New dissipated energy and partial regularity for thin film equations

We study the thin film equation h t = − ( h n h x x x ) x , and prove that the energy ∫ h − p h x 2 d x dissipates for suitable weak solutions when ( n , p ) lies in the whole domain E 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.