# On Totally Characteristic Type Non-linear Partial Differential Equations in the Complex Domain

### Hua Chen

Wuhan University, Wuhan, Hubei, China### Hidetoshi Tahara

Sophia University, Tokyo, Japan

## Abstract

The paper deals with a singular non-linear partial differential equation $t∂u/∂t=F(t,x,u,∂u/∂x)$ with two independent variables $(t,x)∈C_{2}$ under the assumption that $F(t,x,u,v)$ is holomorphic and $F(0,x,0,0)=0$. Set $γ(x)=(∂F/∂v)(0,x,0,0)$. In case $γ(x)=0$ the equation was investigated quite well by Gérard–Tahara [3]. In case $γ(0)=0$ and $Reγ_{′}<0$ the existence of holomorphic solution was proved in Chen–Tahara [2] under a non-resonance condition. The present paper proves the existence of holomorphic solution under the same non-resonance condition but using the following weaker condition: $γ(0)=0$ and $γ_{′}(0)∈C\[0,∞)$. The result is extended to higher order equations.

## Cite this article

Hua Chen, Hidetoshi Tahara, On Totally Characteristic Type Non-linear Partial Differential Equations in the Complex Domain. Publ. Res. Inst. Math. Sci. 35 (1999), no. 4, pp. 621–636

DOI 10.2977/PRIMS/1195143496