Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 95, pp. 1-29.
Title: Heteroclinic orbits, mobility parameters and stability
for thin film type equations
Authors: Richard. S. Laugesen (Univ.Illinois, Urbana, IL, USA)
Mary C. Pugh (Univ. of Toronto, ON, Canada)
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.
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.