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 -\chi u(u+1)^{p-2}\nabla v +\xi u(u+1)^{q-2}\nabla w\big) +f(u), \\[1.05mm] 0=\Delta v+\alpha u-\beta v, \\[1.05mm] 0=\Delta w+\gamma u-\delta w \end{cases} \end{align*} in a bounded domain $\Omega \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) with smooth boundary $\partial\Omega$, where $m, p, q \in \mathbb{R}$, $\chi, \xi, \alpha, \beta, \gamma, \delta>0$ 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 $f(u)=\lambda u-\mu u^{\kappa}$ with $\lambda \ge 0$, $\mu>0$ and $\kappa>1$ sufficiently close to~$1$, in the radially symmetric setting. In the case that $\xi=0$ and $f(u) \equiv 0$, global existence and boundedness have been proved under the condition $p<m+\frac2n$. Also, in the case that $m=1$, $p=q=2$ and $f$ is a function of logistic type, finite-time blow-up has been established by assuming $\chi\alpha-\xi\gamma>0$. This paper classifies boundedness and blow-up into the cases $p<q$ and $p>q$ without any condition for the sign of $\chi\alpha-\xi\gamma$ and the case $p=q$ with $\chi\alpha-\xi\gamma<0$ or $\chi\alpha-\xi\gamma>0$.