Linear Stability of Steady States for Thin Film and Cahn-Hilliard Type Equations

with R. S. Laugesen, Arch. Ration. Mech. Anal. 154(2000)1:3-51.

Abstract

We study the linear stability of smooth steady states of the evolution equation

h_t = -(f(h) h_xxx)_x - (g(h) h_x)_x

under both periodic and Neumann boundary conditions. If a is nonzero 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)=infinity.

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)=\B y^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: for a large range of exponents m and n there cannot be two positive periodic steady states with the same period and volume.

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Laugesen was partially supported by NSF grant number DMS-9970228.
Pugh was partially supported by NSF grant number DMS-9971392, by the MRSEC Program of the NSF under Award Number DMR-9808595, and by the ASCI Flash Center at the University of Chicago under DOE contract B341495.