Explicit Blowing Up Solutions for a Higher Order Parabolic Equation with Hessian Nonlinearity

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.