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

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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

Abstract
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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

References:

[1]	S. Amraoui and H. Labani, Global existence and maximal attractor of facilitated diffusion model,, Forum Math., 15 (2003), 923.
[2]	W. Arendt, "Semigroups and Evolution Equations: Functional Calculus, Regularity and Kernel Estimates,", Evolutionary equations. Vol. I, (2004), 1.
[3]	V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Noordhoff International Publishing, (1976).
[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.
[5]	Ph. Bénilan and H. Labani, "Systèmes de récation-diffusion abstraites" (French),, Publications Math\'ematiques de Besan\ccon. Analyse Non Lin\'eaire, (1994).
[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.
[7]	R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,", Vol. 2, (1988).
[8]	N. Dunford and J. T. Schwartz, "Linear Operators. Parts I and II,", Wiley-Interscience, (1958).
[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.
[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.
[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.
[12]	A. Friedman, "Partial Differential Equations,", New York, (1969).
[13]	D. Gilbard and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Grundl. Math. Wiss. 224, (1983).
[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).
[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.
[16]	R. H. Martin and M. Pierre, Nonlinear reaction-diffusion systems,, In, (1992), 363. doi: 10.1016/S0076-5392(08)62804-0.
[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.
[18]	J. Morgan, Boundedness and decay results for reaction-diffusion systems,, SIAM J. Math. Anal., 21 (1990), 1172. doi: 10.1137/0521064.
[19]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Appl. Math. Sci. 44, (1983).
[20]	F. Rothe, "Global Solutions of Reaction-diffusion Systems,", Lect. Notes in Math. 1072, (1072).
[21]	R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,", Appl. Math. Sci. 68, (1988). doi: 10.1007/978-1-4684-0313-8.

show all references

References:

[1]	S. Amraoui and H. Labani, Global existence and maximal attractor of facilitated diffusion model,, Forum Math., 15 (2003), 923.
[2]	W. Arendt, "Semigroups and Evolution Equations: Functional Calculus, Regularity and Kernel Estimates,", Evolutionary equations. Vol. I, (2004), 1.
[3]	V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Noordhoff International Publishing, (1976).
[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.
[5]	Ph. Bénilan and H. Labani, "Systèmes de récation-diffusion abstraites" (French),, Publications Math\'ematiques de Besan\ccon. Analyse Non Lin\'eaire, (1994).
[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.
[7]	R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,", Vol. 2, (1988).
[8]	N. Dunford and J. T. Schwartz, "Linear Operators. Parts I and II,", Wiley-Interscience, (1958).
[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.
[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.
[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.
[12]	A. Friedman, "Partial Differential Equations,", New York, (1969).
[13]	D. Gilbard and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Grundl. Math. Wiss. 224, (1983).
[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).
[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.
[16]	R. H. Martin and M. Pierre, Nonlinear reaction-diffusion systems,, In, (1992), 363. doi: 10.1016/S0076-5392(08)62804-0.
[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.
[18]	J. Morgan, Boundedness and decay results for reaction-diffusion systems,, SIAM J. Math. Anal., 21 (1990), 1172. doi: 10.1137/0521064.
[19]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Appl. Math. Sci. 44, (1983).
[20]	F. Rothe, "Global Solutions of Reaction-diffusion Systems,", Lect. Notes in Math. 1072, (1072).
[21]	R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,", Appl. Math. Sci. 68, (1988). doi: 10.1007/978-1-4684-0313-8.

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