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Preface
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. | Université Chouaïb Doukkali, Faculté des Sciences, Département de Mathématiques et Informatique, BP20 24000 El Jadida, Morocco |
$(\partial_t+B)(B^{-1}u)=B^{-1}f+(I-B^{-1}A_i)u,$
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 $Hb+O_2 \rightleftharpoons HbO_2$, $Hb+CO_2 \rightleftharpoons HbCO_2$.
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). 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. |
[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). 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. |
[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). 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. |
[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). 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. |
[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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