Blow-up of solutions of semilinear parabolic equations
type: Item
instance of: scholarly article
title: Blow-up of solutions of semilinear parabolic equations
zbMATH Open document ID: 0673.35048
DOI: 10.1016/0022-247X(88)90052-2
author: A. Friedman
author: A. A. Lacey
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: Consider the blow-up behavior of solutions of a semilinear parabolic equation \[ (1)\quad u_ t+uu_ x-u_{xx}=f(u)\quad (-a<x<a,\quad t>0) \] with the initial-boundary conditions \(u(\pm a,t)=0\) \((t>0)\), \(u(x,0)=\phi (x)\) \((-a<x<a)\), where \(f(u)=u^ p\) if \(u\geq 0\), \(p>2\), \(\phi (\pm a)=0\), \(\phi\geq 0\) \((-a<x<a).\) This paper shows that (i) a finite time blow-up occurs if \(\phi\) is large enough; (ii) there is a single point blow-up if \(p>3\) and for a large class of initial values \(\phi\) ; (iii) if \(p>3\) then for a large class of \(\phi\) 's \[ u(x,t)\leq c(T-t)^{-1/(p-1)}\quad (c>0). \] These results can be extended to equations \(u_ t-u_{xx}=F(u,u_ x)\).
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B30
zbMATH DE Number: 4101717
zbMATH Keywords: blow-up of solutions
zbMATH Keywords: semilinear parabolic equations
zbMATH Keywords: blow-up
zbMATH Keywords: semilinear
zbMATH Keywords: initial-boundary conditions
zbMATH Keywords: finite time blow-up
zbMATH Keywords: single point blow-up
zbMATH Keywords: initial values
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4101717