Global dynamics in a fully parabolic chemotaxis system with logistic source

Abstract

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.