The cauchy problem for hyperbolic systems with multiple characteristics

In the present paper we give necessary conditions for the well-posedness of the Cauchy problem for a class of first order differential hyperbolic N x N systems, L = L-1(x, D-x) + L-0 (x), with multiple characteristics. Let p be characteristic point of h (x, xi) = det L-1 (x, xi) of multiplicity r; w e assume that rank L-1 (p) = N - 1. Our result is that there is a scalar hy perbolic differential operator P with principal symbol h, such that, if the Cauchy problem for L is correctly posed, then P must satisfy the Ivrii-Pet kov conditions at p of multiplicity r. (C) Elsevier, Paris.