The Cauchy Problem for Wave Equations with NonLipschitz Coefficients

TL;DR

This paper investigates the well-posedness of second order hyperbolic wave equations with coefficients that are less regular than Lipschitz, specifically Log-Lipschitz, establishing local existence, uniqueness, and finite propagation speed.

Contribution

It extends the theory of hyperbolic equations to include Log-Lipschitz coefficients, which are invariant under variable changes, enabling local analysis of such equations.

Findings

Proved local existence of solutions.

Established local uniqueness of solutions.

Demonstrated finite speed of propagation.

Abstract

In this paper we study the Cauchy problem for second order strictly hyperbolic operators when the coefficients of the principal part are not Lipschitz continuous, but only "Log-Lipschitz" with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy problem.