  We study a class of fourth order nonlinear parabolic equations which include
the thin-film equation and the quantum drift-diffusion model as special cases.
We investigate these equations by first developing functional inequalities of
the type $ \int_\Omega u^{2\gamma-\alpha-\beta}\Delta u^\alpha\Delta u^\beta dx
\geq c\int_\Omega|\Delta u^\gamma |^2dx $, which seem to be of interest on
their own right.
