Abstract:

In this paper, we study the following problem $\left\{\begin{array}{l} u_{t}=\Delta{(\gamma(v)u)}, & x\in\Omega, t>0, \\ v_{t}=\Delta v+wz, & x\in\Omega, t>0, \\ w_{t}=-wz, & x\in\Omega, t>0, \\ z_{t}=\Delta z+ u- z, & x\in\Omega, t>0, \\ \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=\frac{\partial z}{\partial \nu}=0, & x\in\partial\Omega, t>0, \\ (u, v, w, z)(x, 0)=(u_0, v_0, w_0, z_0)(x), &x\in\Omega \end{array}\right.$ in a bounded domain $\Omega\subset\mathbb{R} ^n(1\leqq n\leqq 5)$ with smooth boundary and $\nu$ denotes the outward normal vector of $\partial \Omega$, where $0 <\gamma(v)\in C^3[0, \infty)$. Under suitably regular initial data, we show the existence of global classical solution with uniform-in-time bound under one of the following conditions$ \bullet\; 1\leq n\leq 3$,$ \bullet\; 4\leq n\leq 5$ and $\gamma_2\geq \gamma(v)\geq \gamma_1>0$,$\left|\gamma'(v)\right|\leq \gamma_3, $ $v\in [0, \infty)$ with some constants $\gamma_i>0\ (i=1, 2, 3)$.Moreover, we confirm that the solution $(u, v, w, z)$ will exponentially converge to the homogeneous equilibrium $(\bar{u}_0, \bar{v}_0+\bar{w}_0, 0, \bar{u}_0)$ as $t\rightarrow\infty$, where $\bar{u} _0: =\frac{1}{\left|\Omega\right|}\int_{\Omega}u_0{\rm d}x$, $\bar{v}_0: =\frac{1}{\left|\Omega\right|}\int_{\Omega}v_0{\rm d}x$ and $\bar{w}_0: =\frac{1}{\left|\Omega\right|}\int_{\Omega}w_0{\rm d}x$.

Key words: Chemotaxis model, Global existence, Large time behavior.