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

TL;DR

This paper constructs new finite-time blowup solutions for the Keller-Segel system in 3 and 4 dimensions, revealing a specific blowup profile and answering a longstanding open question in the field.

Contribution

It introduces a novel type I-Log blowup solution for the Keller-Segel system in higher dimensions, expanding understanding of singularity formation.

Findings

Established explicit blowup profiles in 3 and 4 dimensions.

Provided rigorous construction of solutions with logarithmic blowup behavior.

Answered a longstanding open question about blowup solutions in Keller-Segel models.

Abstract

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