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 (tDt–ρ(x))u = ta(x)+_G_2(x)(t, tDtu, u, D1u,...,Dnu). It has been proved by Gerard 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.