Global existence versus blow-up results for a fourth order parabolic PDE involving the Hessian

TL;DR

This paper studies a fourth order parabolic PDE modeling epitaxial growth, analyzing conditions for global existence, blow-up, and convergence to stationary solutions based on boundary conditions and data size.

Contribution

It extends previous results on stationary solutions and provides new criteria for local and global existence, blow-up, and convergence in the PDE model.

Findings

Established existence of stationary solutions under Dirichlet boundary conditions.

Proved local existence of solutions for arbitrary initial data.

Identified conditions for finite time blow-up and convergence to stationary states.

Abstract

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.