  In this work we consider $$ w_t=[(w_{hh}+c_0)^{-3}]_{hh},\qquad w(0)=w^0, $$
which is derived from a thin film equation for epitaxial growth on vicinal
surface. We formulate the problem as the gradient flow of a suitably-defined
convex functional in a non-reflexive space. Then by restricting it to a Hilbert
space and proving the uniqueness of its sub-differential, we can apply the
classical maximal monotone operator theory. The mathematical difficulty is due
to the fact that $w_{hh}$ can appear as a positive Radon measure. We prove the
existence of a global strong solution. In particular, the equation holds almost
everywhere when $w_{hh}$ is replaced by its absolutely continuous part.
