Energy levels of steady states for thin-film-type equations

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. 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 that all positive periodic steady-states are saddles in the energy landscape, with respect to zero-mean perturbations, 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) = yⁿ and g(y) = ℬy^m 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 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 for the energy. 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.