Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop

In this work, the fully parabolic chemotaxis-competition system with loop

$ \begin{eqnarray*} \left\{ \begin{array}{llll} &\partial_{t} u_{1} = d_1\Delta u_{1}-\nabla\cdot(u_{1}\chi_{11}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{1}\chi_{12}(v_{2})\nabla v_{2}) +\mu_{1}u_{1}(1-u_{1}-a_{1}u_{2}),\\ &\partial_{t} u_{2} = d_2\Delta u_{2}-\nabla\cdot(u_{2}\chi_{21}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{2}\chi_{22}(v_{2})\nabla v_{2}) +\mu_{2}u_{2}(1-u_{2}-a_{2}u_{1}), \\ &\partial_t v_1 = d_3\Delta v_{1}-\lambda_{1} v_{1}+h_1(u_{1}, u_{2}), \\ &\partial_t v_2 = d_4\Delta v_{2}-\lambda_{2} v_{2}+h_2(u_{1}, u_{2}) \\ \end{array} \right. \end{eqnarray*} $

is considered under the homogeneous Neumann boundary condition, where $ x\in\Omega, t>0 $, $ \Omega\subset \mathbb{R}^{n} (n\leq 3) $ is a bounded domain with smooth boundary. For any regular nonnegative initial data, it is proved that if the parameters $ \mu_1, \mu_2 $ are sufficiently large, then the system possesses a unique and global classical solution for $ n\leq 3 $. Specifically, when $ n = 2 $, the global boundedness can be attained without any constraints on $ \mu_1, \mu_2 $.