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

Boris Andreianov (1) , Halima Labani (2)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 UCD - Université Chouaib Doukkali

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 $\text{Hb}+\text{O}_2 \rightleftharpoons \text{HbO}_2$. The method uses the $L^p$ techniques developed by Martin and Pierre \cite{MartinPierre} and Bénilan and Labani \cite{BenilanLabani-2x2} and the hint of ``preconditioning operators'': roughly speaking, the study of solutions of $\Bigl( {\ptl_t} +A_i\Bigr)\,u=f$ is reduced to the study of solutions to $$\Bigl( {\ptl_t} +B\Bigr)\,(B^{-1}u)=B^{-1}f+(\text{I}\!-\!B^{-1}\!A_i)\,u,$$ 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 $(B^{-1}\!A_i\!-\!\text{I})$ 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 $\text{Hb}+\text{O}_2 \rightleftharpoons \text{HBO}_2$, $\text{Hb}+\text{CO}_2 \rightleftharpoons \text{HbCO}_2$.

Mots clés: Reaction-diffusion system; Global existence; Maximal attractor; Analytic semigroup; Maximal regularity; Preconditioning operator

Domaines: Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]

Boris Andreianov, Halima Labani. Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems. Communications on Pure and Applied Analysis, 2012, 11 (6), pp. 2179-2199. ⟨10.3934/cpaa.2012.11.2179⟩. ⟨hal-00522783⟩