SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 5036-5065, January 2020.
In this paper, we shall study the parabolic-elliptic Keller--Segel system on the Poincaré disk model of the two-dimensional-hyperbolic space. We shall investigate how the negative curvature of this Riemannian manifold influences the solutions of this system. As in the two-dimensional Euclidean case, under the subcritical condition $\chi M< 8\pi$, we shall prove global well-posedness results with any initial $L^1$-data. More precisely, by using dispersive and smoothing estimates we shall prove Fujita--Kato type theorems for local well-posedness. We shall then use the logarithmic Hardy--Littlewood--Sobolev estimates on the hyperbolic space to prove that the solution cannot blow up in finite time. For larger mass $\chi M> 8\pi$ we shall obtain a blow-up result under an additional condition. According to the exponential growth of the hyperbolic space, we find a suitable weighted moment of exponential type on the initial data for blow-up.

中文翻译：

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二维双曲空间上的Keller-Segel系统

SIAM数学分析杂志，第52卷，第5期，第5036-5065页，2020年1月。
在本文中，我们将在二维双曲空间的Poincaré圆盘模型上研究抛物线-椭圆形Keller-Segel系统。我们将研究黎曼流形的负曲率如何影响该系统的解。与二维欧几里得情况一样，在亚临界条件$ \ chi M <8 \ pi $的情况下，我们将使用任何初始$ L ^ 1 $数据证明全局适定性结果。更确切地说，通过使用色散和平滑估计，我们将证明Fujita-Kato型定理用于局部适定性。然后，我们将使用双曲线空间上的对数Hardy-Littlewood-Sobolev估计来证明解不能在有限时间内爆炸。对于较大的质量\\ chi M> 8 \ pi $，我们将在其他条件下获得爆破结果。