Global existence for a thin film equation with subcritical mass

Jian-Guo Liu Duke University Jinhuan Wang Liaoning University

Analysis of PDEs mathscidoc:1702.03013

discrete and continuous dynamical systems, 22, (4), 1461-1492, 2017.1
In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin lm equation in one-dimensional case ht + @x(h n @xxxh) + @x(h n+2@xh) = 0; where n  1. There exists a critical mass Mc = 2 p 6 3 found by Witelski etal. (2004 Euro. J. of Appl. Math. 15, 223{256) for n = 1. We obtain global existence of a non-negative entropy weak solution if initial mass is less than Mc. For n  4, entropy weak solutions are positive and unique. For n = 1,a nite time blow-up occurs for solutions with initial mass larger than Mc.For the Cauchy problem with n = 1 and initial mass less than Mc, we show that at least one of the following long-time behavior holds: the second moment goes to in nity as the time goes to in nity or h(; tk) * 0 in L1 (R) for some subsequence tk ! 1.
Long-wave instability, free-surface evolution, equilibrium, the Sz. Nagy inequality, long-time behavior.
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  title={Global existence for a thin film equation with subcritical mass},
  author={Jian-Guo Liu, and Jinhuan Wang},
  booktitle={discrete and continuous dynamical systems},
Jian-Guo Liu, and Jinhuan Wang. Global existence for a thin film equation with subcritical mass. 2017. Vol. 22. In discrete and continuous dynamical systems. pp.1461-1492.
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