Title

Travelling Waves Of Auto-Catalytic Chemical Reaction Of General Order-An Elliptic Approach

Keywords

General order auto-catalysis; Minimum speed; Reaction-diffusion; Travelling wave

Abstract

In this paper we study the existence and non-existence of travelling wave to parabolic system of the form at = ax x - a f (b), bt = D bx 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 = 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 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) = bn. 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 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. © 2009 Elsevier Inc. All rights reserved.

Publication Date

4-15-2009

Publication Title

Journal of Differential Equations

Volume

246

Issue

8

Number of Pages

3038-3057

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.jde.2009.01.015

Socpus ID

61849170099 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/61849170099

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