Convergence of anisotropically decaying solutions of a supercritical semilinear heat equation

Peter Poláčik, Eiji Yanagida

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

We consider the Cauchy problem for a semilinear heat equation with a supercritical power nonlinearity. It is known that the asymptotic behavior of solutions in time is determined by the decay rate of their initial values in space. In particular, if an initial value decays like a radial steady state, then the corresponding solution converges to that steady state. In this paper we consider solutions whose initial values decay in an anisotropic way. We show that each such solution converges to a steady state which is explicitly determined by an average formula. For a proof, we first consider the linearized equation around a singular steady state, and find a self-similar solution with a specific asymptotic behavior. Then we construct suitable comparison functions by using the self-similar solution, and apply our previous results on global stability and quasi-convergence of solutions.

Original languageEnglish (US)
Pages (from-to)329-341
Number of pages13
JournalJournal of Dynamics and Differential Equations
Volume21
Issue number2
DOIs
StatePublished - Jun 1 2009

Keywords

  • Anisotropic decay
  • Critical exponent
  • Quasi-convergence
  • Self-similar solution
  • Semilinear parabolic equation

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