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 Bron?tein 1 Bron?tein, M.D. (1982). The Cauchy Problem for hyperbolic operators with characteristic of variable multiplicity. Trudy Moskov. Mat. Obshch. 41:83–99. Translation: Trans. Moscow. Math. Soc. 41:87–103]. Google Scholar]] and Kajitani 9 Kajitani, K. (1986). The Cauchy Problem for Uniformly Diagonalizable Hyperbolic Systems in Gevrey Classes, in Hyperbolic Equations and Related Topics. Proceedings of the Taniguchi International Symposium, Katata and Kyoto, 1984. Boston: Academic Press, pp. 101–123. Google Scholar]] 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 Colombini, F., Jannelli, E., Spagnolo, S. (1983). Well-posedness in the Gevrey classes of the Cauchy problem for a nonstrictly hyperbolic equation with coefficients depending on time. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10:291–312. Google Scholar]]. 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 G^s 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 G^s for s<1+k when the coefficients are C^k, which is sharp following the counterexamples of Tarama 12 Tarama, S. (1994). Une note sur les Systèmes Hyperboliques Uniformément Diagonalisables. Mem. Fac. Eng. Kyoto Univ. 56:9–18. Google Scholar]]. The main new ingredient is the construction, for all hyperbolic matrix A, of a family of approximate symmetrizers, S_𝜀, the coefficients of which are polynomials of 𝜀 and the coefficients of A and A*.