The effects of cross-diffusion and logistic source on the boundedness of solutions to a pursuit-evasion model

We study the following quasilinear pursuit-evasion model: \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} u_{t} = \Delta u-\chi\nabla \cdot (u(u+1)^{\alpha}\nabla w)+u(\lambda_{1}-\mu_{1}u^{r_{1}-1}+ av),\ &\ \ x\in \Omega, \ t>0,\\[2.5mm] v_{t} = \Delta v+\xi\nabla \cdot(v(v+1)^{\beta}\nabla z)+v(\lambda_{2}-\mu_{2}v^{r_{2}-1}-bu), \ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0 = \Delta w-w+v, \ &\ \ x\in \Omega, \ t>0 ,\\[2.5mm] 0 = \Delta z-z+u,\ &\ \ x\in \Omega, \ t>0 , \end{array} \right. \end{equation*} $\end{document} in a smooth and bounded domain $ \Omega\subset\mathbb{R}^{n}(n\geq 1), $ where $ a, b, \chi, \xi, \lambda_{1}, \lambda_{2}, \mu_{1}, \mu_{2} > 0, $ $ \alpha, \beta \in\mathbb{R}, $ and $ r_{1}, r_{2} > 1. $ When $ r_{1} > \max\{1, 1+\alpha\}, r_{2} > \max\{1, 1+\beta\}, $ it has been proved that if $ \min\{(r_{1}-1)(r_{2}-\beta-1), (r_{1}-\alpha-1)(r_{2}-\beta-1)\} > \frac{(n-2)_{+}}{n}, $ then for some suitable nonnegative initial data $ u_{0} $ and $ v_{0}, $ the system admits a unique globally classical solution which is bounded in $ \Omega\times(0, \infty) $.