Existence of traveling waves of auto-catalytic systems with decay

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] .