Singular solutions of partial differential equations modelling chemotactic aggregation

International Congress of Mathematicians, ICM 2006

Volumn 3, Issue , 2006, Pages 321-338

  Velázquez, Juan J L ^a  

^a UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

Author keywords

Chemotaxis; Continuation beyond the blow up; Keller Segel model; Singularity formation

Indexed keywords

BLOW-UP; BLOW-UP TIME; CHEMOTACTIC AGGREGATION; CHEMOTAXIS; KELLER-SEGEL MODEL; MATHEMATICAL PROBLEMS; SINGULAR SOLUTIONS; SINGULARITY FORMATION;

BIOCHEMISTRY;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 77956094237     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (49)

1. Alt, W., Biased random walk models for chemotaxis and related diffusion approximations. J. Math. Biol. 9 (1980), 147-177.
2. Andreucci, D., Herrero, M. A., and Velázquez, J. J. L., The classical one-phase Stefan problem:Acatalogue of interface behaviours. Surveys Math. Indust. 9 (4) (2001), 247-337.
3. Angenent, S. B., and Velázquez, J. J. L., Degenerate neckpinches in mean curvature flow. J. Reine Angew. Math. 482 (1997), 15-66.
4. Berg, H., Random walks in biology, Princeton University Press, Princeton, NJ, 1993.
5. Betterton, D., and Brenner, M. P., Collapsing bacterial cylinders. Phys. Rev. E 64 (2001), 519-534.
6. Biler, P., Local and global solvability of some parabolic systems modeling chemotaxis. Adv. Math. Sci. Appl. 8 (1998), 715-743.
7. Bonner, J. T., The cellular slime molds. Princeton University Press, Princeton, NJ, 1967.
8. Brenner, M. P., Levitov, L. S., and Budrene, E. O., Physical mechanisms for chemotactic pattern formation by bacteria. Biophys. J. 74 (1998), 1677-1693.
9. Brenner, M. P., Constantin, P., Kadanoff, L. P., Schenkel, A., and Venkataramani, S. C., Diffusion, attraction and collapse. Nonlinearity 12 (1999), 1071-1098.
10. Bressan, A., Stable blow-up patterns. J. Differential Equations 98 (1992), 57-75.
11. Bretschneider, T., Vasiev, B., andWeijer, C. J., A model for Dictyostelium slug movement. J. Theor. Biol. 199 (1999), 125-136.
12. Chalub, F. A. C. C., Markowitch, P., Perthame, B., Schmeiser, C., On the derivation of drift-diffusion model for chemotaxis from kinetic equations.ANUMPreprint 14/02, Vienna Technical University 2002.
13. Childress, S., Chemotactic collapse in two dimensions. In Modelling of Patterns in Space and Time, Lecture Notes in Biomath. 55, Springer-Verlag, Berlin 1984, 61-68.
14. Dallon, J. C., and Othmer, H. G., How cellular movement determines the collective force generated by the Dyctiostelium discoideum slug. J. Theor. Biol. 231 (2004), 203-222.
15. Dickinson, R. B., and Tranquillo, R. T., A Stochastic Model for Cell Random Motility and Haptotaxis Based on Adhesion Receptor Binding Fluctuations. J. Math. Biol. 31 (6) 563-600 (1993).
16. Gajewski, H., and Zacharias, K., Global behaviour of a reaction-diffusion system modelling chemotaxis. Math. Nachr. 195 (1998), 77-114.
17. Herrero, M. A., and Velázquez, J. J. L., Singularity patterns in a chemotaxis model. Math. Ann. 306 (1996), 583-623.
18. Herrero, M. A., and Velázquez, J. J. L., Chemotactic collapse for the Keller-Segel model. J. Math. Biol. 35 (1996), 177-194.
19. Herrero, M. A., and Velázquez, J. J. L., On the melting of ice balls. SIAM J. Math. Anal. 1 (1997), 1-32.
20. Herrero, M. A., Medina, E., and Velázquez, J. J. L., Self-similar blow-up for a reaction-diffusion system. J. Comp. Appl. Math. 97 (1-2) (1998), 99-119.
21. Herrero, M. A., Medina, E., and Velázquez, J. J. L., Finite time aggregation into a single point in a reaction-diffusion system. Nonlinearity 10 (1997), 1739-1754.
22. Herrero, M. A., and Velázquez, J. J. L., A blow-up mechanism for a chemotaxis model. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), 633-683.
23. Hillen, T., and Othmer, H. G., The diffusion limit of transport equations derived from velocity jump processes. SIAM J. Appl. Math. 61 (2000), 751-775.
24. Hillen, T., and Othmer, H. G., Chemotaxis equations from the diffusion limit of transport equations. SIAM J. Appl. Math. 62 (2002), 1222-1250.
25. Hillen, T., and Painter, K., Global existence for a parabolic chemotaxis model with prevention of overcrowding. Adv. in Appl. Math. 26 (2001), 280-301.
26. Höfer, T., Sherrat, J. A., and Maini, P. K., Cellular pattern formation during Dictyostelium aggregation. Physica D 85 (1995), 425-444.
27. Horstmann, D., From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I. Jahresber. Deutsch. Math.-Verein. 105 (3) (2003), 103-165.
28. Horstmann, D., From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II. Jahresber. Deutsch. Math.-Verein. 106 (2) (2004), 51-69.
29. Horstmann, D., and Wang, G., Blow-up in a chemotaxis model without symmetry assumptions. Eur. J. Appl. Math. 12 (2001), 159-177.
30. Hwang, H. J., Kang, K., and Stevens, A., Global solutions of nonlinear transport equations for chemosensitive movement. SIAM J. of Math. Analysis 36 (4) (2005), 1177-1199.
31. Jäger W., and Luckhaus, S., On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. 329 (1992), 819-824.
32. Keener, J. P., A geometric theory for spiral waves in excitable media. SIAM J. Appl. Math. 46 (1986), 1039-1056.
33. Keller, E. F., and Segel, L. A., Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970), 399-415.
34. Litcanu, G., and Velázquez, J. J. L., Singular perturbation analysis of cAMP signalling in Dictyostelium discoideum aggregates. J. Math. Biol., to appear.
35. Maree, A. F., and Hogeweg, P., Modelling dictyostelium discoideum morphogenesis: the culmination. Bull. Math. Biol. 64 (2) (2002), 327-353.
36. Martiel, J. L., and Goldbeter, A., Amodel based on receptor desensitization for cyclic AMP signalling in Dictyostelium cells. Biophys. J. 52 (1987), 807-828.
37. Nagai, T., Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5 (1995), 581-601.
38. Patlak, C. S., Random walk with persistence and external bias. Bull. Math. Biol. Biophysics 15 (1953), 311-338.
39. Riley, D. S., Smith, F. T., and Poots, G., The inward solidication of spheres and circular cylinders. Int. J. Heat and Mass Transfer 17 (1974), 1507-1516.
40. Schaaf, R., Stationary solutions of chemotaxis systems.Trans.Amer. Math. Soc. 292 (1985), 531-556.
41. Siegert, F., and Weijer, C. J., Three-dimensional scroll waves organize Dictyostelium slugs. Proc. Natl. Acad. Sci. USA 89 (1992), 6433-6437.
42. Stevens, A., The derivation of chemotaxis equations as limit of moderately interacting stochastic many particle systems. SIAM J. Appl. Math. 61 (2000), 183-212.
43. Tang, Y., and Othmer, H. G., Excitation, oscillations and wave propagation in a G-protein based model of signal transduction in Dictyostelium discoideum. Philos. Trans. Roy. Soc. London Ser. B 349 (1995), 179-195.
44. Tomchik, K. J., and Devreotes, P. N., Cyclic AMP waves in Dictyostelium discoideum: A demonstration by isotope dilution fluorography. Science 212 ( 1981), 443-446.
45. Velázquez, J. J. L., Stability of some mechanisms of chemotactic aggregation. SIAM J. Appl. Math. 62 (5) (2002), 1581-1633.
46. Velázquez, J. J. L., Point dynamics for a singular limit of the Keller-Segel model I: Motion of the concentration regions. SIAM J. Appl. Math. 64 (4) (2004), 1198-1223.
47. Velázquez, J. J. L., Point dynamics for a singular limit of the Keller-Segel model II: Formation of the concentration regions. SIAM J. Appl. Math. 64 (4) (2004), 1224-1248.
48. Velázquez, J. J. L., Well posedness of a model of point dynamics for a limit of the Keller-Segel problem. J. Differential Equations 206 (2) (2004), 315-352.
49. Velázquez, J. J. L., Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (4) (1994), 595-628.