Cauchy problem for effectively hyperbolic operators with triple characteristics

We study a class of third-order effectively hyperbolic operators P   in G={(t,x):0?t?T,x∈U?Rⁿ}$G=\{(t,x):0?t?T,x\in U?{\mathbb{R}}^n\}$ with triple characteristics at ρ=(0,x₀,ξ),ξ∈Rⁿ?{0}$\rho =(0,x_0,\xi ),\xi \in {\mathbb{R}}^n?\{0\}$. V. Ivrii introduced the conjecture that every effectively hyperbolic operator is strongly hyperbolic  , that is the Cauchy problem for P+Q$P+Q$ is locally well posed for any lower-order terms Q. For operators with triple characteristics, this conjecture was established [3] 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$t=0$ or for t=T$t=T$. 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.