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Relaxation to Equilibrium in the One-Dimensional Thin-Film Equation with Partial Wetting and Linear Mobility

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Abstract

We investigate the large time behavior of compactly supported smooth solutions for a one-dimensional thin-film equation with linear mobility in the regime of partial wetting. We show the stability of steady state solutions. Relaxation rates are obtained for initial data which are close to a steady state in a suitable sense. The proof uses the Lagrangian coordinates. Our method is to establish and exploit differential relations between the energy and the dissipation as well as some interpolation inequalities. Our result is different from earlier results because here we consider solutions with finite mass.

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Acknowledgements

The authors thank the reviewers for the careful reading of the manuscript and helpful comments. The work of N. M is supported by NSF Grant DMS-1716466 and by Tamkeen under the NYU Abu Dhabi Research Institute grant of the center SITE.

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Authors and Affiliations

  1. Department of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, P. O. Box 1982, Dammam, Saudi Arabia

    Mohamed Majdoub

  2. Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, 31441, Saudi Arabia

    Mohamed Majdoub

  3. NYUAD Research Institute, New York University Abu Dhabi, PO Box 129188, Abu Dhabi, United Arab Emirates

    Nader Masmoudi

  4. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Nader Masmoudi

  5. Département de mathématiques, Laboratoire équations aux dérivées partielles (LR03ES04), Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092, Tunis, Tunisia

    Slim Tayachi

Authors
  1. Mohamed Majdoub
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  2. Nader Masmoudi
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  3. Slim Tayachi
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Correspondence to Mohamed Majdoub.

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Communicated by C. Mouhot

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Majdoub, M., Masmoudi, N. & Tayachi, S. Relaxation to Equilibrium in the One-Dimensional Thin-Film Equation with Partial Wetting and Linear Mobility. Commun. Math. Phys. 385, 837–857 (2021). https://doi.org/10.1007/s00220-021-04111-0

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  • DOI: https://doi.org/10.1007/s00220-021-04111-0

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