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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
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
Received 11 October 2021
Received revised 25 August 2022
Accepted 28 August 2022
Published 24 March 2023