Heteroclinic orbits, mobility parameters and stability for thin film type equations

We study the phase space of the evolution equation h_t = -(hⁿh_xxx)_x-B(h^mh_x) _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 h^m. 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.