Cauchy problem for nonlinear p-evolution equations

The local solvability of the Cauchy problem in Sobolev spaces is studied for a class of nonlinear p-evolution partial differential equations of order pm, p>1, with real characteristic roots in the principal part in the sense of Petrowski. The non real lower order terms do not exceed some order depending on the order of the operator in the time variable m, on the maximum multiplicity of the characteristic roots,r, and on the type of the evolution p ([2]). The paper extends both results concerning linear equations of p-evolution type, p>1 ( [12-15], [9],[1] ) to nonlinear ones and includes , if p=1, results concerning H^{s}-wellposedness for quasilinear weakly hyperbolic equations with characteristics of constant multiplicity [6]