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# Communications in Mathematical Sciences

## Volume 17 (2019)

### Number 8

### Fractional Keller–Segel equation: Global well-posedness and finite time blow-up

Pages: 2055 – 2087

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n8.a1

#### Authors

#### Abstract

This article studies the aggregation diffusion equation\[\partial t \rho = \Delta^{\frac{\alpha}{2}} \rho+ \lambda \operatorname{div}((K \times \rho ) \rho) \; \textrm{,}\]where $\Delta^{\frac{\alpha}{2}}$ denotes the fractional Laplacian and $K= \frac{x}{{\lvert x \rvert}^\beta}$ is an attractive kernel. This equation is a generalization of the classical Keller–Segel equation, which arises in the modelling of the motion of cells. In the *diffusion dominated* case $\beta \lt \alpha$, we prove global well-posedness for an $L^1_k$ initial condition, and in the *fair competition* case $\beta=\alpha$ for an $L^1_k \cap L \operatorname{ln} L$ initial condition with small mass. In the *aggregation dominated* case $\beta \gt \alpha$, we prove global or local well-posedness for an $L^p$ initial condition, depending on some smallness condition on the $L^p$ norm of the initial condition. We also prove that finite time blow-up of even solutions occurs under some initial mass concentration criteria.

#### Keywords

fractional diffusion with drift, fractional Laplacian, aggregation diffusion, mean field equation

#### 2010 Mathematics Subject Classification

35A01, 35A02, 35B40, 35B44, 35R11

The second author was supported by the Fondation des Sciences Mathématiques de Paris, and by Paris Sciences & Lettres Université.

Received 13 August 2018

Accepted 1 July 2019

Published 3 February 2020