###
Properties of steady states for thin film equations

* with R. S. Laugesen, EJAM 11(2000)3:293-351.
*

### Abstract

We consider nonnegative steady-state solutions of the
evolution equation
*h*_{t} = -(*f*(*h*) *h*_{xxx})_{x} - (*g*(*h*) *h*_{x})_{x}.

Our class of coefficients *f*, *g* allows degeneracies at *h*=0, such as *f*(0)=0, as well as divergences like .
We first construct steady states and study their regularity. For *f*,*g* > 0 we construct
positive periodic steady states, and nonnegative steady states with either zero
or nonzero contact angles. For *f* > 0 and *g* < 0, we prove there are no nonconstant positive periodic steady states or steady states with zero
contact angle, but we do construct nonnegative steady states with
nonzero contact angle.

In considering the volume, length (or period) and contact angle of the
steady states, we find a rescaling identity that enables us to answer
questions such as whether a steady state is uniquely determined by its volume and contact angle. Our tools include an improved monotonicity result for the period function of the nonlinear oscillator.

We also relate the steady states
and their scaling properties to a recent blow-up conjecture of
Bertozzi and Pugh.

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Laugesen was supported by NSF grant number DMS--9622837. Pugh was
partially supported by an NSF post-doctoral fellowship and NSF grant
number DMS--9709128.