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 a(t) = a(xx) - af (b), b(1) = Db(xx) + 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 = D-B/D-A > 0 is the ratio of diffusion coefficients, D-B of B and D-A 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 autocatalytic chemical reaction of order n: A + nB -> (n + 1)B with 1 < n < 2, and D not equal 1 due to different molecular weights and/or sizes of A and B. The resulting nonlinearity is f (b) = b(n). Explicit bounds v(*) and v* that depend on D are derived such that there is a unique travelling wave of every speed v >= v* and there does not exist any travelling wave of speed v < v(*). New to the literature, it is shown that v(*) alpha v* alpha D when D < 1. Furthermore, when D > 1, it is shown rigorously that there exists a v(min) such that there is a travelling wave of speed v if and only if v >= v(min). Estimates on v(min) 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. (C) 2009 Elsevier Inc. All rights reserved.