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 $c<\alpha \le 2$, where $c\in (0,2)$ depends on the equation’s parameters. For an even wider range of ${\alpha }^{\mathrm{\prime }}s$, we prove existence of global in time weak solution for general initial data.