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

$$\partial_t u = \Delta u - \nabla \cdot (u \nabla \mathcal{K}_u), \quad -\Delta \mathcal{K}_u = u \quad \text{in}\;\; \mathbb{R}^d,\; d = 3,4,$$

and derive the final blowup profile

$$u(r,T) \sim c_d \frac{|\log r|^\frac{d-2}{d}}{r^2} \quad \text{as}\;\; r \to 0, \;\; c_d > 0.$$

To our knowledge this provides a new blowup solution for the Keller-Segel system, rigorously answering a question by Brenner et al. in [Brenner, Nonlinearity 12, 1999].