Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity

The fully parabolic Keller-Segel system with logistic source \begin{document}$\begin{equation} \left\{ \begin{array}{llc} \displaystyle u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\kappa u-\mu u^2, &(x,t)\in \Omega\times (0,T),\\ \displaystyle \tau v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T), \end{array} \right.(\star) \end{equation}$\end{document} is considered in a bounded domain $\Omega\subset\mathbb{R}^N$ ($N≥ 1$) under Neumann boundary conditions, where $κ∈\mathbb{R}$, $μ>0$, $χ>0$ and $τ>0$. It is shown that if the ratio $\frac{χ}{μ}$ is sufficiently small, then any global classical solution $(u, v)$ converges to the spatially homogenous steady state $(\frac{κ_+}{μ}, \frac{κ_+}{μ})$ in the large time limit. Here we use an approach based on maximal Sobolev regularity and thus remove the restrictions $τ=1$ and the convexity of $\Omega$ required in [17].