# Blow-Up of Solutions to the Patlak-Keller-Segel Equation in Dimension $\nu\geq2$

## TL;DR

This paper establishes a new blow-up criterion for solutions to the Patlak-Keller-Segel equation in multiple dimensions, extending known results in two dimensions to higher dimensions, with implications for understanding solution behavior.

## Contribution

It introduces a novel blow-up criterion applicable in dimensions three and higher, expanding the theoretical understanding of the Patlak-Keller-Segel equation.

## Key findings

- Blow-up occurs if total mass exceeds a critical value in 2D.
- New blow-up criterion established for dimensions ≥ 3.
- In 2D, the criterion matches known results by Dolbeault and Perthame.

## Abstract

We prove a blow-up criterion for the solutions to the $\nu$-dimensional Patlak-Keller-Segel equation in the whole space. The condition is new in dimension three and higher. In dimension two it is exactly Dolbeault's and Perthame's blow-up condition, i.e., blow-up occurs if total mass exceeds $8\pi$ .