Boundedness and stabilization in the 3D minimal attraction–repulsion chemotaxis model with logistic source

In this paper, we consider the fully parabolic attraction–repulsion chemotaxis system with logistic source in a three-dimensional bounded domain with smooth boundary. We first derive an explicit formula μ∗=μ∗(3,d1,d2,d3,β1,β2,χ,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _*=\mu _*(3,d_1,d_2,d_3,\beta _1,\beta _2,\chi ,\xi )$$\end{document} for the logistic damping rate μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} such that the system has no blowups whenever μ>μ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu >\mu _*$$\end{document}. In addition, the asymptotic behavior of the solutions is discussed; we obtain the other explicit formula μ∗=μ∗(d1,d2,d3,α1,α2,β1,β2,χ,ξ,λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu ^*=\mu ^*(d_1,d_2,d_3,\alpha _1,\alpha _2,\beta _1,\beta _2,\chi ,\xi ,\lambda )$$\end{document} for the logistic damping rate so that the convergence rate is expressed explicitly whenever μ>μ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu >\mu ^*$$\end{document}. Our results generalize and improve partial previously known ones.