Boundedness of solutions in a two-species chemotaxis system

In this paper, we consider an initial-Neumann boundary value problem for a two-species chemotaxis system$ \begin{equation*} \left\{ \begin{array}{ll} \frac{\partial u}{\partial t} = \Delta u-\chi\nabla \cdot (u\nabla w)+u(a_{1}-b_{1}u^{m-1}+c_{1}v),\ &\ \ (x,t) \in \Omega\times(0,T_{\max}), \\[2.5mm] \frac{\partial v}{\partial t} = \Delta v-\xi\nabla \cdot (v\nabla w)+v(a_{2}-b_{2}v^{l-1}-c_{2}u), \ &\ \ (x,t) \in \Omega\times(0,T_{\max}), \\[2.5mm] \frac{\partial w}{\partial t} = \Delta w-(u^{\alpha}+v^{\beta})w, \ &\ \ (x,t) \in \Omega\times(0,T_{\max}), \end{array} \right. \end{equation*} $where the domain $ \Omega\subset\mathbb{R}^{n}(n\geq 2) $ is bounded and smooth, $ T_{\max}\in (0, \infty], $ and parameters $ a_{i}, b_{i}, c_{i}, m, l, \alpha, $ $ \beta, \chi, \xi > 0 $ with $ m, l > 1, i = 1, 2. $ In the current work, we provide a sufficient condition of global classical solvability to the above system. More precisely, for some suitable initial data, if $ m > \max\{\frac{\alpha(n+2)}{2}, 1 \} $ and $ l > \max\{\frac{\beta(n+2)}{2}, 1 \}, $ then the system has a global classical solution. Compared to previous work, the existence result established here is more generalized, depending only on the nonlinear power exponents and spatial dimensions.