SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 3385-3419, January 2021.
We study a chemotaxis model describing the space and time evolution in a smooth and bounded domain of $\mathbb{R}^2$ of the densities $u$ and $v$ of subpopulations of moving and static individuals of some species and the concentration $w$ of a chemoattractant. We prove that, in an appropriate functional setting, all solutions exist globally in time. Moreover, we establish the existence of a critical mass $M_c>0$ of the whole population $u+v$ such that, for $M \in (0, M_c)$, any solution is bounded, while, for almost all $M > M_c$, there exist solutions blowing up in infinite time. The building block of the analysis is the construction of a Liapunov functional. As far as we know, this is the first result of this kind when the mass conservation includes the two subpopulations and not only the moving one.

中文翻译：

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具有分裂种群的趋化模型中无限时间膨胀的质量阈值

SIAM 数学分析杂志，第 53 卷，第 3 期，第 3385-3419 页，2021 年 1 月。
我们研究了一个趋化性模型，该模型描述了某些物种的移动和静态个体的密度 $u$ 和 $v$ 的平滑有界域 $\mathbb{R}^2$ 中的空间和时间演化以及浓度$w$ 的化学引诱剂。我们证明，在适当的功能设置中，所有解决方案都在全球范围内及时存在。此外，我们建立了整个群体 $u+v$ 的临界质量 $M_c>0$ 的存在，使得对于 $M \in (0, M_c)$，任何解都是有界的，而对于几乎所有的 $ M > M_c$，存在无限时间内爆炸的解。分析的构建块是构建 Liapunov 泛函。据我们所知，当质量守恒包括两个亚群而不仅仅是移动的亚群时，这是第一个这样的结果。