Fast reaction limit of reaction diffusion systems with nonlinear diffusion

TL;DR

This paper investigates the fast-reaction limits of reaction-diffusion systems with nonlinear diffusion, establishing convergence to a scalar nonlinear diffusion problem on unbounded domains.

Contribution

It introduces a novel approach to characterize the fast-reaction limit in systems with nonlinear diffusion, extending existing methods to more complex diffusion operators.

Findings

Proves convergence of reaction-diffusion systems with nonlinear diffusion as reaction rate tends to infinity.

Identifies the limit problem as a scalar nonlinear diffusion equation.

Handles systems with two equations or mixed reaction-diffusion and ODE systems on unbounded domains.

Abstract

In this paper, we present an approach to characterising fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains. Here, we replace the terms of the form uxx in usual reaction-diffusion equation, which represent linear diffusion, by terms of form phi(u)xx, representing nonlinear diffusion. We prove the convergence as k tends to infinity to a limit that is determined by the unique solution of a certain scalar nonlinear diffusion limit problem.