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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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