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

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 $C^{\mathrm{\infty }}$ and in ${\mathcal{D}}^{\mathrm{\prime }}$. We prove that the analyticity of the coefficients combined with suitable hypotheses on the eigenvalues guarantee the $C^{\mathrm{\infty }}$ 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}.