Counterexamples to $ C^{\infty} $ well posedness for some hyperbolic operators with triple characteristics

TL;DR

This paper demonstrates that certain non-effectively hyperbolic operators with smooth triple characteristics have well-posed Cauchy problems in the Gevrey 2 class, which is optimal and extends beyond the generic Gevrey 3/2 class.

Contribution

It establishes the well-posedness in Gevrey 2 for a class of hyperbolic operators with triple characteristics, showing this regularity is optimal.

Findings

Well-posedness in Gevrey 2 class for specific operators

Optimality of the Gevrey 2 regularity

Extension beyond the generic Gevrey 3/2 class

Abstract

In this paper we prove that for a class of non-effectively hyperbolic operators with smooth triple characteristics the Cauchy problem is well posed in the Gevrey 2 class, beyond the generic Gevrey class $ 3/2 $ (see e.g. \cite{Bro}). Moreover we show that this value is optimal.