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 $0<\alpha\leq 2$. 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 $\alpha<1$ and the initial configuration of cells is sufficiently concentrated. On the opposite, global existence holds true for $\alpha\leq1$ if the initial density is small enough in the sense of the $L^{1/\alpha}$ norm.