Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion

This paper is concerned with a chemotaxis system ut=Δum−∇⋅(χ1(w)u∇w)+μ1u(1−u−a1v),x∈Ω,t>0,vt=Δvn−∇⋅(χ2(w)v∇w)+μ2v(1−a2u−v),x∈Ω,t>0,wt=Δw−(αu+βv)w,x∈Ω,t>0,\left\{\begin{array}{ll}{u}_{t}=\Delta {u}^{m}-\nabla \cdot \left({\chi }_{1}\left(w)u\nabla w)+{\mu }_{1}u\left(1-u-{a}_{1}v),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ {v}_{t}=\Delta {v}^{n}-\nabla \cdot \left({\chi }_{2}\left(w)v\nabla w)+{\mu }_{2}v\left(1-{a}_{2}u-v),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ {w}_{t}=\Delta w-\left(\alpha u+\beta v)w,& x\in \Omega ,\hspace{0.33em}t\gt 0,\end{array}\right. under homogeneous Neumann boundary conditions in a bounded domain Ω⊂R3\Omega \subset {{\mathbb{R}}}^{3} with smooth boundary, where μ1,μ2>0{\mu }_{1},{\mu }_{2}\gt 0, a1,a2>0{a}_{1},{a}_{2}\gt 0, α,β>0\alpha ,\beta \gt 0, and the chemotactic sensitivity function χi∈C1([0,∞)){\chi }_{i}\in {C}^{1}({[}0,\infty )), χi′≥0{\chi }_{i}^{^{\prime} }\ge 0. It is proved that for any large initial data, for any m,n>1m,n\gt 1, the system admits a global weak solution, which is uniformly bounded.