CROSS DIFFUSION AND NONLINEAR DIFFUSION PREVENTING BLOW UP IN THE KELLER–SEGEL MODEL
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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.
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.
SponsorsJ.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.
PublisherWorld Scientific Pub Co Pte Lt