  The dynamics of a one-dimensional two-component Fermi gas in the presence of
a quasi-periodic optical lattice (OL) is investigated by means of a Density
Functional Theory approach. Inspired by the protocol implemented in recent
cold-atom experiments, designed to identify the many-body localization
transition, we analyze the relaxation of an initially prepared imbalance
between the occupation number of odd and of even sites. For quasi-disorder
strength beyond the Anderson localization transition, the imbalance survives
for long times, indicating the inability of the system to reach local
equilibrium. The late time value diminishes for increasing interaction
strength. Close to the critical quasi-disorder strength corresponding to the
noninteracting (Anderson) transition, the interacting system displays an
extremely slow relaxation dynamics, consistent with sub-diffusive behavior. The
amplitude of the imbalance fluctuations around its running average is found to
decrease with time, and such damping is more effective with increasing
interaction strengths. While our study addresses the setup with two equally
intense OLs, very similar effects due to interactions have been observed also
in recent cold-atom experiments performed in the tight-binding regime, i.e.
where one of the two OLs is very deep and the other is much weaker.
