Context :

Aims : 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 .

Methods : First , we analytically probe the time evolution of the column densities of the shell and envelope outside the H II 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 H II region , Sharpless 219 ( Sh219 ) .

Results : 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 H II 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 H II 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 H II 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 H II 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 H II region in Sh219 .

Conclusions :