Blow-up and boundedness in quasilinear attraction-repulsion systems with nonlinear signal production.

In this paper, we consider the quasilinear parabolic-elliptic-elliptic attraction-repulsion system $ \begin{equation} \nonumber \left{ \begin{split} &u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w),&\qquad &x\in\Omega,,t>0, \ & 0 = \Delta v-\mu_{1}(t)+f_{1}(u),&\qquad &x\in\Omega,,t>0, \ &0 = \Delta w-\mu_{2}(t)+f_{2}(u),&\qquad &x\in\Omega,,t>0 \end{split} \right. \end{equation} $ under homogeneous Neumann boundary conditions in a smooth bounded domain $ \Omega\subset\mathbb{R}^n, \ n\geq2 $. The nonlinear diffusivity $ D $ and nonlinear signal productions $ f_{1}, f_{2} $ are supposed to extend the prototypes $ \begin{equation} \nonumber D(s) = (1+s)^{m-1},\ f_{1}(s) = (1+s)^{\gamma_{1}},\ f_{2}(s) = (1+s)^{\gamma_{2}},\ s\geq0,\gamma_{1},\gamma_{2}>0,m\in\mathbb{R}. \end{equation} $ We proved that if $ \gamma_{1} > \gamma_{2} $ and $ 1+\gamma_{1}-m > \frac{2}{n} $, then the solution with initial mass concentrating enough in a small ball centered at origin will blow up in finite time. However, the system admits a global bounded classical solution for suitable smooth initial datum when $ \gamma_{2} < 1+\gamma_{1} < \frac{2}{n}+m $.