  We consider two types of non linear fast diffusion equations in R^N:(1)
External drift type equation with general external potential. It is a natural
extension of the harmonic potential case, which has been studied in many
papers. In this paper we can prove the large time asymptotic behavior to the
stationary state by using entropy methods.(2) Mean-field type equation with the
convolution term. The stationary solution is the minimizer of the free energy
functional, which has direct relation with reverse Hardy-Littlewood-Sobolev
inequalities. In this paper, we prove that for some special cases, it also
exists large time asymptotic behavior to the stationary state.
