Formation of stationary interfaces in the fast reaction limit

TL;DR

This paper analyzes a reaction-diffusion system with a large reaction term, showing that solutions develop a stationary interface separating reactive and nonreactive regions, with convergence to the heat equation away from the interface.

Contribution

It introduces a novel analysis using barrier functions and comparison arguments to characterize stationary interfaces in the fast reaction limit.

Findings

Solutions converge to the heat equation in nonreactive regions

A stationary interface is formed based on initial conditions

The method handles asymmetric reaction terms effectively

Abstract

We study a two-component reaction-diffusion system in which one of the reaction terms becomes singularly large. Assuming that the initial data are nonnegative and mutually segregated, we prove that the solution converges to that of the heat equation in the nonreactive region, while it remains zero elsewhere. The analysis is based on explicitly constructed barrier functions and a comparison argument adapted to systems with asymmetric reaction terms, which together provide control near the interface. As a result, the limiting behavior exhibits a stationary phase interface determined by the initial support of the reactive component.