  We study the effect of strong electron-phonon interactions on the damping of
the Higgs amplitude mode in superconductors by means of non-equilibrium
dynamical mean-field simulations of the Holstein model. In contrast to the BCS
dynamics, we find that the damping of the Higgs mode strongly depends on the
temperature, becoming faster as the systen approaches the transition
temperature. The damping at low temperatures is well described by a power-law,
while near the transition temperature the damping shows exponential-like
behavior. We explain this crossover by a temperature-dependent quasiparticle
lifetime caused by the strong electron- phonon coupling, which smears the
superconducting gap edge and makes the relaxation of the Higgs mode into
quasiparticles more efficient at elevated temperatures. We also reveal that the
phonon dynamics can soften the Higgs mode, which results in a slower damping.
