Global boundedness of solutions to a two-species chemotaxis system

In this paper, we consider the chemotaxis system of two species which are attracted by the same signal substance ut=Δu-∇·(uχ1(w)∇w)+μ1u(1-u-a1v),x∈Ω,t>0,vt=Δv-∇·(vχ2(w)∇w)+μ2v(1-a2u-v),x∈Ω,t>0,wt=Δw-w+u+v,x∈Ω,t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{\begin{array}{lll}u_t = \Delta u - \nabla \cdot (u \chi_1(w)\nabla w) + \mu_1 u(1 - u - a_1 v), \qquad x \in \Omega, \, t >0,\\ v_t = \Delta v - \nabla \cdot (v \chi_2(w) \nabla w) + \mu_2 v(1 - a_2u - v),\qquad x \in \Omega, \, t >0,\\ w_t = \Delta w - w + u + v, \qquad \qquad \qquad \qquad \qquad \qquad\,\,\, x \in \Omega,\, t >0 \end{array}\right.$$\end{document}under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega \subset \mathbb{R}^n}$$\end{document}. We prove that if the nonnegative initial data (u0,v0)∈(C0(Ω¯))2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(u_0, v_0) \in \big(C^0(\bar{\Omega})\big)^2}$$\end{document} and w0∈W1,r(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${w_0 \in W^{1, r}(\Omega)}$$\end{document} for some r > n, the system possesses a unique global uniformly bounded solution under some conditions on the chemotaxis sensitivity functions χ1(w), χ2(w) and the logistic growth coefficients μ1, μ2.