On Keller-Segel systems with fractional diffusion

Abstract

In this note we review some recent results on the parabolic-elliptic and parabolic-parabolic Keller-Segel model with fractional diffusion. Furthermore, we consider with particular emphasis the parabolic-hyperbolic case with a fractional Laplacian. In particular we prove certain new results. Namely, first, we prove the existence of solution for analytic initial data and then we establish the finite time singularity formation for initial data in this class. Finally, we obtain a new equation describing unidirectional wave propagation for the parabolic-hyperbolic Keller-Segel system.