Stability of planar waves in the Allen-Cahn equation

Hiroshi Matano, Mitsunori Nara, Masaharu Taniguchi

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36 Citations (Scopus)

Abstract

We study the asymptotic stability of planar waves for the Allen-Cahn equation on ℝn, where n ≥ 2. Our first result states that planar waves are asymptotically stable under any-possibly large-initial perturbations that decay at space infinity. Our second result states that the planar waves are asymptotically stable under almost periodic perturbations. More precisely, the perturbed solution converges to a planar wave as t → ∞. The convergence is uniform in ℝn. Lastly, the existence of a solution that oscillates permanently between two planar waves is shown, which implies that planar waves are not asymptotically stable under more general perturbations.

Original languageEnglish
Pages (from-to)976-1002
Number of pages27
JournalCommunications in Partial Differential Equations
Volume34
Issue number9
DOIs
Publication statusPublished - Sep 1 2009

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Keywords

  • Allen-Cahn equation
  • Almost periodic
  • Asymptotic stability
  • Planar wave

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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