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

Abstract

We construct finite time blowup solutions to the parabolic-elliptic Keller-Segel system ${\partial }_{t}u=\Delta u-\nabla \cdot (u\nabla {\mathcal{K}}_u),-\Delta {\mathcal{K}}_u=u\text{in}{\mathbb{R}}^d,d=3,4,$ and derive the final blowup profile $u(r,T)\sim c_d\frac{|\log r{|}^{\frac{d-2}{d}}}{r^2}\text{as}r\to 0,c_d>0.$ 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).