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

We investigate the one-dimensional Keller-Segel model where the diffusion is replaced by a non-local operator, namely the fractional diffusion with exponent $0<α\leq 2$. 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 $α<1$ and the initial configuration of cells is sufficiently concentrated. On the opposite, global existence holds true for $α\leq1$ if the initial density is small enough in the sense of the $L^{1/α}$ norm.