Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source

In this paper, we study an attraction-repulsion Keller-Segel chemotaxis model with logistic source \begin{document}$\begin{cases} u_{t}=Δ u-χ\nabla·(u\nabla v)+ξ\nabla·(u\nabla w)+f(u), x\;\;\;(*)$ \end{document} in a smooth bounded domain \begin{document}$Ω \subset \mathbb{R}^n(n≥ 1)$\end{document} , with homogeneous Neumann boundary conditions and nonnegative initial data \begin{document}$(u_0,v_0,w_0)$\end{document} satisfying suitable regularity, where \begin{document}$χ≥ 0,ξ≥ 0,α, β, γ, δ>0$\end{document} and \begin{document}$f$\end{document} is a smooth growth source satisfying \begin{document}$f(0)≥ 0$\end{document} and \begin{document}$f(u)≤ a-bu^θ, \ \ u≥ 0,\ \ \mathrm{with~some} \ \ a≥ 0,b>0,θ≥1.$ \end{document} When \begin{document}$χα=ξγ$\end{document} (i.e. repulsion cancels attraction), the boundedness of classical solution of system (*) is established if the dampening parameter \begin{document}$θ$\end{document} and the space dimension \begin{document}$n$\end{document} satisfy \begin{document}$\begin{cases} θ > \max\{1,3-\frac6n\}, &\text{when }\ \ 1≤ n≤ 5,\\ θ≥ 2, &\text{when }\ \ 6≤ n≤ 9,\\ θ>1+\frac{2(n-4)}{n+2}, &\text{when} \ \ \ n≥10.\\\end{cases}$ \end{document} Furthermore, when \begin{document}$f(u)=μ u(1-u)$\end{document} and repulsion cancels attraction, by constructing appropriate Lyapunov functional, we show that if \begin{document}$μ>\frac{χ^2α^2(β-δ)^2}{8δβ^2}$\end{document} , the solution \begin{document}$(u,v,w)$\end{document} exponentially stabilizes to the constant stationary solution \begin{document}$(1,\frac{α}{β},\frac{γ}{δ})$\end{document} in the case of \begin{document}$1≤ n≤ 9$\end{document} . Our results implies that when repulsion cancels attraction the logistic source play an important role on the solution behavior of the attraction-repulsion chemotaxis system.