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

We prove existence of global weak solutions to the chemotaxis systemut=Δu−∇⋅(u∇v)+κu−μu2ut=Δu−∇⋅(u∇v)+κu−μu2vt=Δv−v+uvt=Δv−v+u under homogeneous Neumann boundary conditions in a smooth bounded convex domain Ω⊂RnΩ⊂Rn, for arbitrarily small values of μ>0μ>0.Additionally, we show that in the three-dimensional setting, after some time, these solutions become classical solutions, provided that κ   is not too large. In this case, we also consider their large-time behaviour: We prove decay if κ≤0κ≤0 and the existence of an absorbing set if κ>0κ>0 is sufficiently small.