The Cauchy problem for hyperbolic systems with multiple characteristics

AbstractIn the present paper we give necessary conditions for the well-posedness of the Cauchy problem for a class of first order differential hyperbolic N × N systems, L = L1 (x, Dx) + LO (x), with multiple characteristics. Let ρ be characteristic point of h(x, ξ) = det L1 (x, ξ) of multiplicity r; we assume that rank L1 (ρ) = N − 1. Our result is that there is a scalar hyperbolic differential operator P with principal symbol h, such that, if the Cauchy problem for L is correctly posed, then P must satisfy the Ivrii-Petkov conditions at p of multiplicity r