Blow-up in a reactive-diffusive medium with a moving heat source
type: Item
instance of: scholarly article
title: Blow-up in a reactive-diffusive medium with a moving heat source
zbMATH Open document ID: 0994.35068
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2002-05-21T00:00:00Z
review text: The authors find sufficient conditions for finite time blow-up and global existence of solutions of the Cauchy problem \(u_t-u_{xx}=\delta(x-x_0)F(u(x_0,t))\) in \(\mathbb R\times\mathbb R^+\), \(u(x,0)=\widehat u(x)\geq 0\), where \(\widehat u\in C_0(\mathbb R)\), \(x_0=x_0(t)\) is a prescribed continuous function, \(F\geq 0\) is a smooth increasing convex function and \(\delta\) is the Dirac distribution. It is shown that if the speed \(x_0'(t)\) of the heat source remains sufficiently high then the solution exists globally. On the other hand, if this speed remains sufficiently low and \(\int^\infty(1/F(t)) dt<\infty\) then the solution cannot exist globally. The results are obtained by studying an integral equation satisfied by \(u(x_0(t),t)\). Some special examples are examined.
reviewed by: Pavol Quittner
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 1743199
zbMATH Keywords: nonlinear heat equation
zbMATH Keywords: global existence
author: C. M. Kirk
author: W. E. Olmstead
MaRDI profile type: Publication
full work available at URL: s00033-002-8147-6
OpenAlex ID: W1646969508
DOI: 10.1007/S00033-002-8147-6
description: scientific article; zbMATH DE number 1743199