Local solvability for nonlinear partial differential equations (Q2746998)

This paper deals with the local solvability of several classes of nonlinear Partial Differential Equations (PDE). At first a semilinear PDE with a quasihomogeneous principal term satisfying the quasihomogeneous version of the famous (P) condition of Nirenberg-Treves is considered, and its local solvability is established in the corresponding Sobolev spaces.NEWLINENEWLINENEWLINEIn the second part of the paper the local solvability in Gevrey spaces of several classes of semilinear PDE having homogeneous principal symbols with multiple characteristics is proved. A model exampel in the last case is given by a nonlinear perturbation of the \(m\)-th power of the Mizohata operator. Assuming the right-hand side to belong to an appropriate Gevrey space the authors show the local existence of a classical solution \(u\in \mathbb{C}^m\).