  We prove existence of global weak solutions to the chemotaxis system
  $ u_t=\Delta u - \nabla\cdot (u\nabla v) +\kappa u -\mu u^2 $
  $ v_t=\Delta v-v+u $
  under homogeneous Neumann boundary conditions in a smooth bounded convex
domain $\Omega\subset R^n$, 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 $\kappa\leq 0$ and the existence of an absorbing set if $\kappa>0$ is
sufficiently small.
  Keywords: chemotaxis, logistic source, existence, weak solutions, eventual
smoothness
