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Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects

Yuan Gao ORCID logo , Xin Yang Lu EMAIL logo and Chong Wang

Abstract

We study the following parabolic nonlocal 4-th order degenerate equation:

ut=-[2πH(ux)+ln(uxx+a)+32(uxx+a)2]xx,

arising from the epitaxial growth on crystalline materials. Here H denotes the Hilbert transform, and a>0 is a given parameter. By relying on the theory of gradient flows, we first prove the global existence of a variational inequality solution with a general initial datum. Furthermore, to obtain a global strong solution, the main difficulty is the singularity of the logarithmic term when uxx+a approaches zero. Thus we show that, if the initial datum u0 is such that (u0)xx+a is uniformly bounded away from zero, then such property is preserved for all positive times. Finally, we will prove several higher regularity results for this global strong solution. These finer properties provide a rigorous justification for the global-in-time monotone solution to the epitaxial growth model with nonlocal elastic effects on vicinal surface.

MSC 2010: 35K65; 35R06; 49J40

Communicated by Irene Fonseca


Funding statement: The second author is very grateful for the support of NSERC Discovery Grants and acknowledges the support of his Lakehead University internal funding.

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Received: 2020-04-06
Revised: 2021-02-28
Accepted: 2021-03-02
Published Online: 2021-04-16

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