In this paper, we study the following problem 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.
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