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 $(t{D}_t–\rho (x))u=ta(x)+G_2(x)(t,t{D}_{t}u,u,D_{1}u,...,D_{n}u)$.

It has been proved by Gérard and Tahara that there exists a unique holomorphic solution with $u(0,x)\equiv 0$ if the characteristic exponent $\rho (x)$ avoids positive integral values. In the present paper we consider what happens if $\rho (x)$ takes a positive integral value at $x=0$. Genetically, the solution $u(t,x)$ is singular along the analytic set $\{t=0,\rho (x)\in {\mathbf{N}}^{\ast }\}$, ${\mathbf{N}}^{\ast }=\{1,2,...\}$, and we investigate how far it can be analytically continued.