Cauchy problem for hyperbolic operators with triple characteristics of variable multiplicity

TL;DR

This paper proves that certain third-order hyperbolic operators with triple characteristics at initial time are strongly hyperbolic, ensuring well-posedness of the Cauchy problem even with lower order perturbations.

Contribution

It establishes strong hyperbolicity for a class of third-order hyperbolic operators with variable multiplicity and triple characteristics, extending well-posedness results.

Findings

Proves strong hyperbolicity of the class of operators studied.

Shows well-posedness of the Cauchy problem with lower order terms.

Analyzes the eigenvalues of the principal symbol's fundamental matrix.

Abstract

We study a class of third order hyperbolic operators $P$ in $G = \Omega \cap \{0 \leq t \leq T\},\: \Omega \subset \R^{n+1}$ with triple characteristics on $t = 0$. We consider the case when the fundamental matrix of the principal symbol for $t = 0$ has a couple of non vanishing real eigenvalues and $P$ is strictly hyperbolic for $t > 0.$ We prove that $P$ is strongly hyperbolic, that is the Cauchy problem for $P + Q$ is well posed in $G$ for any lower order terms $Q$.