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


We study numerically the phase space of the evolution equation

ht = -(h^n hxxx)x - B (h^m hx)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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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.