A generalized Levi condition for weakly hyperbolic Cauchy problems with coefficients low regular in time and smooth in space

TL;DR

This paper introduces a generalized Levi condition for weakly hyperbolic PDEs with low-regularity time coefficients and smooth space coefficients, establishing well-posedness criteria linking coefficient regularity, Levi conditions, and solution spaces.

Contribution

It proposes a new generalized Levi condition that allows more flexible modeling of multiple characteristics in weakly hyperbolic equations with irregular coefficients.

Findings

Established sufficient conditions for well-posedness based on coefficient regularity and Levi conditions.

Showed independence of Levi condition influence and coefficient regularity on solution space.

Linked regularity measures of coefficients to the structure of solution spaces.

Abstract

We consider the Cauchy problem for weakly hyperbolic $m$-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that in general one has to impose Levi conditions to get $C^\infty$ or Gevrey well-posedness even if the coefficients are smooth. 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. Furthermore, we propose a generalized Levi condition that models the influence of multiple characteristics more freely. We establish sufficient conditions for the well-posedness of the Cauchy problem, that link the Levi condition as well as the modulus of continuity and the weight sequence of the coefficients to the weight function of the solution space. Additionally, we obtain that the influences of the Levi condition and the low regularity of coefficients on the weight function of the solution space are independent of each other.