In this work we consider 1$$\begin{aligned} w_t=\left[ \left( w_{hh}+c_0\right) ^{-3}\right] _{hh},\qquad w(0)=w^0, \end{aligned}$$ wt=whh+c0-3hh,w(0)=w0, 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 $$w_{hh}$$ whh can appear as a positive Radon measure. We prove the existence of a global strong solution with hidden singularity. In particular, (1) holds almost everywhere when $$w_{hh}$$ whh is replaced by its absolutely continuous part.