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

Vincent Calvez (1, 2) , Lucilla Corrias (3) , Mohammed Abderrahman Ebde (4)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 NUMED - Numerical Medicine
3 Département de Mathématiques (Evry)
4 ICP - Institut CNRS-PAULI

This paper is devoted to the analysis of the classical Keller-Segel system over $\mathbb{R}^d$, $d\geq 3$. 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 when dealing with the parabolic-elliptic case are: local existence without smallness assumption on the initial density, global existence under an improved smallness condition and comparison of blow-up criteria. A new concentration phenomenon criteria for the fully parabolic case is also given. The analysis is completed by a visualization tool based on the reduction of the parabolic-elliptic system to a finite-dimensional dynamical system of gradient flow type, sharing features similar to the infinite-dimensional system.

Mots clés: Chemotaxis; parabolic systems; global weak solutions; local weak solutions; blow-up; energy methods

Domaines: Equations aux dérivées partielles [math.AP]

Vincent Calvez, Lucilla Corrias, Mohammed Abderrahman Ebde. Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension. 2010. ⟨hal-00462450⟩