Nonlinear Singular First Order Partial Differential Equations Whose Characteristic Exponent Takes a Positive Integral Value

Abstract

We consider nonlinear singular partial differential equations of the form (tD_t−ρ(x))u=ta(x)+G₂(x) (t, tD_tu, u, D₁u, .., D_nu).
It has been proved by Gérard and Tahara that there exists a unique holomorphic solution with u(0, x)≡0 if the characteristic exponent ρ(x) avoids positive integral values. In the present paper we consider what happens if ρ(x) takes a positive integral value at x=0. Genetically, the solution u(t, x) is singular along the analytic set {t=0, ρ(x)∈N^*}, N^*={1, 2, ..}, and we investigate how far it can be analytically continued.