Explosion in a diffusive strip due to a concentrated nonlinear source
type: Item
instance of: scholarly article
title: Explosion in a diffusive strip due to a concentrated nonlinear source
zbMATH Open document ID: 0838.35062
author: C. A. Roberts
author: W. E. Olmstead
published in: Methods and Applications of Analysis
publication date: 1996-05-30T00:00:00Z
review text: The semilinear heat equation \(v_t= v_{xx}+ F(x, v)\) for \(0< x< \ell\), \(t> 0\), is studied for the temperature \(v(x, t)\), where the nonlinear source has a strongly concentrated and localized spatial dependence of the form \(F(x, v)= \delta(x- a) g(v)\) for \(0< a< \ell\), with \(\delta(x- a)\) being the Dirac delta distribution. The initial condition \(v(x, 0)= v_0(x)\), is imposed, and homogeneous Dirichlet or Neumann boundary conditions. It is assumed that \(g(v)\) is a smooth function and has the properties \(g(v)> 0\), \(g'(v)> 0\), \(g''(v)> 0\) for \(v> 0\), and \(\int^\infty_{h_0} dv/g(v)< \infty\) for some \(h_0> 0\). The authors prove that blow-up always occurs for the Neumann problem, whereas for the Dirichlet problem it depends upon the length \(\ell\) of the strip and proximity of the source site \(a\) to the boundary. For those situations in which blow-up does occur the growth rate is determined for certain class of nonlinearities. The proofs are achieved by reducing the initial-boundary value problem to a nonlinear Volterra equation for \(v(a, t)\), and then studying it for blow-up solutions and the growth rate of such blow-up.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 45M05
Mathematics Subject Classification ID: 35R05
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 818499
zbMATH Keywords: concentrated nonlinear source
zbMATH Keywords: semilinear heat equation
zbMATH Keywords: blow-up
zbMATH Keywords: growth rate
zbMATH Keywords: nonlinear Volterra equation
MaRDI profile type: Publication
full work available at URL: maa.1994.v1.n4.a5
OpenAlex ID: W185230949
DOI: 10.4310/MAA.1994.V1.N4.A5
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description: scientific article; zbMATH DE number 818499