New dissipated energy and partial regularity for thin film equations

We study the thin film equation ht=−(hnhxxx)xht=−(hnhxxx)x, and prove that the energy ∫h−phx2dx dissipates for suitable weak solutions when (n,p)(n,p) lies in the whole domain EE defined by Laugesen, which extends partially the corresponding results of Laugesen, Chugunova and Taranets, and then improve some previous results of partial regularity by Chou and the first author.