  In this work we consider a nonlinear parabolic higher order partial
differential equation that has been proposed as a model for epitaxial growth.
This equation possesses both global-in-time solutions and solutions that blow
up in finite time, where this blow-up is mediated by its Hessian nonlinearity.
Herein we further analyze its blow-up behaviour by building explicit solutions
in the square, the disc, and the plane. Some of these solutions show complete
blow-up either in finite or infinite time. Finally, we refine a blow-up
criterium that has been proven for this evolution equation. Still, existent
blow-up criteria based on a priori estimates do not reflect the singular
character of these explicit blowing up solutions.
