Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model

We consider the classical parabolic-parabolic Keller-Segel system {u(t) = Delta u -del . (u del v), x is an element of Omega, t > 0, v(t) = Delta v - v + u, x is an element of , t > 0, under homogeneous Neumann boundary conditions in a smooth bounded domain Omega subset of R-n. It is proved that in space dimension n >= 3, for each q > n/2 and p > n one can find epsilon(0) > 0 Such that if the initial data (u(0), v(0)) satisfy parallel to u(0)parallel to L-q (Omega) < epsilon and parallel to del v(0)parallel to L-q (Omega) < epsilon then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic equations. In particular, (u, v) approaches the steady state (m, m) as t -> infinity, where m is the total mass m := integral Omega u(0) of the population. Moreover, we shall show that if Omega is a ball then for arbitrary prescribed m > 0 there exist unbounded solutions emanating from initial data (u(0), v(0)) having total mass integral Omega u(0) = m. (C) 2010 Elsevier Inc. All rights reserved.