Cauchy problem for effectively hyperbolic operators with triple characteristics

We consider the Cauchy problem for hyperbolic operators with characteristics of variable multiplicities r ≤ 3 assuming that the fundamental matrix of the principal sym- bol has two non-vanishing real eigenvalues. The last condition is necessary for the Cauchy problem to be well posed for every choice of lower order terms. The operators with this pro- perty are called strongly hyperbolic and it was conjectured that every effectively hyperbolic operator is strongly hyperbolic. In this talk we present a survey of the results in the case r = 3. The proofs are based on the energy estimates with a big loss of derivatives depending of lower order terms. This is a joint work with T. Nishitani.