  In this work, we study a fourth order exponential equation, $u_t=\Delta
e^{-\Delta u},$ derived from thin film growth on crystal surface in multiple
space dimensions. We use the gradient flow method in metric space to
characterize the latent singularity in global strong solution, which is
intrinsic due to high degeneration. We define a suitable functional, which
reveals where the singularity happens, and then prove the variational
inequality solution under very weak assumptions for initial data. Moreover, the
existence of global strong solution is established with regular initial data.
