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

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 $\frac{7}{4}$ < 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.