Threshold Behavior and Non-quasiconvergent Solutions with Localized Initial Data for Bistable Reaction–Diffusion Equations

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Abstract

We consider bounded solutions of the semilinear heat equation ut= ux x+ f(u) on R, where f is of the unbalanced bistable type. We examine the ω-limit sets of bounded solutions with respect to the locally uniform convergence. Our goal is to show that even for solutions whose initial data vanish at x= ± ∞, the ω-limit sets may contain functions which are not steady states. Previously, such examples were known for balanced bistable nonlinearities. The novelty of the present result is that it applies to a robust class of nonlinearities. Our proof is based on an analysis of threshold solutions for ordered families of initial data whose limits at infinity are not necessarily zeros of f.

Original languageEnglish (US)
Pages (from-to)605-625
Number of pages21
JournalJournal of Dynamics and Differential Equations
Volume28
Issue number3-4
DOIs
StatePublished - Sep 1 2016

Keywords

  • Asymptotic behavior
  • Bistable reaction–diffusion equation
  • Localized initial data
  • Nonconvergent solutions
  • Threshold solutions

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