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 fractional diffusion of order α∈(0,2) and a logistic term. In the case of an analogous problem with standard diffusion, introduction of the logistic term, well motivated by biological applications, results in global smoothness of solutions (i.e. suppression of blowup), compare Tello & Winkler [48]. We show that this phenomenon extends into potentially less regular case of fractional diffusions. Namely, we obtain existence of global in time regular solutions emanating from initial data with no size restrictions for 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.