  Motivated by the ``universe as a brane'' idea, we investigate the motion of a
$(D-2)$-brane (or domain wall) that couples to bulk matter. Usually one would
expect the spacetime outside such a wall to be time dependent however we show
that in certain cases it can be static, with consistency of the Israel
equations yielding relationships between the bulk metric and matter that can be
used as ans\"atze to solve the Einstein equations. As a concrete model we study
a domain wall coupled to a bulk dilaton with Liouville potentials for the
dilaton both in the bulk and on the wall. The bulk solutions we find are all
singular but some have black hole or cosmological horizons, beyond which our
solutions describe domain walls moving in time dependent bulks. A significant
period of world volume inflation occurs if the potential on the wall is not too
steep; in some cases the bulk also inflates (with the wall comoving) while in
others the wall moves relative to a non-inflating bulk. We apply our method to
obtain cosmological solutions of Ho\v{r}ava-Witten theory compactified on a
Calabi-Yau space. tive to a non-inflating bulk. We apply our method to obtain
cosmological solutions of Ho\v{r}ava-Witten theory compactified on a Calabi-Yau
space.
