This paper studies traveling front solutions of pyramidal shapes in the Allen-Cahn equation in $\mathbb{R}^{N}$ with $N\geq 3$.It is well known that two-dimensional V-form traveling fronts and three-dimensional pyramidal traveling fronts exist and are stable. The aim of this paper is to show that for $N\geq 4$,there exist $N$-dimensional pyramidal traveling fronts.We construct a supersolution and a subsolution, and find a pyramidaltraveling front solution between them. For the construction ofa supersolution we use a multi-scale method.
各種検索
サポート
ヘルプ