Global existence and boundedness in an attraction-repulsion chemotaxis system with nonlocal logistic source and sublinear productions

This paper deals with the following attraction-repulsion chemotaxis system with nonlocal logistic source and sublinear productions $$\{\begin{array}{rrll}& & u_t=d_1\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (u^k\mathrm{\nabla }v)+\xi \mathrm{\nabla }\cdot (u^k\mathrm{\nabla }w)+\mu u^m(1-{\int }_{\mathrm{\Omega }}u(x,t)\mathrm{d}x),& x\in \mathrm{\Omega },t>0,\\ & & v_t=d_2\mathrm{\Delta }v-\alpha v+f(u),& x\in \mathrm{\Omega },t>0,\\ & & w_t=d_3\mathrm{\Delta }w-\beta w+f(u),& x\in \mathrm{\Omega },t>0,\\ & & \frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }=\frac{\mathrm{\partial }v}{\mathrm{\partial }\nu }=\frac{\mathrm{\partial }w}{\mathrm{\partial }\nu }=0,& x\in \mathrm{\partial }\mathrm{\Omega },t>0,\\ & & u(x,0)=u_0,v(x,0)=v_0,w(x,0)=w_0,& x\in \mathrm{\Omega },\end{array}$$ in an open, bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$ with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$. Assume the parameters $d_1$, $d_2$, $d_3$, $\chi$, $\xi$, $\alpha$, $\beta$ and $\mu$ are positive constants, initial data $(u_0,v_0,w_0)$ are nonnegative and the function $f(u)\le K{u}^l\in C^1([0,\mathrm{\infty }))$ for some $K,l>0$. Under appropriate conditions on the parameter $k$, $l$ and $m$ we show that the above problem admits a unique globally bounded classical solution.