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 $2$--$5times10^{-10}(Phi_{rm EUV}/10^{41}{,rm s}^{-1})^{1/2},M_odot,{rm yr}^{-1}$, 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.
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