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Positivity and time behavior of a linear reaction–diffusion system, non‐local in space and time
Author(s) -
Khrabustovskyi Andrii,
Stephan Holger
Publication year - 2008
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.998
Subject(s) - mathematics , reaction–diffusion system , manifold (fluid mechanics) , diffusion , limit (mathematics) , mathematical analysis , space (punctuation) , constant (computer programming) , simple (philosophy) , zero (linguistics) , riemannian manifold , order (exchange) , diffusion equation , physics , thermodynamics , mechanical engineering , linguistics , philosophy , economy , epistemology , computer science , engineering , economics , programming language , service (business) , finance
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.