  Many-body localization is a fascinating theoretical concept describing the
intricate interplay of quantum interference, i.e. localization, with many-body
interaction induced dephasing. Numerous computational tests and also several
experiments have been put forward to support the basic concept. Typically,
averages of time-dependent global observables have been considered, such as the
charge imbalance. We here investigate within the disordered spin-less Hubbard
($t-V$) model how dephasing manifests in time dependent variances of
observables. We find that after quenching a N\'eel state the local charge
density exhibits strong temporal fluctuations with a damping that is sensitive
to disorder $W$: variances decay in a power law manner, $t^{-\zeta}$, with an
exponent $\zeta(W)$ strongly varying with $W$. A heuristic argument suggests
the form, $\zeta\approx\alpha(W)\xi_\text{sp}$, where $\xi_\text{sp}(W)$
denotes the noninteracting localization length and $\alpha(W)$ characterizes
the multifractal structure of the dynamically active volume fraction of the
many-body Hilbert space. In order to elucidate correlations underlying the
damping mechanism, exact computations are compared with results from the
time-dependent Hartree-Fock approximation. Implications for experimentally
relevant observables, such as the imbalance, will be discussed.
