On hyperbolic systems with time dependent H\"older characteristics

TL;DR

This paper investigates the well-posedness of weakly hyperbolic systems with time-dependent coefficients that are H"older continuous, extending previous results to ultradistribution spaces through a reduction to block Sylvester form and energy estimates.

Contribution

It improves existing Gevrey well-posedness results for hyperbolic systems with H"older eigenvalues and introduces well-posedness in ultradistribution spaces using a novel reduction and energy estimate approach.

Findings

Enhanced well-posedness results for systems with H"older eigenvalues

Extension to ultradistribution spaces

Reduction to block Sylvester form facilitates analysis

Abstract

In this paper we study the well-posedness of weakly hyperbolic systems with time dependent coefficients. We assume that the eigenvalues are low regular, in the sense that they are H\"older with respect to $t$. In the past these kind of systems have been investigated by Yuzawa \cite{Yu:05} and Kajitani \cite{KY:06} by employing semigroup techniques (Tanabe-Sobolevski method). Here, under a certain uniform property of the eigenvalues, we improve the Gevrey well-posedness result of \cite{Yu:05} and we obtain well-posedness in spaces of ultradistributions as well. Our main idea is a reduction of the system to block Sylvester form and then the formulation of suitable energy estimates inspired by the treatment of scalar equations in \cite{GR:11}