  Spatial distribution and growth of dust in a clumpy protoplanetary disk
subject to vigorous gravitational instability and fragmentation is studied
numerically with sub-au resolution using the FEOSAD code. Hydrodynamics
equations describing the evolution of self-gravitating and viscous
protoplanetary disks in the thin-disk limit were modified to include a dust
component consisting of two parts: sub-micron-sized dust and grown dust with a
variable maximum radius. The conversion of small to grown dust, dust growth,
friction of dust with gas, and dust self-gravity were also considered. We found
that the disk appearance is notably time-variable with spiral arms, dusty
rings, and clumps, constantly forming, evolving, and decaying. As a
consequence, the total dust-to-gas mass ratio is highly non-homogeneous
throughout the disk extent, showing order-of-magnitude local deviations from
the canonical 1:100 value. Gravitationally bound clumps formed through
gravitational fragmentation have a velocity pattern that deviates notably from
the Keplerian rotation. Small dust is efficiently converted into grown dust in
the clump interiors, reaching a maximum radius of several decimeters.
Concurrently, grown dust drifts towards the clump center forming a massive
compact central condensation (70-100 $M_\oplus$). We argue that protoplanets
may form in the interiors of inward migrating clumps before they disperse
through the action of tidal torques. We foresee the formation of protoplanets
at orbital distances of several tens of au with initial masses of gas and dust
in the protoplanetary seed in the (0.25-1.6) $M_{\rm Jup}$ and (1.0-5.5)
$M_\oplus$ limits, respectively. The final masses of gas and dust in the
protoplanets may however be much higher due to accretion from surrounding
massive metal-rich disks/envelopes.
