Fractional Keller-Segel Equation: Global Well-posedness and Finite Time Blow-up

TL;DR

This paper investigates a fractional Keller-Segel equation, establishing conditions for global solutions and finite time blow-up, thus advancing understanding of cell motion models with fractional diffusion and attractive kernels.

Contribution

It generalizes the classical Keller-Segel model to fractional diffusion, providing new results on well-posedness and blow-up criteria based on initial data and parameters.

Findings

Global well-posedness in diffusion dominated case for $L^1_k$ initial data.

Existence of finite time blow-up under initial mass concentration.

Conditions for local and global solutions depending on parameters and initial data.

Abstract

This article studies the aggregation diffusion equation \[ \partial_t\rho = \Delta^\frac{\alpha}{2} \rho + \lambda\,\mathrm{div}((K*\rho)\rho), \] where $\Delta^\frac{\alpha}{2}$ denotes the fractional Laplacian and $K = \frac{x}{|x|^\beta}$ is an attractive kernel. This equation is a generalization of the classical Keller-Segel equation, which arises in the modeling of the motion of cells. In the diffusion dominated case $\beta < \alpha$ we prove global well-posedness for an $L^1_k$ initial condition, and in the fair competition case $\beta = \alpha$ for an $L^1_k\cap L\ln L$ initial condition with small mass. In the aggregation dominated case $\beta > \alpha$, 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 data. We also prove that finite time blow-up of even solutions occurs under some initial mass concentration criteria.