The Lax-Mizohata theorem for Kirchhoff-type equations

摘要

In this paper we prove the necessity of the hyperbolicity to the Cauchy problem in the Kirchhoff-type equations to be locally (in a neighbourhood of the given solution and local in time) well-posed. The method of proof is based on the gauge invariance of the Kirchhoff-type equations. which allows to use high frequency components of solutions. and on some ideas from the Lyapunov stability theory. (C) 2001 Academic Press. [References: 26]