On Gevrey solutions of threefold singular nonlinear partial differential equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:255

Article

Lastra, Alberto¹  Malek, Stephane²  Sanz, Javier³ 

[1] Univ Alcala, Dept Fis & Matemat, Madrid, Spain

[2] Univ Lille 1, UFR Math Pures & Appl, F-59655 Villeneuve Dascq, France

[3] Univ Valladolid, Fac Ciencias, Dept Algebra Anal Matemat Geometria & Topol, E-47011 Valladolid, Spain

关键词: Nonlinear partial differential equations;    Singular perturbations;    Formal power series;    Borel-Laplace transform;    Borel summability;    Gevrey asymptotic expansions;   

DOI  :  10.1016/j.jde.2013.07.031

来源: Elsevier

【 摘 要 】

We study Gevrey asymptotics of the solutions to a family of threefold singular nonlinear partial differential equations in the complex domain. We deal with both Fuchsian and irregular singularities, and allow the presence of a singular perturbation parameter. By means of the Borel-Laplace summation method, we construct sectorial actual holomorphic solutions which turn out to share a same formal power series as their Gevrey asymptotic expansion in the perturbation parameter. This result rests on the Malgrange-Sibuya theorem, and it requires to prove that the difference between two neighboring solutions is exponentially small, what in this case involves an asymptotic estimate for a particular Dirichlet-like series. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free