On the Cauchy problem for $p$-evolution equations with variable coefficients: a necessary condition for Gevrey well-posedness

TL;DR

This paper establishes necessary decay conditions on variable coefficients of $p$-evolution equations to ensure well-posedness in Gevrey spaces, advancing understanding of their analytical properties.

Contribution

It introduces necessary decay rate conditions on coefficients for Gevrey well-posedness of $p$-evolution equations with variable coefficients.

Findings

Necessary decay conditions for coefficients are identified.

Conditions are crucial for well-posedness in Gevrey spaces.

Results apply to arbitrary order $p$-evolution equations.

Abstract

In this paper we consider a class of $p$-evolution equations of arbitrary order with variable coefficients depending on time and space variables $(t,x)$. We prove necessary conditions on the decay rates of the coefficients for the well-posedness of the related Cauchy problem in Gevrey spaces.