The Cauchy problem for $3$-evolution equations with data in Gelfand-Shilov spaces

TL;DR

This paper studies the well-posedness and behavior of solutions to a third-order evolution PDE with coefficients depending on time and space, assuming initial data in Gelfand-Shilov spaces with exponential decay.

Contribution

It establishes existence and regularity results for solutions with initial data in Gelfand-Shilov spaces, extending the understanding of such PDEs with complex lower order terms.

Findings

Existence of solutions with Gevrey regularity matching initial data

Description of solution behavior as |x| approaches infinity

Conditions on coefficient decay ensuring well-posedness

Abstract

We consider the Cauchy problem for a $3$-evolution operator $P$ with $(t,x)$-depending coefficients and complex valued lower order terms. We assume the initial data to be Gevrey regular and to admit an exponential decay at infinity, that is, the data belong to some Gelfand-Shilov spaces of type $\mathscr{S}$. Under suitable assumptions on the decay at infinity of the imaginary parts of the coefficients of $P$ we prove the existence of a solution with the same Gevrey regularity of the data and we describe its behavior for $|x| \to\infty$.