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

摘要

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 Mc exist globally while the solution blows up in finite time for a large class of initial data with mass greater than Mc. Unlike in space dimension 2, finite mass self-similar blowing-up solutions are shown to exist in space dimension d ≥ 3.