Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 483 - 512, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.483

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Kelei Wang - Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, Wuhan 430071, China (email)

Abstract: In this paper we study the singularly perturbed parabolic system of competing species. This problem exhibit a ``phase separation" phenomena when the interaction between different species is very strong. We are concerned with the case where different species may have different diffusion rates. We identify its singular limit with the heat flow (i.e. gradient flow) of harmonic maps into a metric space with non-positive curvature, by establishing the system of differential inequalities satisfied by this heat flow and uniqueness of the solution to the corresponding initial-boundary value problem.

Keywords:  Phase separation, singular perturbation, free boundary problems, gradient flows.
Mathematics Subject Classification:  Primary: 35B25, 35R35; Secondary: 49N60.

Received: August 2013;      Revised: June 2014;      Available Online: August 2014.