Global boundedness in a quasilinear two-species attraction-repulsion chemotaxis system with two chemicals

This paper studies the quasilinear attraction-repulsion chemotaxis system of two-species with two chemicals \begin{document}$ u_{t} = \nabla\cdot( D_1(u)\nabla u)-\nabla\cdot( \Phi_1(u)\nabla v) $\end{document} , \begin{document}$ 0 = \Delta v-v+w^{\gamma_1} $\end{document} , \begin{document}$ w_{t} = \nabla\cdot( D_2(w)\nabla w)+\nabla\cdot( \Phi_2(w)\nabla z) $\end{document} , \begin{document}$ 0 = \Delta z-z+u^{\gamma_2} $\end{document} , subject to the homogeneous Neumann boundary conditions in a bounded domain \begin{document}$ \Omega\subset\mathbb{R}^N $\end{document} ( \begin{document}$ N\geq2 $\end{document} ) with smooth boundary, where \begin{document}$ \gamma_i>0 $\end{document} , \begin{document}$ D_i,\Phi_i\in C^2[0,+\infty) $\end{document} , \begin{document}$ D_i(s)\ge(s+1)^{p_i},\; \Phi_i(s)\ge0 $\end{document} for \begin{document}$ s\ge 0 $\end{document} , and \begin{document}$ \Phi_i(s)\le\chi_i s^{q_i} $\end{document} for \begin{document}$ s>s_0 $\end{document} with \begin{document}$ \chi_i>0 $\end{document} , \begin{document}$ p_i,q_i\in\mathbb{R} $\end{document}\begin{document}$ (i = 1,2) $\end{document} , \begin{document}$ s_0>1 $\end{document} . It is shown that if \begin{document}$ \gamma_1<\frac{2}{N} $\end{document} (or \begin{document}$ \gamma_2<\frac{4}{N} $\end{document} with \begin{document}$ \gamma_2\le1 $\end{document} ), the global boundedness of solutions are guaranteed by the self-diffusion dominance of \begin{document}$ u $\end{document} (or \begin{document}$ w $\end{document} ) with \begin{document}$ p_1>q_1+\gamma_1-1-\frac{2}{N} $\end{document} (or \begin{document}$ p_2>q_2+\gamma_2-1-\frac{4}{N} $\end{document} ); if \begin{document}$ p_j\ge q_i+\gamma_i- 1-\frac{2}{N} $\end{document} , \begin{document}$ i,j = 1,2 $\end{document} (i.e. the self-diffusion of \begin{document}$ u $\end{document} and \begin{document}$ w $\end{document} are dominant), then the solutions are globally bounded; in particular, different from the results of the single-species chemotaxis system, for the critical case \begin{document}$ p_j = q_i+\gamma_i- 1-\frac{2}{N} $\end{document} , the global boundedness of the solutions can be obtained.