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

$$\{\begin{array}{ll}\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (u\mathrm{\nabla }w)+u(a_1-b_1{u}^{m-1}+c_{1}v),& (x,t)\in \mathrm{\Omega }\times (0,T_{max}),\\ \frac{\mathrm{\partial }v}{\mathrm{\partial }t}=\mathrm{\Delta }v-\xi \mathrm{\nabla }\cdot (v\mathrm{\nabla }w)+v(a_2-b_2{v}^{l-1}-c_{2}u),& (x,t)\in \mathrm{\Omega }\times (0,T_{max}),\\ \frac{\mathrm{\partial }w}{\mathrm{\partial }t}=\mathrm{\Delta }w-(u^{\alpha }+v^{\beta })w,& (x,t)\in \mathrm{\Omega }\times (0,T_{max}),\end{array}$$

where the domain $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 2)$ is bounded and smooth, $T_{max}\in (0,\mathrm{\infty }],$ 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.