Construction of type I-Log blowup for the Keller-Segel system in dimensions $3$ and $4$

We construct finite time blowup solutions to the parabolic-elliptic Keller-Segel system ${\mathrm{\partial }}_{t}u=\mathrm{\Delta }u-\mathrm{\nabla }\cdot (u\mathrm{\nabla }{\mathcal{K}}_u),-\mathrm{\Delta }{\mathcal{K}}_u=u\text{in}{\mathbb{R}}^d,d=3,4,$ and derive the final blowup profile $u(r,T)\sim c_d\frac{\mathrm{\mid }\log r{\mathrm{\mid }}^{\frac{d-2}{d}}}{r^2}\text{as}r\to 0,c_d>0.$ To our knowledge this provides a new blowup solution for the Keller-Segel system, rigorously answering a question by Brenner, Constantin, Kadanoff, Schenkel, and Venkataramani (Nonlinearity, 1999).