Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

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

Pages: 3369 - 3378, Volume 22, Issue 9, November 2017      doi:10.3934/dcdsb.2017141

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Xinru Cao - Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China (email)

Abstract: The fully parabolic Keller-Segel system with logistic source \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} is considered in a bounded domain $\Omega\subset\mathbb{R}^N$ ($N\ge 1$) under Neumann boundary conditions, where $\kappa\in\mathbb{R}$, $\mu>0$, $\chi>0$ and $\tau>0$. It is shown that if the ratio $\frac{\chi}{\mu}$ is sufficiently small, then any global classical solution $(u,v)$ converges to the spatially homogenous steady state $(\frac{\kappa_+}{\mu},\frac{\kappa_+}{\mu})$ in the large time limit. Here we use an approach based on maximal Sobolev regularity and thus remove the restrictions $\tau=1$ and the convexity of $\Omega$ required in [17].

Keywords:  Chemotaxis, asymptotic behavior, stability.
Mathematics Subject Classification:  Primary: 35B40, 35K45.

Received: October 2016;      Revised: January 2017;      Available Online: April 2017.