Doubly nonlinear thin-film equations in one space dimension

We consider a free-boundary problem for a class of fourth-order nonlinear parabolic equations which are degenerate both with respect to the unknown and to its third derivative. The problem is relevant in the description of the surface-tension driven spreading of a non-Newtonian liquid over a solid surface in the "complete wetting" regime. Relying solely on global and local energy estimates and on Bernis' inequalities, we prove existence of solutions to this problem, and obtain sharp upper bounds for the propagation of their support. A necessary condition for the occurrence of waiting-time phenomena is also derived.