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

Abstract The quasilinear chemotaxis system ( ⋆ ) { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) , v t = Δ v − v + u , is considered under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n , 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 ) ≤ C s α for all  s ≥ 0 with some C > 0 and α 2 n , for all suitably regular initial data satisfying an essentially explicit smallness assumption on the total mass ∫ Ω u 0 , 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 > 2 n , this confirms criticality of the exponent α = 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.