  We study the dynamics and stability in a strongly interacting resonantly
driven two-band model. Using exact numerical simulations, we find a stable
regime at large driving frequencies where the time evolution is governed by a
local Floquet Hamiltonian that is approximately conserved out to very long
times. For slow driving, on the other hand, the system becomes unstable and
heats up to infinite temperature. While thermalization is relatively fast in
these two regimes (but to different "temperatures"), in the crossover between
them we find slow nonthermalizing time evolution: temporal fluctuations become
strong and temporal correlations long lived. Microscopically, we trace back the
origin of this nonthermalizing time evolution to the properties of rare Floquet
many-body resonances, whose proliferation at lower driving frequency removes
the approximate energy conservation, and thus produces thermalization to
infinite temperature.
