Energy levels of steady states for thin film type equations
with R. S. Laugesen. Journal of Differential Equations
We study the phase space of the evolution equation
ht = -(f(h) hxxx)x - (g(h) hx)x
by means of a dissipated energy (a Liapunov function). Here h(x,t)
>= 0 , and at h=0 the coefficient functions f>0 and
g can either degenerate to 0, or blow up to infinity,
or tend to a nonzero constant.
We first show all positive periodic steady states are `energy
unstable' fixed points for the evolution (meaning the energy decreases
under some zero--mean perturbation) if (g/f)'' >= 0 or if the
perturbations are allowed to have period longer than that of the
For power law coefficients (f(y) = y^n and g(y) = B y^m
for some B > 0) we analytically determine the relative energy
levels of distinct steady states. For example, with 1 <= m-n < 2
and for suitable choices of the period and mean value, we
find three fundamentally different steady states. The first is a
constant steady state that is nonlinearly stable and is a local
minimum of the energy. The second is a positive periodic steady state
that is linearly unstable and has higher energy than the constant
steady state; it is a saddle point. The third is a periodic collection
of `droplet' (compactly supported) steady states having lower energy
than either the positive steady state or the constant one. Since the
energy must decrease along every orbit, these results significantly
constrain the dynamics of the evolution equation.
Our results suggest that heteroclinic connections could exist between
certain of the steady states, for example from the periodic steady
state to the droplet one. In a companion article we perform numerical
simulations to confirm their existence.
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The authors are grateful to Andrew Bernoff for stimulating comments on the energy landscape of phase space.
Laugesen was partially supported by NSF grant number DMS-9970228, and
a grant from the University of Illinois Research Board. He is grateful
for the hospitality of the Department of Mathematics at Washington
University in St. Louis.
Pugh was partially supported by NSF grant number DMS-9971392, by the
MRSEC Program of the NSF under Award Number DMR-9808595, by the ASCI
Flash Center at the University of Chicago under DOE contract B341495,
and by an Alfred P. Sloan fellowship. Some of the computations were
done using a network of workstations paid for by an NSF SCREMS grant,
DMS-9872029. Part of the research was conducted while enjoying the
hospitality of the Mathematics Department and the James Franck
Institute of the University of Chicago.