Blow-up, Concentration Phenomenon and Global Existence for the Keller-Segel Model in High Dimension

This article is devoted to the analysis of the classical Keller-Segel system over R-d, d >= 3. We describe as much as possible the dynamics of the system characterized by various criteria, both in the parabolic-elliptic case and in the fully parabolic case. The main results in the parabolic-elliptic case are: local existence without smallness assumption on the initial density and a quantified blow-up rate, global existence under an improved smallness condition and comparison of blow-up criteria. A new concentration phenomenon for the fully parabolic case is also given.