Criterion on Initial Energy for Finite-time Blowup in Parabolic-parabolic Keller--Segel System

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 $.