WIAS Preprint No. 1555, (2010)

Asymptotics for the spectrum of a thin film equation in a singular limit


  • Kitavtsev, Georgy
  • Recke, Lutz
  • Wagner, Barbara
    ORCID: 0000-0001-8306-3645

2010 Mathematics Subject Classification

  • 76D08 34E057 35B35 35P15 35P20


  • spectrum analysis, lubrication equation, asymptotic analysis




In this paper the linear stability properties of the steady states of a no-slip lubrication equation are studied. The steady states are configurations of droplets and arise during the late-phase dewetting process under the influence of both destabilizing van der Waals and stabilizing Born intermolecular forces, which in turn give rise to the minimum thickness $eps$ of the remaining film connecting the droplets. The goal of this paper is to give an asymptotic description of the eigenvalues and eigenfunctions of the problem, linearized about the one-droplet solutions, as $epsto 0$. For this purpose, corresponding asymptotic eigenvalue problems with piecewise constant coefficients are constructed, such that their eigenvalue asymptotics can be determined analytically. A comparison with numerically computed eigenvalues and eigenfunctions shows good agreement with the asymptotic results and the existence of a spectrum gap to a single exponentially small eigenvalue for sufficiently small $eps$.

Appeared in

  • SIAM J. Appl. Dyn. Syst., 11(4) (2012), pp. 1425--1457.

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