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

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.

Show me the PDF file (1265K), TEX file, and other files for this article.

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

Return to the EJDE web page