  The central region of a circumstellar disk is difficult to resolve in global
numerical simulations of collapsing cloud cores, but its effect on the
evolution of the entire disk can be significant. We use numerical hydrodynamics
simulations to model the long-term evolution of self-gravitating and viscous
circumstellar disks in the thin-disk limit. Simulations start from the
gravitational collapse of prestellar cores of 0.5--1.0~$M_\odot$ and both
gaseous and dusty subsystems were considered, including a model for dust
growth. The inner unresolved 1.0 au of the disk is replaced with a central
"smart" cell (CSC) -- a simplified model that simulates physical processes that
may occur in this region. We found that the mass transport rate through the CSC
has an appreciable effect on the evolution of the entire disk. Models with slow
mass transport form more massive and warmer disks and they are more susceptible
to gravitational instability and fragmentation, including a newly identified
episodic mode of disk fragmentation in the T Tauri phase of disk evolution.
Models with slow mass transport through the CSC feature episodic accretion and
luminosity bursts in the early evolution, while models with fast transport are
characterized by a steadily declining accretion rate with low-amplitude
flickering. Dust grows to a larger, decimeter size in the slow transport models
and efficiently drifts in the CSC, where it accumulates reaching the limit when
streaming instability becomes operational. We argue that gravitational
instability, together with streaming instability likely operating in the inner
disk regions, constitute two concurrent planet-forming mechanisms, which may
explain the observed diversity of exoplanetary orbits (Abridged).
