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

Abstract

We study nonnegative solutions of parabolic-parabolic Keller-Segel minimal-chemotaxis-growth systems with prototype given byin a smooth bounded smooth but not necessarily convex domain Ω ⊂ R n (n ≥ 3) with nonnegative initial data u0, v0 and homogeneous Neumann boundary data, where d1, d2, α, β, µ > 0, χ, κ ∈ R.We provide quantitative and qualitative descriptions of the competition between logistic damping and other ingredients, especially, chemotactic aggregation to guarantee boundedness and convergence. Specifically, we first obtain an explicit formula µ0 = µ0(n, d1, d2, α, χ) for the logistic damping rate µ such that the system has no blow-ups whenever µ > µ0. In particular, for Ω ⊂ R 3 , we get a clean formula for µ0:2000 Mathematics Subject Classification. Primary: 35K59, 35K51, 35K57, 92C17; Secondary: 35B44, 35A01.

Sub-logistic source can prevent blow-up in the 2D minimal Keller-Segel chemotaxis system

“…As for the boundary integral in (3.28), there are a couple of known ways to bound it in terms of the boundedness of ∇v L 2 , cf. [14,28,41]; the final outcome is Proof of Theorem 1.1. In light of the uniform L 2 -boundedness of u provided by Lemmas 3.3 and 3.4, the quite known L n 2 + -boundedness criterion with n = 2 in [1,40] obtained via Moser type iteration technique shows T m = ∞ and the uniform boundedness as stated in (1.7).…”

Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller–Segel model

“…This bound µ 0 was further improved by Lin and Mu [18] in 3-D, wherein they replaced the logistic source in (1.1) by thedamping term u − µu r with r ≥ 2 to derive the boundedness under µ 1 r−1 > 20χ. Very recently, for a full-parameter version of (1.1), we calculate out the explicit formula for µ 0 in terms of the involving parameters, which states that µ > 9 √ 10−2 χ = (7.743416 · · · )χ ensures boundedness and global existence for (1.1) in 3-D [39]. Yet, it is a big open challenging problem whether or not blow-up occurs in (1.1) for small µ > 0, even though the existence of global weak solutions is available in convex 3-D domains for µ > 0 [16].…”

Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations

“…In the case of doubly parabolic Keller-Segel system, the question of stability of the homogeneous solution was addressed by Lin & Mu [60], Winkler [78], Xiang [80] and Zheng [85], see also Tello & Winkler [72]. For conditions forcing the solutions to vanish, compare Lankeit [45].…”