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Uniqueness of the regular waiting-time type solution of the thin film equation

Published online by Cambridge University Press:  16 April 2012

MARINA CHUGUNOVA ,
JOHN R. KING  and
ROMAN M. TARANETS
MARINA CHUGUNOVA
Affiliation:
School of Mathematical Sciences, Claremont Graduate University, 710 N. College Avenue, Claremont, CA 91711, USA email: chugunovamar@yahoo.ca
JOHN R. KING
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK email: john.king@nottingham.ac.uk, taranets_r@yahoo.com
ROMAN M. TARANETS
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK email: john.king@nottingham.ac.uk, taranets_r@yahoo.com
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Abstract

The main result of this paper is the proof of uniqueness of non-negative entropy solutions of the thin film equation ht + (|h|nhxxx)x = 0 for < n < 4. The uniqueness proved under assumptions that the initial data satisfy a finite β-entropy condition for some negative enough exponent β and that the solution is locally monotone at the touchdown point. The new dissipated functional recently constructed by Laugesen (Commun. Pure Appl. Anal., 4(3):613–634, 2005) is used to prove an auxiliary energy equality, and then Grönwall's lemma leads to uniqueness.

Keywords

Thin film equationUniquenessEntropy solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 4 , August 2012 , pp. 537 - 554
Copyright
Copyright © Cambridge University Press 2012

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