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 - \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 
Johannes Lankeit
math.AP