Global and exponential attractor of the repulsive Keller–Segel model with logarithmic sensitivity

Abstract

We consider a Keller–Segel model that describes the cellular chemotactic movement away from repulsive chemical subject to logarithmic sensitivity function over a confined region in ${{\mathbb{R}}^n},\,n \le 2$ . This sensitivity function describes the empirically tested Weber–Fecher’s law of living organism’s perception of a physical stimulus. We prove that, regardless of chemotaxis strength and initial data, this repulsive system is globally well-posed and the constant solution is the global and exponential in time attractor. Our results confirm the ‘folklore’ that chemorepulsion inhibits the formation of non-trivial steady states within the logarithmic chemotaxis model, hence preventing cellular aggregation therein.

A repertoire of repulsive Keller–Segel models with logarithmic sensitivity: Derivation, traveling waves, and quasi‐stationary dynamics

“…Indeed, it is shown that the strong damping effect due to chemotactic repulsion contributes to prevent blowup. The long‐time convergence of this system to the (unique) constant solution is proved in Chen et al, 51 which reinforces the idea of chemorepulsion as a mechanism that inhibits the formation of nontrivial patterns.…”

Global existence in a chemotaxis system with singular sensitivity and signal production

“…Zhao and Zheng [56] generalized their own work [54] to the higher dimensional case. For more recent outcomes, one can see [5,6,10,11,36,39,41,49,53,57].…”

Stability, free energy and dynamics of multi-spikes in the minimal Keller-Segel model

“…Here and in the sequel, the Big-O only depends on L. Now that (U, V)(L − x) is an increasing steady state of (1.1), any nonconstant monotone steady state of (1.1) is expected to be one of the pair when κ is sufficiently large. Before presenting our main results, it is worthwhile noting that a few recent studies (e.g., [6,9,20]) prove that chemorepulsion, i.e., bacteria moves against repulsive chemical, tends to stabilize the constant solution for the Keller-Segel systems with linear or logarithmic sensitivities. This also motivates us to study the pattern formation for stable spike steady states when chemoattraction is strong.…”