  An elastic sheet that deforms near a solid substrate in a viscous fluid is a
situation relevant to various dynamical processes in biology, geophysics and
engineering. Here, we study the relaxation dynamics of an elastic plate resting
on a thin viscous film that is supported by a solid substrate. By combining
scaling analysis, numerical simulations and experiments, we identify asymptotic
regimes for the elastohydrodynamic leveling of a surface perturbation of the
form of a bump, when the flow is driven by either the elastic bending of the
plate or thermal fluctuations. In both cases, two distinct regimes are
identified when the bump height is either much larger or much smaller than the
thickness of the pre-wetted viscous film. Our analysis reveals a distinct
crossover between the similarity exponents with the ratio of the perturbation
height to the film height.
