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

Abstract

The quasilinear chemotaxis system (⋆){ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),vt=Δv−v+u, is considered under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn, n≥2, with smooth boundary, where the focus is on cases when herein the diffusivity D(s) decays exponentially as s→∞.It is shown that under the subcriticality condition that (0.1)S(s)D(s)≤Csαfor all s≥0 with some C>0 and α<2n, for all suitably regular initial data satisfying an essentially explicit smallness assumption on the total mass ∫Ωu0, the corresponding Neumann initial–boundary value problem for (⋆) 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 α>2n, this confirms criticality of the exponent α=2n 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.