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