Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion

2009/11/04 by Adrien Blanchet, Philippe Laurençot, Philippe Laurencot
Economics, Econometrics and Finance · Mathematics · #Blowing up #Critical mass (sociodynamics) #Diffusion #Dimension (graph theory) #Mathematical Biology Tumor Growth #Mathematical analysis #Mathematics #Physics #Poisson distribution #Pure mathematics #Space (punctuation) #Statistics #Stochastic processes and financial applications #Thermodynamics #advanced mathematical theories #math.AP #msc:34C10 #msc:35K65 #msc:92B99

paper · pdf · doi:10.3934/cpaa.2012.11.47

published as Communications on Pure and Applied Analysis 11, 1 (2012) 47-60

arxiv created 2009/11/04 · openalex publication_date 2011/09/16 · arxiv updated 2012/07/10 · openalex created_date 2021/02/01 · openalex updated_date 2026/08/05

Abstract

For a specific choice of the diffusion, the parabolic-elliptic Patlak-Keller-Segel system with non-linear diffusion (also referred to as the quasi-linear Smoluchowski-Poisson equation) exhibits an interesting threshold phenomenon: there is a critical mass M_c>0 such that all solutions with initial data of mass smaller or equal to M_c exist globally while the solution blows up in finite time for a large class of initial data with mass greater than M_c. Unlike in space dimension 2, finite mass self-similar blowing-up solutions are shown to exist in space dimension d≥ 3.

Citations

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