  The exact description of the time evolution of open correlated quantum
systems remains one of the major challenges of the condensed matter theory,
specially for asymptotic long times where most numerical methods fail. Here,
the post-quench dynamics of the $N$-component Bose-Anderson impurity model is
studied in the $N\to\infty$ limit. The equilibrium phase diagram is similar to
that of the Bose-Hubbard model in that it contains local versions of the Mott
and Bose phases. Using a numerically exact procedure we are able to study the
real time evolution including asymptotic long time regimes. The formation of
long-lived transient phases is observed for quench paths crossing foreign
phases. For quenches inside the local Bose condensed phase, a dynamical phase
transition is reported, that separates the evolution towards a new equilibrium
state and a regime characterized at large times by a persistent phase rotation
of the order parameter. We explain how such non-decaying mode can exist in the
presence of a dissipative bath. We discuss the extension of our results to the
experimental relevant finite-$N$ case and their implication for the existence
of non-decaying modes in generic quantum systems in the presence of a bath.
