  We investigate the dynamics of Mott-insulating regions of a trapped bosonic
gas as the interaction strength is changed linearly with time. The bosonic gas
considered is loaded into an optical lattice and confined to a parabolic
trapping potential. Two situations are addressed: the formation of Mott domains
in a superfluid gas as the interaction is increased, and their melting as the
interaction strength is lowered. In the first case, depending on the local
filling, Mott-insulating barriers can develop and hinder the density and energy
transport throughout the system. In the second case, the density and local
energy adjust rapidly whereas long range correlations require longer time to
settle. For both cases, we consider the time evolution of various observables:
the local density and energy, and their respective currents, the local
compressibility, the local excess energy, the heat and single particle
correlators. The evolution of these observables is obtained using the
time-dependent density-matrix renormalization group technique and comparisons
with time-evolutions done within the Gutzwiller approximation are provided.
