Herrero, Miguel A. and Velázquez, J.J. L.
(1996)
*Chemotactic collapse for the Keller-Segel model.*
Journal of Mathematical Biology, 35
(2).
pp. 177-194.
ISSN 0303-6812

PDF
Restricted to Repository staff only until 31 December 2020. 307kB |

Official URL: http://www.springerlink.com/content/n1f8vbefpr5g1jnu/

## Abstract

This work is concerned with the system (S) {u(t)=Delta u-chi del(u del upsilon) for x is an element of Omega, t>0 Gamma upsilon(t)=Delta upsilon=Delta upsilon+(u-1) for x is an element of Omega, t>0 where Gamma; chi are positive constants and Omega is a bounded and smooth open set in IR(2). On the boundary delta Omega, we impose no-flux conditions: (N)partial derivative u/partial derivative n=partial derivative upsilon/partial derivative n=0 for x is an element of partial derivative Omega, t>0 Problem (S), (N) is a classical model to describe chemotaxis corresponding to a species of concentration u(x, t) which tends to aggregate towards high concentrations of a chemical that the species releases. When completed with suitable initial values at t=0 for u(x, t), upsilon(x, t), the problem under consideration is known to be well posed, locally in time. By means of matched asymptotic expansions techniques, we show here that there exist radial solutions exhibiting chemotactic collapse. By this we mean that u(r, t)-->A delta(y) as t-->T for some T <infinity, where A is the total concentration of the species.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Chemotaxis; advection-diffusion systems; matched asymptotic expansions; blow-up; asymptotic behaviour; equations |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 17184 |

References: | S. B. Angenent and J. J. L. Velázquez, Asymptotic shape of cusp singularities in curve shortening. Duke Math. Journal 77(1) (1995) 71-110 Adimurthi and S. L. Yadava, Existence and nonexistence of positive radial solutions of Neumann problems with critical Sobolev exponents. Archive Rat. Mech. Anal. 115 (1991) 275-296 S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis. Math. Biosci. 56 (1981) 217-237 Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations. Comm. Pure Appl. Math. 38 (1985) 297-319 W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. vol. 239, nº 2 (1992) 819-824 M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model. To appear in Mathematische Annalen M. A. Herrero and J. J. L. Velázquez, On the melting of ice balls. To appear in SIAM Journal of Mathematical Analysis M. A. Herrero and J. J. L. Velázquez, Explosion de solutions d’equations paraboliques semilineaires supercritiques. C.R. Acad. Sci. Paris, t. 319, Serie I (1994) 141-145 E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970) 399-415 C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system. Journal of Differential Equations 72, 1 (1988) 1-23 T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. and Appl. (1995) 1-21 V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology. J. Theor. Biol. 42 (1973) 63-105 J. J. L. Velázquez, Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow. Annali Scuola Normale Superiore di Pisa, Serie IV, vol. XXI, Fasc. 4 (1994) 595-628 |

Deposited On: | 23 Nov 2012 11:53 |

Last Modified: | 07 Feb 2014 09:43 |

Repository Staff Only: item control page