Biblioteca de la Universidad Complutense de Madrid

Finite-time aggregation into a single point in a reaction-diffusion system


Herrero, Miguel A. y Medina Reus, Elena y 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

[img] PDF
Restringido a Sólo personal autorizado del repositorio hasta 31 Diciembre 2020.


URL Oficial:


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.

Tipo de documento:Artículo
Palabras clave:Chemotaxis; equations; singularities; clusters
Materias:Ciencias > Matemáticas > Ecuaciones diferenciales
Código ID:16970

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

Depositado:05 Nov 2012 11:29
Última Modificación:07 Feb 2014 09:39

Sólo personal del repositorio: página de control del artículo