Two nonlinear diffusion equations for thin film epitaxy, with or without slope selection, are
studied in this work. The nonlinearity models the Ehrlich–Schwoebel effect – the kinetic
asymmetry in attachment and detachment of adatoms to and from terrace boundaries. Both
perturbation analysis and numerical simulation are presented to show that such an atomistic
effect is the origin of a nonlinear morphological instability, in a rough-smooth-rough pattern,
that has been experimentally observed as transient in an early stage of epitaxial growth
on rough surfaces. Initial-boundary-value problems for both equations are proven to be
well-posed, and the solution regularity is also obtained. Galerkin spectral approximations are
studied to provide both a priori bounds for proving the well-posedness and numerical schemes
for simulation. Numerical results are presented to confirm part of the analysis and to explore
the difference between the two models on coarsening dynamics.