  Isolated quantum many-body systems which thermalize under their own dynamics
are expected to act as their own thermal baths, thereby bringing their local
subsystems to thermal equilibrium. Here we show that the infinite-dimensional
limit of a quantum lattice model, as described by Dynamical Mean-Field theory
(DMFT), provides a natural framework to understand this self-consistent
thermalization process. Using the Fermi-Hubbard model as working example, we
demonstrate that the emergence of a self-consistent bath thermalising the
system is characterized by a sharp thermalization front, moving balistically
and separating the initial condition from the long-time thermal fixed point. We
characterize the full DMFT dynamics through an effective temperature for which
we derive a travelling-wave equation of the
Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type. This equation allows to
predict the asymptotic shape of the front and its velocity, which match
perfectly the full DMFT numerics. Our results provide a new angle to understand
the onset of quantum thermalisation in closed isolated systems.
