Self-similar fast-reaction limits for reaction-diffusion systems on unbounded domains

TL;DR

This paper characterizes the self-similar limits of reaction-diffusion systems on unbounded domains as the reaction rate becomes infinitely fast, extending previous work and describing substance penetration in chemical reactions.

Contribution

It introduces a unified approach to analyze fast-reaction limits for various reaction-diffusion systems on unbounded domains, including cases with different diffusion components and domain types.

Findings

Solutions converge to self-similar profiles as reaction rate increases

Long-time behavior also approaches these self-similar profiles

Results generalize earlier models to broader classes of reaction-diffusion systems

Abstract

We present a unified approach to characterising fast-reaction limits of systems of either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains, motivated by models of fast chemical reactions where either one or both reactant(s) is/are mobile. For appropriate initial data, solutions of four classes of problems each converge in the fast-reaction limit $k \to \infty$ to a self-similar limit profile that has one of four forms, depending on how many components diffuse and whether the spatial domain is a half or whole line. For fixed $k$, long-time convergence to these same self-similar profiles is also established, thanks to a scaling argument of Kamin. Our results generalise earlier work of Hilhorst, van der Hout and Peletier to a much wider class of problems, and provide a quantitative description of the penetration of one substance into another in both the fast-reaction and long-time regimes.