Weakly hyperbolic equations with non-analytic coefficients and lower order terms

TL;DR

This paper investigates the well-posedness of weakly hyperbolic equations with time-dependent coefficients and lower order terms across various regularity classes, extending existing results to include discontinuous lower order terms and arbitrary dimensions.

Contribution

It introduces a novel approach to include lower order terms in weakly hyperbolic equations and considers arbitrary space dimensions, broadening the scope of well-posedness results.

Findings

Gevrey well-posedness under $C^k$ regularity of coefficients

$C^ abla$ well-posedness for analytic coefficients

Results on ultradistributional and distributional well-posedness

Abstract

In this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time-dependent coefficients and lower order terms. We prove the Gevrey well-posedness of the Cauchy problem under $C^k$-regularity of coefficients of the principal part and natural Levi conditions on lower order terms which may be only continuous. In the case of analytic coefficients in the principal part we establish the $C^\infty$ well-posedness. The proofs are based on using the quasi-symmetriser for the corresponding system. The main novelty compared to the existing literature is the possibility to include lower order terms to the equation as well as considering any space dimensions. We also give results on the ultradistributional and distributional well-posedness of the problem, and we look at new effects for equations with discontinuous lower order terms.