Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems

Boris Andreianov; Halima Labani

doi:10.3934/cpaa.2012.11.2179

Article Contents

2012, Volume 11, Issue 6: 2179-2199. Doi: 10.3934/cpaa.2012.11.2179

This issue Previous Article Preface Next Article Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup

Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems

Boris Andreianov^1, and
Halima Labani²

1.
Laboratoire de Mathématiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex
2.
Université Chouaïb Doukkali, Faculté des Sciences, Département de Mathématiques et Informatique, BP20 24000 El Jadida, Morocco

Received: September 30, 2010

Revised: March 31, 2011

Published: October 2012

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Abstract

We suggest an approach for proving global existence of bounded solutions and existence of a maximal attractor in $L^\infty$ for a class of abstract $3\times 3$ reaction-diffusion systems. The motivation comes from the concrete example of ``facilitated diffusion'' system with different non-homogeneous boundary conditions modelling the blood oxigenation reaction $Hb+O_2 \rightleftharpoons HbO_2$.
The method uses classical tools of linear semigroup theory, the $L^p$ techniques developed by Martin and Pierre [16] and B\'enilan and Labani [6] and the hint of ``preconditioning operators'': roughly speaking, the study of solutions of $(\partial_t +A_i)u=f$ is reduced to the study of solutions to

$(\partial_t+B)(B^{-1}u)=B^{-1}f+(I-B^{-1}A_i)u,$

with a conveniently chosen operator $B$. In particular, we need the $L^\infty-L^p$ regularity of $B^{-1}A_i$ and the positivity of the operator $(B^{-1}A_i-I)$ on the domain of $A_i$.
The same ideas can be applied to systems of higher dimension. To give an example, we prove the existence of a maximal attractor in $L^\infty$ for the $5\times 5$ system of facilitated diffusion modelling the coupled reactions $Hb+O_2 \rightleftharpoons HbO_2$, $Hb+CO_2 \rightleftharpoons HbCO_2$.

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Mathematics Subject Classification: Primary: 35B41, 35K57, 35Q92; Secondary: 35K90.

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