Sharp Estimates on Minimum Travelling Wave Speed of Reaction Diffusion Systems Modelling Autocatalysis

Abstract

This article studies propagating wave fronts in an isothermal chemical reaction $ A + 2B \rightarrow 3B$ involving two chemical species, a reactant A and an autocatalyst B, whose diffusion coefficients, $D_A$ and $D_B$, are unequal due to different molecular weights and/or sizes. Explicit bounds $v_*$ and $v^*$ that depend on $D_B/D_A$ are derived such that there is a unique travelling wave of every speed $v \geq v^*$ and there does not exist any travelling wave of speed $v<v_*$. New to the literature, it is shown that $v_*\propto v^* \propto D_B/ D_A$ when $D_B \leq D_A $. Furthermore, when $D_A \leq D_B$, it is shown rigorously that there exists a $v_{\min}$ such that there is a travelling wave of speed v if and only $v \geq v_{\min}$. Estimates on $v_{\min}$ significantly improve those of early works. The framework is built upon general isothermal autocatalytic chemical reactions $A + n B \rightarrow (n+1) B$ of arbitrary order $n \geq 1$.