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

Abstract

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 $\kappa \le 0$ and the existence of an absorbing set if $\kappa >0$ is sufficiently small. Keywords: chemotaxis, logistic source, existence, weak solutions, eventual smoothness