The Cauchy problem for $p$-evolution equations with initial data in Gelfand-Shilov spaces

TL;DR

This paper investigates the well-posedness of a class of linear evolution equations of arbitrary order with variable coefficients in Gelfand-Shilov spaces, under decay conditions on the coefficients.

Contribution

It establishes well-posedness results for $p$-evolution equations with initial data in Gelfand-Shilov spaces, extending previous theories to more general variable coefficient cases.

Findings

Proves well-posedness in Gelfand-Shilov spaces for certain evolution equations.

Shows decay conditions on coefficients are sufficient for well-posedness.

Extends classical results to equations with variable coefficients and higher order.

Abstract

We study the Cauchy problem for a class of linear evolution equations of arbitrary order with coefficients depending both on time and space variables. Under suitable decay assumptions on the coefficients of the lower order terms for $|x|$ large, we prove a well-posedness result in Gelfand-Shilov spaces.