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

International audienceWe 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 $\text{Hb}+\text{O}_2 \rightleftharpoons \text{HBO}_2$, $\text{Hb}+\text{CO}_2 \rightleftharpoons \text{HbCO}_2$