Local Solutions to Quasi-linear Weakly Hyperbolic Differential Equations
Local Solutions to Quasi-linear Weakly Hyperbolic Differential Equations
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DOI：
10.1007/978-3-0348-8073-2_3
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发表时间：
2003
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影响因子：
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通讯作者：
M. Dreher
中科院分区：
文献类型：
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作者：
M. Dreher
本文的目的是研究空间变量和时间变量退化的弱双曲方程。这些退化以及尖锐的Levi条件的Coo型制定通过一定的权重函数。对于这类拟线性弱双曲型方程的Cauchy问题，研究了Sobolev空间和Coo空间中解的局部存在性、爆破准则、依赖域和Coo正则性.主要的工具是高阶方程的一阶系统，微积分的pseudodiomatic算子与非光滑符号，和推广的Gronwall引理微分不等式与奇异系数的变换。
The purpose of this paper is to investigate weakly hyperbolic equations with degeneracies in the space and time variables. These degeneracies as well as the sharp Levi conditions of Coo type are formulated by means of certain weight functions. For Cauchy problems to such quasi-linear weakly hyperbolic equations, the following subjects are studied: local existence of solutions in Sobolev spaces and Coo, a blow-up criterion, domains of dependence, and Coo regularity. The main tools are the transformation of the higher-order equation to a first-order system, a calculus for pseudodifferential operators with non-smooth symbols, and a generalization of Gronwall's lemma to differential inequalities with a singular coefficient.