Abstract
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. ©
| 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 | |
| Publication status | Published - Oct 2011 |
| Externally published | Yes |
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ASJC Scopus subject areas
- Mathematics(all)
Cite this
Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations. / Kurokawa, Yu; Taniguchi, Masaharu.
In: Proceedings of the Royal Society of Edinburgh Section A: Mathematics, Vol. 141, No. 5, 10.2011, p. 1031-1054.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations
AU - Kurokawa, Yu
AU - Taniguchi, Masaharu
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. ©
UR - http://www.scopus.com/inward/record.url?scp=80054939501&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=80054939501&partnerID=8YFLogxK
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 -