Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay

Abstract

The reaction-diffusion systems which are based on an isothermal autocatalytic chemical reaction involving both an autocatalytic step of the ( m + 1 ) (m+1) th order ( A + m B → ( m + 1 ) B A+mB\rightarrow (m+1)B ) and a decay step of the same order ( B → C B\rightarrow C ) have very rich and interesting dynamics. Previous studies in the literature indicate that traveling waves play a key role in understanding these interesting dynamical phenomena. However, there is a lack of rigorous proof of the existence of traveling waves to this system. Here we generalize this isothermal autocatalytic chemical reaction model and provide a rigorous proof of the existence of traveling waves for the resulting reaction-diffusion system which also includes the systems arising from epidemiology and the microbial growth in a flow reactor.