A convolution model for interfacial motion: the generation and propagation of internal layers in higher space dimensions

Abstract

Properties of solutions of a bistable nonlinear convolution equation in higher space dimensions are studied. The nonlinearity is the derivative of a double well function. The theory of traveling waves for this equation was given in a previous publication ([4]). Here we consider spreading phenomena for state regions, in some cases by means of the motion of domain walls, which are modeled by internal layers. These phenomena are analogous to those known for the bistable nonlinear diffusion equation, and in particular, for the Allen--Cahn equation, which is a model for the motion of some grain boundaries in solid materials. Cases when the two wells have unequal depth are considered, as well as when they have equal depth. In the latter case a motion-by-curvature law is derived formally in two space dimensions.