  An approach to some "optimal" (more precisely, non-improvable) regularity of
solutions of the thin film equation u_{t} = -\nabla \cdot(|u|^{n} \nabla \D u)
in \ren \times \re_+, u(x,0)=u_0(x) in \re^N, where n in (0,2) is a fixed
exponent, with smooth compactly supported initial data u_0(x), in dimensions $N
\geq 2$ is discussed. Namely, a precise exponent for the H\"older continuity
with respect to the spatial radial variable $|x|$ is obtained by construction
of a Graveleau-type focusing self-similar solution. As a consequence, optimal
regularity of the gradient $\nabla u$ in certain $L^p$ spaces, as well as a
H\"older continuity property of solutions with respect to x and t, are derived,
which cannot be obtained by classic standard methods of integral
identities-inequalities. Several profiles for the solutions in the cases n=0
and n>0 are also plotted.
  In general, we claim that, even for arbitrarily small n>0 and positive
analytic initial data u_0(x), the solutions u(x,t) cannot be better than
$C_x^{2-\e}$-smooth, where $\e(n)=O(n)$ as $n \to 0$.
