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In this paper we study the chemotaxis-system \begin{document}$\begin{equation*}\begin{cases}u_{t}=Δ u-χ \nabla · (u\nabla v)+g(u)x'>defined in a convex smooth and bounded domain \begin{document}$Ω$\end{document} of \begin{document}$\mathbb{R}^n$\end{document} , \begin{document}$n≥ 1$\end{document} , with \begin{document}$χ>0$\end{document} and endowed with homogeneous Neumann boundary conditions. The source \begin{document}$g$\end{document} behaves similarly to the logistic function and satisfies \begin{document}$g(s)≤ a -bs^α$\end{document} , for \begin{document}$s≥ 0$\end{document} , with \begin{document}$a≥ 0$\end{document} , \begin{document}$b>0$\end{document} and \begin{document}$α>1$\end{document} . Continuing the research initiated in [ 33 ], where for appropriate \begin{document}$1 and \begin{document}$(u_0,v_0) ∈ C^0(\bar{Ω})× C^2(\bar{Ω})$\end{document} the global existence of very weak solutions \begin{document}$(u,v)$\end{document} to the system (for any \begin{document}$n≥ 1$\end{document} ) is shown, we principally study boundedness and regularity of these solutions after some time. More precisely, when \begin{document}$n=3$\end{document} , we establish that -for all \begin{document}$τ>0$\end{document} an upper bound for \begin{document}$\frac{a}{b}, ||u_0||_{L^1(Ω)}, ||v_0||_{W^{2,α}(Ω)}$\end{document} can be prescribed in a such a way that \begin{document}$(u,v)$\end{document} is bounded and Holder continuous beyond \begin{document}$τ$\end{document} ; -for all \begin{document}$(u_0,v_0)$\end{document} , and sufficiently small ratio \begin{document}$\frac{a}{b}$\end{document} , there exists a \begin{document}$T>0$\end{document} such that \begin{document}$(u,v)$\end{document} is bounded and Holder continuous beyond \begin{document}$T$\end{document} . Finally, we illustrate the range of dynamics present within the chemotaxis system in one, two and three dimensions by means of numerical simulations.

Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth