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 degenerate diffusion, $$ \textstyle\begin{cases} u_{t} = \Delta u^{m_{1}} - \chi _{1} \nabla \cdot (u\nabla w) + \mu u (1-u-a_{1}v), &x\in \Omega ,\ t>0, \\ v_{t} v^{m_{2}} _{2} (v\nabla v (1-a_{2}u-v), 0 w +u+v-\overline{M}(t), \end{cases} $\int _{\Omega }w(x,t)\,dx=0$ , $t>0$ where $\Omega := B_{R}(0) \subset \mathbb{R}^{n}$ $(n\ge 5)$ a ball some $R>0$ ; $m_{1},m_{2}>1$ $\chi _{1},\chi _{2},\mu _{1},\mu _{2},a_{1},a_{2}>0$ $\overline{M}(t)$ spatial average of $u+v$ . In this paper, we show that if m_{1}< 2-\frac{4}{n}, \quad _{1}>\frac{n(2-m_{1})}{n(2-m_{1})-4} \max \{ 1,a_{1} \}\mu \text{and} _{2}>\mu _{2}a_{2} or m_{2}< _{1}>\mu _{1}a_{1}\quad _{2}>\frac{n(2-m_{2})}{n(2-m_{2})-4}\cdot 1,a_{2} _{2}, then there exist radially symmetric initial data such weak solution blows up in finite time sense $\widetilde{T}_{\mathrm{max}}\in (0,\infty )$ \limsup _{t \nearrow \widetilde{T}_{\mathrm{max}}}\, (\|u(t)\|_{L^{\infty }( )} \|v(t)\|_{L^{\infty }(\Omega )})=\infty To obtain result, apply method previous (Discrete Contin. Dyn. Syst., Ser. B 28(1):262–286, 2023) to derive an integral inequality for moment-type functional, which was introduced by Winkler (Z. Angew. Math. Phys. 69(2):69, 2018). Moreover, before proving blow-up solutions above model, give result on finite-time under same conditions $m_{1}$ $m_{2}$ _{1}$ and _{2}$ terms $\Delta u^{m_{1}}$ v^{m_{2}}$ replaced nondegenerate diffusion (u+{\varepsilon })^{m_{1}}$ (v+{\varepsilon })^{m_{2}}$ ${\varepsilon }\in (0,1]$