On the well-posedness for Keller-Segel system with fractional diffusion

Communicated by M. Costabel In this paper, we study the Cauchy problem for the Keller-Segel system with fractional diffusion generalizing the Keller-Segel model of chemotaxis for the initial data (u₀,v₀) in critical Fourier-Herz spaces B•q2-2αRn× B•q2-2αRn with q a [2, ∞], where 1 < α 2. Making use of some estimates of the linear dissipative equation in the frame of mixed time-space spaces, the Chemin "mono-norm method", the Fourier localization technique and the Littlewood-Paley theory, we get a local well-posedness result and a global well-posedness result with a small initial data. In addition, ill-posedness for "doubly parabolic" models is also studied.