  Lattice growth models where uncorrelated random deposition competes with some
aggregation dynamics that generates correlations are studied with rates of the
correlated component decreasing as a power law. These models have anomalous
roughening, with anomalous exponents related to the normal exponents of the
correlated dynamics, to an exponent characterizing the aggregation mechanism
and to that power law exponent. This is shown by a scaling approach extending
the Family-Vicsek relation previously derived for the models with
time-independent rates, thus providing a connection of normal and anomalous
growth models. Simulation results for several models support those conclusions.
Remarkable anomalous effects are observed even for slowly decreasing rates of
the correlated component, which may correspond to feasible temperature changes
in systems with activated dynamics. The scaling exponents of the correlated
component can be obtained only from the estimates of three anomalous exponents,
without knowledge of the aggregation mechanism, and a possible application is
discussed. For some models, the corresponding Edwards-Wilkinson and
Kardar-Parisi-Zhang equations are also discussed.
