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

TL;DR

This paper analyzes the Keller-Segel system in high dimensions, providing new criteria for blow-up, concentration, and global existence, along with a visualization approach linking PDE dynamics to finite-dimensional systems.

Contribution

It introduces improved conditions for global existence, new blow-up and concentration criteria, and a novel visualization method connecting PDE behavior to finite-dimensional dynamical systems.

Findings

Global existence under smallness conditions on initial data

New criteria for blow-up and concentration phenomena

A visualization tool reducing PDE dynamics to finite-dimensional systems

Abstract

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.