  We study how the interplay of dissipation and interactions affects the
dynamics of a bosonic many-body quantum system. In the presence of both
dissipation and strongly repulsive interactions, observables such as the
coherence and the compressibility display three dynamical regimes: an initial
exponential variation followed by a power-law regime and finally a slow
exponential convergence to their asymptotic values corresponding to the
infinite temperature state. These very long-time scales arise as dissipation
forces the population of states disfavored by interactions. The long-time,
strong coupling dynamics are understood by performing a mapping onto a
classical diffusion process displaying non-Brownian behavior. While both
dissipation and strong interactions tend to suppress coherence when acting
separately, we find that strong interaction impedes the decoherence process
generated by the dissipation.
