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(O) = 0, as well as divergences like g(0) = 00. We first construct steady states and study their regularity. For f, g > 0 we construct positive periodic steady states, and non-negative steady states with either zero or nonzero contact angles. For f > 0 and g < 0, we prove there are no non-constant positive periodic steady states or steady states with zero contact angle, but we do construct non-negative 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.
ASJC Scopus subject areas
- Applied Mathematics