The blow-up properties of solutions to semilinear heat equations with nonlinear boundary conditions
type: Item
instance of: scholarly article
title: The blow-up properties of solutions to semilinear heat equations with nonlinear boundary conditions
zbMATH Open document ID: 0926.35062
author: Zhigui Lin
author: Mingxin Wang
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1999-11-16T00:00:00Z
review text: The authors consider the problem \(u_t=u_{xx}+u^p\) in \((0,1)\times(0,T)\), \(u_x(0,t)=0\), \(u_x(1,t)=u^q(1,t)\), \(t\in(0,T)\), \(u(x,0)=u_0(x)\geq 0\), \(x\in[0,1]\), where \(p,q>0\) and \(u_0\not\equiv 0\). If \(\max(p,q)>1\) then the solution of this problem blows up in a finite time \(T\) in the \(L^\infty\)-norm. The authors study the corresponding blow-up rate and blow-up set. Denote \(U(t)=\sup_x| u(x,t)| \) and \(\alpha=1/\max(p-1,2q-2)\). Under some additional assumptions on \(u_0\) it is shown that \(0<c\leq U(t)(T-t)^\alpha\leq C<\infty\) and that blow-up occurs only at \(x=1\).
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 1320486
zbMATH Keywords: blow-up rate
zbMATH Keywords: blow-up set
reviewed by: Pavol Quittner
MaRDI profile type: Publication
full work available at URL: s000330050023
OpenAlex ID: W1974948562
DOI: 10.1007/S000330050023
description: scientific article; zbMATH DE number 1320486