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

### Hideshi Yamane

University of Tokyo, Japan

## Abstract

We consider nonlinear singular partial differential equations of the form $(tD_{t}–ρ(x))u=ta(x)+G_{2}(x)(t,tD_{t}u,u,D_{1}u,...,D_{n}u)$.

It has been proved by Gérard and Tahara that there exists a unique holomorphic solution with $u(0,x)≡0$ 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.

## Cite this article

Hideshi Yamane, Nonlinear Singular First Order Partial Differential Equations Whose Characteristic Exponent Takes a Positive Integral Value. Publ. Res. Inst. Math. Sci. 33 (1997), no. 5, pp. 801–811

DOI 10.2977/PRIMS/1195145018