# A Maillet Type Theorem for First Order Singular Nonlinear Partial Differential Equations

### Akira Shirai

Nagoya University, Nagoya, Chikusa-Ku, Japan

## Abstract

We shall study the ﬁrst order singular nonlinear partial diﬀerential equation of the form *f*(*x*, *u*(*x*), ∂_x__u_(*x*)) = 0 with *u*(0) = 0, where *x* ∈ ℂ_n_ and *f*(*x*, *u*, ξ) is holomorphic in a neighborhood of the origin. This equation is said to be singular if *f*(0, 0, ξ) ≡ 0 for all ξ ∈ ℂ_n_. The purpose of this paper is to study the Maillet type theorem which means to determine the Gevrey order of a formal power series solution which may diverge.

## Cite this article

Akira Shirai, A Maillet Type Theorem for First Order Singular Nonlinear Partial Differential Equations. Publ. Res. Inst. Math. Sci. 39 (2003), no. 2, pp. 275–296

DOI 10.2977/PRIMS/1145476104