# American Institute of Mathematical Sciences

November  2017, 22(9): 3369-3378. doi: 10.3934/dcdsb.2017141

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

 1 Institute of Mathematical Sciences, Renmin University, Beijing 100872, China 2 Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

* Corresponding author

Received  October 2016 Revised  January 2017 Published  April 2017

Fund Project: XC is supported by the Research Funds of Renmin University of China (15XNLF21), and by China Postdoctoral Science Foundation (2016M591319)

The fully parabolic Keller-Segel system with logistic source
 $$$\left\{ \begin{array}{llc} \displaystyle u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\kappa u-\mu u^2, &(x,t)\in \Omega\times (0,T),\\ \displaystyle \tau v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T), \end{array} \right.(\star)$$$
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].
Citation: Xinru Cao. Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3369-3378. doi: 10.3934/dcdsb.2017141
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