Linear Stability of Steady States for Thin Film and Cahn-Hilliard Type Equations |
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Authors: | R S Laugesen and M C Pugh |
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Institution: | (1) Dept. of Mathematics?Univ. of Illinois?Urbana?IL 61801 ?e-mail: laugesen@math.uiuc.edu, US;(2) Dept. of Mathematics?Univ. of Pennsylvania?Philadelphia?PA 19104?e-mail: mpugh@math.upenn.edu, US |
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Abstract: | We study the linear stability of smooth steady states of the evolution equation
under both periodic and Neumann boundary conditions. If a≠ 0 we assume f≡ 1. In particular we consider positive periodic steady states of thin film equations, where a=0 and f, g might have degeneracies such as f(0)=0 as well as singularities like g(0)=+∞.
If a≤ 0, we prove each periodic steady state is linearly
unstable with respect to volume (area) preserving perturbations whose period is an integer multiple of the steady state's
period. For area-preserving perturbations having the same period as the steady state, we prove linear instability for all a if the ratio g/f is a convex function. Analogous results hold for Neumann boundary conditions.
The rest of the paper concerns the special case of a=0 and power-law coefficients f(y)=y
n
and g(y)=ℬy
m
. We characterize the linear stability of each positive periodic steady state under perturbations of the same period. For
steady states that do not have a linearly unstable direction, we find all neutral directions. Surprisingly, our instability
results imply a nonexistence result: there is a large range of exponents m and n for which there cannot be two positive periodic steady states with the same period and volume.
Accepted October 1, 1999?Published online July 12, 2000 |
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Keywords: | |
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