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In this paper, we present an approach to characterising self-similarfast-reaction limits of systems with nonlinear diffusion. For appropriateinitial data, in the fast-reaction limit as k tends to infinithy,spatialsegregation results in the two components of the original systems converging tothe positive and negative parts of a self-similar limit profile f (n), wheren=xt^(1/2) that satisfies one of four ordinary differential systems. Theexistence of these self-similar solutions of the k tends to infinity limitproblems is proved by using shooting methods which focus on a, the position ofthe free boundary which separates the regions where the solution is positiveand where it is negative, and g, the derivative of -phi(f) at n = a. Theposition of the free boundary gives us intuition about how one substancepenetrates into the other, and for specific forms of nonlinear diffusion, therelationship between the given form of the nonlinear diffusion and the positionof the free boundary is also studied.

Self-similar solutions of fast reaction limit problem with nonlinear diffusion