| Chen Shuxing.[J].数学年刊A辑,1980,1(3-4):511~521 |
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| ON THE INITIAL BOUNDARY VALUE PROBLEMS FORQUASILINEAR SYMMETRIC HYPERBOILC SYSTEMAND THEIR APPLICATIONS |
| Received:December 14, 1979 |
| DOI： |
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| In this paper we discuss the initial-boundary value problems for qnasilinear
gymmetrio hyperbolic system and their applications. It is proved that
Theorem 1, Suppose \(\Omega \) is a bomded domain, its boundary \(\partial \Omega \) is sufficient smooth. We consider the quasilinear symmetric hyperbolic system
\[\sum\limits_{i = 0}^n {{a^i}(x,u)\frac{{\partial u}}{{\partial {x_i}}}} = f(x,u)\]
in the domain \([0,h] \times \Omega \). The initial-boimda/ry conditions
\[\begin{array}{l}
{\left. u \right|_{{x_0} = 0}} = 0\{\left. {Mu} \right|_{\partial \Omega }} = 0
\end{array}\]
are given. If \({a^0}\) is positive definite，\(\partial \Omega \) is noncharaGieristic, \(Mu = 0\) is stable admissible
and all coefficients are smooth enough, some of derivatives of \(f(x,0)\) at \({{x_0} = 0}\) vanish., then the smooth solution of (1), (2) uniquely exists, if h is sufficiently small.
Theorem 2. We consider the semi-Unear symmetric hyperbolic system
\[\sum\limits_{i = 0}^n {{a^i}(x,u)\frac{{\partial u}}{{\partial {x_i}}}} = f(x,u)\]
The initial-boundary conditions are still
\[\begin{array}{l}
{\left. u \right|_{{x_0} = 0}} = 0\{\left. {Mu} \right|_{\partial \Omega }} = 0
\end{array}\]
If the bowndary \(\partial \Omega \) is a regular characteristic, \(Mu = 0\) is normally admissible and other conditions is the same as that in the theorem 1., then the smooth solution of (3), (4) still wriiquely exists if hM sufficiently small. |
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