  Debris disks are classically considered to be gas-less systems, but recent
(sub)millimeter observations have detected tens of those with rich gas content.
The origin of the gas component remains unclear; namely, it can be
protoplanetary remnants and/or secondary products deriving from large bodies.
In order to be protoplanetary in origin, the gas component of the parental
protoplanetary disk is required to survive for $\gtrsim10{\,\rm Myr}$. However,
previous models predict $\lesssim 10{\,\rm Myr}$ lifetimes because of efficient
photoevaporation at the late stage of disk evolution. In the present study, we
investigate photoevaporation of gas-rich, optically-thin disks around
intermediate-mass stars at a late stage of the disk evolution. The evolved
system is modeled as those where radiation force is sufficiently strong to
continuously blow out small grains ($\lesssim 4 {\,\rm \mu m}$), which are an
essential component for driving photoevaporation via photoelectric heating
induced by stellar far-ultraviolet (FUV). We find that the grain depletion
reduces photoelectric heating, so that FUV photoevaporation is not excited.
Extreme-ultraviolet (EUV) photoevaporation is dominant and yields a mass-loss
rate of $2$--$5\times10^{-10}(\Phi_{\rm EUV}/10^{41}{\,\rm
s}^{-1})^{1/2}\,M_\odot\,{\rm yr}^{-1}$, where $\Phi_{\rm EUV}$ is the EUV
emission rate. The estimated lifetimes of the gas component are $\sim 50
(M_{\rm disk}/10^{-2}\,M_\odot)(\Phi_{\rm EUV}/10^{41}\,{\rm
s}^{-1})^{1/2}\,{\rm Myr}$ and depend on the ``initial'' disk mass at the point
small grains have been depleted in the system. With an order estimation, we
show that the gas component can survive for a much longer time around A-type
stars than lower-mass stars. This trend is consistent with the higher frequency
of gas-rich debris disks around A-type stars, implying the possibility of the
gas component being protoplanetary remnants.
