Finite-Time Blow-up in a Two-Species Chemotaxis-Competition Model with Degenerate Diffusion

Abstract

This paper is concerned with the two-species chemotaxis-competition model with degenerate diffusion,

$$ \textstyle\begin{cases} u_{t} = \Delta u^{m_{1}} - \chi _{1} \nabla \cdot (u\nabla w) + \mu _{1} u (1-u-a_{1}v), &x\in \Omega ,\ t>0, \\ v_{t} = \Delta v^{m_{2}} - \chi _{2} \nabla \cdot (v\nabla w) + \mu _{2} v (1-a_{2}u-v), &x\in \Omega ,\ t>0, \\ 0 = \Delta w +u+v-\overline{M}(t), &x\in \Omega ,\ t>0, \end{cases} $$

with $\int _{\Omega }w(x,t)\,dx=0$, $t>0$, where $\Omega := B_{R}(0) \subset \mathbb{R}^{n}$ $(n\ge 5)$ is a ball with some $R>0$; $m_{1},m_{2}>1$, $\chi _{1},\chi _{2},\mu _{1},\mu _{2},a_{1},a_{2}>0$; $\overline{M}(t)$ is the spatial average of $u+v$. In this paper, we show that if

$$ m_{1}< 2-\frac{4}{n}, \quad \chi _{1}>\frac{n(2-m_{1})}{n(2-m_{1})-4} \cdot \max \{ 1,a_{1} \}\mu _{1} \quad \text{and} \quad \chi _{2}>\mu _{2}a_{2} $$

or

$$ m_{2}< 2-\frac{4}{n}, \quad \chi _{1}>\mu _{1}a_{1}\quad \text{and} \quad \chi _{2}>\frac{n(2-m_{2})}{n(2-m_{2})-4}\cdot \max \{ 1,a_{2} \}\mu _{2}, $$

then there exist radially symmetric initial data such that the weak solution blows up in finite time in the sense that there is $\widetilde{T}_{\mathrm{max}}\in (0,\infty )$ such that

$$ \limsup _{t \nearrow \widetilde{T}_{\mathrm{max}}}\, (\|u(t)\|_{L^{\infty }( \Omega )} + \|v(t)\|_{L^{\infty }(\Omega )})=\infty . $$

To obtain this result, we apply the method in the previous paper (Discrete Contin. Dyn. Syst., Ser. B 28(1):262–286, 2023) to derive an integral inequality for a moment-type functional, which was introduced by Winkler (Z. Angew. Math. Phys. 69(2):69, 2018). Moreover, before proving blow-up of solutions in the above model, we give a result on finite-time blow-up under the same conditions for $m_{1}$, $m_{2}$, $\chi _{1}$ and $\chi _{2}$ in the model with the terms $\Delta u^{m_{1}}$, $\Delta v^{m_{2}}$ replaced with the nondegenerate diffusion terms $\Delta (u+{\varepsilon })^{m_{1}}$, $\Delta (v+{\varepsilon })^{m_{2}}$, where ${\varepsilon }\in (0,1]$.