Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball
type: Item
instance of: scholarly article
title: Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball
zbMATH Open document ID: 0974.35055
published in: Studia Mathematica
publication date: 2001-10-14T00:00:00Z
full work available at URL: https://eudml.org/doc/216790
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35Q55
zbMATH DE Number: 1570922
zbMATH Keywords: nonlinear heat equation
zbMATH Keywords: fractional Laplacian
zbMATH Keywords: first initial-boundary value problem
zbMATH Keywords: long time asymptotics
author: V. V. Varlamov
MaRDI profile type: Publication
OpenAlex ID: W849899646
DOI: 10.4064/SM-142-1-71-99
review text: The aim of the paper is to study the nonlinear heat equation with a fractional Laplacian NEWLINE\[NEWLINEu_t+ (-\Delta)^{\alpha/2}n= u^2,\quad 0< \alpha\leq 2,NEWLINE\]NEWLINE in a unit ball. Homogeneous boundary conditions and small initial conditions are considered. The paper concentrates on the higher-order long time asymptotics. The first initial-boundary value problem is discussed in a unit ball and its solutions are constructed in the form of an eigenfunction expansion series, proving the well-posedness. The higher-order long time asymptotics are determined by means of nonlinear iterations. It is pointed out that the success of constructing solutions depends greatly on the convergence of the spatial eigenfunctions series and this convergence is rather poor.
reviewed by: István Ecsedi
description: scientific article