  We study a $4$th order PDE with a $2$nd order correction that is a
modification of one suggested by Peter Smereka using the tools of Krug et al
(Z. Phys. B., 1994). First, we provide a non-rigorous derivation of the full
model derived with a quadratic interaction potential using the rates and
methods of the last author with Jonathan Weare (PRE, 2013). Then, we prove the
existence of a global in time weak solution by regularizing the equation in a
way that allows us to apply the tools of Bernis-Friedman (JDE, 1990). The
methods developed here can be applied to a large number of $4$th order
degenerate models. In an appendix, we also discuss the global smooth solution
with small data in the Weiner algebra framework following recent developments
using tools of the second author with Robert Strain (IFB, 2018).
