A necessary condition for $H^\infty$ well-posedness of p-evolution equations

TL;DR

This paper establishes a necessary decay condition on the imaginary part of the subprincipal coefficient for the $H^$ well-posedness of p-evolution equations with complex coefficients, advancing understanding of their stability criteria.

Contribution

It identifies a fundamental decay condition on coefficients that must be satisfied for well-posedness of p-evolution equations with complex coefficients.

Findings

Necessary decay condition proven for the imaginary part of the subprincipal coefficient

Clarifies stability criteria for p-evolution equations with complex coefficients

Advances theoretical understanding of $H^$ well-posedness in PDEs

Abstract

We consider p-evolution equations, for $p\geq2$, with complex valued coefficients. We prove that a necessary condition for $H^\infty$ well-posedness of the associated Cauchy problem is that the imaginary part of the coefficient of the subprincipal part (in the sense of Petrowski) satisfies a decay estimate as x goes to infinity.