Location of blow-up set for a semilinear parabolic equation with large diffusion
type: Item
instance of: scholarly article
title: Location of blow-up set for a semilinear parabolic equation with large diffusion
zbMATH Open document ID: 1049.35022
author: Kazuhiro Ishige
author: Noriko Mizoguchi
published in: Mathematische Annalen
publication date: 2004-07-07T00:00:00Z
review text: The paper deals with the problem \[ \begin{cases} u_t=D\Delta u+u^p &\text{in}\;\Omega\times(0,T_D),\\ \displaystyle {{\partial u}\over {\partial \nu}}(x,t)=0 &\text{on}\;\partial\Omega\times(0,T_D),\\ u(x,0)=\phi(x)\geq0 &\text{in}\;\Omega, \end{cases} \] where \(\Omega\) is a bounded and smooth domain in \({\mathbb R}^N,\) \(D>0,\) \(T_D>0\) and \(p>1\) with \((N-2)p\leq N+2.\) Set \(P_2\) for the projection of \(L^2(\Omega)\) onto the second Neumann eigenspace. The authors prove that, if \(P_2\phi\not\equiv0\) in \(\Omega\) and if \(D\) is large enough then the solution \(u(x,t)\) of the above problem blows-up only near the set \({\mathcal M}\cup \partial \Omega\), where \({\mathcal M}=\{x\in\overline\Omega: (P_2\phi)(x)=\max_{y\in\overline\Omega} (P_2\phi)(y)\}.\)
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 2078341
zbMATH Keywords: semilinear parabolic equation
zbMATH Keywords: initial-boundary value problem
zbMATH Keywords: blow-up
zbMATH Keywords: large diffusion
zbMATH Keywords: second Neumann eigenspace
MaRDI profile type: Publication
full work available at URL: s00208-003-0463-4
OpenAlex ID: W2047688363
DOI: 10.1007/S00208-003-0463-4
description: scientific article; zbMATH DE number 2078341