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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 $Hb+O_2 \rightleftharpoons HbO_2$, $Hb+CO_2 \rightleftharpoons HbCO_2$.
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show all references
References:
[1] |
Forum Math., 15 (2003), 923-933. |
[2] |
Evolutionary equations. Vol. I, 1-85, Handb. Differ. Equ., North-Holland, Amsterdam, 2004. |
[3] |
Noordhoff International Publishing, Leiden, 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] |
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. |
[7] |
Vol. 2, Springer-Verlag, Berlin, 1988. |
[8] |
Wiley-Interscience, 1958. |
[9] |
Math. Z., 193 (1986), 41-66.
doi: 10.1007/BF01163353. |
[10] |
J. Math. Anal. Appl., 181 (1994), 462-472.
doi: 10.1006/jmaa.1994.1035. |
[11] |
M3AS Math. Models Meth. Appl. Sci., 17 (1994), 155-169.
doi: 10.1002/mma.1670170302. |
[12] |
New York, 1969, 2008. |
[13] |
Grundl. Math. Wiss. 224, Springer-Verlag, Berlin, 1983, 1998, 2001. |
[14] |
Thèse d'État, Marrakesh, 2002. Google Scholar |
[15] |
J. Funct. Anal., 72 (1987), 252-262.
doi: 10.1016/0022-1236(87)90088-7. |
[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. |
[17] |
Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1053-1066.
doi: 10.1017/S0308210500026883. |
[18] |
SIAM J. Math. Anal., 21 (1990), 1172-1189.
doi: 10.1137/0521064. |
[19] |
Appl. Math. Sci. 44, Springer-Verlag, New York, 1983. |
[20] |
Lect. Notes in Math. 1072, Springer-Verlag, Berlin, 1984. |
[21] |
Appl. Math. Sci. 68, Springer-Verlag, New York, 1988, 1997.
doi: 10.1007/978-1-4684-0313-8. |
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