Existence of traveling waves of auto-catalytic systems with decay

Abstract

This article establishes the existence of traveling waves of a class of reaction–diffusion systems which model the pre-mixed isothermal autocatalytic chemical reaction of order m (m>1) between two chemical species, a reactant and an auto-catalyst, and a linear decay. Moreover, our result shows that the set of speed is contained in a bounded interval for any fixed initial value at x=−∞. This is in strong contrast to either the reaction–diffusion systems of autocatalytic chemical reaction of the order m without decay, or to the systems which have the same order of decay, which were shown by various authors (e.g. [8,13,17,26]) that the set of traveling wave speeds contains [c⁎,∞) for some c⁎>0.The same systems also appear in a mathematical model of microbial growth and competition in a flow reactor; see [2,24].