Blanchet, Adrien; Laurençot, Philippe Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. (English) Zbl 1264.35057 Commun. Pure Appl. Anal. 11, No. 1, 47-60 (2012). Summary: For a specific choice of the diffusion, the parabolic-elliptic Patlak-Keller-Segel system with nonlinear diffusion (also referred to as the quasilinear 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 \(d \geq 3\). Cited in 10 Documents MSC: 35B44 Blow-up in context of PDEs 35K65 Degenerate parabolic equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 92C17 Cell movement (chemotaxis, etc.) 35R09 Integro-partial differential equations 35C06 Self-similar solutions to PDEs Keywords:backward self-similar solutions; Patlak-Keller-Segel model; degenerate diffusion; parabolic elliptic system; Smoluchowski-Poisson equation PDFBibTeX XMLCite \textit{A. Blanchet} and \textit{P. Laurençot}, Commun. Pure Appl. Anal. 11, No. 1, 47--60 (2012; Zbl 1264.35057) Full Text: DOI arXiv