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

Abstract In this paper we study the existence and non-existence of travelling wave to parabolic system of the form a t = a x x − a f ( b ) , b t = D b x x + a f ( 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 auto-catalytic chemical reaction of order n : A + n B → ( n + 1 ) B with 1 n 2 , and D ≠ 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 ∗ ∝ v ∗ ∝ 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.

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