TY - JOUR

T1 - Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations

AU - Kurokawa, Yu

AU - Taniguchi, Masaharu

N1 - Funding Information:
The authors express their sincere gratitude to Professor Hirokazu Ninomiya of Meiji University, Professor Eiji Yanagida of Tokyo Institute of Technology, Professor Wei-Ming Ni of University of Minnesota and Professor Hiroshi Matano of University of Tokyo for discussions and encouragement. This work was supported by Grant-in-Aid for Scientific Research (C) 18540208, Japan Society for the Promotion of Science.

PY - 2011/10

Y1 - 2011/10

N2 - We study travelling-front solutions of pyramidal shapes in the Allen-Cahn equation in RN 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. ©

AB - We study travelling-front solutions of pyramidal shapes in the Allen-Cahn equation in RN 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. ©

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U2 - 10.1017/S0308210510001253

DO - 10.1017/S0308210510001253

M3 - Article

AN - SCOPUS:80054939501

VL - 141

SP - 1031

EP - 1054

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 5

ER -