Large time behavior of a quasilinear two-species attraction-repulsion chemotaxis system with two chemicals

This paper deals with the quasilinear two-species attraction-repulsion chemotaxis system with two chemicals: $u_t=\nabla \cdot \left(D_1\right(u\left)\nabla u\right)-\nabla \cdot \left({\Phi }_1\right(u\left)\nabla v\right)+r_{1}u-{\mu }_1{u}^{k_1}$, $0=\Delta v-v+w$, $w_t=\nabla \cdot \left(D_2\right(w\left)\nabla w\right)+\nabla \cdot \left({\Phi }_2\right(w\left)\nabla z\right)+r_{2}w-{\mu }_2{w}^{k_2}$, $0=\Delta z-z+u$, subject to the homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset R^N$($N\ge 2$) with smooth boundary, where the parameters $r_i,{\mu }_i>0$, $k_i>1$ and $D_i\left(s\right)={(s+1)}^{p_i},{\Phi }_i\left(s\right)={\chi }_{i}s{(s+1)}^{q_i-1}$ with ${\chi }_i>0$, $p_i,q_i\in R$, $i=1,2$. The interactions among the diffusion, attraction, repulsion, and logistic sources in the system determine the behavior of solutions. It is showed that when $N<4$, as long as the diffusion mechanism of population w dominates with $q_2-p_2<\frac{4}{N}$, global boundedness of solutions can be guaranteed; if $\max \{q_1,q_2\}\le \min \{p_1+\frac{2}{N},p_2+\frac{2}{N}\}$ or $\max \{q_1,q_2\}\le \min \{k_1-1,k_2-1\}$, i.e. the diffusion mechanisms or the logistic source terms of populations u and w are both dominant, the solutions are globally bounded; when the diffusion mechanism of u (or w) and the logistic source term of w (or u) dominate with $\max \{q_1,q_2\}\le \min \{k_2-1,p_1+\frac{2}{N}\}$ (or $\max \{q_1,q_2\}\le \min \{k_1-1,p_2+\frac{2}{N}\}$), the solutions are globally bounded. Also, we have proved that when the logistic source term of either u or w dominates, the global boundedness of the solutions can be obtained. Moreover, we give the large time behavior of the globally bounded solutions.