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

Abstract

We study the phase space of the evolution equation $$ h_t = -(h^n h_{xxx})_x - mathcal{B} (h^m h_x)_x , $$ where $h(x,t) geq 0$. The parameters $n>0$, $m in mathbb{R}$, and the Bond number $mathcal{B}>0$ 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, $h^n$ 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.

Keywords

• Nonlinear PDE of parabolic type
• heteroclinic orbits
• stability problems
• lubrication theory.