Free Access
Math. Model. Nat. Phenom.
Volume 5, Number 1, 2010
Cell migration
Page(s) 123 - 147
Published online 03 February 2010
  1. R. A. Adams. Sobolev spaces. Academic Press, New York, 1975.
  2. M. Alber, R. GejjiB. Kaźmierczak. Existence of global solutions of a macroscopic model of cellular motion in a chemotactic field. Applied Mathematics Letters., 22 (2009), No. 11, 1645–1648 [CrossRef] [MathSciNet]
  3. B. Ainsebaa, M. Bendahmaneb, A. Noussairc. A reaction–diffusion system modeling predator–prey with prey-taxis. Nonlinear Anal. R. World Appl., 9 (2008), No. 5, 2086–2105. [CrossRef]
  4. H. Amann. Dynamic theory of quasilinear parabolic systems III. Global existence. Math. Z., 202 (1989), No. 2, 219–250.
  5. H. Amann. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems.9–126, in: (H. Triebel, H.J. Schmeisser., eds.), Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte Math., 133, Teubner, Stuttgart, 1993.
  6. D. G. Aronson. The porous medium equation., in: (A.Fasano, M.Primicerio.,eds.) Some Problems in Nonlinear Diffusion. Lecture Notes in Mathematics., 1224, Springer, Berlin, 1986.
  7. M. Bendahmane, K. H. Karlsen, J. M. Urbano. On a two-sidedly degenerate chemotaxis model with volume-filling effect. Math. Models Methods Appl. Sci., 17 (2007), No. 2, 783–804. [CrossRef] [MathSciNet]
  8. P. Biler. Local and global solvability of some parabolic systems modelling chemotaxis. Adv. Math. Sci. Appl. Nachr., 195 (1998), No. 8, 76–114
  9. M. P. Brenner, L. S. Levitov and E. O. Budrene. Physical mechanism for chemotactic pattern formation by bacteria. Biophys. J., 74 (1998), No. 4, 1677–1693. [CrossRef] [PubMed]
  10. H. M. ByrneM. R. Owen. A new interpretation of the Keller-Segel model based on multiphase modelling. J. Math. Biol., 49 (2004), No. 6, 604–626 [CrossRef] [MathSciNet] [PubMed]
  11. F. A. C. C. ChalubJ. F. Rodrigues. A class of kinetic models for chemotaxis with threshold to prevent overcrowding. Portugaliae Math., 26 (2006), No. 2, 227–250
  12. V. CalvezJ. A. Carillo. Volume effects in the KellerSegel model: energy estimates preventing blow-up. J. Math. Pures Appl., 86 (2006), No. 2, 155–175 [CrossRef] [MathSciNet]
  13. T. Cieślak . The solutions of the quasilinear Keller-Segel system with the volume filling effect do not blow up whenever the Lyapunov functional is bounded from below. 127–132, in: Self-similar solutions of nonlinear PDE, Banach Center Publ., 74, Warsaw, 2006.
  14. T. Cieślak. Quasilinear nonuniformly parabolic system modelling chemotaxis. J. Math. Anal. Appl., 326 (2007), No. 2, 1410–1426 [CrossRef] [MathSciNet]
  15. T. CieślakC. Morales-Rodrigo. Quasilinear non-uniformly parabolic-elliptic system modelling chemotaxis with volume filling effect. Existence and uniqueness of global-in-time solutions. Topol. Methods Nonlinear Anal., 29 (2007), No. 2, 361–381 [MathSciNet]
  16. T. CieślakM. Winkler. Finite-time blow-up in a quasilinear system of chemotaxis. Nonlinearity., 21 (2008), No. 5, 1057–1076 [CrossRef] [MathSciNet]
  17. Y. S. ChoiZ. A. Wang. Prevention of blow up by fast diffusion in chemotaxis. J. Math. Anal. Appl., 362 (2010), No. 2, 553-564 [CrossRef] [MathSciNet]
  18. M. DiFrancescoJ. Rosado. Fully parabolic Keller-Segel model for chemotaxis with prevention of overcrowding. Nonlinearity., 21 (2008), No. 11, 2715–2730 [CrossRef] [MathSciNet]
  19. Y. DolakC. Schmeiser. The Keller-Segel model with logistic sensitivity function and small diffusivity. SIAM J. Appl. Math., 66 (2005), No. 1 Cell migration, 286–308 [CrossRef] [MathSciNet]
  20. E. Feireisl, Ph. LaurençotH. Petzeltova. On convergence to equilibria for the Keller-Segel chemotaxis model. J.Diff.Equations., 236 (2007), No. 2, 551–569 [CrossRef]
  21. H. GajewskiK. Zacharias. Global behavior of a reaction-diffusion system modelling chemotaxis. Math. Nachr., 195 (1998), No. 1 Cell migration, 77–114 [CrossRef] [MathSciNet]
  22. D. Henry. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1981.
  23. M. A. HerreroJ. J. L Velázquez. A blow-up mechanism for a chemotaxis model. Ann. Scuola Norm. Sup. Pisa., 24 (1997), No. 4, 633–683
  24. M. A. HerreroJ. J. L Velázquez. Chemotactic collapse for the Keller-Segel model. J. Math. Biol., 35 (1996), No. 2, 583–623
  25. T. HillenK. J. Painter. A user’s guide to PDE models for chemotaxis. J. Math. Biol., 58 (2009), No. 1–2, 183–217 [CrossRef] [MathSciNet] [PubMed]
  26. T. HillenK. Painter. Global existence for a parabolic chemotaxis model with prevention of overcrowding. Adv. Appl. Math., 26 (2001), No. 4, 280–301 [CrossRef] [MathSciNet]
  27. D. Horstmann. Lyapunov functions and Lp;-estimates for a class of reaction-diffusion systems. Colloq. Math., 87 (2001), No. 1 Cell migration, 113–127 [CrossRef] [MathSciNet]
  28. D. Horstmann. From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein., 105 (2003), No. 3, 103–165 [MathSciNet]
  29. D. Horstmann. From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein., 106 (2004), No. 2, 51–69 [MathSciNet]
  30. J. JiangY. Zhang. On Convergence to equilibria for a Chemotaxis Model with Volume filling effect. Asymptotic Analysis., 65 (2009), No. 1–2, 79–102 [MathSciNet]
  31. E. Keller and L. Segel. Initiation of slime mold aggregation viewed as an instability. J. Theor. Biology. 26 (1970), No. 3, 399–415. [CrossRef] [PubMed]
  32. R. Kowalczyk, A. Gamba and L. Preciosi. On the stability of homogeneous solutions to some aggregation models. Discrete Contin. Dynam. Systems-Series B. 4 (2004), No. 1 Cell migration, 204–220.
  33. Ph. Laurençot, D. Wrzosek. A chemotaxis model with threshold density and degenerate diffusion. 273-290 in: Progress in Nonlinear Differential Equations and Their Applications., 64, Birkhäuser, Basel, 2005.
  34. J.-L. Lions. Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris, 1969.
  35. P. M. Lushnikov, N. Chen and M. Alber. Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact. Phys. Rev. E., 78 (2008), No. 6, 061904. [CrossRef]
  36. T. Nagai. Blow-up of radially symetric solutions to a chemotaxis system. Adv. Math. Sci. Appl., 5 (1995), No. 2, 581–601 [MathSciNet]
  37. T. Nagai, T. SenbaT. Suzuki. Chemotaxis collapse in a parabolic system of mathematical biology. Hiroshima Math. J., 30 (2000), No. 3, 463–497 [MathSciNet]
  38. K. OsakiA. Yagi. Finite dimensional attractors for one dimensional Keller-Segel equations. Funkcial. Ekvac., 44 (2001), No. 3, 441–469 [MathSciNet]
  39. K. Osaki, A. Yagi. Global existence for a chemotaxis-growth system in ℝ2. Adv. Math. Sci. Appl., 12 (2002), No. 2, 587–606. [MathSciNet]
  40. K. Osaki, T. Tsujikawa, A. YagiM. Mimura. Exponential attractor for a chemotaxis-growth system of equations. Nonlinear Anal., 51 (2002), No. 1 Cell migration, 119–144 [CrossRef] [MathSciNet]
  41. K. PainterT. Hillen Volume-filling and quorum-sensing in models for chemosensitive movement. Canadian Appl. Math. Q., 10 (2002), No. 4, 501–543
  42. C. S. Patlak. Random walk with persistence and external bias. Bull. Math. Biol. Biophys., 15 (1953), No. 3, 311–338 [CrossRef]
  43. B. PerthameA. -L. Dalibard. Existence of solutions of the hyperbolic Keller-Segel model. Trans. Amer. Math. Soc., 361 (2008), No. 5, 2319–2335 [CrossRef] [MathSciNet]
  44. A. B. PotapovT. Hillen. Metastability in Chemotaxis Models. J. Dyn. Diff. Eq., 17 (2005), No. 2, 293-330 [CrossRef]
  45. R. Schaaf. Stationary solutions of Chemotaxis systems. Trans. Am. Math. Soc., 292 (1985), No. 2, 531-556 [CrossRef]
  46. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer- Verlag, New York, 1988.
  47. J. J. L Velázquez. Point dynamics in a singular limit of the Keller-Segel model 1: motion of the concentration regions. SIAM J. Appl. Math., 64 (2004), No. 4, 1198–1223 [CrossRef] [MathSciNet]
  48. M. Winkler. Does a volume filling effect always prevent chemotactic colapse. Math. Meth. Appl. Sci., 33 (2010), No. 1 Cell migration, 12–24
  49. Z.A. WangT. Hillen. Classical solutions and pattern formation for a volume filling chemotaxis model. Chaos., 17 (2007), No. 3, 037108–037121 [CrossRef] [MathSciNet] [PubMed]
  50. D. Wrzosek. Global attractor for a chemotaxis model with prevention of overcrowding. Nonlinear Anal. TMA., 59 (2004), No. 8, 1293–1310
  51. D. Wrzosek. Long time behaviour of solutions to a chemotaxis model with volume filling effect. Proc. Roy. Soc. Edinburgh., 136A (2006), No. 2, 431–444 [CrossRef]
  52. D. Wrzosek. Chemotaxis models with a threshold cell density. in: Parabolic and Navier-Stokes equations. Part 2, 553–566, Banach Center Publ., 81, Warsaw, 2008.
  53. D. Wrzosek. Model of chemotaxis with threshold density and singular diffusion. Nonlinear Anal. TMA.. to appear.
  54. Y. Zhang, S. Zheng. Asymptotic Behavior of Solutions to a Quasilinear Nonuniform Parabolic System Modelling Chemotaxis. J. Diff. Equations. in press.

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