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This paper is devoted to the analysis of the classical Keller-Segel systemover $\mathbb{R}^d$, $d\geq 3$. We describe as much as possible the dynamics ofthe system characterized by various criteria, both in the parabolic-ellipticcase and in the fully parabolic case. The main results when dealing with theparabolic-elliptic case are: local existence without smallness assumption onthe initial density, global existence under an improved smallness condition andcomparison of blow-up criteria. A new concentration phenomenon criteria for thefully parabolic case is also given. The analysis is completed by avisualization tool based on the reduction of the parabolic-elliptic system to afinite-dimensional dynamical system of gradient flow type, sharing featuressimilar to the infinite-dimensional system.

Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension