Two-scale homogenization of nonlinear   reaction-diffusion systems with slow diffusion

Alexander Mielke; Sina Reichelt; Marita Thomas

doi:10.3934/nhm.2014.9.353

Article Contents

2014, Volume 9, Issue 2: 353-382. Doi: 10.3934/nhm.2014.9.353

This issue Previous Article A revisit to the consensus for linearized Vicsek model under joint rooted leadership via a special matrix Next Article

Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion

1.
Weierstrass Institute, Mohrenstraße 39, 10117 Berlin
2.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin

Received: October 31, 2013

Revised: March 31, 2014

Published: May 2014

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Abstract

We derive a two-scale homogenization limit for reaction-diffusion systems where for some species the diffusion length is of order 1 whereas for the other species the diffusion length is of the order of the periodic microstructure. Thus, in the limit the latter species will display diffusion only on the microscale but not on the macroscale. Because of this missing compactness, the nonlinear coupling through the reaction terms cannot be homogenized but needs to be treated on the two-scale level. In particular, we have to develop new error estimates to derive strong convergence results for passing to the limit.

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Mathematics Subject Classification: 35B25, 35A01, 35K57, 35K65, 35M10.

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