Author:
LAUGESEN R. S.,PUGH M. C.
Abstract
We consider nonnegative steady-state solutions of the evolution equationformula hereOur class of coefficients f, g allows degeneracies at h = 0,
such as f(0) = 0, as well as divergences like g(0) = ±∞. 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.
Publisher
Cambridge University Press (CUP)
Cited by
63 articles.
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