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| GLOBAL CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS WITH WEAK LINEAR DEGENERACY |
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Citation： |
ZHOU Yi.GLOBAL CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS WITH WEAK LINEAR DEGENERACY[J].Chinese Annals of Mathematics B,2004,25(1):37~56 |
| Page view： 1185
Net amount： 719 |
Authors： |
ZHOU Yi; |
Foundation： |
Project supported by the National Natural Science Foundation of China (No.10225102), the 973 Project
of the Ministry of Science and Technology of China and the Doctoral Programme Foundation of the
Ministry of Education of China. |
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| Abstract： |
Consider the following Cauchy problem for the first order
quasilinear strictly hyperbolic system
$$\frac{\partial u}{\partial t}+A(u)\frac{\partial u}{\partial x}=0,$$
$$t=0:\quad u=f(x).$$
We let
$$M=\sup_{x\in R}|f'(x)|<+\infty.$$
The main result of this paper is that under the assumption that
the system is weakly linearly degenerated, there exists a
positive constant $\e$ independent of $M$,
such that the above Cauchy
problem admits a unique global $C^1$ solution $u=u(t,x)$ for all
$t\in R$, provided that
\begin{align*}
\int_{-\infty}^{+\infty}|f'(x)|dx&\le \e,\\int_{-\infty}^{+\infty}|f(x)|dx&\le \frac{\e}{M}.
\end{align*} |
Keywords： |
Global classical solutions, Cauchy problems, Weak linear degeneracy |
Classification： |
35L45, 35L60 |
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