CROSS DIFFUSION AND NONLINEAR DIFFUSION PREVENTING BLOW UP IN THE KELLER–SEGEL MODEL

Handle URI:
http://hdl.handle.net/10754/597896
Title:
CROSS DIFFUSION AND NONLINEAR DIFFUSION PREVENTING BLOW UP IN THE KELLER–SEGEL MODEL
Authors:
CARRILLO, JOSÉ ANTONIO; HITTMEIR, SABINE; JÜNGEL, ANSGAR
Abstract:
A parabolic-parabolic (Patlak-)Keller-Segel model in up to three space dimensions with nonlinear cell diffusion and an additional nonlinear cross-diffusion term is analyzed. The main feature of this model is that there exists a new entropy functional, yielding gradient estimates for the cell density and chemical concentration. For arbitrarily small cross-diffusion coefficients and for suitable exponents of the nonlinear diffusion terms, the global-in-time existence of weak solutions is proved, thus preventing finite-time blow up of the cell density. The global existence result also holds for linear and fast diffusion of the cell density in a certain parameter range in three dimensions. Furthermore, we show L∞ bounds for the solutions to the parabolic-elliptic system. Sufficient conditions leading to the asymptotic stability of the constant steady state are given for a particular choice of the nonlinear diffusion exponents. Numerical experiments in two and three space dimensions illustrate the theoretical results. © 2012 World Scientific Publishing Company.
Citation:
CARRILLO JA, HITTMEIR S, JÜNGEL A (2012) CROSS DIFFUSION AND NONLINEAR DIFFUSION PREVENTING BLOW UP IN THE KELLER–SEGEL MODEL. Mathematical Models and Methods in Applied Sciences 22: 1250041. Available: http://dx.doi.org/10.1142/S0218202512500418.
Publisher:
World Scientific Pub Co Pte Lt
Journal:
Mathematical Models and Methods in Applied Sciences
Issue Date:
Dec-2012
DOI:
10.1142/S0218202512500418
Type:
Article
ISSN:
0218-2025; 1793-6314
Sponsors:
J.A.C. was partially supported by the project MTM2011-27739-C04/-02 DGI (Spain) and 2009-SGR-345 from AGAUR-Generalitat de Catalunya. The work of S. H. was supported by Award No. KUK-I1-007-43, funded by King Abdullah University of Science and Technology (KAUST). S. H. and A.J. acknowledge partial support from the Austrian Science Fund (FWF), grants P20214, P22108, and I395; the Austrian-Croatian Project HR 01/2010 and the Austrian-French Project FR 07/2010 of the Austrian Exchange Service (OAD). All authors acknowledge support from the Austrian-Spanish Project ES 08/2010 of the OAD.
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Full metadata record

DC FieldValue Language
dc.contributor.authorCARRILLO, JOSÉ ANTONIOen
dc.contributor.authorHITTMEIR, SABINEen
dc.contributor.authorJÜNGEL, ANSGARen
dc.date.accessioned2016-02-25T12:58:31Zen
dc.date.available2016-02-25T12:58:31Zen
dc.date.issued2012-12en
dc.identifier.citationCARRILLO JA, HITTMEIR S, JÜNGEL A (2012) CROSS DIFFUSION AND NONLINEAR DIFFUSION PREVENTING BLOW UP IN THE KELLER–SEGEL MODEL. Mathematical Models and Methods in Applied Sciences 22: 1250041. Available: http://dx.doi.org/10.1142/S0218202512500418.en
dc.identifier.issn0218-2025en
dc.identifier.issn1793-6314en
dc.identifier.doi10.1142/S0218202512500418en
dc.identifier.urihttp://hdl.handle.net/10754/597896en
dc.description.abstractA parabolic-parabolic (Patlak-)Keller-Segel model in up to three space dimensions with nonlinear cell diffusion and an additional nonlinear cross-diffusion term is analyzed. The main feature of this model is that there exists a new entropy functional, yielding gradient estimates for the cell density and chemical concentration. For arbitrarily small cross-diffusion coefficients and for suitable exponents of the nonlinear diffusion terms, the global-in-time existence of weak solutions is proved, thus preventing finite-time blow up of the cell density. The global existence result also holds for linear and fast diffusion of the cell density in a certain parameter range in three dimensions. Furthermore, we show L∞ bounds for the solutions to the parabolic-elliptic system. Sufficient conditions leading to the asymptotic stability of the constant steady state are given for a particular choice of the nonlinear diffusion exponents. Numerical experiments in two and three space dimensions illustrate the theoretical results. © 2012 World Scientific Publishing Company.en
dc.description.sponsorshipJ.A.C. was partially supported by the project MTM2011-27739-C04/-02 DGI (Spain) and 2009-SGR-345 from AGAUR-Generalitat de Catalunya. The work of S. H. was supported by Award No. KUK-I1-007-43, funded by King Abdullah University of Science and Technology (KAUST). S. H. and A.J. acknowledge partial support from the Austrian Science Fund (FWF), grants P20214, P22108, and I395; the Austrian-Croatian Project HR 01/2010 and the Austrian-French Project FR 07/2010 of the Austrian Exchange Service (OAD). All authors acknowledge support from the Austrian-Spanish Project ES 08/2010 of the OAD.en
dc.publisherWorld Scientific Pub Co Pte Lten
dc.subjectBlow upen
dc.subjectChemotaxisen
dc.subjectCross-diffusionen
dc.subjectDegenerate diffusionen
dc.subjectGlobal existence of solutionen
dc.subjectKeller-Segel modelen
dc.titleCROSS DIFFUSION AND NONLINEAR DIFFUSION PREVENTING BLOW UP IN THE KELLER–SEGEL MODELen
dc.typeArticleen
dc.identifier.journalMathematical Models and Methods in Applied Sciencesen
dc.contributor.institutionUniversidad Autonoma de Barcelona, Barcelona, Spainen
dc.contributor.institutionTechnische Universitat Wien, Vienna, Austriaen
dc.contributor.institutionImperial College London, London, United Kingdomen
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