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

### Hidetoshi Tahara

Sophia University, Tokyo, Japan### Hua Chen

Wuhan University, Wuhan, Hubei, China

## 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*) ∈ ℂ2 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 Gerard-Tahara [3]. In case *γ*(0) = 0 and Re_γ_' < 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

Hidetoshi Tahara, Hua Chen, 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