  We consider a partial differential equation that arises in the coarse-grained
description of epitaxial growth processes. This is a parabolic equation whose
evolution is governed by the competition between the determinant of the Hessian
matrix of the solution and the biharmonic operator. This model might present a
gradient flow structure depending on the boundary conditions. We first extend
previous results on the existence of stationary solutions to this model for
Dirichlet boundary conditions. For the evolution problem we prove local
existence of solutions for arbitrary data and global existence of solutions for
small data. By exploiting the boundary conditions and the variational structure
of the equation, according to the size of the data we prove finite time blow-up
of the solution and/or convergence to a stationary solution for global
solutions.
