Global existence for the nonlinear heat equation in the Fujita subcritical case with initial value having zero mean value

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Abstract

In this paper we prove for 1<p<1+2/N+k, where k is an integer in [1, N], the existence of an initial value ψ, odd with respect to the k first coordinates, and with ∫ R{double-struck}Nx 1...x kψdx 1...dx N≠0, such that the resulting solution of u t-δu=|u| p-1u is global. In the case k=1 and 1<p<1+1N+1, it is known that the solution u with the initial value u(0)=λψ blows up in finite time if λ>0 either sufficiently small or sufficiently large. The result in this paper extends a similar result of Cazenave, Dickstein, and Weissler in the case k=0, i.e. with ∫ R{double-struck}Nψ≠0 and 1<p<1+2/N.

Original languageEnglish (US)
Pages (from-to)562-568
Number of pages7
JournalJournal of Mathematical Analysis and Applications
Volume389
Issue number1
DOIs
StatePublished - May 1 2012

Keywords

  • Blow-up
  • Global existence
  • Invariant manifold
  • Local existence
  • Nonlinear heat equation
  • Semiflow
  • Sign-changing solutions

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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