Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Short-time pattern formation in thin film equations

Pages: 867 - 885, Volume 23, Issue 3, March 2009      doi:10.3934/dcds.2009.23.867

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Hyung Ju Hwang - Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, South Korea (email)
Thomas P. Witelski - Mathematics Department, Duke University, Box 90320, Durham, NC 27708-0320, United States (email)

Abstract: We study the early stages of the nonlinear dynamics of pattern formation for unstable generalized thin film equations. For unstable constant steady states, we obtain rigorous estimates for the short- to intermediate-time nonlinear evolution which extends the mathematical characterization for pattern formation derived from linear analysis: formation of patterns can be bounded by the finitely many dominant growing eigenmodes from the initial perturbation.

Keywords:  Thin film equation, nonlinear stability, pattern formation
Mathematics Subject Classification:  35K25 35B35 76A20 35G25 35K55

Received: December 2007;      Revised: July 2008;      Available Online: November 2008.