Motion of interfaces by an Allen–Cahn type equation with multiple-well potentials

Article type: Research Article

Authors: Ohtsuka, Takeshi

Affiliations: Department of Mathematics, Faculty of Science and Technology, Sophia University, 7-1, Kioi-cho, Chiyoda-ku, Tokyo 102-8554, Japan. E-mail: ohtsuka@mm.sophia.ac.jp

Abstract: We consider the singular limit of an Allen–Cahn type equation with a periodic nonlinear term. In this case we can find a multi-step internal transition layer when the interface thickness parameter tends to zero. We give a rigorous proof of the convergence of internal transition layers to interfaces which move under a mean curvature flow with a driving force, even if no traveling wave solutions connecting two nonadjacent stable equilibria exist.

Keywords: Allen–Cahn equation, multiple-well potential, mean curvature flow, viscosity solution

Journal: Asymptotic Analysis, vol. 56, no. 2, pp. 87-123, 2008