The one-dimensional Keller-Segel model with fractional diffusion of cells

Abstract

We investigate the one-dimensional Keller-Segel model where the diffusion is replaced by a non-local operator, namely the fractional diffusion with exponent $0<\alpha \le 2$. We prove some features related to the classical two-dimensional Keller-Segel system: blow-up may or may not occur depending on the initial data. More precisely a singularity appears in finite time when $\alpha <1$ and the initial configuration of cells is sufficiently concentrated. On the opposite, global existence holds true for $\alpha \le 1$ if the initial density is small enough in the sense of the $L^{1/\alpha }$ norm.

12 pages

References in corpus (3)

• An extension problem related to the fractional Laplacian
• The fractional Keller-Segel model
• Blowup of solutions to a diffusive aggregation model

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