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

Laurent Lafleche (Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique, Centre National de la Recherche Scientifique (CNRS), Université Paris-Saclay, Palaiseau, France; and CEREMADE, Université Paris-Dauphine, Paris, France)

Samir Salem (Centre de Recherche en Mathématiques de la Décision (CEREMADE), UMR CNRS 7534, Université Paris-Dauphine, PSL Research University, Paris, France)

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

Received 13 August 2018

Accepted 1 July 2019

Published 3 February 2020