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Uniqueness for the thin-film equation with a Dirac mass as initial data


Authors: Mohamed Majdoub, Nader Masmoudi and Slim Tayachi
Journal: Proc. Amer. Math. Soc. 146 (2018), 2623-2635
MSC (2010): Primary 74K35, 76A20, 35K65, 35K25, 35A02, 28D20, 35C06
DOI: https://doi.org/10.1090/proc/13935
Published electronically: February 14, 2018
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Abstract: We show the uniqueness of strong solutions for the thin-film equation $ u_t + (u u_{xxx})_x =0$ with initial data $ u(0)=m\delta ,\; m>0$, where $ \delta $ is the Dirac mass at the origin. In particular, the solution is the source type one obtained by Smyth and Hill. The argument is based on an entropy estimate for the equation in self-similar variables.


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Additional Information

Mohamed Majdoub
Affiliation: Department of Mathematics, Imam Abdulrahman Bin Faisal University, College of Science, Dammam, Kingdom of Saudi Arabia
Email: mmajdoub@iau.edu.sa

Nader Masmoudi
Affiliation: The Courant Institute for Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012-1185
Email: masmoudi@courant.nyu.edu

Slim Tayachi
Affiliation: Département de mathématiques, Université de Tunis El Manar, Faculté des Sciences de Tunis, Laboratoire équations aux dérivées partielles (LR03ES04), 2092 Tunis, Tunisie
Email: slim.tayachi@fst.rnu.tn

DOI: https://doi.org/10.1090/proc/13935
Keywords: Thin-film equation, zero contact angle, uniqueness, self-similar variables, entropy estimates.
Received by editor(s): July 29, 2016
Received by editor(s) in revised form: August 5, 2017
Published electronically: February 14, 2018
Communicated by: Catherine Sulem
Article copyright: © Copyright 2018 American Mathematical Society

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