Fast reaction limit with nonmonotone reaction function

TL;DR

This paper investigates the behavior of a reaction-diffusion system with a nonmonotone reaction function as the reaction speed becomes infinite, revealing oscillations and convergence properties of the system.

Contribution

It precisely characterizes the Young measure of the non-diffusing component and extends analysis techniques for forward-backward parabolic equations using kinetic functions.

Findings

Identification of the Young measure for the non-diffusing component

Proof of strong convergence of the diffusing component

Relaxed assumptions and new weak formulation for the system

Abstract

We analyse fast reaction limit in the reaction-diffusion system with nonmonotone reaction function and one non-diffusing component. As speed of reaction tends to infinity, the concentration of non-diffusing component exhibits fast oscillations. We identify precisely its Young measure which, as a by-product, proves strong convergence of the diffusing component, a result that is not obvious from a priori estimates. Our work is based on analysis of regularization for forward-backward parabolic equations by Plotnikov. We rewrite his ideas in terms of kinetic functions which clarifies the method, brings new insights, relaxes assumptions on model functions and provides a weak formulation for the evolution of the Young measure.