### Impacto

Herrero, Miguel A. and Medina Reus, Elena and Velázquez, J.J. L.
(1997)
*Finite-time aggregation into a single point in a reaction-diffusion system.*
Nonlinearity, 10
(6).
pp. 1739-1754.
ISSN 0951-7715

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Official URL: http://iopscience.iop.org/0951-7715/10/6/016

## Abstract

We consider the following system: [GRAPHICS] which has been used as a model for various phenomena, including motion of species by chemotaxis and equilibrium of self-attracting clusters. We show that, in space dimension N = 3, (S) possess radial solutions that blow-up in a finite time. The asymptotic behaviour of such solutions is analysed in detail. In particular, we obtain that the profile of any such solution consists of an imploding, smoothed-out shock wave that collapses into a Dine mass when the singularity is formed. The differences between this type of behaviour and that known to occur for blowing-up solutions of (S) in the case N = 2 are also discussed.

Item Type: | Article |
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Uncontrolled Keywords: | Chemotaxis; equations; singularities; clusters |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 16970 |

References: | Andreucci D, Herrero M A and Velázquez J J L 1997 Liouville theorems and blow-up behaviour in semilinear reaction-diffusion systems Ann. l’Institute Henri Poincaré 14 1–53 Angenent S B and Velázquez J J L 1995 Asymptotic shape of cusp singularities in curve shortening Duke Math. J. 77 71–110 Childress S and Percus J K 1981 Nonlinear aspects of chemotaxis Math. Biosc. 56 217–37 Escobedo M, Herrero M A and Velázquez J J L 1996 A nonlinear Fokker–Planck equation modelling the approach to thermal equilibrium in a homogeneous plasma Trans. Amer. Maths Soc. submitted Filippas S and Kohn R V 1992 Refined asymptotics for the blow-up of ut − Δu = up Comm. Pure Appl.Math. 45 821–69 Jäger W and Luckhaus S 1992 On explosions of solutions to a system of partial differential equations modelling chemotaxis Trans. Am. Math. Soc. 329 819–24 Keller E F and Segel L A 1970 Initiation of slime mold aggregation viewed as an instability J. Theor. Biol. 26 399–415 Herrero M A and Velázquez J J L 1996 Singularity patterns in a chemotaxis model Math. Ann. 306 583–623 Herrero M A and Velázquez J J L 1996 Chemotactic collapse for the Keller–Segel model J. Math. Biol. 35 177–96 Herrero M A and Velázquez J J L A blow-up mechanism for a chemotaxis model Ann. Scuola Normale Sup.Pisa. to appear Nagai T 1995 Blow-up of radially symmetric solutions to a chemotaxis system Adv. Math. Sci. Appl. 1–21 Nanjundiah V 1973 Chemotaxis, signal relaying and aggregation morphology J. Theor. Biol. 42 63–105 Velázquez J J L 1993 Classification of singularities for blowing-up solutions in higher dimensions Trans. Am. Math. Soc. 338 441–64 Velázquez J J L 1992 Higher dimensional blow-up for semilinear parabolic equations Commun. PDE 17 1567–96 Velázquez J J L 1994 Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow Annali Scuola Normale Sup. Pisa 21 595–628 Wolansky G 1992 On steady distributions of self-attracting clusters under friction and fluctuations Arch.Rational Mech. Anal. 119 355–91 Wolansky G 1992 On the evolution of self-interacting clusters and applications to semilinear equations with exponential nonlinearity J. Anal. Math. 59 251–72 |

Deposited On: | 05 Nov 2012 11:29 |

Last Modified: | 07 Feb 2014 09:39 |

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