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September 2016, 36(9): 5025-5046. doi: 10.3934/dcds.2016018
Global dynamics in a fully parabolic chemotaxis system with logistic source
Ke Lin 1, and Chunlai Mu 1,
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
Received July 2015 Revised January 2016 Published May 2016
In this paper, we consider a fully parabolic chemotaxis system \begin{eqnarray*}\label{1} \left\{ \begin{array}{llll} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+u-\mu u^r,\quad &x\in \Omega,\quad t>0,\\ v_t=\Delta v-v+u,\quad &x\in\Omega,\quad t>0,\\ \end{array} \right. \end{eqnarray*} with homogeneous Neumann boundary conditions in an arbitrary smooth bounded domain $\Omega\subset R^n(n=2,3)$, where $\chi>0, \mu>0$ and $r\geq 2$.
For the dimensions $n=2$ and $n=3$, we establish results on the global existence and boundedness of classical solutions to the corresponding initial-boundary problem, provided that $\chi$, $\mu$ and $r$ satisfy some explicit conditions. Apart from this, we also show that if $\frac{\mu^{\frac{1}{r-1}}}{\chi}>20$ and $r\geq 2$ and $r\in \mathbb{N}$ the solution of the system approaches the steady state $\left(\mu^{-\frac{1}{r-1}}, \mu^{-\frac{1}{r-1}}\right)$ as time tends to infinity.
Keywords: logistic source, Chemotaxis, boundedness, asymptotic stability..
Mathematics Subject Classification: Primary: 35K35, 92C17; Secondary: 35K55, 35B3.
Citation: Ke Lin, Chunlai Mu. Global dynamics in a fully parabolic chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5025-5046. doi: 10.3934/dcds.2016018
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Ke Lin Chunlai Mu