An (N - 1)-dimensional convex compact set gives an N-dimensional traveling front in the Allen-Cahn equation

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21 Citations (Scopus)

Abstract

This paper studies traveling fronts to the Allen-Cahn equation in ℝN for N ≥ 3. Let (N - 2)-dimensional smooth surfaces be the boundaries of compact sets in ℝN - 1 and assume that all principal curvatures are positive everywhere. We define an equivalence relation between them and prove that there exists a traveling front associated with a given surface and that it is asymptotically stable for given initial perturbation. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation.

Original languageEnglish
Pages (from-to)455-476
Number of pages22
JournalSIAM Journal on Mathematical Analysis
Volume47
Issue number1
DOIs
Publication statusPublished - 2015

Fingerprint

Allen-Cahn Equation
Travelling Fronts
Compact Convex Set
Equivalence relation
Principal curvature
Smooth surface
Asymptotically Stable
Compact Set
Phase Transition
Phase transitions
If and only if
Perturbation

Keywords

  • Allen-Cahn equation
  • Nonsymmetric
  • Traveling front

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Computational Mathematics

Cite this

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AB - This paper studies traveling fronts to the Allen-Cahn equation in ℝN for N ≥ 3. Let (N - 2)-dimensional smooth surfaces be the boundaries of compact sets in ℝN - 1 and assume that all principal curvatures are positive everywhere. We define an equivalence relation between them and prove that there exists a traveling front associated with a given surface and that it is asymptotically stable for given initial perturbation. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation.

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