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

Abstract

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.

arXiv comment34 pages, a second revised version; detected typos were amended

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• Chemotaxis effect vs logistic damping on boundedness in the 2-D minimal Keller-Segel model
• A class of chemotaxis systems with growth source and nonlinear secretion
• Global solvability in a two-species chemotaxis system with signal production
• Emergence of lager densities in chemotaxis system with indirect signal production and non-radial symmetry case
• Possible points of blow-up in chemotaxis systems with spatially heterogeneous logistic source
• Immediate smoothing and global solutions for initial data in $L^1\times W^{1,2}$ in a Keller-Segel system with logistic terms in 2D
• Global solvability and asymptotical behavior in a two-species chemotaxis model with signal absorption
• Boundedness and asymptotically stability to chemotaxis system with competitive kinetics and nonlocal terms
• Negligibility of haptotaxis effect in a chemotaxis-haptotaxis model