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Asymptotic profiles of a decaying contaminant in transport through a porous medium

Published online by Cambridge University Press:  26 September 2008

M. Escobedo  and
R. E. Grundy
M. Escobedo
Affiliation:
Department of Mathematics, Universidad del Pais Vasco, Bilbao, Spain
R. E. Grundy
Affiliation:
Department of Mathematical and Computational Sciences, University of St Andrews, Scotland
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Abstract

In this paper we construct formal large-time solutions of a model equation, with initial data possessing bounded support, describing transport of a reacting and decaying contaminant in a porous medium. This we do in one, two and three space dimensions where, depending on the reaction model used, the solution may or may not have bounded support for all time. In the former case, working with what we call the reduced equation, we prove convergence, in one space dimension, to an outer limit as t → ∞. The outer solution has to be supplemented by inner solutions valid near the edges of the support. These inner solutions take the form of decaying travelling waves which we analyse using phase plane methods. Using the travelling waves as sub- and super-solutions, we establish the large-time behaviour of the interfaces which we refine using asymptotic matching. These ideas can be formally extended to higher space dimensions where we deduce the shape of the support of the large-time profiles which turns out to be ellipsoidal. For reaction models where the support is unbounded we prove convergence of the solution of the reduced equation to a travelling and decaying fundamental solution of the linear heat equation with convection and absorption. Finally, we indicate how the results for the reduced equations can be formally embedded in an asymptotic analysis of the original model equation.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 7 , Issue 4 , August 1996 , pp. 395 - 416
Copyright
Copyright © Cambridge University Press 1996

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