  We study the phase space of the evolution equation h_t = -(f(h) h_{xxx})_x -
(g(h) h_x)_x by means of a dissipated energy (a Liapunov function). Here h(x,t)
is nonnegative, 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 is convex or if the perturbations are allowed to have
period longer than that of the steady state.
  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 m-n in [1,2) 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.
