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Energy Levels of Steady States for Thin-Film-Type Equations
- Source :
-
Journal of Differential Equations . 2002, Vol. 182 Issue 2, p377. 39p. - Publication Year :
- 2002
-
Abstract
- We study the phase space of the evolution equation<f>ht=−(f(h)hxxx)x−(g(h)hx)x</f> by 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 ∞, 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)″&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)=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. [Copyright &y& Elsevier]
- Subjects :
- *PHASE space
*EVOLUTION equations
*ENERGY dissipation
*PERTURBATION theory
Subjects
Details
- Language :
- English
- ISSN :
- 00220396
- Volume :
- 182
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Journal of Differential Equations
- Publication Type :
- Academic Journal
- Accession number :
- 8507456
- Full Text :
- https://doi.org/10.1006/jdeq.2001.4108