Suppression of blow up by a logistic source in $2$D 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 $\alpha$. 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. 
Jan Burczak and Rafael Granero-Belinch\'on
math.AP