  In this paper we construct the time dependent boundary states describing the
``rolling D-brane solutions'' in the NS5 background discovered recently by
Kutasov by means of the classical DBI analysis. We first survey some aspects of
non-compact branes in the NS5 background based on known boundary states in the
N=2 Liouville theory. We consider two types of non-compact branes, one of which
is BPS and the other is non-BPS but stable. Then we clarify how to Wick-rotate
the non-BPS one appropriately. We show that the Wick-rotated boundary state
realizes the correct trajectory of rolling D-brane in the classical limit, and
leads to well behaved spectral densities of open strings due to the existence
of non-trivial damping factors of energy. We further study the cylinder
amplitudes and the emission rates of massive closed string modes.
