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

$\begin{cases} u_{t}=Δ u-χ\nabla·(u\nabla v)+ξ\nabla·(u\nabla w)+f(u), &x∈Ω,\ t>0,\\ v_{t}=Δ v+α u-β v, &x∈Ω,\ t>0,\\ w_{t}=Δ w+γ u-δ w, &x∈Ω,\ t>0,\\\end{cases}\;\;\;\;(*)$

in a smooth bounded domain $Ω \subset \mathbb{R}^n(n≥ 1)$, with homogeneous Neumann boundary conditions and nonnegative initial data $(u_0,v_0,w_0)$ satisfying suitable regularity, where $χ≥ 0,ξ≥ 0,α, β, γ, δ>0$ and $f$ is a smooth growth source satisfying $f(0)≥ 0$ and

$f(u)≤ a-bu^θ, \ \ u≥ 0,\ \ \mathrm{with~some} \ \ a≥ 0,b>0,θ≥1.$

When $χα=ξγ$ (i.e. repulsion cancels attraction), the boundedness of classical solution of system (*) is established if the dampening parameter $θ$ and the space dimension $n$ satisfy

$\begin{cases} θ > \max\{1,3-\frac6n\}, &\text{when }\ \ 1≤ n≤ 5,\\ θ≥ 2, &\text{when }\ \ 6≤ n≤ 9,\\ θ>1+\frac{2(n-4)}{n+2}, &\text{when} \ \ \ n≥10.\\\end{cases}$

Furthermore, when $f(u)=μ u(1-u)$ and repulsion cancels attraction, by constructing appropriate Lyapunov functional, we show that if $μ>\frac{χ^2α^2(β-δ)^2}{8δβ^2}$, the solution $(u,v,w)$ exponentially stabilizes to the constant stationary solution $(1,\frac{α}{β},\frac{γ}{δ})$ in the case of $1≤ n≤ 9$. 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.