Herrero, Miguel A. and Medina Reus, Elena and Velázquez, J.J. L.
(1998)
*Self-similar blow-up for a reaction-diffusion system.*
Journal of Computational and Applied Mathematics, 97
(1-2).
pp. 99-119.
ISSN 0377-0427

PDF
Restricted to Repository staff only until 31 December 2020. 950kB |

Official URL: http://www.sciencedirect.com/science/article/pii/S0377042798001046

## Abstract

This work is concerned with the following system: [GRAPHICS] which is a model to describe several phenomena in which aggregation plays a crucial role as, for instance, motion of bacteria by chemotaxis and equilibrium of self-attracting clusters. When the space dimension N is equal to three, we show here that (S) has radial solutions with finite mass that blow-up in finite time in a self-similar manner. When N = 2, however, no radial solution with finite mass may give rise to self-similar blow-up.

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

Uncontrolled Keywords: | Reaction-diffusion systems; blow-up; self-similar behaviour; matched asymptotic expansions; chemotaxis; equations; aggregation; clusters; model |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 16946 |

References: | P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., to appear. S. Childress, J.K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci. 56 (1981) 217-237. J.W. Dold, Analysis of the early stage of thermal runaway, Quart. J. Mech. Appl. Math. 38 (1985) 361-387. Y. Giga, R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985) 297-319. V.A. Galaktionov, S.A. Posashkov, Application of new comparison theorems in the investigation of unbounded solutions of nonlinear parabolic equations, Differential Equations 22 (7) (1986) 1165-1173. M.A. Herrero, J.J.L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann. 306 (3) (1996) 583-623. M.A. Herrero, J.J.L. Velázquez, Chemotactic collapse for the Keller-Segel model. J. Math. Biol. 35 (1996) 177-194. M.A. Herrero, J.J.L. Velázquez, A blow-up mechanism for a chemotaxis problem, Annali Scuola Normale Sup. Pisa, to appear. M.A. Herrero, E. Medina, J.J.L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity 10 (1997) 1739-1754. W. Jäger, S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (2) (1992) 819-824. E.F. Keller, L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970) 399-415. A.A.Lacey, The form of blow-up for nonlinear parabolic equations, Proc. Roy. Soc. Edinburg A 98 (1984) 183-202. T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl. (1995) 1-21. V. Nanjundiah, Chemotaxis signal relaying and aggregation morphology, J. Theor. Biology 42 (1973) 63-105. J.J.L. Velázquez, Classification of singularities for blowing-up solutions in higher dimensions, Trans. Amer. Math. Soc. 338 (1) (1993) 441-464. G. Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations, Arch. Rat. Mech. Anal. 119 (1992) 355-391. G. Wolansky, On the evolution of self-interacting clusters and applications to semilinear equations with exponential nonlinearity, J. Anal. Math. 59 (1992) 251-272. |

Deposited On: | 31 Oct 2012 09:24 |

Last Modified: | 07 Feb 2014 09:38 |

Repository Staff Only: item control page