Counterexamples to the well posedness of the Cauchy problem for hyperbolic systems

This paper is concerned with the well posedness of the Cauchy problem for first order symmetric hyperbolic systems in the sense of Friedrichs. The classical theory says that if the coefficients of the system and if the coefficients of the symmetrizer are Lipschitz continuous, then the Cauchy problem is well posed in $L^2$. When the symmetrizer is Log-Lipschtiz or when the coefficients are analytic or quasi-analytic, the Cauchy problem is well posed $C^\infty$. In this paper we give counterexamples which show that these results are sharp. We discuss both the smoothness of the symmetrizer and of the coefficients.