  We consider the classical molecular beam epitaxy (MBE) model with logarithmic
type potential known as no-slope-selection. We employ a third order backward
differentiation (BDF3) in time with implicit treatment of the surface diffusion
term. The nonlinear term is approximated by a third order explicit
extrapolation (EP3) formula. We exhibit mild time step constraints under which
the modified energy dissipation law holds. We break the second Dahlquist
barrier and develop a new theoretical framework to prove unconditional uniform
energy boundedness with no size restrictions on the time step. This is the
first unconditional result for third order BDF methods applied to the MBE
models without introducing any stabilization terms or fictitious variables. A
novel theoretical framework is also established for the error analysis of high
order methods.
