Stabilization in a higher-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity

The quasilinear chemotaxis system {u(t) = del . (D(u)del u) - del. (S(u)del v), v(t) = Delta v - v + u, (star) T is considered under homogeneous Neumann boundary conditions in a bounded domain Omega subset of R-n, n >= 2, with smooth boundary, where the focus is on cases when herein the diffusivity D(s) decays exponentially as s -> infinity. It is shown that under the subcriticality condition that S(s)/D(s) <= Cs-alpha for all s >= 0 (0.1) with some C > 0 and alpha < 2/n, for all suitably regular initial data satisfying an essentially explicit smallness assumption on the total mass integral(Omega)u(0), the corresponding Neumann initial-boundary value problem for (star) possesses a globally defined bounded classical solution which moreover approaches a spatially homogeneous steady state in the large time limit. Viewed as a complement of known results on the existence of small-mass blow-up solutions in cases when in (0.1) the reverse inequality holds with some alpha > 2/n, this confirms criticality of the exponent n = 2/n in (0.1) with regard to the singularity formation also for arbitrary n >= 2, thereby generalizing a recent result on unconditional global boundedness in the two-dimensional situation. As a by-product of our analysis, without any restriction on the initial data, we obtain boundedness and stabilization of solutions to a so-called volume-filling chemotaxis system involving jump probability functions which decay at sufficiently large exponential rates. (C) 2016 Elsevier Ltd. All rights reserved.