Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions

We consider the following attraction-repulsion Keller-Segel system:

$\begin{equation*}\begin{cases}u_t=\nabla· (D(u) \nabla u)-χ\nabla·( u\nabla v)+ξ\nabla·( u\nabla w), &x∈ Ω, ~~t>0,\\ v_t=Δ v+α u-β v,& x∈ Ω, ~~t>0,\\0=Δ w+γ u-δ w, &x∈ Ω, ~~t>0,\\u(x,0)=u_0(x),~v(x,0)=v_0(x),&x∈ Ω,\end{cases}\end{equation*}$

with homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n(n>2)$ with smooth boundary. Here all the parameters $χ, ξ, α, β, γ$ and $δ$ are positive. The smooth diffusion $D(u)$ satisfies $D(u)≥ d u^θ, u>0$ for some $d>0, θ∈\mathbb{R}$. It is recently known from [25] that boundedness of solutions is ensured whenever $θ>1-\frac{2}{n}$. Here, it is shown, if repulsion dominates or cancels attraction in the sense either $\{ξγ> χα\}$ or $\{ξγ=χα, β≥ δ\}$, the corresponding initial-boundary value problem possesses a unique global classical solution which is uniformly-in-time bounded for large initial data provided $θ>1-\frac{4}{n+2}$. In this way, the range of $θ>1-\frac{2}{n}$ of boundedness is enlarged and thus the repulsion effect on boundedness is exhibited.