In this paper, we investigate a two-species competitive chemotaxis Keller-Segel system equipped with a free boundary. The local existence and global existence of classical solutions are established, and then the asymptotic behavior of the solutions is described. Some spreading-vanishing dichotomies have been obtained both for the strong competition case: $0<a_1<1\le a_2$, and weak competition case: $0<a_1$, $a_2<1$. Finally, an obstructed spreading case has been found, which could happen due to the effect of the chemotaxis.

中文翻译：

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具有自由边界的两种竞争Keller-Segel趋化系统的动力学

在本文中，我们研究了具有自由边界的两种种群竞争趋化性Keller-Segel系统。建立经典解的局部存在和全局存在，然后描述解的渐近性质。在激烈的竞争中，都获得了一些消失的二分法：$0<{\mathrm{一种}}_{\mathrm{1个}}<\mathrm{1个}\le {\mathrm{一种}}_{\mathrm{2个}}$，以及竞争较弱的案例： $0<{\mathrm{一种}}_{\mathrm{1个}}$， ${\mathrm{一种}}_{\mathrm{2个}}<\mathrm{1个}$。最后，发现了一个受阻的扩散病例，这可能是由于趋化性的影响而发生的。