Degenerate 3-evolution equations in Gevrey classes

TL;DR

This paper investigates the well-posedness of third-order evolution equations with variable coefficients that degenerate at initial time, establishing conditions for solutions in various functional spaces including Gevrey classes.

Contribution

It provides new sufficient conditions on lower order coefficients ensuring well-posedness in Gevrey and other function spaces for degenerate third-order evolution equations.

Findings

Established well-posedness criteria in Gevrey spaces.

Derived conditions on coefficients near degeneracy point.

Extended results to $L^2$ and $H^{ abla}( abla)$ spaces.

Abstract

We consider the Cauchy problem for third-order evolution differential operators with variable coefficients, depending on time $t\in [0,T]$ and space $x\in\mathbb{R}$, where the leading coefficient $a_3(t)$ vanishes at $t = 0$ with finite order. We establish sufficient conditions on the behavior of the lower order coefficients $a_j(t,x)$ $j=1,2$ as $t \to 0^{+}$ and $|x| \to \infty$ that ensure well-posedness in $L^2(\mathbb{R})$, $H^{\infty}(\mathbb{R})$ and Gevrey-type spaces.