Vanishing, moving and immovable interfaces in fast reaction limits

Abstract

We consider a type of singular limit problem called the fast reaction limit. The problem of the fast reaction limit involves studying the behaviour of solutions of reaction–diffusion systems when the reaction speeds are very fast. Fast reaction limits of two-component systems have been studied in recent decades. In most of these systems, the fast reaction terms of each component are represented by the same function. Fast reaction limits of systems with different fast reaction terms are still far from being well understood. In this paper, we focus on a reaction–diffusion system for which the reaction terms consist of monomial functions of various powers. The behaviour of interfaces arising in the fast reaction limit of this system is studied. Depending on the powers, three types of behaviour are observed: (i) the initial interface vanishes instantaneously, (ii) the interface propagates at a finite speed, and (iii) the interface does not move.