The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations

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58 Citations (Scopus)

Abstract

This paper studies the uniqueness and the asymptotic stability of a pyramidal traveling front in the three-dimensional whole space. For a given admissible pyramid we prove that a pyramidal traveling front is uniquely determined and that it is asymptotically stable under the condition that given perturbations decay at infinity. For this purpose we characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces. Moreover we characterize the pyramidal traveling front in another way, that is, we write it as a combination of two-dimensional V-form waves on the edges. This characterization also uniquely determines a pyramidal traveling front.

Original languageEnglish
Pages (from-to)2103-2130
Number of pages28
JournalJournal of Differential Equations
Volume246
Issue number5
DOIs
Publication statusPublished - Mar 1 2009
Externally publishedYes

Keywords

  • Allen-Cahn equation
  • Pyramidal traveling wave
  • Stability

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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