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

Claudia Garetto; Michael Ruzhansky

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

Analysis of PDEs 2016-01-12 v2

Authors: Claudia Garetto , Michael Ruzhansky

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}.

Keywords

hyperbolic systems well-posedness multiplicities Cauchy problem analytic coefficients weakly hyperbolic

Cite

@article{arxiv.1512.06243,
  title  = {On $C^\infty$ well-posedness of hyperbolic systems with multiplicities},
  author = {Claudia Garetto and Michael Ruzhansky},
  journal= {arXiv preprint arXiv:1512.06243},
  year   = {2016}
}

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