Singular Limits of Porous Media Equations with Bistable Reactions

Abstract: We consider a porous media equation with balanced bistable reactions, equipped with some general nonlinear boundary condition. When the coefficient of the reaction term is much larger than that of the diffusion term, we see that, besides the possible free boundary, sharp interfaces appear between two stable steady states. By using the method of matched asymptotic expansions, we derive the motion law of each interface, which is a mean curvature flow (may depends on normal direction of the interface). In addition, the original boundary condition reduces to Robin ones at the points where the interface contacts the domain boundary.