  We study the expansion of the ionization and dissociation fronts (DFs) in a
radially stratified molecular cloud, whose density distribution is represented
as n(r) \propto r^-w. We focus on cases with w \leq 1.5, when the ionization
front is ``trapped'' in the cloud and expands with the preceding shock front.
The simultaneous evolution of the outer photodissociation region (PDR) is
examined in detail. First, we analytically probe the time evolution of the
column densities of the shell and envelope outside the HII region, which are
key physical quantities for the shielding of dissociating photons. Next, we
perform numerical calculations, and study how the thermal/chemical structure of
the outer PDR changes with different density gradients. We apply our numerical
model to the Galactic HII region, Sharpless 219 (Sh219). The time evolution of
the column densities of the shell and outer envelope depends on w, and
qualitatively changes across w = 1. In the cloud with w < 1, the shell column
density increases as the HII region expands. The DFs are finally trapped in the
shell, and the molecular gas gradually accumulates in the shell. The molecular
shell and envelope surround the HII region. With w > 1, on the other hand, the
shell column density initially increases, but finally decreases. The column
density of the outer envelope also quickly decreases as the HII region swells
up. It becomes easier and easier for the dissociating photons to penetrate the
shell and envelope. The PDR broadly extends around the trapped HII region. A
model with w = 1.5 successfully explains the observational properties of Sh219.
Our model suggests that a density-bounded PDR surrounds the photon-bounded HII
region in Sh219.
