In this paper we consider the global existence, regularizing-decay rate
and asymptotic behavior of mild solutions to the Cauchy problem of
fractional drift diffusion system with power-law nonlinearity. Using the
properties of fractional heat semigroup and the classical estimates of
fractional heat kernel, we first prove the global-in-time existence and
uniqueness of the mild solutions in the frame of mixed time-space Besov
space with multi-linear continuous mappings. Then we show the asymptotic
behavior and regularizing-decay rate estimates of the solution to
equations with power-law nonlinearity by the method of multi-linear
operator and the classical Hardy-Littlewood-Sobolev inequality.