  In the present work, we study well-posedness and regularity of the
multidimensional thin film equation with linear mobility in a neighborhood of
the self-similar Smyth--Hill solutions. To be more specific, we perform a von
Mises change of dependent and independent variables that transforms the thin
film free boundary problem into a parabolic equation on the unit ball. We show
that the transformed equation is well-posed and that solutions are smooth and
even analytic in time and angular direction. The latter entails the analyticity
of level sets of the original equation, and thus, in particular, of the free
boundary.
