Energy Levels of Steady States for Thin-Film-Type Equations

Authors: Laugesen R.S.¹; Pugh M.C.²

Source: Journal of Differential Equations, Volume 182, Number 2, July 2002 , pp. 377-415(39)

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Abstract:

We study the phase space of the evolution equationh_t=-(f(h)h_xxx)_x-(g(h)h_x)_xby means of a dissipated energy. Here h(x,t) ges 0, and at h=0 the coefficient functions f>0 and g can either degenerate to 0, or blow up to infin , 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) Prime ges 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)= Bscr y^m for some Bscr >0) we analytically determine the relative energy levels of distinct steady-states. For example, with m-n isin [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. © 2002 Elsevier Science (USA).

Language: English

Document Type: Research article

Affiliations: 1: Department of Mathematics, University of Illinois, Urbana, Illinois, 61801 2: Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 3G3, Canada

Publication date: 2002-07-01