Travelling waves of auto-catalytic chemical reaction of general order—An elliptic approach

In this paper we study the existence and non-existence of travelling wave to parabolic system of the form at=axx−af(b), bt=Dbxx+af(b), with f a degenerate nonlinearity. In the context of an auto-catalytic chemical reaction, a is the density of a chemical species called reactant A, b that of another chemical species B called auto-catalyst, and D=DB/DA>0 is the ratio of diffusion coefficients, DB of B and DA of A, respectively. Such a system also arises from isothermal combustion. The nonlinearity is called degenerate, since f(0)=f′(0)=0. One case of interest in this article is the propagating wave fronts in an isothermal auto-catalytic chemical reaction of order with 11, it is shown rigorously that there exists a vmin such that there is a travelling wave of speed v if and only if v⩾vmin. Estimates on vmin improve significantly that of early works. Another case in which two different orders of isothermal auto-catalytic chemical reactions are involved is also studied with interesting new results proved.