SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5840-5864, January 2020.
We consider a parabolic-parabolic Keller--Segel system in a ball of $ \mathbb{R}^N $ under the Neumann boundary condition. This was introduced as a model of aggregation of bacteria. The aggregation is mathematically defined as finite-time blowup. When $ N = 2 $, an optimal criterion for finite-time blowup was obtained in [N. Mizoguchi and M. Winkler, Boundedness of Global Solutions in the Two-Dimensional Parabolic Keller--Segel System, preprint]. On the other hand, there has been no criterion for finite-time blowup for $ N \geq 3 $ though existence of radial solutions blowing up in finite time was known due to [M. Winkler, J. Math. Pures Appl., 100 (2013), pp. 748--767]. In this paper, focusing on common nature in all dimensions, we give a criterion for finite-time blowup for $ N = 2, 3, 4 $.

中文翻译：

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抛物线-抛物线型Keller-Segel系统中有限时间爆燃的初始能量标准

SIAM数学分析期刊，第52卷，第6期，第5840-5864页，2020年1月。
我们考虑在Neumann边界条件下$ \ mathbb {R} ^ N $的球中的抛物-抛物型Keller-Segel系统。这是作为细菌聚集的模型引入的。聚合在数学上定义为有限时间爆炸。当$ N = 2 $时，在[N. Mizoguchi和M. Winkler，二维抛物线Keller-Segel系统中整体解的有界性，预印本]。另一方面，尽管由于[M。]已知在有限时间内爆炸的径向解的存在，但对于$ N \ geq 3 $，没有关于有限时间爆炸的判据。Winkler，J。Math。Pures Appl。，100（2013），第748--767页]。在本文中，着眼于所有维度的共同性质，我们给出了$ N = 2、3、4 $的有限时间爆破准则。