Electron. J. Diff. Eqns., Vol. 2002(2002), No. 95, pp. 129.
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
where
.
The parameters
,
,
and the Bond number
are given.
We find numerically, for some ranges of
and
, 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,
and
.
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 touchdown 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
can affect the number of touchdown 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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