  Black holes in asymptotically anti-de Sitter (AdS) spacetimes have been the
subject of intense scrutiny, including detailed frequency-domain analysis and
full nonlinear evolutions. Remarkably, studies of linearized perturbations in
the time-domain are scarce or non-existing. We close this gap by evolving
linearized scalar wavepackets in the background of rotating BHs in AdS
spacetimes. Our results show a number of interesting features. Small BHs in AdS
behave as asymptotically flat BHs for early/intermediate times, displaying the
same ringdown modes and power-law tails. As the field bounces back and forth
between the horizon and the timelike boundary it "thermalizes" and the modes of
AdS settle in. Finally, we have indications that wavepackets in the vicinity of
fastly spinning BHs grow exponentially in time, signalling a superradiant
instability of the geometry previously reported through a frequency-domain
analysis.
