Cauchy problem for effectiively hyperbolic operators with triple characteristics

We study a class of third-order effectively hyperbolic operators P in G = { x \in U, 0 \leq t \leq T} with triple characteristics at ρ = (0, x_0, ξ), ξ ∈ R^n \ {0}. V. Ivrii introduced the conjecture that every effectively hyperbolic operator is strongly hyperbolic, that is the Cauchy problem for P + Q is locally well posed for any lower-order terms Q . For operators with triple characteristics, this conjecture was established by Ivrii in the case when the principal symbol of P admits a factorization as a product of two symbols of principal type. A strongly hyperbolic operator in G could have triple characteristics in G only for t = 0 or for t = T . The operators that we investigate have a principal symbol which in general is not factorizable and we prove that these operators are strongly hyperbolic if T is small enough