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 <= 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 other hand, global existence holds true for alpha <= 1 if the initial density is small enough in the sense of the L-1/alpha norm.