Abstract:  
The nonlinear Cahn-Hilliard equation (CH) is a celebrated continuum model for the description of the dynamics of pattern formation in phase separation. We consider boundary value problems for higher-order version (HCH) of CH with additional convective term to model faceting of a growing crystal surface, caused by strongly anisotropic surface tension, driven by surface diffusion and accompanied by deposition. Stationary shapes and dynamics of faceted pyramidal structures are studied by means of matched asymptotic expansions and numerically. In the 1D case it is shown that a solitary hill as well as periodic hill-and-valley solutions are unique, while solutions in the form of a solitary valley form a one-parameter family. We also used HCH to study the interplay between faceting and the film/substrate wetting interactions as a mechanism for the formation of quantum dots.