College of Mathematics and Information, China West Normal University, Nanchong 637009, China.
Math Biosci Eng. 2023 Jan 10;20(3):5243-5267. doi: 10.3934/mbe.2023243.
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 $.
本文研究了齐次 Neumann 边界条件下光滑有界区域 Ω⊂Rn（n≥2）中一类拟线性抛物-椭圆-椭圆吸引-排斥系统：
[
\begin{align*}
u_t&=\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w),&&x\in\Omega,t>0,\
0&=\Delta v-\mu_{1}(t)+f_{1}(u),&&x\in\Omega,t>0,\
0&=\Delta w-\mu_{2}(t)+f_{2}(u),&&x\in\Omega,t>0.
\end{align*}
]
其中非线性扩散系数$D$和非线性信号产生函数$f_{1},f_{2}$扩展了原型函数：
[
\begin{align*}
D(s)&=(1+s)^{m-1},\
f_{1}(s)&=(1+s)^{\gamma_{1}},\
f_{2}(s)&=(1+s)^{\gamma_{2}},
\end{align*}
]
(s\geq0,\gamma_{1},\gamma_{2}>0,m\in\mathbb{R})。
我们证明了，若(\gamma_{1}>\gamma_{2})且(1+\gamma_{1}-m>\frac{2}{n})，则具有足够初始质量集中在原点附近的小球的解将在有限时间内爆炸。然而，当(\gamma_{2}<1+\gamma_{1}<\frac{2}{n}+m)时，对于合适的光滑初始数据，系统可以允许全局有界经典解。
