This paper considers the following Keller-Segel-type fully parabolic system \(\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta (u\phi (v))+ru-\mu u^\alpha ,&x\in \Omega ,t>0,\\&v_t=d_v\Delta v-vw,&x\in \Omega ,t>0,\\&w_t=d_w\Delta w-w+u,&x\in \Omega ,t>0, \end{aligned} \right. \end{aligned}\) under no-flux boundary conditions in a smoothly bounded domain \(\Omega \subset \mathbb {R}^n\) , \(n\ge 1\) , where the parameters r, \(\mu \) , \(d_v\) , \(d_w\) are positive constants and \(\alpha >1\) . If the motility function enjoys \(\phi \in C^3((0,\infty ))\) with \(\phi (s)>0\) for all \(s>0\) , it is shown that the system admits a global classical solution for any appropriately regular initial value when \(\alpha >\max \bigl \{\frac{n+2}{4},1\bigr \}\) . Additionally, if we exclude the singular at \(s=0\) , i.e., \(\phi \in C^3([0,\infty ))\) , \(\phi >0\) on \([0,\infty )\) , then the smooth classical solution is globally bounded when any of the following conditions are met: (i) \(n\le 5\) , \(\alpha >1\) ; (ii) \(n\ge 6\) , \(\alpha >2\) ; (iii) \(n\ge 6\) , \(\alpha =2\) and \(\mu >\mu _*\) , where \(\mu _*\) is a positive constant independent of t, and further, such bounded solution will be stable at the constant \(\bigl ((\frac{r}{\mu })^\frac{1}{\alpha -1}, 0, (\frac{r}{\mu })^\frac{1}{\alpha -1}\bigr )\) with exponential decay rate. Finally, in the case of \(n\ge 6\) and \(1<\alpha \le 2\) we also showed that the system has at least one global weak solution which will become smooth after some waiting time.
