Fractional Keller-Segel Equation: Global Well-posedness and Finite Time Blow-up
This article studies the aggregation diffusion equation \[ \partial_tρ= Δ^\fracα{2} ρ+ λ\,\mathrm{div}((K*ρ)ρ), \] where $Δ^\fracα{2}$ denotes the fractional Laplacian and $K = \frac{x}{|x|^β}$ 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 $β< α$ we prove global well-posedness for an $L^1_k$ initial condition, and in the fair competition case $β= α$ for an $L^1_k\cap L\ln L$ initial condition with small mass. In the aggregation dominated case $β> α$, 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.