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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

Boris Andreianov 1, and  Halima Labani 2,

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

Keywords: preconditioning operator, global existence, maximal regularity, Reaction-diffusion system, maximal attractor, analytic semigroup.
Mathematics Subject Classification: Primary: 35B41, 35K57, 35Q92; Secondary: 35K90.
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
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show all references

References:
[1]

Forum Math., 15 (2003), 923-933.  Google Scholar

[2]

Evolutionary equations. Vol. I, 1-85, Handb. Differ. Equ., North-Holland, Amsterdam, 2004.  Google Scholar

[3]

Noordhoff International Publishing, Leiden, 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]

Publications Mathématiques de Besançon. Analyse Non Linéaire, vol.14, 1994. Google Scholar

[6]

In "Nonlinear Evolution Equations and Related Topics," 771-784, Birkhäuser, Basel, 2004. doi: 10.1007/978-3-0348-7924-8_36.  Google Scholar

[7]

Vol. 2, Springer-Verlag, Berlin, 1988.  Google Scholar

[8]

Wiley-Interscience, 1958.  Google Scholar

[9]

Math. Z., 193 (1986), 41-66. doi: 10.1007/BF01163353.  Google Scholar

[10]

J. Math. Anal. Appl., 181 (1994), 462-472. doi: 10.1006/jmaa.1994.1035.  Google Scholar

[11]

M3AS Math. Models Meth. Appl. Sci., 17 (1994), 155-169. doi: 10.1002/mma.1670170302.  Google Scholar

[12]

New York, 1969, 2008.  Google Scholar

[13]

Grundl. Math. Wiss. 224, Springer-Verlag, Berlin, 1983, 1998, 2001.  Google Scholar

[14]

Thèse d'État, Marrakesh, 2002. Google Scholar

[15]

J. Funct. Anal., 72 (1987), 252-262. doi: 10.1016/0022-1236(87)90088-7.  Google Scholar

[16]

In "Nonlinear Equations in the Applied Sciences," 363-398, Math. Sci. Engrg., 185, Academic Press, Boston, 1992. doi: 10.1016/S0076-5392(08)62804-0.  Google Scholar

[17]

Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1053-1066. doi: 10.1017/S0308210500026883.  Google Scholar

[18]

SIAM J. Math. Anal., 21 (1990), 1172-1189. doi: 10.1137/0521064.  Google Scholar

[19]

Appl. Math. Sci. 44, Springer-Verlag, New York, 1983.  Google Scholar

[20]

Lect. Notes in Math. 1072, Springer-Verlag, Berlin, 1984.  Google Scholar

[21]

Appl. Math. Sci. 68, Springer-Verlag, New York, 1988, 1997. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

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