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Positivity and time behavior of a linear reaction–diffusion system, non-local in space and time

✍ Scribed by Andrii Khrabustovskyi; Holger Stephan

Publisher
John Wiley and Sons
Year
2008
Tongue
English
Weight
330 KB
Volume
31
Category
Article
ISSN
0170-4214

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✦ Synopsis

Abstract

We consider a general linear reaction–diffusion system in three dimensions and time, containing diffusion (local interaction), jumps (nonlocal interaction) and memory effects. We prove a maximum principle and positivity of the solution and investigate its asymptotic behavior. Moreover, we give an explicit expression of the limit of the solution for large times. In order to obtain these results, we use the following method: We construct a Riemannian manifold with complicated microstructure depending on a small parameter. We study the asymptotic behavior of the solution to a simple diffusion equation on this manifold as the small parameter tends to zero. It turns out that the homogenized system coincides with the original reaction–diffusion system. Using this and the facts that the diffusion equation on manifolds satisfies the maximum principle and its solution converges to a easily calculated constant, we can obtain analogous properties for the original system. Copyright © 2008 John Wiley & Sons, Ltd.

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