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For a specific choice of the diffusion, the parabolic-ellipticPatlak-Keller-Segel system with non-linear diffusion (also referred to as thequasi-linear Smoluchowski-Poisson equation) exhibits an interesting thresholdphenomenon: there is a critical mass $M_c>0$ such that all the solutions withinitial data of mass smaller or equal to $M_c$ exist globally while thesolution blows up in finite time for a large class of initial data with massgreater than $M_c$. Unlike in space dimension 2, finite mass self-similarblowing-up solutions are shown to exist in space dimension $d?3$.

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