Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations

Yu Kurokawa; Masaharu Taniguchi

doi:10.1017/S0308210510001253

Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations

Yu Kurokawa, Masaharu Taniguchi

Research output: Contribution to journal › Article

33 Citations (Scopus)

Abstract

We study travelling-front solutions of pyramidal shapes in the Allen-Cahn equation in R^N with N 3. It is well known that two-dimensional V-form travelling fronts and three-dimensional pyramidal travelling fronts exist and are stable. The aim of this paper is to show that for N 4 there exist N-dimensional pyramidal travelling fronts. We construct a supersolution and a subsolution, and find a pyramidal travelling-front solution between them. For the construction of a supersolution we use a multi-scale method. ©

Original language	English
Pages (from-to)	1031-1054
Number of pages	24
Journal	Proceedings of the Royal Society of Edinburgh Section A: Mathematics
Volume	141
Issue number	5
DOIs	https://doi.org/10.1017/S0308210510001253
Publication status	Published - Oct 1 2011
Externally published	Yes

ASJC Scopus subject areas

Mathematics(all)

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Kurokawa, Y., & Taniguchi, M. (2011). Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 141(5), 1031-1054. https://doi.org/10.1017/S0308210510001253

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