## Abstract

We study the phase space of the evolution equation h_{t} = -(h^{n}h_{xxx})_{x}-B(h^{m}h_{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, 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.

Original language | English (US) |
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Pages (from-to) | XCXCIII-XCXCIV |

Journal | Electronic Journal of Differential Equations |

Volume | 2002 |

State | Published - 2002 |

## Keywords

- Heteroclinic orbits
- Lubrication theory
- Nonlinear pde of parabolic type
- Stability problems

## ASJC Scopus subject areas

- Analysis