Multisummability of formal power series solutions of nonlinear partial differential equations in complex domains.

*(English)*Zbl 1152.35015This paper deals with the multisummability of formal solutions of some nonlinear partial differential equations. It is a continuation of the author’s previous work on the multisummability of formal solutions of linear partial differential equations [S. Ōuchi, J. Differ. Equations 185, 513–549 (2002; Zbl 1020.35018)]. Roughly speaking, a nonlinear partial differential equation discussed here is an equation which can be regarded as a perturbation of an ordinary differential equation.

A very rough sketch of the proof of the multisummability of formal solutions is as follows: First, the author converts a partial differential equation in question into convolution equations by applying the Borel transformation. Then, after introducing appropriate majorant functions and preparing some estimates, he shows (i) the existence of solutions of the convolution equations on compact sets, (ii) their holomorphic extensibility, and (iii) their growth estimate.

A very rough sketch of the proof of the multisummability of formal solutions is as follows: First, the author converts a partial differential equation in question into convolution equations by applying the Borel transformation. Then, after introducing appropriate majorant functions and preparing some estimates, he shows (i) the existence of solutions of the convolution equations on compact sets, (ii) their holomorphic extensibility, and (iii) their growth estimate.

Reviewer: Yoshitsugu Takei (Kyoto)