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

* with R. S. Laugesen *
Electronic Journal of Differential Equations, 2002(2002)95:1-29

### Abstract

We study numerically 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* is a real
number, and the Bond number *B* are given.

For example, 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, while 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.

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the article.

Laugesen was partially supported by NSF grant number DMS-9970228, by a
grant from the University of Illinois Research Board, and by a
fellowship from the University of Illinois Center for Advanced Study.
He is grateful for the hospitality of the Department of Mathematics at
Washington University in St. Louis.

Pugh was partially supported by NSF grant number DMS-9971392, by the
MRSEC Program of the NSF under Award Number DMR-9808595, by the ASCI
Flash Center at the University of Chicago under DOE contract B341495,
and by an Alfred P. Sloan fellowship. The computations were done using
a network of workstations paid for by an NSF SCREMS grant,
DMS-9872029. Part of the research was conducted while enjoying the
hospitality of the Mathematics Department and the James Franck
Institute of the University of Chicago.

Pugh thanks Todd Dupont and Bastiaan Braams for illuminating
conversations regarding numerical issues.