Stabilization in the logarithmic Keller–Segel system

Abstract

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.