# American Institute of Mathematical Sciences

November  2012, 11(6): 2179-2199. doi: 10.3934/cpaa.2012.11.2179

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

 1 Laboratoire de Mathématiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex 2 Université Chouaïb Doukkali, Faculté des Sciences, Département de Mathématiques et Informatique, BP20 24000 El Jadida, Morocco

Received  October 2010 Revised  April 2011 Published  April 2012

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 $Hb+O_2 \rightleftharpoons HbO_2$, $Hb+CO_2 \rightleftharpoons HbCO_2$.

Citation: Boris Andreianov, Halima Labani. Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2179-2199. doi: 10.3934/cpaa.2012.11.2179
##### References:
 [1] S. Amraoui and H. Labani, Global existence and maximal attractor of facilitated diffusion model,, Forum Math., 15 (2003), 923.   Google Scholar [2] W. Arendt, "Semigroups and Evolution Equations: Functional Calculus, Regularity and Kernel Estimates,", Evolutionary equations. Vol. I, (2004), 1.   Google Scholar [3] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Noordhoff International Publishing, (1976).   Google Scholar [4] P. Baras, J.-C. Hassan and L. Véron, Compacité de l'opérateur définissant la solution d'une équation d'évolution non homogéne,, C.R. Acad. Sci. Paris S\'er. A, 284 (): 799.   Google Scholar [5] Ph. Bénilan and H. Labani, "Systèmes de récation-diffusion abstraites" (French),, Publications Math\'ematiques de Besan\ccon. Analyse Non Lin\'eaire, (1994).   Google Scholar [6] Ph. Bénilan and H. Labani, Existence of attractors in $L^\infty(\Omega)$ for a class of reaction-diffusion systems,, In, (2004), 771.  doi: 10.1007/978-3-0348-7924-8_36.  Google Scholar [7] R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,", Vol. 2, (1988).   Google Scholar [8] N. Dunford and J. T. Schwartz, "Linear Operators. Parts I and II,", Wiley-Interscience, (1958).   Google Scholar [9] W. Ebel, Existence and asymptotic behaviour of solutions in a system of reaction diffusion equations,, Math. Z., 193 (1986), 41.  doi: 10.1007/BF01163353.  Google Scholar [10] W. Feng, Global existence and boundedness of the solution for a blood oxigenation model,, J. Math. Anal. Appl., 181 (1994), 462.  doi: 10.1006/jmaa.1994.1035.  Google Scholar [11] W. Feng, Stability and asymptotic behaviour in a reaction-diffusion system,, M3AS Math. Models Meth. Appl. Sci., 17 (1994), 155.  doi: 10.1002/mma.1670170302.  Google Scholar [12] A. Friedman, "Partial Differential Equations,", New York, (1969).   Google Scholar [13] D. Gilbard and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Grundl. Math. Wiss. 224, (1983).   Google Scholar [14] H. Labani, "Comportement asymptotique de certaines équations de réaction-diffusion et d'une classe d'équations des ondes" (French),, Th\ese d'\'Etat, (2002).   Google Scholar [15] D. Lamberton, Équations d'évolution lináires associés à des semi-groupes de contractions dans les espaces $L^p$, (French), J. Funct. Anal., 72 (1987), 252.  doi: 10.1016/0022-1236(87)90088-7.  Google Scholar [16] R. H. Martin and M. Pierre, Nonlinear reaction-diffusion systems,, In, (1992), 363.  doi: 10.1016/S0076-5392(08)62804-0.  Google Scholar [17] R. Martin and M. Pierre, Influence of mixed boundary conditions in some reaction-diffusion systems,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1053.  doi: 10.1017/S0308210500026883.  Google Scholar [18] J. Morgan, Boundedness and decay results for reaction-diffusion systems,, SIAM J. Math. Anal., 21 (1990), 1172.  doi: 10.1137/0521064.  Google Scholar [19] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Appl. Math. Sci. 44, (1983).   Google Scholar [20] F. Rothe, "Global Solutions of Reaction-diffusion Systems,", Lect. Notes in Math. 1072, (1072).   Google Scholar [21] R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,", Appl. Math. Sci. 68, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

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##### References:
 [1] S. Amraoui and H. Labani, Global existence and maximal attractor of facilitated diffusion model,, Forum Math., 15 (2003), 923.   Google Scholar [2] W. Arendt, "Semigroups and Evolution Equations: Functional Calculus, Regularity and Kernel Estimates,", Evolutionary equations. Vol. I, (2004), 1.   Google Scholar [3] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Noordhoff International Publishing, (1976).   Google Scholar [4] P. Baras, J.-C. Hassan and L. Véron, Compacité de l'opérateur définissant la solution d'une équation d'évolution non homogéne,, C.R. Acad. Sci. Paris S\'er. A, 284 (): 799.   Google Scholar [5] Ph. Bénilan and H. Labani, "Systèmes de récation-diffusion abstraites" (French),, Publications Math\'ematiques de Besan\ccon. Analyse Non Lin\'eaire, (1994).   Google Scholar [6] Ph. Bénilan and H. Labani, Existence of attractors in $L^\infty(\Omega)$ for a class of reaction-diffusion systems,, In, (2004), 771.  doi: 10.1007/978-3-0348-7924-8_36.  Google Scholar [7] R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,", Vol. 2, (1988).   Google Scholar [8] N. Dunford and J. T. Schwartz, "Linear Operators. Parts I and II,", Wiley-Interscience, (1958).   Google Scholar [9] W. Ebel, Existence and asymptotic behaviour of solutions in a system of reaction diffusion equations,, Math. Z., 193 (1986), 41.  doi: 10.1007/BF01163353.  Google Scholar [10] W. Feng, Global existence and boundedness of the solution for a blood oxigenation model,, J. Math. Anal. Appl., 181 (1994), 462.  doi: 10.1006/jmaa.1994.1035.  Google Scholar [11] W. Feng, Stability and asymptotic behaviour in a reaction-diffusion system,, M3AS Math. Models Meth. Appl. Sci., 17 (1994), 155.  doi: 10.1002/mma.1670170302.  Google Scholar [12] A. Friedman, "Partial Differential Equations,", New York, (1969).   Google Scholar [13] D. Gilbard and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Grundl. Math. Wiss. 224, (1983).   Google Scholar [14] H. Labani, "Comportement asymptotique de certaines équations de réaction-diffusion et d'une classe d'équations des ondes" (French),, Th\ese d'\'Etat, (2002).   Google Scholar [15] D. Lamberton, Équations d'évolution lináires associés à des semi-groupes de contractions dans les espaces $L^p$, (French), J. Funct. Anal., 72 (1987), 252.  doi: 10.1016/0022-1236(87)90088-7.  Google Scholar [16] R. H. Martin and M. Pierre, Nonlinear reaction-diffusion systems,, In, (1992), 363.  doi: 10.1016/S0076-5392(08)62804-0.  Google Scholar [17] R. Martin and M. Pierre, Influence of mixed boundary conditions in some reaction-diffusion systems,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1053.  doi: 10.1017/S0308210500026883.  Google Scholar [18] J. Morgan, Boundedness and decay results for reaction-diffusion systems,, SIAM J. Math. Anal., 21 (1990), 1172.  doi: 10.1137/0521064.  Google Scholar [19] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Appl. Math. Sci. 44, (1983).   Google Scholar [20] F. Rothe, "Global Solutions of Reaction-diffusion Systems,", Lect. Notes in Math. 1072, (1072).   Google Scholar [21] R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,", Appl. Math. Sci. 68, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar
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