European Journal of Applied Mathematics

Table of Contents - August 2012 - Volume 23, Issue 04  

Papers

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

MARINA CHUGUNOVAa1, JOHN R. KINGa2 and ROMAN M. TARANETSa2

a1 School of Mathematical Sciences, Claremont Graduate University, 710 N. College Avenue, Claremont, CA 91711, USA email: chugunovamar@yahoo.ca

a2 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK email: john.king@nottingham.ac.uk, taranets_r@yahoo.com

Abstract

The main result of this paper is the proof of uniqueness of non-negative entropy solutions of the thin film equation ht + (|h|n hxxx)x = 0 for $\frac{7}{4}$ < 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.

(Received August 30 2011)

(Revised March 26 2012)

(Accepted March 27 2012)

(Online publication April 16 2012)

Key words:

  • Thin film equation;
  • Uniqueness;
  • Entropy solutions