Heteroclinic Orbits, Mobility Parameters and Stability for Thin Film Type Equations

We study the phase space of the evolution equation ht = −(hhxxx)x − B(hhx)x, where h(x, t) ≥ 0. The parameters n > 0, m ∈ R, and the Bond number B > 0 are given. We find numerically, for some ranges of n and m, that perturbing the positive periodic steady state in a certain direction yields a solution that relaxes to the constant steady state. Meanwhile perturbing in the opposite direction yields a solution that appears to touch down or ‘rupture’ in finite time, apparently approaching a compactly supported ‘droplet’ steady state. We then investigate the structural stability of the evolution by changing the mobility coefficients, hn and hm. We find evidence that the above heteroclinic orbits between steady states are perturbed but not broken, when the mobilities are suitably changed. We also investigate touch–down singularities, in which the solution changes from being everywhere positive to being zero at isolated points in space. We find that changes in the mobility exponent n can affect the number of touch–down points per period, and affect whether these singularities occur in finite or infinite time.