Cauchy problem of certain equations of Schrodinger type

We consider a 2-evolution operator in the sense of Petrowsky, and we assume that the characteristic roots of the principal polynomial with constant coefficients are real and of constant constant multiplicities. Then we give sufficient conditions so that the Cauchy problem both for the future and for the past is well-posed in the Sobolev spaces. Our conditions are analogous to the Levi conditions and to the decomposition conditions of operators in the hyperbolic case for Kowalewskian operators.