| Summary: | This article is concerned with the model
$$\displaylines{
u_t=\Delta u-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w),\quad
x\in \Omega,\; t>0,\cr
0=\Delta v+\alpha u-\beta v,\quad x\in\Omega,\; t>0,\cr
0=\Delta w+\gamma u-\delta w,\quad x\in\Omega,\; t>0
}$$
with homogeneous Neumann boundary conditions in a bounded domain
$\Omega\subset \mathbb{R}^{n}\;(n=2,3)$. Under the critical condition
$\chi \alpha-\xi \gamma=0$, we show that the system possesses a unique
global solution that is uniformly bounded in time. Moreover, when $n=2$,
by some appropriate smallness conditions on the initial data, we
assert that this solution converges to
($\bar{u}_0$, $\frac{\alpha}{\beta}\bar{u}_0$,
$\frac{\gamma}{\delta}\bar{u}_0$) exponentially,
where $\bar{u}_0:=\frac{1}{|\Omega|}\int_{\Omega}u_0$.
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