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

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

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