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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.