Symmetrization techniques on unbounded domains: Application to a chemotaxis system on R-N
| dc.contributor.author | Díaz Díaz, Jesús Ildefonso | |
| dc.contributor.author | Nagai, Toshitaka | |
| dc.contributor.author | Rakotoson, Jean Michel Theresien | |
| dc.date.accessioned | 2023-06-20T16:54:18Z | |
| dc.date.available | 2023-06-20T16:54:18Z | |
| dc.date.issued | 1998-05-01 | |
| dc.description.abstract | The authors study the parabolic-elliptic system on RN: ∂u/∂t=∇⋅(∇u−χu∇v), 0=Δv−γv+αu, u(0,⋅)=u0, a version of the mathematical model of chemotaxis proposed by Keller and Segel. A differential inequality for the quantity ∫s0u∗(t,σ)dσ, where u∗ is the decreasing rearrangement of the solution u(t,⋅) with respect to the spatial variable, is obtained. As a consequence, they obtain e.g. Lp-bounds of the solution (u,v) on R2 and global-in-time existence of solutions under the condition αχ∫R2u0<8π. This result is sharp. It is also proved that if u0 is radially symmetric and αχ∫R2u0>8π, then the solution (u,v) blows up in a finite time. Compared to the previous work of Díaz Díaz and Nagai [Adv. Math. Sci. Appl. 5 (1995), no. 2, 659--680; MR1361010 (96j:35246)], where this problem has been considered on bounded domains of RN, there are some additional technical difficulties connected with the regularity of the derivative ∂u∗/∂t. | |
| dc.description.department | Depto. de Análisis Matemático y Matemática Aplicada | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | TRUE | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/15695 | |
| dc.identifier.doi | 10.1006/jdeq.1997.3389 | |
| dc.identifier.issn | 0022-0396 | |
| dc.identifier.officialurl | http://zv4fy5pr5l.scholar.serialssolutions.com/?sid=google&auinit=JI&aulast=Diaz&atitle=Symmetrization+Techniques+on+Unbounded+Domains:+Application+to+a+Chemotaxis+System+on+UN&title=Journal+of+Differential+Equations&volume=145&issue=1&date=1998&spage=156 | |
| dc.identifier.relatedurl | http://www.sciencedirect.com/ | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/57385 | |
| dc.issue.number | 1 | |
| dc.journal.title | Journal of Differential Equations | |
| dc.language.iso | eng | |
| dc.page.final | 183 | |
| dc.page.initial | 156 | |
| dc.publisher | Elsevier | |
| dc.rights.accessRights | restricted access | |
| dc.subject.cdu | 517.518.28 | |
| dc.subject.keyword | weighted monotone rearrangement | |
| dc.subject.keyword | measure space application | |
| dc.subject.keyword | relative rearrangement | |
| dc.subject.keyword | regularity. | |
| dc.subject.ucm | Análisis funcional y teoría de operadores | |
| dc.title | Symmetrization techniques on unbounded domains: Application to a chemotaxis system on R-N | |
| dc.type | journal article | |
| dc.volume.number | 145 | |
| dcterms.references | N. D. Alikakos, L p bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations 4 (1979), 827–868. C. Bandle, "Isoperimetric Inequalities and Applications," Pitman, London, 1980. C. Bandle and I. Stakgold, The formation of the dead core in parabolic reaction-diffusion equations, Trans. Amer. Math. Soc. 286 (1984), 275–293. P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Report No. 80, Math. Inst., Univ. of Wroclaw, 1996. S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci. 56 (1981), 217–237. S. Childress, "Chemotactic Collapse in Two Dimensions," Lecture Notes in Biomath., Vol. 55, pp. 61–66, Springer-Verlag, Berlin/New York, 1984. K. M. Chong and N. M. Rice, "Equimeasurable Rearrangements of Functions," Queen's University, 1971. J. I. Díaz, "Nonlinear Partial Differential Equations and Free Boundaries, Vol 1. Elliptic Equations," Research Notes in Math., Vol. 106, Pitman, London, 1985. J. I. Díaz, Simetrización de problemas parabólicos no lineales: Aplicación a ecuaciones de reacción-difusión, Mem. Real Acad. Cien. Exac., Fís., Natur. Madrid 27 (1991). J. I. Díaz and J. Mossino, Isoperimetric inequalities in the parabolic obstacle problems, J. Math. Pures Appl. 71 (1992), 233–266. J. I. Díaz and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Adv. Math. Sci. Appl. 5 (1995), 659–680. I. Ekeland and R. Temam, "Convex Analysis and Variational Problems," Am. Elsevier, New York, 1976. H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Preprint No. 232, Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, 1996. B. Gusfsson and J. Mossino, Isoperimetric inequalities for the Stefan problem, SIAM J. Math. Anal. 20 (1989), 1095–1108. D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Math., Vol. 840, Springer-Verlag, Berlin/New York, 1981. M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996), 583–623. M. A. Herrero and J. J. L. Velázquez, Chemotaxis collapse for the Keller-Segel model, J. Math. Biol. 35 (1996), 177–194. M. A. Herrero and J. J. L. Velázquez, A blow up mechanism for a chemotaxis model, preprint. W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), 819–824. E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970), 399–415. P. J. Laurent, "Approximation et Optimisation," Hermann, Paris, 1972. J. Mossino and J. M. Rakotoson, Isoperimetric inequalities in parabolic equations, Ann. Scuola Norm. Sup. Pisa El. Sci. (4) 13 (1986), 51–73. T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl. 5 (1995), 581–601. T. Nagai, T. Senba, and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, to appear. V. Nanjundiah, Chemotaxis, signal relaying, and aggregation morphology, J. Theor. Biol. 42 (1973), 63–105. J. F. Padial and M. L. Tello del Castillo, Introduction to the monotone and relative rearrangements and applications, Course by J. M. Rakotoson, Madrid, June, 1993. J. M. Rakotoson, Une nouvelle démonstration de la dérivée directionnelle u↦u∗pour un domaine quelconque de RN, unpublished. J. M. Rakotoson and B. Simon, Relative rearrangement on a measure space, application to the regularity of weighted monotone rearrangement, Part I, Appl. Math. Lett. 6 (1993), 75--78; Part II, Appl. Math. Lett. 6 (1993), 79–82. J. M. Rakotoson and B. Simon, Relative rearrangement on a finite measure space, application to the regularity of weighted monotone rearrangement and to P.D.E., Academia de Ciencias de Madrid, submitted. B. Simon, "Réarrangement relatif sur un espace mesuré et applications," Thesis, Univ. de Poitiers, April, 1994. B. Simon, Article, in preparation. | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | 34ef57af-1f9d-4cf3-85a8-6a4171b23557 | |
| relation.isAuthorOfPublication.latestForDiscovery | 34ef57af-1f9d-4cf3-85a8-6a4171b23557 |
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