  We study the dynamics of a spatially inhomogeneous quantum $\lambda \phi^4$
field theory in 1+1 dimensions in the Hartree approximation. In particular, we
investigate the long-time behavior of this approximation in a variety of
controlled situations, both at zero and finite temperature. The observed
behavior is much richer than that in the spatially homogeneous case.
Nevertheless, we show that the fields fail to thermalize in a canonical sense,
as expected from analogous results in closely related (mean field) transport
theory. We argue that this dynamical approximation is best suited as a means to
study the short-time decay of spatially inhomogeneous fields and in the
dynamics of coherent quasi-classical inhomogeneous configurations (e.g.
solitons) in a background of dynamical self-consistent quantum fluctuations.
