Boundedness in prey-taxis system with rotational flux terms

This paper investigates prey-taxis system with rotational flux terms

$ \begin{equation*} \begin{cases} u_t = \Delta u-\nabla\cdot(uS(x,u,v)\nabla v)+uv-\rho u,& x\in\Omega,\ t>0,\\ v_t = \Delta v-\xi uv+\mu v(1-\alpha v),&x\in\Omega,\ t>0, \end{cases} \end{equation*} $

under no-flux boundary conditions in a bounded domain $ \Omega\subset\mathbb{R}^n\; (n\geq1) $ with smooth boundary. Here the matrix-valued function $ S\in C^2(\bar{\Omega}\times[0,\infty)^2;\mathbb{R}^{n\times n}) $ fulfills $ |S(x,u,v)|\leq\frac{S_0(v)}{(1+u)^\theta}(\theta\geq0) $ for all $ (x,u,v)\in\bar{\Omega}\times[0,\infty)^2 $ with some nondecreasing function $ S_0 $. It is proved that for nonnegative initial data $ u_0\in C^0(\overline{\Omega}) $ and $ v_0\in W^{1,q}(\Omega) $ with some $ q>\max\{n,2\} $, if one of the following assumptions holds: (i) $ n = 1 $, (ii) $ n\geq2, \theta = 0 $ and $ S_0(m)m<\frac{2}{\sqrt{3n(11n+2)}} $, (iii) $ \theta>0 $, then the model possesses a global classical solution that is uniformly bounded. Where $ m: = \max\{\|v_0\|_{L^\infty(\Omega)}, \frac{1}{\alpha}\} $.