Communications in Mathematical Sciences

Volume 21 (2023)

Number 4

Existence theorems for a fourth-order exponential PDE related to crystal surface relaxation

Pages: 949 – 966

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n4.a3

Authors

Brock C. Price (Department of Mathematics and Statistics, Mississippi State University, Mississippi State, Miss., U.S.A.)

Xiangsheng Xu (Department of Mathematics and Statistics, Mississippi State University, Mississippi State, Miss., U.S.A.)

Abstract

In this article we prove the global existence of a unique strong solution to the initial boundary-value problem for a fourth-order exponential PDE. The equation we study was originally proposed to study the evolution of crystal surfaces, and was derived by applying a nonstandard scaling regime to a microscopic Markov jump process with Metropolis rates. Our investigation here finds that compared to the PDEs which use Arrhenius rates (and also have a fourth-order exponential nonlinearity), the hyperbolic sine nonlinearity in our equation can offer much better control over the exponent term even in high dimensions.

Keywords

crystal surface models, exponential nonlinearity, existence, nonlinear fourth-order parabolic equations

2010 Mathematics Subject Classification

35A01, 35D30, 35Q99

Full Text (PDF format)

Received 11 October 2021

Received revised 25 August 2022

Accepted 28 August 2022

Published 24 March 2023