On the parabolic-elliptic limit of the doubly parabolic keller-segel system modelling chemotaxis

Studia Mathematica

Volumn 193, Issue 3, 2009, Pages 241-261

  Biler, Piotr ^a   Brandolese, Lorenzo ^b  

^a UNIVERSITÉ LYON 1   (France)

^b UNIVERSITY OF WROC AW   (Poland)

Author keywords

Blowup of solutions; Chemotaxis; Decay of solutions; Global in time solutions; Keller Segel model

Indexed keywords

EID: 69249193119     PISSN: 00393223     EISSN: None     Source Type: Journal    
DOI: 10.4064/sm193-3-2     Document Type: Article

References (25)

1. P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles, III, Colloq. Math. 68 (1995), 229-239.
2. -, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math. 114 (1995), 181-205.
3. -, Local and global solvability of parabolic systems modelling Chemotaxis, Adv. Math. Sci. Appl. 8 (1998), 715-743.
4. -, A note on the paper of Y. Naito, in: Self-Similar Solutions of Nonlinear PDE, Banach Center Publ. 74, Inst. Math., Polish Acad. Sci., Warszawa, 2006, 33-40.
5. P. Biler and L. Brandolese, Global existence versus blow up for some models of interacting particles, Colloq. Math. 106 (2006), 293-303.
6. P. Biler, M. Cannone, I. Guerra and G. Karch, Global regular and singular solutions for a model of gravitating particles, Math. Ann. 330 (2004), 693-708.
7. P. Biler, L. Corrias and J. Dolbeault, in preparation.
8. P. Biler, G. Karch, Ph. Laurençot and T. Nadzieja, The 8π-problem for radially symmetric solutions of a Chemotaxis model in a disc, Topol. Methods Nonlinear Anal. 27 (2006), 133-144.
9. -, The 8π-problem for radially symmetric solutions of a Chemotaxis model in the plane, Math. Methods Appl. Sci. 29 (2006), 1563-1583.
10. 2, Comm. Pure Appl. Math. 61 (2007), 1449-1481.
11. A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations 2006, no. 44, 33 pp.
12. 2, Comm. Math. Sci. 6 (2008), 417-447.
13. N, Adv. Differential Equations 10 (2005), 361-398.
14. F. Filbet, A finite volume scheme for the Patlak-Keller-Segel Chemotaxis model, Numer. Math. 104 (2006), 457-488.
15. G. Furioli and E. Terraneo, Molecules of the Hardy space and the Navier-Stokes equations, Funkcial. Ekvac. 45 (2002), 141-160.
16. M. Herrero and J. J. L. Velazquez, A blow-up mechanism for a Chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), 633-683.
17. 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.
18. n) and its applications to self-similar solutions, Indiana Univ. Math. J. 57 (2008), 1467-1500.
19. P.-G. Lemarié-Rieusset, Recent Developments in the Navier - Stokes Problem, Chapman & Hall/CRC Press, Boca Raton, FL, 2002.
20. Y. Meyer, Wavelets, paraproducts and Navier-Stokes equations, in: Current Developments in Mathematics, Int. Press, Cambridge, MA, 1997, 105-212.
21. n, Funkcial. Ekvac. 41 (1998), 383-434.
22. n, ibid. 43 (2000), 541-557.
23. S. Montgomery-Smith, Finite time blow up for a Navier-Stokes like equation, Proc. Amer. Math. Soc. 129 (2001), 3025-3029.
24. Y. Naito, Asymptotically self-similar solutions for the parabolic system modelling Chemotaxis, in: Self-Similar Solutions of Nonlinear PDE, Banach Center Publ. 74, Inst. Math., Polish Acad. Sci., Warszawa, 2006, 149-160.
25. A. Raczyński, Stability property of the two-dimensional Keller-Segel model, Asymptot. Anal. 61 (2009), 35-59.