(s;t,x)$ satisfying that

$\displaystyle\left\\{\begin{array}[]{ll}R(t,x)=R(0,x_{-}(0))+\int_{0}^{t}N_{1}(u,R,S)(s,x_{-}(s))+L(u,R,S)(s,x_{-}(s))ds,\\\
S(t,x)=S(0,x_{+}(0))+\int_{0}^{t}N_{2}(v,R,S)(s,x_{+}(s))+L(u,R,S)(s,x_{+}(s))ds,\end{array}\right.$
(3.40) $\displaystyle u(t,x)=u_{0}(x)+\int_{0}^{t}\frac{R+S}{2}ds$ (3.41)

and

$\displaystyle
x_{\pm}(s;t,x)=x\pm\int_{t}^{s}u^{a}(\tau,x_{\pm}(\tau;t,x))d\tau.$ (3.42)

###### Proof.

We fix $K\geq 1$ arbitrarily. From Proposition 3.1, we can define a sequence
$\\{u_{n},R_{n},S_{n}\\}_{n\in\mathbb{N}}$ in $X_{\alpha}\times
Y_{\beta,1}\times Y_{\beta,2}$ such that

$\displaystyle(u_{n+1},R_{n+1},S_{n+1})=\Phi(u_{n},R_{n},S_{n})$

with initial term $(u_{0},S_{0},R_{0})$. By Proposition 3.2,
$(u_{n},R_{n},S_{n})$ converges the fixed point $(u,R,S)$ in the topology of
$L^{\infty}$. While we can define a sequence of the characteristic curves
$\\{x_{\pm,n}(\cdot;t,x)\\}_{n\in\mathbb{N}}$. We note that the characteristic
curves can be defined uniquely on $[0,T]$ with arbitrarily fixed $(t,x)$ by
the Lipschitz continuity and the boundedness of $u_{n}^{a}$. For arbitrarily
fixed $K\geq 1$, we see that $\\{x_{\pm,n}(\cdot;t,x)\\}_{n\in\mathbb{N}}$ is
the uniform equicontinuous and uniform bounded from Lemma 2.1 and (2.15). Thus
the Arzelá-Ascoli theorem implies that there exists a subsequence of
$\\{x_{\pm,n}(\cdot;t,x)\\}_{n\in\mathbb{N}}$ (we use the same suffix as in
the original sequence) such that $x_{\pm,n}(\cdot)$ converges $x_{\pm}(\cdot)$
uniformly on $[0,T]\times[0,T]\times[-K,K]$ as $n\rightarrow\infty$. Note that
this choice of the subsequence is depending on $K$. However, from Cantor’s
diagonal argument, we can reselect a subsequence independently of $K$ such
that the convergence holds on $[0,T]\times[0,T]\times[-K^{\prime},K^{\prime}]$
with any $K^{\prime}\geq 1$. From (3.33) and (3.34), we see that as
$n\rightarrow\infty$

$\displaystyle(u_{n}(t,x_{\pm,n}(t)),R_{n}(t,x_{\pm,n}(t)),S_{n}(t,x_{\pm,n}(t)))\rightarrow(u(t,x_{\pm}(t)),R(t,x_{\pm}(t)),S(t,x_{\pm}(t))).$

Hence (3.40) and (3.42) are satisfied. Now we check that $(u,R,S)\in
X_{\alpha}\times Y_{\beta,1}\times Y_{\beta,2}.$ It is obvious that the
properties (3.2) and (3.5) are satisfied. From the Lipschitz continuity,
$u,R,S$ are differentiable almost everywhere. In the same way as in the proof
of Proposition 3.1, we can obtain the boundedness of
$\braket{x}^{\beta}u^{a}u_{x}$ and $\braket{x}^{\beta}u_{t}$, since the
constant $A_{2}$ is taken independently of $A_{4}$ and $A_{5}$. Thus we have
$u\in X_{\alpha}$. To show the boundedness of $\braket{x}^{\gamma}R_{x}$ and
$\braket{x}^{\gamma}S_{x}$, differentiating the both side of (3.40) with $x$,
we obtain that

$\displaystyle V(t,x)$
$\displaystyle=V_{0}(x_{-}(0;t,x))\partial_{x}x_{-}(0;t,x)$ $\displaystyle+$
$\displaystyle\int_{0}^{t}\partial_{x}x_{-}(s;t,x)\left(N_{1u}u_{x}+N_{1R}V+N_{1S}W\right)(t,x_{-}(s;t,x))ds$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\partial_{x}x_{-}(s;t,x)\left(L_{u}u_{x}+L_{R}V+L_{S}W\right)(t,x_{-}(s;t,x))ds$
(3.43)

and

$\displaystyle W(t,x)$
$\displaystyle=W_{0}(x_{-}(0;t,x))\partial_{x}x_{+}(0;t,x)$ $\displaystyle+$
$\displaystyle\int_{0}^{t}\partial_{x}x_{+}(s;t,x)\left(N_{2u}u_{x}+N_{2R}W+N_{2S}V\right)(t,x_{+}(s;t,x))ds$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\partial_{x}x_{+}(s;t,x)\left(L_{u}u_{x}+L_{R}V+L_{S}W\right)(t,x_{+}(s;t,x))ds.$
(3.44)

In the same way as in the proof of (3.24) and (3.25), we achieve the
boundedness of $\braket{x}^{\gamma}W$ and $\braket{x}^{\gamma}V$. The
estimates of $\braket{x}^{\gamma}R_{t}$ and $\braket{x}^{\gamma}S_{t}$ can be
shown by similarity to in (3.26). Thus we have $(R,S)\in Y_{\beta,1}\times
Y_{\beta,2}.$ The uniqueness can be shown in the same way as in the proof of
Proposition 3.2. ∎

In the discussions so far, we do not assume any relations between $u_{0}$ and
$R_{0},S_{0}$. To show $u$ in (3.41) is a solution of (1.4), we assume that

$\displaystyle\left\\{\begin{array}[]{ll}u^{\prime}_{0}=\frac{R_{0}-S_{0}}{2u^{a}_{0}},\\\
u_{1}=\frac{R_{0}+S_{0}}{2}.\end{array}\right.$ (3.47)

Moreover, we improve the regularity of the solution if $(u_{0},u_{1})\in
C^{2}\times C^{1}$. The following proposition completes the proof of Theorem
1.1.

######  Proposition 3.4.

Addition to the assumption of Proposition 3.3, we assume for $(u_{0},u_{1})\in
C^{2}\times C^{1}_{b}$ that (3.47) is satisfied. Then the function $u$ defined
in 3.41 is $C^{2}$ on $[0,T]\times\mathbb{R}$ and is the classical solution of
(1.4).

###### Proof.

From the Lipschitz continuity of $R,S$, these are differentiable almost
everywhere and satisfy that

$\displaystyle\left\\{\begin{array}[]{ll}R_{t}-u^{a}R_{x}=N_{1}(u,R,S)+L(u,R,S),\\\
S_{t}+u^{a}S_{x}=N_{2}(u,R,S)+L(u,R,S).\end{array}\right.$ (3.50)

Since $u$ is also differentiable almost everywhere, differentiating (3.41), we
have that

$\displaystyle\partial_{x}u=u^{\prime}_{0}(x)+\int_{0}^{t}\frac{R_{x}+S_{x}}{2}ds.$
(3.51)

From the first and third equation of (3.50), we have that

$\displaystyle\int_{0}^{t}R_{x}+S_{x}ds=$
$\displaystyle\int_{0}^{t}\frac{1}{u^{a}}\left(N_{2}(u,R,S)-N_{1}(u,R,S)+R_{t}-S_{t}\right)ds$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\frac{1}{u^{a}}\left(\frac{a}{2u}(S^{2}-R^{2})+R_{t}-S_{t}\right)ds.$
(3.52)

From the integration by parts, $u_{t}=\frac{R+S}{2}$ and (3.47), it follows
that

$\displaystyle\int_{0}^{t}\frac{1}{u^{a}}\left(R_{t}-S_{t}\right)ds=$
$\displaystyle-2u^{\prime}_{0}(x)+\frac{R-S}{u^{a}}+\int_{0}^{t}\frac{au_{t}}{u^{a+1}}\left(R-S\right)ds$
$\displaystyle=$ $\displaystyle-2u^{\prime}_{0}(x)+\frac{R-S}{u^{a}}$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\frac{a}{2u^{a+1}}\left(R^{2}-S^{2}\right)ds.$
(3.53)

From (3.51), (3.52) and (3.53), we have that

$\displaystyle\partial_{x}u=\frac{R-S}{2u^{a}}.$ (3.54)

Combining (3.50), (3.41) and (3.54), we have that the function $u$ satisfies
(1.4). Lastly, applying Theorem 4 in Douglis [5], we obtain the continuity of
the $W=R_{x}$ and $V=S_{x}$. From the equations of $R,S$, we see that $R_{t}$
and $S_{t}$ are also continuous. Hence we have the continuity of
$u_{xx},u_{tx},u_{tt}$. Therefore we have that $u\in
C^{2}([0,T]\times\mathbb{R})$. ∎

##  4 Concluding remarks

###  4.1 Physical background

We set a function $G$ as a primitive function of $F$ such that $G(0)=0$.
Integrating with $x$ over $[-\infty,x]$, we formally obtain that

$\displaystyle\int_{-\infty}^{x}u_{tt}dx=\partial_{x}\left(\frac{u^{a+1}}{a+1}\right)+G(u).$

Setting $v=\int_{-\infty}^{x}u_{t}dx$ and $\sigma(u)=u^{a+1}/a+1$, we have the
following 1st order hyperbolic equation:

$\displaystyle\left\\{\begin{array}[]{ll}u_{t}-v_{x}=0,\\\
v_{t}-\partial_{x}(\sigma(u))=G(u).\end{array}\right.$ (4.3)

This equations govern the motion for one dimensional elastic waves with the
case that the density of material is equal to $1$. Unknown functions $u$ and
$v$ describe the differentiations of the displacement $X$ with $x$ and $t$
respectively. Namely $u=X_{x}(t,x)$ and $v=X_{t}(t,x)$. The first equation
means the relation $u_{t}=X_{xt}=X_{tx}=v_{x}$. The second equation is
Newton’s second since $v_{t}$ is the acceleration. From the definition of $u$,
$u$ is the strain (more precisely, $(1,1)$ component of the stain matrix) and
$\sigma(u)$ is so-called stress-strain relation. $G$ is a external force term
depending only on the strain. The detailed derivation with $G\equiv 0$ is
given in Cristescu’s book [4].

###  4.2 On the generalization of the main theorem

The our existence theorem is also applicable to
$\partial^{2}_{t}u=(c(u)^{2}u_{x})_{x}+F(u)u_{x}$ under the following
assumptions on $c(\cdot)\in C([0,\infty))\cap C^{2}((0,\infty))$ and $F\in
C([0,\infty))\cap C^{1}((0,\infty))$

$\displaystyle C_{1.K}\theta^{a}\leq c(\theta)\leq C_{2,K},$ (4.4)
$\displaystyle|c^{\prime}(\theta)|\leq C_{3,K}\theta^{a-1},$ (4.5)
$\displaystyle|c^{\prime\prime}(\theta)|\leq C_{4,K}\theta^{a-2},$ (4.6)

and

$\displaystyle|F(\theta)|\leq C_{5,K}\theta^{a},$ (4.7)
$\displaystyle|F^{\prime}(\theta)|\leq C_{6,K}\theta^{a-1},$ (4.8)

where $a\geq 0$, $\theta\in[0,K]$ for $K>0$ and $C_{j,K}$ are positive
constants depending on $K$ for $j=1,\ldots,6$. For this equation, the unknown
valuable $R$ and $S$ are defined by

$\displaystyle R=u_{t}+c(u)u_{x},$ $\displaystyle S=u_{t}-c(u)u_{x}$

and $R$ and $S$ satisfy that

$\displaystyle\left\\{\begin{array}[]{ll}\partial_{t}R-u^{a}\partial_{x}R=\frac{c^{\prime}}{2c}(RS-S^{2})+F(u)\frac{R-S}{2c},\\\
\partial_{t}S+u^{a}\partial_{x}S=\frac{c^{\prime}}{2c}(RS-R^{2})+F(u)\frac{R-S}{2c}.\end{array}\right.$
(4.11)

Since we have that $\frac{|c^{\prime}(u)|}{c(u)}\leq C\braket{x}^{\alpha}$
from the assumption on $c$ and initial data, we can obtain weighted
$L^{\infty}$ estimated for $R$ and $S$. The assumption (4.6) is used in the
proof of the construction of the contraction mapping.

###  4.3 Finite time blow-up or degeneracy

We define $T^{*}$ as the maximal existence time of the solution constructed by
Theorem 1.1. When $T^{*}<\infty$, we have the following criterion of the
break-down:

$\displaystyle\limsup_{t\rightarrow
T^{*}}\|\braket{x}^{\beta}u_{t}\|_{L^{\infty}}+\|\braket{x}^{\beta}u_{x}\|_{L^{\infty}}=\infty$
(4.12)

or

$\displaystyle\liminf_{t\rightarrow
T^{*}}\inf_{\mathbb{R}}\braket{x}^{\alpha}u(t,x)=0.$ (4.13)

We call (4.12) and (4.13) the blow-up and the degeneracy respectively. In the
case that $F\equiv 0$, we can obtain the non-trivial solutions blow up in
finite time, if $R(0,x)$ and $S(0,x)$ are non-negative. In fact, we can show
that the non-negativity of $R(0,x)$ and $S(0,x)$ preserves as time goes by,
from which we have $u_{t}(t,x)\geq 0$. Thus we find that (4.13) does not
occurs in finite time. Therefore, using the method of Lax [13] or [23] (see
also Chen [1]), we have the conclusion. While, in the case that $F\equiv 0$,
we can apply main theorems to the equation in (1.4) and find that (4.13) can
occur in finite time for non-trivial solution, if $R(0,x)$ and $S(0,x)$ are
non-positive. Sufficient conditions for the occurrence of (4.13) have been
studied in the author’s papers [20, 21].

###  4.4 Multi-dimensional case

The multi-dimensional version of the equation in (1.4) is

$\displaystyle\partial^{2}_{t}u=u^{2a}\Delta u+F(u)\cdot\nabla u=0.$

The method of characteristic (and Riemann invariant) does not work, even with
radial initial data. In the forthcoming paper, we deal this problem via a
local-energy argument.

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