JournalsprimsVol. 35, No. 4pp. 621–636

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