On the Gevrey strong hyperbolicity

TL;DR

This paper investigates the Gevrey strong hyperbolicity index for certain differential operators, establishing its relation to bicharacteristic behaviors and conditions for well-posedness in Gevrey classes.

Contribution

It introduces the concept of the Gevrey strong hyperbolicity index and analyzes its maximal value in relation to the operator's bicharacteristics.

Findings

Defined the Gevrey strong hyperbolicity index.

Connected the index with the behavior of bicharacteristics.

Provided conditions for well-posedness in Gevrey classes.

Abstract

In this paper we are concerned with a homogeneous differential operator $p$ of order $m$ of which characteristic set of order $m$ is assumed to be a smooth manifold. We define the Gevrey strong hyperbolicity index as the largest number $s$ such that the Cauchy problem for $p+Q$ is well-posed in the Gevrey class of order $s$ for any differential operator $Q$ of order less than $m$. We study the case of the largest index and we discuss in which way the Gevrey strong hyperbolicity index relates with behaviors of bicharacteristics of $p$ near the characteristic manifold.