×

Exponential attractor for a chemotaxis-growth system of equations. (English) Zbl 1005.35023

This paper is concerned with an initial-boundary value problem for a quasilinear parabolic system of equations describing the density of a biological population and the concentration of a chemical substance. The existence of an exponential attractor for the system is investigated.

MSC:

35B41 Attractors
92C17 Cell movement (chemotaxis, etc.)
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adler, J., Chemotaxis in bacteria, Science, 153, 708-716 (1966)
[2] Alt, W.; Lauffenburger, D. A., Transient behavior of a chemotaxis system modelling certain types of tissue inflammation, J. Math. Biol., 24, 691-722 (1985) · Zbl 0609.92020
[3] Berg, H. C.; Turner, L., Chemotaxis of bacteria in glass capillary, Biophys. J., 58, 919-930 (1990)
[4] Biler, P.; Hebisch, W.; Nadzieja, T., The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23, 1189-1209 (1994) · Zbl 0814.35054
[5] Brezis, H., Analyse fonctionnelle, théorie et applications (1983), Masson: Masson Paris · Zbl 0511.46001
[6] Budrene, E. O.; Berg, H. C., Complex patterns formed by motile cells of Escherichia coli, Nature, 349, 630-633 (1991)
[7] Dalquist, F. W.; Lovely, P.; Koshland, D. E., Quantitative analysis of bacterial migration in chemotaxis, Nature New Biol., 236, 120-123 (1972)
[8] Diaz, J. I.; Nagai, T., Symmetrization in a parabolic-elliptic system related to chemotaxis, Adv. Math. Sci. Appl., 5, 659-680 (1995) · Zbl 0859.35004
[9] Diaz, J. I.; Nagai, T.; Rakotoson, J. M., Symmetrization techniques on unbounded domains: application to a chemotaxis system on \(R^N\), J. Differential Equations, 145, 156-183 (1998) · Zbl 0908.35016
[10] Eden, A.; Foias, C.; Nicolaenko, B.; Temam, R., Exponential attractors for dissipative evolution equations, Research in Applied Mathematics, Vol. 37 (1994), Wiley: Wiley New York · Zbl 0842.58056
[11] Ford, R. M.; Lauffenburger, D. A., Analysis of chemotactic bacterial distributions in population migration assays using a mathematical model applicable to steep or shallow attractant gradients, Bull. Math. Biol., 53, 721-749 (1991) · Zbl 0729.92029
[12] Friedman, A., Partial Differential Equations (1969), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York
[13] Herrero, M. A.; Velázquez, J. J.L., Chemotactic collapse for the Keller-Segel model, J. Math. Biol., 35, 177-194 (1996) · Zbl 0866.92009
[14] Jäger, W.; Luckhaus, S., On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329, 819-824 (1992) · Zbl 0746.35002
[15] K. Kawasaki, N. Shigesada, Modeling pattern formation of chemotactic bacteria, in preparation.; K. Kawasaki, N. Shigesada, Modeling pattern formation of chemotactic bacteria, in preparation.
[16] Keller, E. F.; Segel, L. A., Initiation of slime mold aggregation viewed as instability, J. Theor. Biol., 26, 399-415 (1970) · Zbl 1170.92306
[17] Lauffenburger, D. A.; Kennedy, C. R., Localized bacterial infection in a distributed model for tissue inflammation, J. Math. Biol., 16, 141-163 (1983) · Zbl 0537.92007
[18] Lions, J. L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires (1969), Dunod/Gauthier-Villars: Dunod/Gauthier-Villars Paris · Zbl 0189.40603
[19] Lions, J. L.; Magenes, E., Problèms aux limites non homogenes et applications, Vol. 1 (1968), Dunod: Dunod Paris · Zbl 0165.10801
[20] Mimura, M.; Tsujikawa, T., Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230, 499-543 (1996)
[21] Murray, J. D., Mathematical Biology (1989), Springer: Springer Berlin Heidelberg · Zbl 0682.92001
[22] Myerscough, M. R.; Murray, J. D., Analysis of propagating pattern in a chemotaxis system, Bull. Math. Biol., 54, 77-94 (1992) · Zbl 0733.92002
[23] Nagai, T.; Senba, T.; Yoshida, K., Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40, 411-433 (1997) · Zbl 0901.35104
[24] S.-U. Ryu, A. Yagi, Optimal control of Keller-Segel equations, J. Math. Anal. Appl. 256 (2001) 45-66.; S.-U. Ryu, A. Yagi, Optimal control of Keller-Segel equations, J. Math. Anal. Appl. 256 (2001) 45-66. · Zbl 0982.49006
[25] A. Stevens, Trail following and aggregation of myxobacteria, Proceedings of the second ECMBM, World Scientific, Lyon.; A. Stevens, Trail following and aggregation of myxobacteria, Proceedings of the second ECMBM, World Scientific, Lyon.
[26] H. Tanabe, Equation of Evolution, Iwanami, Tokyo, 1975 (in Japanese); English translation, Pitman, London, 1979.; H. Tanabe, Equation of Evolution, Iwanami, Tokyo, 1975 (in Japanese); English translation, Pitman, London, 1979. · Zbl 0327.15012
[27] Tanabe, H., Functional Analytic Methods for Partial Differential Equations (1997), Marcel Dekker: Marcel Dekker New York · Zbl 0867.35003
[28] Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics (1997), Springer: Springer New York · Zbl 0871.35001
[29] Tsujikawa, T., Singular limit analysis of planar equilibrium solutions to a chemotaxis model equation with growth, Method Appl. Anal., 3, 401-431 (1996) · Zbl 0892.35084
[30] Woodward, D. E.; Tyson, R.; Myerscough, M. R.; Murray, J. D.; Budrene, E. O.; Berg, H. C., Spatio-temporal patterns generated by Salmonella typhimurium, Biophys. J., 68, 2181-2189 (1995)
[31] Yagi, A., Norm behavior of solutions to a parabolic system of chemotaxis, Math. Japonica, 45, 241-265 (1997) · Zbl 0910.92007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.