  A nonlinear parabolic equation of the fourth order is analyzed. The equation
is characterized by a mobility coefficient that degenerates at 0. Existence of
at least one weak solution is proved by using a regularization procedure and
deducing suitable a-priori estimates. If a viscosity term is added and
additional conditions on the nonlinear terms are assumed, then it is proved
that any weak solution becomes instantaneously strictly positive. This in
particular implies uniqueness for strictly positive times and further
time-regularization properties. The long-time behavior of the problem is also
investigated and the existence of trajectory attractors and, under more
restrictive conditions, of strong global attractors is shown.
