Singular limit of a spatially inhomogeneous Lotka-Volterra competition-diffusion system

We discuss the generation and the motion of internal layers for a Lotka-Volterra competition-diffusion system with spatially inhomogeneous coefficients. We assume that the corresponding ODE system has two stable equilibria ((u) over bar, 0) and (0, (upsilon) over bar) with equal strength of attraction in the sense to be specified later. The equation involves a small parameter epsilon, which reflects the fact that the diffusion is very small compared with the reaction terms. When the parameter epsilon is very small, the solution develops a clear transition layer between the region where the u species is dominant and the one where the upsilon species is dominant. As epsilon tends to zero, the transition layer becomes a sharp interface, whose motion is subject to a certain law of motion, which is called the "interface equation". A formal asymptotic analysis suggests that the interface equation is the motion by mean curvature coupled with a drift term. We will establish a rigorous mathematical theory both for the formation of internal layers at the initial stage and for the motion of those layers in the later stage. More precisely, we will show that, given virtually arbitrary smooth initial data, the solution develops an internal layer within the time scale of O(epsilon(2) log epsilon) and that the width of the layer is roughly of O(epsilon). We will then prove that the motion of the layer converges to the formal interface equation as epsilon -> 0. Our results also give an optimal convergence rate, which has not been known even for spatially homogeneous problems.