Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions

dc.contributor.authorBlanchet, Adrien
dc.contributor.authorDolbeault, Jean
dc.contributor.authorPerthame, Benoit
dc.date.accessioned2021-07-16T14:24:44Z
dc.date.available2021-07-16T14:24:44Z
dc.date.issued2006-04-06
dc.description.abstractThe Keller-Segel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. It is known that, in two space dimensions, for small initial mass, there is global existence of solutions and for large initial mass blow-up occurs. In this paper we complete this picture and give a detailed proof of the existence of weak solutions below the critical mass, above which any solution blows-up in finite time in the whole Euclidean space. Using hypercontractivity methods, we establish regularity results which allow us to prove an inequality relating the free energy and its time derivative. For a solution with sub-critical mass, this allows us to give for large times an "intermediate asymptotics'' description of the vanishing. In self-similar coordinates, we actually prove a convergence result to a limiting self-similar solution which is not a simple reflect of the diffusion.
dc.description.departmentMathematics
dc.formatText
dc.format.extent33 pages
dc.format.medium1 file (.pdf)
dc.identifier.citationBlanchet, A., Dolbeault, J., & Perthame, B. (2006). Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions. <i>Electronic Journal of Differential Equations, 2006</i>(44), pp. 1-33.
dc.identifier.issn1072-6691
dc.identifier.urihttps://hdl.handle.net/10877/13917
dc.language.isoen
dc.publisherTexas State University-San Marcos, Department of Mathematics
dc.rightsAttribution 4.0 International
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/
dc.sourceElectronic Journal of Differential Equations, 2006, San Marcos, Texas: Texas State University-San Marcos and University of North Texas.
dc.subjectKeller-Segel model
dc.subjectExistence
dc.subjectWeak solutions
dc.subjectFree energy
dc.subjectEntropy method
dc.subjectLogarithmic Hardy-Littlewood-Sobolev inequality
dc.subjectCritical mass
dc.subjectAubin-Lions compactness method
dc.subjectHypercontractivity
dc.subjectLarge time behavior
dc.subjectTime-dependent rescaling
dc.subjectSelf-similar variables
dc.subjectIntermediate asymptotics
dc.titleTwo-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions
dc.typeArticle

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