  We prove locally in time the existence of the unique smooth solution
(including smooth interface) to the multidimensional free boundary problem for
the thin film equation in the case of partial wetting. We also obtain the
Schauder estimates and solvability for the Dirichlet and the Neumann problem
for a linear degenerate parabolic equation of fourth order. The final expanded
version of this paper is available at AIMS Journals, Discrete and Continuous
Dynamical Systems - A at
http://aimsciences.org/article/doi/10.3934/dcds.2017156
