SUPPRESSION OF BLOW UP BY A LOGISTIC SOURCE IN 2D KELLER-SEGEL SYSTEM WITH FRACTIONAL DISSIPATION

We consider a two dimensional parabolic-elliptic Keller-Segel equation with a logistic forcing and a fractional diffusion of order α. We obtain existence of global in time regular solution for arbitrary initial data with no size restrictions and c < α ≤ 2, where c ∈ (0, 2) depends on the equation's parameters. For an even wider range of α s, we prove existence of global in time weak solution for general initial data