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

Abstract

It is well-known that the Neumann initial-boundary value problem for the minimal-chemotaxis-logistic systemin a bounded smooth domain Ω ⊂ R 2 doesn't have any blow-ups for any a ∈ R, τ ≥ 0, χ > 0 and b > 0. Here, we obtain the same conclusion by replacing the logistic source au−bu 2 with a kinetic term fη>0 sup{f (s) + ηs : s > 0} η . In this setup, it is shown that this problem doesn't have any blow-up by ensuring all solutions are global-in-time and uniformly bounded. Clearly, f covers super-, logistic, sub-logistic sources like f (s) = as − bs θ with b > 0 and θ ≥ 2, f (s) = as− bs 2 ln γ (s+1) with b > 0 and γ ∈ (0, 1), and f (s) = as− bs 2 ln(ln(s+e)) with b > 0 etc. This indicates that logistic damping is not the weakest damping to guarantee boundedness for the 2D Keller-Segel minimal chemotaxis model. 2000 Mathematics Subject Classification. Primary: 35K59, 35K51, 35K57, 92C17; Secondary: 35B44, 35A01.

Approaching optimality in blow-up results for Keller–Segel systems with logistic-type dampening

“…Moreover, for n ≥ 3 and μ = n−2 n (and again τ = 0 and at least λ ≥ 0) solutions to (1.2) are global in time [9], but to the best of our knowledge it is unknown whether these are also always bounded. For the parabolic-parabolic case, that is, for τ > 0, the situation is similar: In the two-dimensional setting, assuming merely μ > 0 suffices to guarantee global existence of classical solutions [19], even for dampening terms growing slightly slower then quadratically [41]. Moreover, for higher dimensional convex domains, global classical solutions have been constructed for μ > μ 0 for some μ 0 > 0 in [28], where explicit upper bounds of μ 0 then have been derived in [16,40] and the convexity assumption has been removed in [39] at the cost of worsening the condition on μ.…”

An n-dimensional chemotaxis system with signal-dependent motility and generalized logistic source: global existence and asymptotic stabilization

“…In [8], the global classical solution and large time behaviour are considered with l = 2. In [35], the author shows the uniform-in-time boundedness for the corresponding 2D Neumann initial-boundary value problem in a large class of cell kinetics including sub-logistic sources. Besides, many authors are interested in qualitative convergence of the solution [31,32] and large time behaviour [3,30] for such kind of model.…”

Upper bound on the solution for a class of 2‐D Chemotaxis model with generalized logistic damping

“…If , then logistic damping effect is stronger than that of chemotactic aggregation. Even if is small enough, it can still prevent the blow‐ups, which ensures that the solution of () is global‐in‐time and uniformly bounded [6, 14, 23, 25]. While when and if is sufficiently large, for all sufficiently smooth initial values, () has unique classical solution[21].…”