Strictly Hyperbolic Equations with Coefficients Low-Regular in Time and Smooth in Space

TL;DR

This paper investigates the well-posedness of strictly hyperbolic PDEs with coefficients that are low-regular in time and smooth in space, using moduli of continuity and weighted function spaces to generalize existing results.

Contribution

It introduces a unified framework employing moduli of continuity and weight functions to analyze well-posedness for hyperbolic equations with various low-regularity coefficients.

Findings

Established conditions linking coefficient regularity to solution space weights.

Recovered classical well-posedness results for Lipschitz, Log-Lipschitz, and H"older coefficients.

Provided a generalized approach for coefficients with arbitrary moduli of continuity.

Abstract

We consider the Cauchy problem for strictly hyperbolic $m$-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that the problem is $L^2$ well-posed in the case of Lipschitz continuous coefficients in time, $H^s$ well-posed in the case of Log-Lipschitz continuous coefficients in time (with an, in general, finite loss of derivatives) and Gevrey well-posed in the case of H\"older continuous coefficients in time (with an, in general, infinite loss of derivatives). Here, we use moduli of continuity to describe the regularity of the coefficients with respect to time, weight sequences for the characterization of their regularity with respect to space and weight functions to define the solution spaces. We establish sufficient conditions for the well-posedness of the Cauchy problem, that link the modulus of continuity and the weight sequence of the coefficients to the weight function of the solution space. The well-known results for Lipschitz, Log-Lipschitz and H\"older coefficients are recovered.