Explicit blowing up solutions for a higher order parabolic equation with Hessian nonlinearity

TL;DR

This paper constructs explicit solutions demonstrating blow-up behavior in a higher order nonlinear parabolic PDE modeling epitaxial growth, providing insights into the blow-up mechanisms and refining existing criteria.

Contribution

It introduces explicit solutions for a complex PDE, revealing detailed blow-up behaviors and improving the understanding of blow-up criteria in such equations.

Findings

Explicit solutions exhibit finite and infinite time blow-up.

Refined blow-up criteria better capture the singular behavior.

Solutions constructed in various geometries like square, disc, and plane.

Abstract

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, for which this blow-up is mediated by its Hessian nonlinearity. Herein, we further analyze its blow-up behaviour by means of the construction of explicit solutions in the square, the disc, and the plane. Some of these solutions show complete blow-up in either finite or infinite time. Finally, we refine a blow-up criterium that was proved for this evolution equation. Still, existent blow-up criteria based on a priori estimates do not completely reflect the singular character of these explicit blowing up solutions.