Global Existence for a Parabolic Chemotaxis Model with Prevention of Overcrowding

Advances in Applied Mathematics

Volumn 26, Issue 4, 2001, Pages 280-301

  Hillen, T ^a   Painter, K ^b  

^a UNIVERSITY OF TÜBINGEN   (Germany)

^b HERIOT WATT UNIVERSITY   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001404830     PISSN: 01968858     EISSN: None     Source Type: Journal    
DOI: 10.1006/aama.2001.0721     Document Type: Article

References (31)

1. R. A. Adams, "Sobolev Spaces," Academic Press, San Diego, 1975.
2. H. Amann, Quasilinear evolution equations and parabolic systems, Trans. Amer. Math. Soc. 293 (1986), 191-227.
3. P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl. 8, No. 2 (1998), 715-743.
4. P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl. 9, No. 1 (1999), 347-359.
5. S. Childress, Chemotactic collapse in two dimensions, Lecture Notes in Biomath. 55 (1984), 61-68.
6. S. Childress, and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci. 56 (1981), 217-237.
7. H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr. 159 (1998), 77-114.
8. M. Herrero and J. Velázquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol. 35 (1996), 177-194.
9. M. Herrero and J. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996), 583-623.
10. T. Hillen and A. Stevens, Hyperbolic models for chemotaxis in 1-D, Nonlinear Anal. Real World Appl. 1 (2000), 409-433.
11. W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), 819-824.
12. E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970), 399-415.
13. O. A. Ladyžhenskaja, V. A. Solonnikov, and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Am. Math. Soc., Providence, 1968.
14. H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math. 57 (1997), 683-730.
15. T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl. 5 (1995), 581-601.
16. T. Nagai and T. Senba, Behavior of radially symmetric solutions of a system related to chemotaxis, Nonlinear Anal. 30 (1997), 3837-3842.
17. T. Nagai and T. Senba, Global existence and blow-up of radially solutions to a parabolic-elliptic system of chemotaxis, Adv. Math. Sci. Appl. 8, No. 1 (1998), 145-156
18. T. Nagai, T. Senba, and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac. 40, No. 3 (1997), 411-433.
19. T. Nagai, T. Senba, and K. Yoshida, Global existence of solutions to the parabolic system of chemotaxis, RIMS Kokyuroku 1009 (1997), 22-28.
20. V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theoret. Biol. 42 (1973), 63-105.
21. H. G. Othmer and A. Stevens, Aggregation, blowup and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math. 57 (1997), 1044-1081.
22. K. Painter and T. Hillen, Prevention of overcrowding in chemotaxis models: modelling and pattern formation (2001), in preparation.
23. C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys. 15 (1953), 311-338.
24. K. Post, A system of non-linear partial differential equations modeling chemotaxis with sensitivity functions, Ph.D. thesis, Humboldt University of Berlin, 1999.
25. M. Rascle and C. Ziti, Finite-time blow-up in some models of chemotaxis, J. Math. Biol. 33 (1995), 388-414.
26. J. Rietdorf, F. Siegert, and C. J. Weijer, Induction of optical density waves and chemotactic cell movement in Dictyostelium discoideum by microinjection of camp pulses, Dev. Biol. 204 (1998), 525-536.
27. R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc. 292, No. 2 (1985), 531-556.
28. T. Senba, Blow-up of radially symmetric solutions to some systems of partial differential equations modelling chemotaxis, Adv. Math. Sci. Appl. 7 (1997), 79-92.
29. F. Siegert and C. J. Weijer, Spiral and concentric waves organize multicellular Dictyostelium mounds, Curr. Biol. 5 (1995), 937-943.
30. M. E. Taylor, "Partial Differential Equations III," Springer-Verlag, New York, 1996.
31. A. Yagi, Norm behavior of solutions to the parabolic system of chemotaxis, Math. Japon. 45 (1997), 241-265.