Vanishing interfaces in an asymmetric fast reaction limit

TL;DR

This paper analyzes the rapid reaction limit in a two-component reaction-diffusion system with asymmetric reactions, showing the initial interface disappears instantly and the diffusive component follows the heat equation.

Contribution

It proves the instantaneous vanishing of interfaces and convergence to heat equation solutions in asymmetric reaction-diffusion systems with explicit barrier methods.

Findings

Initial interface vanishes instantaneously for nonnegative, segregated data.

Diffusive component converges uniformly to the heat equation solution.

Non-diffusive component vanishes away from initial time.

Abstract

We study the fast reaction limit for a two-component reaction-diffusion system with asymmetric reaction terms, where only one component diffuses. For nonnegative and mutually segregated initial data, we prove that the initial interface vanishes instantaneously. More precisely, the diffusive component converges uniformly to the solution of the heat equation, while the non-diffusive component vanishes away from the initial time. The proof is based on explicit barriers and a comparison argument, and applies under both Dirichlet and Neumann boundary conditions.