On the Cauchy problem for $D_t^2-D_x(b(t)a(x))D_x$

TL;DR

This paper investigates the well-posedness of a second order differential operator's Cauchy problem with variable coefficients, establishing conditions under which solutions exist uniquely in certain Gevrey classes.

Contribution

It proves the well-posedness of the Cauchy problem for a specific second order differential operator with variable coefficients in Gevrey classes, extending previous results.

Findings

Well-posedness in Gevrey classes for certain parameters

Conditions on coefficient regularity for solution existence

Extension of classical well-posedness results

Abstract

We consider the Cauchy problem for second order differential operators with two independent variables $P=D_t^2-D_x(b(t)a(x))D_x$. Assume that $b(t)$ is a nonnegative $C^{n,alpha}$ function and $a(x)$ is a nonnegative Gevrey function of order $s>1$ we prove that the Cauchy problem for $P$ is well-posed in the Gevrey class of any order $s<s'<1+(n+alpha)/2$.