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

Abstract

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

$h_t = - (f(h)_{xxx} )_x - (g(h)h_x )_x - ah$

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.