Stabilization in the logarithmic Keller-Segel system

The Keller-Segel system {ut = D Delta u - D chi del center dot(u/v del v), x is an element of Omega, t > 0, vt = D Delta v - v + u, x is an element of Omega, t >0, is considered in a bounded domain Omega subset of R-n, n >= 2, with smooth boundary, where chi > 0 and D > 0. The main results identify a condition on the parameters chi < root 2/n and D > 0, essentially reducing to the assumption that chi(2)/D 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) over bar (0), (u) over bar (0)) at an exponential rate with respect to the norm in (L-infinity(Omega))(2) as t -> infinity, where (u) over bar (0) := 1/vertical bar Omega vertical bar integral(Omega) u(center dot, 0). As a particular consequence, this entails a convergence statement of the above flavor in the normalized system with D = 1 and fixed chi < root 2/n, provided that Omega satisfies a certain smallness condition. (c) 2018 Elsevier Ltd. All rights reserved.