Local existence theorems and blow-up of solutions for quasi-linear hyperbolic Cauchy problems in a domain with characteristic boundary generated by initial data
type: Item
instance of: scholarly article
title: Local existence theorems and blow-up of solutions for quasi-linear hyperbolic Cauchy problems in a domain with characteristic boundary generated by initial data
zbMATH Open document ID: 0713.35054
author: Reiko Sakamoto
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1989-01-01T00:00:00Z
review text: The quasilinear Cauchy problem: \[ (1)\quad A(t,x;\phi^{m- 1}u;\partial)u\equiv \partial^ m_ tu+\sum_{j+| v| \leq m,j\leq m-1}a_{jv}(t,x;\phi^{m-1}u)\partial^ j_ t\partial^ v_ xu=f(t,x;\partial^{m-1}u) \] in (0,T)\(\times \Omega\), \[ \partial^ j_ tu=\phi_ j(x),\quad j=0,1,...,m-1\text{ on } \{t=0\}\times \Omega, \] where the simple notations: \[ \partial^ ku=\{\partial^ j_ t\partial^ v_ xu| \quad j:| v| =k\},\quad \phi^ ku=\{\partial^ j_ t\partial^ v_ xu| \quad j+| v| \leq k\}=(u,\partial u,...,\partial^ ku) \] are used, is studied. It is proved that if initial data are ``degenerate-structure generating'', there is a unique classical solution in a small time of (1). By proving existence of the solution a standard iterative method is used. Some examples of equations with solutions blowing-up on the boundary are given.
reviewed by: K.Zlateva
Mathematics Subject Classification ID: 35L70
zbMATH DE Number: 4173513
zbMATH Keywords: quasilinear Cauchy problem
zbMATH Keywords: unique classical solution
zbMATH Keywords: small time
zbMATH Keywords: blowing-up
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195173350
OpenAlex ID: W1990047553
cites work: Q3808560
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DOI: 10.2977/PRIMS/1195173350
description: scientific article; zbMATH DE number 4173513