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

TL;DR

This paper characterizes self-similar solutions in fast-reaction limits of systems with nonlinear diffusion, revealing how solutions segregate and how free boundaries evolve, using shooting methods for existence proofs.

Contribution

It introduces a method to analyze self-similar solutions in fast-reaction limits with nonlinear diffusion, including existence proofs and boundary behavior insights.

Findings

Self-similar solutions converge to positive and negative parts in the fast-reaction limit.

The free boundary's position indicates how substances penetrate each other.

Relationship between nonlinear diffusion form and free boundary position is established.

Abstract

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