# 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 (*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.

## 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