The Lax–Mizohata Theorem for Kirchhoff-Type Equations

In this paper we prove the necessity of the hyperbolicity to the Cauchy problem for 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