The asymptotic behavior of singular solutions of some nonlinear partial differential equations in the complex domain
type: Item
instance of: scholarly article
title: The asymptotic behavior of singular solutions of some nonlinear partial differential equations in the complex domain
zbMATH Open document ID: 1176.35004
author: Sunao Ōuchi
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2009-04-14T00:00:00Z
review text: Let \(u(t, x)\) \(((t, x)\in C\times C^d)\) be a solution of nonlinear partial differential equation \[ Lu= L(t,x,(t\partial_t)^{\alpha_0}u(t, x); (\alpha_0, \alpha')\in \mathbb{N}\times \mathbb{N}^d,|\alpha|\leq m)= 0 \] in a neighborhood of the origin, which is not necessary holomorphic on \(\{t=0\}\). Then under some conditions on \(L\) there exist \(u_n(t, x)\) represented by Mellin type integral and y > 0, vo such that \[ |u_n(,x)|\leq C_0 C^m\Gamma\Biggl({p_n\over\gamma}+ 1\Biggr)|t|^{p_n+ \nu_0}, \] \[ \Biggl| u(t,x)- \sum^{N-1}_{n=0} u_n(t, x)\Biggr|\leq AB^N\Gamma\Biggl({p_N\over\gamma}+ 1\Biggr)|t|^{p_n+ \nu_0}, \] where \(0= p_0< p_1<\cdots< p_n<\to +\infty\).
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35C20
zbMATH DE Number: 5543220
zbMATH Keywords: singular solution
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: Mellin transform
reviewed by: Kunihiko Kajitani
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1231263777
OpenAlex ID: W2098167712
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DOI: 10.2977/PRIMS/1231263777
description: scientific article; zbMATH DE number 5543220