Intermediate dynamics of internal layers for a nonlocal reaction-diffusion equation

Abstract

A singular perturbation problem for a reaction-di¤usion equation with a nonlocal term is treated. We derive an interface equation which describes the dynamics of internal layers in the intermediate time scale, i.e., in the time scale after the layers are generated and before the interfaces are governed by the volume-preserving mean curvature flow. The unique existence of solutions for the interface equation is demonstrated. A continuum of equilibria for the interface equation are identified and the stability of the equilibria is established. We rigorously prove that layer solutions of the nonlocal reaction-di¤usion equation converge to solutions of the interface equation on a finite time interval as the singular perturbation parameter tends to zero.