Intermediate dynamics of internal layers for a nonlocal reaction-diffusion equation. (English) Zbl 1097.35017

Summary: A singular perturbation problem for a reaction-diffusion 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-diffusion equation converge to solutions of the interface equation on a finite time interval as the singular perturbation parameter tends to zero.


35B25 Singular perturbations in context of PDEs
35K57 Reaction-diffusion equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations