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

Abstract

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 ∈ [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. Copyright © 2011 John Wiley & Sons, Ltd.