This paper investigates a multi-dimensional attraction–repulsion system $\left\{\begin{array}{cc}{u}_t=\Delta u-\chi \nabla \cdot (u\nabla v)+\xi \nabla \cdot (u\nabla w),\hfill & x\in \Omega ,t>0,\hfill \\ 0=\Delta v-{\mathrm{\mu }}_1\left(t\right)+f_1\left(u\right),\hfill & x\in \Omega ,t>0,\hfill \\ 0=\Delta w-{\mathrm{\mu }}_2\left(t\right)+f_2\left(u\right),\hfill & x\in \Omega ,t>0,\hfill \\ \hfill \end{array}\right.$where ${\mathrm{\mu }}_1\left(t\right)=\frac{1}{\left|\Omega \right|}{\int }_{\Omega }{f}_1\left(u\right)\mathrm{d}x$, ${\mathrm{\mu }}_2\left(t\right)=\frac{1}{\left|\Omega \right|}{\int }_{\Omega }{f}_2\left(u\right)\mathrm{d}x$, $\Omega =B_R\left(0\right)\subset {\mathbb{R}}^n(n\ge 2)$ and $f_1$ and $f_2$ are suitably regular functions generalizing the prototype determined by $f_1\left(s\right)=s^{{\gamma }_1}$ and $f_2\left(s\right)=s^{{\gamma }_2}$, $s\ge 0$, with ${\gamma }_1,{\gamma }_2>0$. Under homogeneous boundary conditions of Neumann type for $u$, $v$ and $w$, it is proved that, among other things, if ${\gamma }_1>{\gamma }_2$ and ${\gamma }_1>\frac{2}{n}$, the solution with initial mass concentrating enough in a small ball centered at origin will blow up in finite time, for any ${\gamma }_1,{\gamma }_2$, if ${\gamma }_1<\frac{2}{n}$, then for suitable smooth initial data $(u_0,v_0,w_0)$, the system possesses a unique global bounded classical solution. We point out that there exists a gap in the parameter regime: if ${\gamma }_1<{\gamma }_2$, does the solution exist globally?

本文研究了多维吸引-排斥系统 $\left\{\begin{array}{cc}{ü}_{Ť}=\Delta ü-\chi \nabla \cdot （ü\nabla v）+\xi \nabla \cdot （ü\nabla w），\hfill & X\in \Omega ，Ť>0，\hfill \\ 0=\Delta v-{\mathrm{\mu }}_{\mathrm{1个}}（Ť）+F_{\mathrm{1个}}（ü），\hfill & X\in \Omega ，Ť>0，\hfill \\ 0=\Delta w-{\mathrm{\mu }}_{\mathrm{2个}}（Ť）+F_{\mathrm{2个}}（ü），\hfill & X\in \Omega ，Ť>0，\hfill \\ \hfill \end{array}\right.$在哪里 ${\mathrm{\mu }}_{\mathrm{1个}}（Ť）=\frac{\mathrm{1个}}{\left|\Omega \right|}{\int }_{\Omega }{F}_{\mathrm{1个}}（ü）\mathrm{d}X$， ${\mathrm{\mu }}_{\mathrm{2个}}（Ť）=\frac{\mathrm{1个}}{\left|\Omega \right|}{\int }_{\Omega }{F}_{\mathrm{2个}}（ü）\mathrm{d}X$， $\Omega ={乙}_{\mathrm{[R}}（0）\subset {\mathbb{[R}}^{ñ}（ñ\ge \mathrm{2个}）$ 和 $F_{\mathrm{1个}}$ 和 $F_{\mathrm{2个}}$ 是适当的常规函数​​，用于概括由...确定的原型 $F_{\mathrm{1个}}（s）=s^{{\gamma }_{\mathrm{1个}}}$ 和 $F_{\mathrm{2个}}（s）=s^{{\gamma }_{\mathrm{2个}}}$， $s\ge 0$， 和 ${\gamma }_{\mathrm{1个}}，{\gamma }_{\mathrm{2个}}>0$。在Neumann型的齐次边界条件下$ü$， $v$ 和 $w$，事实证明，除其他外，如果 ${\gamma }_{\mathrm{1个}}>{\gamma }_{\mathrm{2个}}$ 和 ${\gamma }_{\mathrm{1个}}>\frac{\mathrm{2个}}{ñ}$，初始质量集中在以原点为中心的小球中的溶液将在有限时间内爆炸，对于任何情况 ${\gamma }_{\mathrm{1个}}，{\gamma }_{\mathrm{2个}}$， 如果 ${\gamma }_{\mathrm{1个}}<\frac{\mathrm{2个}}{ñ}$，然后获得合适的平滑初始数据 $（{ü}_0，v_0，w_0）$，该系统拥有独特的全局有界经典解决方案。我们指出参数体系中存在一个空白：${\gamma }_{\mathrm{1个}}<{\gamma }_{\mathrm{2个}}$，解决方案是否存在于全球？