Cauchy problem for hyperbolic operators with triple characteristics of variable multiplicity

We study a class of third order hyperbolic operators $P$ in $G = \Omega \cap {0 \leq t \leq T},: \Omega \subset \R^{n+1}$withtriplecharacteristicson$t = 0$. We consider the case when the fundamental matrix of the principal symbol for $t=0$ has a couple of non vanishing real eigenvalues and $P$ is strictly hyperbolic for $t>0.$ We prove that $P$ is strongly hyperbolic, that is the Cauchy problem for $P+Q$ is well posed in $G$ for any lower order terms $Q$.