* with A. Bertozzi, Nonlinearity, 7(1994)1535-1564.
*

We consider the effect of a second order `porous media' term on the evolution of weak solutions of the fourth order degenerate diffusion equation in one space dimension. The equation without the second order term is derived from a `lubrication approximation' and models surface tension dominated motion of thin viscous films and spreading droplets. Here

For simplicity we consider periodic boundary conditions which has the
physical interpretation of modeling a periodic array of droplets. In
a previous work we studied the above equation without the
second order `porous media' term. In particular we showed the
existence of nonnegative weak solutions with increasing support for
0<*n*<3 but the techniques failed for . This is consistent
with the fact that, in this case, nonnegative self-similar source-type
solutions do not exist for .

In this work,
we discuss a physical justification for the `porous media' term
when *n*=3 and 1<*m*<2.
We propose such behavior as a cut off of the singular
`disjoining pressure' modeling long range Van der Waals interactions.

For all *n*>0 and 1<*m*<2 we discuss possible
behavior at the edge of the support of the solution
via leading order asymptotic analysis
of traveling wave solutions.
This analysis predicts a certain `competition' between
the second and fourth order terms.
We present rigorous weak existence theory
for equation (1) for all *n*>0 and 1<*m*<2.
In particular, the presence of a second order `porous media' term in
equation (1) yields *nonnegative* weak solutions
that converge to their mean as and that
have additional regularity.
Moreover, we show that there exists a time *T ^{*}* after which the weak
solution is a

Finally, we present several numerical computations of solutions.
The simulations
use a weighted implicit-explicit scheme on a dynamically adaptive mesh.
The numerics suggest that the weak solution
described by our existence theory has
compact support with a finite speed of propagation.
The data confirms the local `power law'
behavior at the edge of the support predicted by
asymptotics.

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Some of the computations presented in this paper were performed at the
Advanced Computing Research Facility, Mathematics and Computer Science
Division, Argonne National Laboratory.

Both authors are supported by NSF postdoctoral fellowships. AB is
also partially supported by the Materials Research Laboratory at the
Univ. of Chicago and the DOE and MP by NSF grant DMS-9305996.