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

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 $k\to \infty$, spatial segregation results in the two components of the original systems converging to the positive and negative parts of a self-similar limit profile $f\left(\eta \right)$, where $\eta =\frac{x}{\sqrt{t}}$, that satisfies one of four ordinary differential systems. The existence of these self-similar solutions of the $k\to \infty$ 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 γ, the derivative of $-\varphi \left(f\right)$ at $\eta =a$. 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.