Electron. J. Diff. Eqns., Vol. 2002(2002), No. 95, pp. 1-29.

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

Richard. S. Laugesen & Mary C. Pugh

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

Submitted February, 28, 2002. Published November 5, 2002
Math Subject Classifications: 35K55, 37C29, 37L15, 76D08.
Key Words: Nonlinear PDE of parabolic type, heteroclinic orbits, stability problems, lubrication theory.

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Richard. S. Laugesen
Department of Mathematics
University of Illinois
Urbana, IL 61801, USA
email: laugesen@math.uiuc.edu
  Mary C. Pugh
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3, Canada
email: mpugh@math.toronto.edu

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