Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source

We prove existence of global weak solutions to the chemotaxis system u(t) = Delta u - del . (u del v) + kappa u - mu u(2) v(t) = Delta v - v + u under homogeneous Neumann boundary conditions in a smooth bounded convex domain Omega subset of R-n, for arbitrarily small values of mu > 0. Additionally, we show that in the three-dimensional setting, after some time, these solutions become classical solutions, provided that kappa is not too large. In this case, we also consider their large-time behaviour: We prove decay if kappa <= 0 and the existence of an absorbing set if kappa > 0 is sufficiently small. (C) 2014 Elsevier Inc. All rights reserved.