TY - JOUR

T1 - Energy levels of steady states for thin-film-type equations

AU - Laugesen, R. S.

AU - Pugh, M. C.

N1 - Funding Information:
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. Some of 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.
Funding Information:
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.

PY - 2002/7/1

Y1 - 2002/7/1

N2 - We study the phase space of the evolution equation ht = -(f(h)hxxx)x - (g(h)hx)x by means of a dissipated energy. Here h(x, t) ≥0, and at h = 0 the coefficient functions f > 0 and g can either degenerate to 0, or blow up to ∞, or tend to a nonzero constant. We first show that all positive periodic steady-states are saddles in the energy landscape, with respect to zero-mean perturbations, if (g/f)″ ≥ 0 or if the perturbations are allowed to have period longer than that of the steady-state. For power-law coefficients (f(y) = yn and g(y) = ℬym for some ℬ > 0) we analytically determine the relative energy levels of distinct steady-states. For example, with m - n ∈ [1, 2) and for suitable choices of the period and mean value, we find three fundamentally different steady-states. The first is a constant steady-state that is stable and is a local minimum of the energy. The second is a positive periodic steady-state that is linearly unstable and has higher energy than the constant steady-state; it is a saddle point for the energy. The third is a periodic collection of 'droplet' (compactly supported) steady-states having lower energy than either the positive steady-state or the constant one. Since the energy must decrease along every orbit, these results significantly constrain the dynamics of the evolution equation.

AB - We study the phase space of the evolution equation ht = -(f(h)hxxx)x - (g(h)hx)x by means of a dissipated energy. Here h(x, t) ≥0, and at h = 0 the coefficient functions f > 0 and g can either degenerate to 0, or blow up to ∞, or tend to a nonzero constant. We first show that all positive periodic steady-states are saddles in the energy landscape, with respect to zero-mean perturbations, if (g/f)″ ≥ 0 or if the perturbations are allowed to have period longer than that of the steady-state. For power-law coefficients (f(y) = yn and g(y) = ℬym for some ℬ > 0) we analytically determine the relative energy levels of distinct steady-states. For example, with m - n ∈ [1, 2) and for suitable choices of the period and mean value, we find three fundamentally different steady-states. The first is a constant steady-state that is stable and is a local minimum of the energy. The second is a positive periodic steady-state that is linearly unstable and has higher energy than the constant steady-state; it is a saddle point for the energy. The third is a periodic collection of 'droplet' (compactly supported) steady-states having lower energy than either the positive steady-state or the constant one. Since the energy must decrease along every orbit, these results significantly constrain the dynamics of the evolution equation.

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U2 - 10.1006/jdeq.2001.4108

DO - 10.1006/jdeq.2001.4108

M3 - Article

AN - SCOPUS:0036647446

SN - 0022-0396

VL - 182

SP - 377

EP - 415

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 2

ER -