Communications in Information and Systems

Volume 13 (2013)

Number 1

Special Issue in Honor of Marshall Slemrod: Part 1 of 4

Asymptotic behavior of minima and mountain pass solutions for a class of Allen-Cahn models

Pages: 75 – 95

DOI: https://dx.doi.org/10.4310/CIS.2013.v13.n1.a3

Authors

Jaeyoung Byeon (Department of Mathematical Sciences, KAIST, Yuseong-gu, Daejeon, Korea)

Paul H. Rabinowitz (Department of Mathematics, University of Wisconsin, Madison, Wisc., U.S.A.)

Abstract

In an earlier paper, the authors studied a class of Allen-Cahn models for which the solution was near $1$ on a prescribed set, $T +[0, 1]^n$ where $T \subset \mathbb{Z}^n$, and near $0$ on its complement. In this note, when $T$ is finite and consists of two widely spaced subsets, $T_1$ and $l + T_2$ with $l \in \mathbb{Z}^n$, we study the asymptotic behavior of two special families of solutions as $l \to \infty$.

Published 28 May 2014