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This paper is concerned with the fractional Keller-Segel system in thetemporal and spatial variables. We consider fractional dissipation for thephysical variables including a fractional dissipation mechanism for thechemotactic diffusion, as well as a time fractional variation assumed in theCaputo sense. We analyze the fractional heat semigroup obtaining time decay andintegral estimates of the Mittag-Leffler operators in critical Besov spaces,and prove a bilinear estimate derived from the nonlinearity of the Keller-Segelsystem, without using auxiliary norms. We use these results in order to provethe existence of global solutions in critical homogeneous Besov spacesemploying only the norm of the natural persistence space, including theexistence of self-similar solutions, which constitutes a persistence result inthis framework. In addition, we prove a uniqueness result without assuming anysmallness condition of the initial data.

On the fractional heat semigroup and product estimates in Besov spaces  and applications in theoretical analysis of the fractional Keller-Segel  system