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

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.