WIAS Preprint No. 1264, (2007)

Positivity and time behavior of a general linear evolution system, non-local in space and time


  • Stephan, Holger
    ORCID: 0000-0002-6024-5355
  • Khrabustovskyi, Andrii

2010 Mathematics Subject Classification

  • 35B27 35K60


  • diffusion-reaction systems, positive solutions, maximum principle, homogenization, Riemannian manifold




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 explicite 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 of 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 what allows us to investigate its properties.

Appeared in

  • Math. Methods Appl. Sci., 31 (2008) pp. 1809--1834.

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