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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 $α$. We obtain existence of global in time regular solution for arbitrary initial data with no size restrictions and $c<α\leq 2$, where $c \in (0,2)$ 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.