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Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. (English) Zbl 1264.35057

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 \(d \geq 3\).

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
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