For any finite set of points in the half-plane of axial symmetry, there exists a 3D Keller-Segel solution whose mass concentrates at those points with a precisely quantified finite-time blow-up rate.
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Finite-time blow-up for the three dimensional axially symmetric Keller-Segel system
For any finite set of points in the half-plane of axial symmetry, there exists a 3D Keller-Segel solution whose mass concentrates at those points with a precisely quantified finite-time blow-up rate.