Energy Levels of Steady States for Thin Film Type 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 ∞, 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 steady state. For power law coefficients (f(y) = y and g(y) = Bym for some B > 0) we analytically determine the relative energy levels of distinct steady states. For example, with m−n ∈ [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.