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

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, w+\gamma u-\delta w \end{cases} \end{align*} in a bounded domain $\Omega \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) smooth boundary $\partial\Omega$, where $m, p, q \mathbb{R}$, $\chi, \xi, \alpha, \beta, \gamma, \delta>0$ are constants. Moreover, it is supposed that function $f$ satisfies $f(u)\equiv0$ study of boundedness, whereas, when considering blow-up, assumed $m>0$ and logistic type such as $f(u)=\lambda u-\mu u^{\kappa}$ $\lambda \ge 0$, $\mu>0$ $\kappa>1$ sufficiently close to~$1$, radially symmetric setting. In case $\xi=0$ $f(u) \equiv global existence boundedness have been proved under condition $p0$. classifies into cases $pq$ without any for sign $\chi\alpha-\xi\gamma$ $p=q$ $\chi\alpha-\xi\gamma<0$ or