A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation

Peter Poláčik, Pavol Quittner

Research output: Contribution to journalArticlepeer-review

33 Scopus citations

Abstract

We consider the semilinear parabolic equation ut=Δu+up on RN, where the power nonlinearity is subcritical. We first address the question of existence of entire solutions, that is, solutions defined for all x∈RN and t∈R. Our main result asserts that there are no positive radially symmetric bounded entire solutions. Then we consider radial solutions of the Cauchy problem. We show that if such a solution is global, that is, defined for all t≥0, then it necessarily converges to 0, as t→∞, uniformly with respect to x∈RN.

Original languageEnglish (US)
Pages (from-to)1679-1689
Number of pages11
JournalNonlinear Analysis, Theory, Methods and Applications
Volume64
Issue number8
DOIs
StatePublished - Apr 15 2006

Bibliographical note

Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

Keywords

  • Decay of global solutions
  • Entire solutions
  • Liouville theorem
  • Subcritical semilinear heat equation

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