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 = Δu^{m_1} - χ_1 \nabla\cdot(u\nabla w) + μ_1 u (1-u-a_1v), &x\inΩ,\ t>0,\\% v_t = Δv^{m_2} - χ_2 \nabla\cdot(v\nabla w) + μ_2 v (1-a_2u-v), &x\inΩ,\ t>0,\\% 0 = Δw +u+v-\overline{M}(t), &x\inΩ,\ t>0, \end{cases} with ${\int }_{\mathrm{\Omega }}w(x,t)dx=0$, $t>0$, where $\mathrm{\Omega }:=B_R(0)\subset {\mathbb{R}}^n$ $(n\ge 5)$ is a ball with some $R>0$; $m_1,m_2>1$, ${\chi }_1,{\chi }_2,{\mu }_1,{\mu }_2,a_1,a_2>0$; $\bar{M}(t)$ 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 ${\tilde{T}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\in (0,\mathrm{\infty })$ such that $$\underset{t\nearrow {\tilde{T}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{\mathrm{lim\hspace{0.17em}sup}}(\mathrm{\parallel }u(t){\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}+\mathrm{\parallel }v(t){\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })})=\mathrm{\infty }$$ 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 $\mathrm{\Delta }{u}^{m_1}$, $\mathrm{\Delta }{v}^{m_2}$ replaced with the nondegenerate diffusion terms $\mathrm{\Delta }(u+\delta {)}^{m_1}$, $\mathrm{\Delta }(v+\delta {)}^{m_2}$, where $\delta \in (0,1]$.