  In any number of space variables, we study the Cauchy problem related to the
thin-film equation in the simplest case of a linearly degenerate mobility. This
equation, derived from a lubrication approximation, also models the surface
tension dominated flow of a thin viscous film in the Hele-Shaw cell. Our focus
is on uniqueness of weak solutions in the "complete wetting" regime, when a
zero contact angle between liquid and solid is imposed. In this case, we
transform the problem by zooming into the free boundary and look at small
Lipschitz perturbations of a quadratic stationary solution. In the
perturbational setting, the main difficulty is to construct scale invariant
function spaces in such a way that they are compatible with the structure of
the nonlinear equation. Here we rely on the theory of singular integrals in
spaces of homogeneous type to obtain linear estimates in these functions spaces
which provide "optimal" conditions on the initial data under which a unique
solution exists. In fact, this solution can be used to define a class of
functions in which the original initial value problem has a unique (weak)
solution. Moreover, we show that the interface between empty and wetted regions
is an analytic hypersurface in time and space.
