Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains

J. Földes, P. Poláčik

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the Dirichlet problem for a class of semilinear parabolic equations on a bounded domain which is reflectionally symmetric about a hyperplane H. The equations consist of a symmetric time-autonomous part and a nonsymmetric perturbation which decays to zero as time approaches infinity. In our first theorem, we prove the asymptotic symmetry of each bounded positive solution of this asymptotically symmetric problem. The novelty of this result is that the solutions considered are not assumed uniformly positive, which prevents one from applying common techniques based on the method of moving hyperplanes. In our second main theorem, we classify the positive entire solutions of the unperturbed time-autonomous problems. In particular, we characterize all entire solutions, which are not symmetrically decreasing in the direction orthogonal to H, as connecting orbits from an equilibrium with a nontrivial nodal set to another invariant set.

Original languageEnglish (US)
Pages (from-to)1859-1888
Number of pages30
JournalJournal of Differential Equations
Volume258
Issue number6
DOIs
StatePublished - Mar 15 2015

Keywords

  • Asymptotic symmetry
  • Classification of entire solutions
  • Equilibria with a nontrivial nodal set
  • Morse decomposition
  • Semilinear parabolic equations

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