Well-posedness of hyperbolic systems with multiplicities and smooth coefficients

TL;DR

This paper establishes well-posedness results for hyperbolic systems with multiplicities and smooth coefficients, extending previous work to non-analytic and analytic cases with Gevrey and $C^ abla$ regularity.

Contribution

It introduces a transformation to block Sylvester form for hyperbolic systems with multiplicities, enabling well-posedness proofs in various regularity classes.

Findings

Well-posedness in Gevrey classes for smooth coefficients.

$C^ abla$ well-posedness for analytic coefficients.

Extension of previous results to systems with multiplicities.

Abstract

We study hyperbolic systems with multiplicities and smo\-oth coefficients. In the case of non-analytic, smooth coefficients, we prove well-posedness in any Gevrey class and when the coefficients are analytic, we prove $C^\infty$ well-posedness. The proof is based on a transformation to block Sylvester form introduced by D'Ancona and Spagnolo in Ref. 9 which increases the system size but does not change the eigenvalues. This reduction introduces lower order terms for which appropriate Levi-type conditions are found. These translate then into conditions on the original coefficient matrix. This paper can be considered as a generalisation of Ref. 12, where weakly hyperbolic higher order equations with lower order terms were considered.