Well posedness of the Cauchy problem for nonlinear weakly hyperbolic equations

The authors study well-posedness of the Cauchy problem for several classes of nonlinear (semilinear) weakly hyperbolic equations. It is assumed that the principal part of the operator possesses real characteristic roots of constant multiplicity, and that Levi-type conditions and Levi conditions of nonlinear type [respectively, Gevrey-Levi conditions and nonlinear Gevrey-Levi-type conditions] are satisfied in C1 [respectively, in Gevrey categories] with respect to the space variables. Local existence and uniqueness results in the time variable t are proved both in C1 and Gevrey classes.