Existence of Traveling Waves of General Gray-Scott Models

Abstract

This work gives a rigorous proof of the existence of propagating traveling waves of a nonlinear reaction–diffusion system which is a general Gray-Scott model of the pre-mixed isothermal autocatalytic chemical reaction of order m ( $$m > 1$$ ) between two chemical species, a reactant A and an auto-catalyst B, $$ A + m B \rightarrow (m+1) B$$ , and a super-linear decay of order $$ n > 1$$ , $$ B \rightarrow C$$ , where $$ 1< n < m$$ . Here C is an inert product. Moreover, we establish that the speed set for existence must lie in a bounded interval for a given initial value $$u_0$$ at $$ - \infty $$ . The explicit bound is also derived in terms of $$u_0$$ and other parameters. The same system also appears in a mathematical model of SIR type in infectious diseases.