On C ∞ well-posedness of hyperbolic systems with multiplicities

In this paper, we study first-order hyperbolic systems of any order 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∞C∞ and in D′D′ . We prove that the analyticity of the coefficients combined with suitable hypotheses on the eigenvalues guarantees the C∞C∞ well-posedness of the corresponding Cauchy problem.