Linear stability of steady states for thin film and Cahn-Hilliard type equations
The authors consider equations of the form \[ h_t = -(f(h)h_{xxx})_x -(g(h) h_x)_x -ah, \] where \(a \in \mathbb{R},a\leq 0\), with either periodic or Neumann boundary conditions, and discuss linear stability of stationary solutions for various forms of the functions \(f\) and \(g\) (if \(a \neq 0\), \(f(h) \equiv 1\)). Equations of this form appear in many situations involving thin film flow, but have also been implicated in population dynamics models. To derive the results in this paper, the authors linearize around a steady state and rewrite the corresponding eigenvalue problem in a selfadjoint form. This allows them to define a Rayleigh quotient and then to decide the stability issue by a judicious choice of trial functions (usually derivatives of the steady state). A lot of information is also obtained by a careful analysis of the second-order nonlinear oscillator equation satisfied by the steady states in the case \(a=0\). The methods also allow the authors to derive in some situations uniqueness results. We just mention two from the vast array of results proved in the paper. Theorem 3: If \(a \leq 0\) and \((f/g)'' \geq 0\), then any periodic steady state is linearly unstable to perturbations having the same period. This theorem covers many important cases of thin film flow. Theorem 12: If \(a=0\), \(f(h)=h^n\) and \(g(h)=b h^m\), \(b > 0\), then if \(m<n\) and \(m \geq n+1\), there exists at most one (up to translation) positive periodic steady state having a given period and area.
scientific article; zbMATH DE number 1529605