Finite-time blow-up in a two species chemotaxis-competition model with degenerate diffusion

This paper is concerned with the two-species chemotaxis-competition model with degenerate diffusion, \begin{cases} u_t = \Delta u^{m_1} - \chi_1 \nabla\cdot(u\nabla w) + \mu_1 u (1-u-a_1v), &x\in\Omega,\ t>0,\\% v_t = \Delta v^{m_2} - \chi_2 \nabla\cdot(v\nabla w) + \mu_2 v (1-a_2u-v), &x\in\Omega,\ t>0,\\% 0 = \Delta w +u+v-\overline{M}(t), &x\in\Omega,\ t>0, \end{cases} with $\int_\Omega w(x,t)\,dx=0$, $t>0$, where $\Omega := B_R(0) \subset \mathbb{R}^n$ $(n\ge5)$ is a ball with some $R>0$; $m_1,m_2>1$, $\chi_1,\chi_2,\mu_1,\mu_2,a_1,a_2>0$; $\overline{M}(t)$ is the spatial average of $u+v$. The purpose of this paper is to show finite-time blow-up in the sense that there is $\widetilde{T}_{\rm max}\in(0,\infty)$ such that $$\limsup_{t \nearrow \widetilde{T}_{\rm max}} (\|u(t)\|_{L^\infty(\Omega)} + \|v(t)\|_{L^\infty(\Omega)})=\infty$$ for the above model within a concept of weak solutions fulfilling a moment inequality which leads to blow-up. To this end, we also give a result on finite-time blow-up in the above model with the terms $\Delta u^{m_1}$, $\Delta v^{m_2}$ replaced with the nondegenerate diffusion terms $\Delta (u+\delta)^{m_1}$, $\Delta (v+\delta)^{m_2}$, where $\delta\in(0,1]$.