Properties of steady states for thin film equations

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


We consider nonnegative steady-state solutions of the evolution equation

ht = -(f(h) hxxx)x - (g(h) hx)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.