The Cauchy problem for weakly hyperbolic systems

We consider the well-posedness of the Cauchy problem in Gevrey spaces for N×N first-order weakly hyperbolic systems. The question is to know whether the general results of Bronštein [1] and Kajitani [9] can be improved when the coefficients depend only on time and are smooth, as it has been done for the scalar wave equation in [3]. The answer is no for general systems, and yes when the system is uniformly diagonalizable: in this case, we show that the Cauchy problem is well posed in all Gevrey classes Gs when the coefficients are C∞. Moreover, for 2×2 systems and some other special cases, we prove that the Cauchy problem is well posed in Gs for s<1+k when the coefficients are Ck, which is sharp following the counterexamples of Tarama [12]. The main new ingredient is the construction, for all hyperbolic matrix A, of a family of approximate symmetrizers, S, the coefficients of which are polynomials of and the coefficients of A and A*.