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

Abstract

We study chemotaxis effect vs logisticdamping on boundedness for the two-dimensional minimal Keller-Segel model with logistic source:

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

“…While when and if is sufficiently large, for all sufficiently smooth initial values, () has unique classical solution[21]. When , Jin and Xiang [10, 24] show how the upper bound of the solution of the Chemotaxis model depends on the parameters in the equation. However, to the best of our knowledge, no results are available for the problem () with general logistic source.…”

Global dynamics in a chemotaxis model describing tumor angiogenesis with/without mitosis in any dimensions

“…We now use similar sprits as used in [16,44] to derive the L ∞ -bound of u. To that purpose, we use the variation-of-constants formula to the u-equation in (1.4) to write…”

Global solvability in a three-dimensional Keller-Segel-Stokes system involving arbitrary superlinear logistic degradation

“…Mathematically, the introduction of such logistic e ects can rule out the possibility of unboundedness phenomena in the style of those known to occur for (1.1); indeed, in the most prototypical framework related to the choice α = , corresponding Neumann-type initial-boundary value problems admit bounded smooth solutions for essentially arbitrary initial data if either the spatial dimension N satis es N = and µ is any positive number ( [13]), or N ≥ and µ ≥ µ with some suitably large µ = µ (ρ, Ω) > ( [25]; cf. also [7], [34], [35] and the references therein for some more studies concerned with global classical solvability of (1.2)).…”