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

Abstract

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