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

Abstract

This article studies the aggregation diffusion equation \[ \partial_tρ= Δ^\fracα{2} ρ+ λ\,\mathrm{div}((K*ρ)ρ), \] 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 $L_k^1$ initial condition, and in the fair competition case $\beta =\alpha$ for an $L_k^1\cap L\ln L$ 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.

30 pages, 3 figures

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• The derivation of Swarming models: Mean-Field Limit and Wasserstein distances
• Propagation of chaos for the VPFP equation with a polynomial cut-off
• Blowup of solutions to a diffusive aggregation model
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Cited by in corpus (1)

• p-Laplacian Keller-Segel Equation: Fair Competition and Diffusion Dominated Cases