Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems doi:10.3934/cpaa.2012.11.2179
Boris Andreianov - Laboratoire de Mathématiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France (email) 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 $Hb+O_2 \rightleftharpoons HbO_2$, $Hb+CO_2 \rightleftharpoons HbCO_2$.
Keywords: Reaction-diffusion system, global existence, maximal attractor, analytic semigroup, maximal regularity, preconditioning operator
Received: October 2010; Revised: April 2011; Published: April 2012. |
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