Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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EXISTENCE AND LARGE TIME BEHAVIOR OF SOLUTIONS TO A FOURTH-ORDER DEGENERATE PARABOLIC EQUATION

  • LIANG, BO (SCHOOL OF SCIENCE DALIAN JIAOTONG UNIVERSITY) ;
  • WANG, MEISHAN (SCHOOL OF SCIENCE DALIAN JIAOTONG UNIVERSITY) ;
  • WANG, YING (SCHOOL OF SCIENCE DALIAN JIAOTONG UNIVERSITY)
  • Received : 2013.03.30
  • Published : 2015.07.31
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Abstract

The paper is devoted to studying a fourth-order degenerate parabolic equation, which arises in fluid, phase transformation and biology. Based on the existence and uniqueness of one semi-discrete problem, two types of approximate solutions are introduced. By establishing some necessary uniform estimates for those approximate solutions, the existence and uniqueness of the corresponding parabolic problem are obtained. Moreover, the long time asymptotic behavior is established by the entropy functional method.

Keywords

Acknowledgement

Supported by : National Natural Science Foundation of China

References

  1. R. A. Adams, Sobolev Space, Academic Press, New York, 1975.
  2. F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations 83 (1990), no. 1, 179-206. https://doi.org/10.1016/0022-0396(90)90074-Y
  3. J. M. Cahn and J. E. Hilliard, Free energy of a non-uniform system I. Interfacial free energy, J. Chem. Phys. 28 (1958), 258-367. https://doi.org/10.1063/1.1744102
  4. J. A. Carrillo and G. Toscani, Long-time asymptotics for strong solutions of the thin film equation, Commu. Math. Phys. 225 (2002), no. 3, 551-571. https://doi.org/10.1007/s002200100591
  5. D. Gilbarg and N. S. Trudinger, Elliptic Partial Different Equations of Second Order, Second ed., Springer-Verlag, New York, 1983.
  6. G. Grun, Degenerate parabolic differential equations of fourth order and a plasticity model with nonlocal hardening, Z. Anal. Anwendungen 14 (1995), no. 3, 541-574. https://doi.org/10.4171/ZAA/639
  7. B. Liang and S. Zheng, Existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth-order, J. Math. Anal. Appl. 348 (2008), no. 1, 234-243. https://doi.org/10.1016/j.jmaa.2008.07.022
  8. T. G. Myers, Thin films with high surface tension, SIAM Rev. 40 (1998), no. 3, 441-462. https://doi.org/10.1137/S003614459529284X
  9. J. Simon, Compact sets in the space $L^p$(0, T;B), Ann. Math. Pura. Appl. 146 (1987), 65-96.
  10. M. Xu and S. Zhou, Existence and uniqueness of weak solutions for a generalized thin film equation, Nonlinear Anal. 60 (2005), no. 4, 755-774. https://doi.org/10.1016/j.na.2004.01.013
  11. M. Xu and S. Zhou, Stability and regularity of weak solutions for a generalized thin film equation, J. Math. Anal. Appl 337 (2008), no. 1, 49-60. https://doi.org/10.1016/j.jmaa.2007.03.075