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A note on the small-time development of the solution to a coupled, nonlinear, singular reaction-diffusion system

Published online by Cambridge University Press:  17 February 2009

J. A. Leach
J. A. Leach
Affiliation:
Department of Mathematics, University of Reading, Whiteknights, Reading, Berkshire RG6 6AX, UK; e-mail: [email protected].
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Abstract

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In this paper, we consider a coupled, nonlinear, singular (in the sense that the reaction terms in the equations are not Lipschitz continuous) reaction-diffusion system, which arises from a model of fractional order chemical autocatalysis and decay, with positive initial data. In particular, we consider the cases when the initial data for the the dimensionless concentration of the autocatalyst, β, is of (a) O(x−λ) or (b) O(e−σ x) at large x (dimensionless distance), where σ > 0 and λ are constants. While initially the dimensionless concentration of the reactant, α, is identically unity, we establish, by developing the small-t (dimensionless time) asymptotic structure of the solution, that the support of β(x, t) becomes finite in infinitesimal time in both cases (a) and (b) above. The asymptotic form for the location of the edge of the support of β as t → 0 is given in both cases.

Keywords

reaction-diffusionmatched asymptotic expansions.
Type
Research Article
Information
The ANZIAM Journal , Volume 47 , Issue 4 , April 2006 , pp. 581 - 591
Copyright
Copyright © Australian Mathematical Society 2006

References

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