Cauchy problem for effectively hyperbolic operators with triple characteristics of variable multiplicity

We study a class of third order hyperbolic operators $P$ in $G = {(t, x):0 \leq t \leq T, x \in U \Subset {\mathbb R}^{n}}$ with triple characteristics at $\rho =(0,x_0,\xi ),\xi \in {\mathbb{R}}^n\setminus \{0\}$. We consider the case when the fundamental matrix of the principal symbol of $P$ at $\rho$ has a couple of non-vanishing real eigenvalues. Such operators are called {\it effectively hyperbolic}. V. Ivrii introduced the conjecture that every effectively hyperbolic operator is {\it strongly hyperbolic}, that is the Cauchy problem for $P+Q$ is locally well posed for any lower order terms $Q$. This conjecture has been solved for operators having at most double characteristics and for operators with triple characteristics in the case when the principal symbol admits a factorization. A strongly hyperbolic operator in $G$ could have triple characteristics in $G$ only for $t=0$ or for $t=T$. We prove that the operators in our class are strongly hyperbolic if $T$ is small enough. Our proof is based on energy estimates with a loss of regularity.