Existence of globally bounded solutions for a predator-prey system with indirect taxis-driven pursuit-evasion and parabolic-elliptic signal

In this paper, we consider a cross-diffusive model for pursuit-evasion processes which was described as$ \begin{equation*} \left\{\begin{aligned} & u_{t} = \Delta u-\chi\nabla\cdot(u(u+1)^{r-1}\nabla w)+u(\lambda_{1}-\mu_{1}u^{r_{1}-1}+av), &x\in\Omega, t>0, \\ & v_{t} = \Delta v+\xi\nabla\cdot(v\nabla z)+v(\lambda_{2}-\mu_{2}v^{r_{2}-1}-bu), &x\in\Omega, t>0, \\ & w_{t} = \Delta w-w+v, &x\in\Omega, t>0, \\ & 0 = \Delta z-z+u, &x\in\Omega, t>0 \end{aligned}\right. \end{equation*} $in a smooth bounded domain $ \Omega \subset \mathbb{R}^N $ ($ N \geq 1 $) under homogeneous Neumann boundary conditions. For nonnegative initial data with appropriate regularity, the unique globally bounded classical solution is obtained provided that $ \min \{(r_{1}-1)(r_{2}-1), (r_{1}-r)(r_{2}-1) \} > \frac{(N-2)_{+}}{N} $, where $ r, \lambda_i, \mu_i, \chi, \xi > 0 $ and $ r_i > 1 $($ i = 1, 2 $). This generalizes the results of Li et al. [1], Zheng et al. [2], and Zheng and Zhang [3].