FINITE MASS SELF-SIMILAR BLOWING-UP SOLUTIONS OF A CHEMOTAXIS SYSTEM WITH NON-LINEAR DIFFUSION

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.