Stabilization in the logarithmic Keller-Segel system

The Keller-Segel system ut=DΔu−Dχ∇⋅(uv∇v),x∈Ω,t>0,vt=DΔv−v+u,x∈Ω,t>0,is considered in a bounded domain Ω⊂Rn, n≥2, with smooth boundary, where χ>0 and D>0. The main results identify a condition on the parameters χ<2n and D>0, essentially reducing to the assumption that χ2D be suitably small, under which for all reasonably regular and positive initial data the corresponding classical solution of an associated Neumann initial-boundary value problem, known to exist globally according to previous findings, approaches the homogeneous steady state (u¯0,u¯0) at an exponential rate with respect to the norm in (L∞(Ω))2 as t→∞, where u¯0≔1|Ω|∫Ωu(⋅,0). As a particular consequence, this entails a convergence statement of the above flavor in the normalized system with D=1 and fixed χ<2n, provided that Ω satisfies a certain smallness condition.