A question on the Cauchy problem in the Gevrey classes for weakly hyperbolic equations

TL;DR

This paper investigates the well-posedness of the Cauchy problem for weakly hyperbolic equations with Gevrey coefficients, demonstrating the optimality of the Gevrey order s=2 through specific examples.

Contribution

It provides examples that show the limitations of well-posedness in Gevrey classes for hyperbolic equations, confirming the optimality of the order s=2.

Findings

Well-posedness holds for s<2 in Gevrey classes.

Counterexamples show failure of well-posedness at s=2.

The results establish the optimality of the Gevrey order s=2.

Abstract

For a homogeneous polynomial $p$ in $\xi\in {\bf R}^n$ with Gevrey coefficients, it is known that the Cauchy problem for any realization of $p$ is well-posed in the Gevrey class of order $s<2$ if the characteristic roots are real. In this note, we give examples showing the situation of the converse direction, in particular the optimality of the Gevrey order $s=2$.