How strong a logistic damping can prevent blow-up for the minimal Keller–Segel chemotaxis system?

In this paper, we study the minimal Keller-Segel model with a logistic source and obtain quantitative and qualitative descriptions of the competition between logistic damping and other ingredient, especially, chemotactic aggregation to guarantee boundedness and convergence. More specifically, we establish how precisely strong a logistic source can prevent blow-up, and then we obtain an explicit relationship between logistic damping and other ingredient, especially, chemotactic aggregation so that convergences are ensured and their respective convergence rates are explicitly calculated out. Known results in the literature are completed and refined. Furthermore, our findings provide clues on how to produce blowup solutions for KS chemotaxis models with logistic sources.
Comment: 34 pages, a second revised version; detected typos were amended