Finite-time aggregation into a single point in a reaction-diffusion system
| dc.contributor.author | Herrero, Miguel A. | |
| dc.contributor.author | Medina Reus, Elena | |
| dc.contributor.author | Velázquez, J.J. L. | |
| dc.date.accessioned | 2023-06-20T17:02:28Z | |
| dc.date.available | 2023-06-20T17:02:28Z | |
| dc.date.issued | 1997-11 | |
| dc.description.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. | |
| dc.description.department | Depto. de Análisis Matemático y Matemática Aplicada | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | DGICYT | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/16970 | |
| dc.identifier.doi | 10.1088/0951-7715/10/6/016 | |
| dc.identifier.issn | 0951-7715 | |
| dc.identifier.officialurl | http://iopscience.iop.org/0951-7715/10/6/016 | |
| dc.identifier.relatedurl | http://iopscience.iop.org | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/57672 | |
| dc.issue.number | 6 | |
| dc.journal.title | Nonlinearity | |
| dc.language.iso | eng | |
| dc.page.final | 1754 | |
| dc.page.initial | 1739 | |
| dc.publisher | IOP Publishing Ltd | |
| dc.relation.projectID | Grant PB93-0438 | |
| dc.rights.accessRights | restricted access | |
| dc.subject.cdu | 517.956.4 | |
| dc.subject.cdu | 539.2 | |
| dc.subject.keyword | Chemotaxis | |
| dc.subject.keyword | equations | |
| dc.subject.keyword | singularities | |
| dc.subject.keyword | clusters | |
| dc.subject.ucm | Ecuaciones diferenciales | |
| dc.subject.unesco | 1202.07 Ecuaciones en Diferencias | |
| dc.title | Finite-time aggregation into a single point in a reaction-diffusion system | |
| dc.type | journal article | |
| dc.volume.number | 10 | |
| dcterms.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 | |
| dspace.entity.type | Publication |
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