Global existence and decay to equilibrium for some crystal surface models

Rafael Granero-Belinchón; Martina Magliocca

doi:10.3934/dcds.2019088

Article Contents

2019, Volume 39, Issue 4: 2101-2131. Doi: 10.3934/dcds.2019088

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Global existence and decay to equilibrium for some crystal surface models

Rafael Granero-Belinchón^1, and
Martina Magliocca^2,

1.
Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria. Avda. Los Castros s/n, Santander, Spain
2.
Dipartimento di Matematica, Università degli Studi Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy

Received: May 2018

Revised: September 2018

Published: January 2019

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Abstract

In this paper we study the large time behavior of the solutions to the following nonlinear fourth-order equations

$ \ \ \ \ \ {\partial _t} u = \Delta e^{-\Delta u}, \\ {\partial _t} u = -u^2\Delta ^2(u^3). $

These two PDE were proposed as models of the evolution of crystal surfaces by J. Krug, H.T. Dobbs, and S. Majaniemi (Z. Phys. B, 97,281-291, 1995) and H. Al Hajj Shehadeh, R. V. Kohn, and J. Weare (Phys. D, 240, 1771-1784, 2011), respectively. In particular, we find explicitly computable conditions on the size of the initial data (measured in terms of the norm in a critical space) guaranteeing the global existence and exponential decay to equilibrium in the Wiener algebra and in Sobolev spaces.

Keywords:

Crystal surface model,
nonlinear fourth-order parabolic equations,
global existence,
decay to equilibrium.

Mathematics Subject Classification: 35A01, 35B40, 35G25, 35K30, 35K55.

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References

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