Special solutions to a fourth-order nonlinear parabolic equation in non-divergence form

TL;DR

This paper investigates special solutions to a complex fourth-order nonlinear parabolic equation modeling crystal surfaces, establishing existence results and deriving self-similar solutions that differ from classical linear models.

Contribution

It introduces a novel approach to solving a boundary value problem with degeneracy and constructs explicit self-similar solutions for the crystal surface model.

Findings

Existence of solutions with meaningful boundary conditions remains open.

Constructed positive, unbounded self-similar solutions.

Contrasts with linear biharmonic heat equation behavior.

Abstract

In this paper we study a crystal surface model first proposed by H.~Al Hajj Shehadeh, R.V.~Kohn, and J.~Weare (2011 Physica D, 240,1771-1784). By seeking a solution of a particular function form, we are led to a boundary value problem for a fourth-order nonlinear elliptic equation. The mathematical challenge of the problem is due to the fact that the degeneracy in the equation is directly imposed by one of the two boundary conditions. An existence theorem is established in which a meaningful mathematical interpretation of one of the boundary conditions remains open. Our proof seems to suggest that this is unavoidable. We also obtain self-similar solutions to the crystal surface model which are positive and unbounded. This is in sharp contrast with the linear biharmonic heat equation.