Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic-elliptic attraction-repulsion chemotaxis system

Abstract

This paper deals with the quasilinear attraction-repulsion chemotaxis system \begin{align*} \begin{cases} u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u -χu(u+1)^{p-2}\nabla v +ξu(u+1)^{q-2}\nabla w\big) +f(u), \\[1.05mm] 0=Δv+αu-βv, \\[1.05mm] 0=Δw+γu-δw \end{cases} \end{align*} in a bounded domain $Ω\subset \mathbb{R}^n$ ($n \in \mathbb{N}$) with smooth boundary $\partialΩ$, where $m, p, q \in \mathbb{R}$, $χ, ξ, α, β, γ, δ>0$ are constants. Moreover, it is supposed that the function $f$ satisfies $f(u)\equiv0$ in the study of boundedness, whereas, when considering blow-up, it is assumed that $m>0$ and $f$ is a function of logistic type such as $f(u)=λu-μu^κ$ with $λ\ge 0$, $μ>0$ and $κ>1$ sufficiently close to~$1$, in the radially symmetric setting. In the case that $ξ=0$ and $f(u) \equiv 0$, global existence and boundedness have been proved under the condition $p 0$. This paper classifies boundedness and blow-up into the cases $p q$ without any condition for the sign of $χα-ξγ$ and the case $p=q$ with $χα-ξγ<0$ or $χα-ξγ>0$.