tokens: maximum grain size is determined by the fragmentation barrier (Eq. ( 3 )) and particles can reach centimetre sizes, but small particles are continuously reproduced. For a more massive planet, the location of the pressure maximum (and therefore the peak of mm dust concentration) occurs further away at 30, 44, and 54 AU for 1, 5, and 15 , respectively, as previously shown by Pinilla et al. (2012) . The radial range in which mm-sized grains are concentrated becomes wider. For the case, the second peak is caused by a transient density feature that in reality would be short-lived and would not function as a pressure maximum that can effectively concentrate particles. This issue has no impact on our results since we focus on the inner disk. In the regions inside the planetary gap ( AU) where the gas surface density steeply decreases outwards (and therefore the pressure gradient is negative), the maximum grain size is determined by radial drift (Fig. 2 ), and only close to the snow line ( AU) does the fragmentation barrier drop below the drift barrier due to the change in at . With the decrease in at , the maximum grain size is a few centimetres and the Stokes number of the particles at that location is much lower than unity (), thus they are better coupled to the gas, their radial drift velocities are lower (Eq. ( 2 )), and they move with at a similar speed to the gas (see also Brauer et al. 2008 ). If does not change at , the maximum grain size is dominated by radial drift, reaching values of the Stokes number of around , which would lead to very high drift velocities () depleting the inner disk on very short time scales (100–1000 yr). Therefore, when the inner disk is constantly resupplied from the outer disk, the changes of at allows replenishment of warm, small, and optically thick dust near , for which the drift velocities are slow. In the case of a mass planet, the micron-sized particles (1–10 m), are not perfectly trapped at the outer edge of the planet-gap, and they move continuously through the gap via turbulent diffusion. This partial filtration of dust grains supports a constant replenishment of dust from the outer to the inner disk, which together with variations of at extends the lifetime of a detectable inner disk reaching the life of the disk (5 Myr). In the cases of and mass planets, there is more efficient trapping of particles at the outer edge of the planetary gap, filtering all particle sizes (from 1m, which is the minimum grain size in our setup). The dust inside the planetary gap ( AU) is the dust that was initially within the gap. These particles grow, drift and fragment when they reach the snow line. Although there is a considerable amount of dust in the inner disk for all cases (1, 5, and 15 ) after 1 Myr, after 5 Myr of evolution the inner disk is practically depleted of dust for a disk hosting a and a mass planet. The dust density distribution after 5 Myr of evolution for AU is almost identical for and because the
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