On $C^\infty$ well-posedness of hyperbolic systems with multiplicities

TL;DR

This paper investigates conditions under which first-order hyperbolic systems with multiple characteristics and time-dependent analytic coefficients are well-posed in the space of smooth functions, extending scalar case results to systems.

Contribution

It establishes that analyticity of coefficients and specific eigenvalue conditions ensure $C^{ abla}$ well-posedness for such hyperbolic systems, extending previous scalar results.

Findings

Proves $C^{ abla}$ well-posedness under analyticity and eigenvalue conditions.

Extends scalar equation results to hyperbolic systems.

Provides conditions guaranteeing well-posedness in $C^{ abla}$.

Abstract

In this paper we study first order hyperbolic systems with multiple characteristics (weakly hyperbolic) and time-dependent analytic coefficients. The main question is when the Cauchy problem for such systems is well-posed in $C^{\infty}$ and in $\mathcal{D}'$. We prove that the analyticity of the coefficients combined with suitable hypotheses on the eigenvalues guarantee the $C^\infty$ well-posedness of the corresponding Cauchy problem. This result is an extension to systems of the analogous results for scalar equations recently obtained by Jannelli and Taglialatela in \cite{JT} and by the authors in \cite{GR:13}.