The singular limit problem in a phase separation model with different diffusion rates $^*$

Abstract

In this paper we study the singularly perturbed parabolic system ofcompeting species. This problem exhibit a ``phase separation'phenomena when the interaction between different species is verystrong. We are concerned with the case where different species mayhave different diffusion rates. We identify its singular limit withthe heat flow (i.e. gradient flow) of harmonic maps into a metricspace with non-positive curvature, by establishing the system ofdifferential inequalities satisfied by this heat flow and uniquenessof the solution to the corresponding initial-boundary value problem.