Energy Levels of Steady States for Thin-Film-Type Equations

We study the phase space of the evolution equationht=−(f(h)hxxx)x−(g(h)hx)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)= ym for some >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.