Well-posedness for hyperbolic equations whose coefficients lose regularity at one point

TL;DR

This paper establishes well-posedness results in smooth and Gevrey classes for hyperbolic equations with coefficients that become less regular at a single point, addressing a key challenge in PDE theory.

Contribution

It provides new well-posedness results for hyperbolic equations with irregular coefficients at a point, expanding understanding of such PDEs.

Findings

Proves $C^ abla$ and Gevrey well-posedness under specific coefficient regularity conditions.

Identifies conditions under which hyperbolic equations remain well-posed despite coefficient irregularities.

Extends classical results to cases with localized loss of regularity.

Abstract

We prove some $C^\infty$ and Gevrey well-posedness results for hyperbolic equations whose coefficients lose regularity at one point.