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

We consider the classical parabolic–parabolic Keller–Segel system {ut=Δu−∇⋅(u∇v),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn. It is proved that in space dimension n⩾3, for each q>n2 and p>n one can find e0>0 such that if the initial data (u0,v0) satisfy ‖u0‖Lq(Ω)<e and ‖∇v0‖Lp(Ω)<e 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→∞, where m is the total mass m:=∫Ωu0 of the population. Moreover, we shall show that if Ω is a ball then for arbitrary prescribed m>0 there exist unbounded solutions emanating from initial data (u0,v0) having total mass ∫Ωu0=m.