Traveling fronts of pyramidal shapes in the Allen-Cahn equations

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This paper studies pyramidal traveling fronts in the Allen-Cahn equation or in the Nagumo equation. For the nonlinearity we are concerned mainly with the bistable reaction term with unbalanced energy density. Two-dimensional V-form waves and cylindrically symmetric waves in higher dimensions have been recently studied. Our aim in this paper is to construct truly threedimensional traveling waves. For a pyramid that satisfies a condition, we construct a traveling front for which the contour line has a pyramidal shape. We also construct generalized pyramidal fronts and traveling waves of a hybrid type between pyramidal waves and planar V-form waves. We use the comparison principles and construct traveling fronts between supersolutions and subsolutions.

Original languageEnglish
Pages (from-to)319-344
Number of pages26
JournalSIAM Journal on Mathematical Analysis
Issue number1
Publication statusPublished - 2007
Externally publishedYes



  • Allen-Cahn equation
  • Bistable
  • Pyramidal traveling wave

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis
  • Applied Mathematics
  • Numerical Analysis

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