  Using a combination of numerically exact and renormalization-group techniques
we study the nonequilibrium transport of electrons in an one-dimensional
interacting system subject to a quasiperiodic potential. For this purpose we
calculate the growth of the mean-square displacement as well as the melting of
domain walls. While the system is nonintegrable for all studied parameters,
there is no on finite region default of parameters for which we observe
diffusive transport. In particular, our model shows a rich dynamical behavior
crossing over from superdiffusion to subdiffusion. We discuss the implications
of our results for the general problem of many-body localization, with a
particular emphasis on the rare region Griffiths picture of subdiffusion.
