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Asymtotic behaviour of the viscous Cahn-Hilliard equation

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Analytical solutions for the viscous Cahn-Hilliard equation are considered. Existence and uniqueness of the solution are shown. The exponential decay of the solution inH 2-norm, which is an improvement of the result in Elliott and Zheng[5]. We also compare the early stages of evolution of the viscous Cahn-Hilliard equation with that of the Cahn-Hilliard equation, which has been given as an open question in Novick-Cohen[8].

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  1. 1.

    F. Bai, C. M. Elliott, A. Gardiner, A. Spence and A. M. Stuart,The viscous Cahn-Hilliard equation. Part I: computations, Nonlinearity 18 (1995), 131–160.

  2. 2.

    J. M. Ball,Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl. 42 (1973), 61–90.

  3. 3.

    J. W. Cahn and J. E. Hilliard,Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys. 28 (1958), 258–267.

  4. 4.

    C. M. Elliott and A. M. Stuart,Viscous Cahn-Hilliard equation II. analysis, J. Diff. Eq. 128 (1996), 387–414.

  5. 5.

    C. M. Elliott and S. Zheng,On the Cahn-Hilliard equation, Arch. Rat. Mech. Anal. 96 (1986), 339–357.

  6. 6.

    M. Grinfeld and A. Novick-Cohen,The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor, Trans. Amer, Math. Sci. 351 (1999), 2375–2406.

  7. 7.

    E. Jabbari and N. A. Peppas,A model for interdiffusion at interfaces of polymers with dissimilar physical properties, Polymer 36 (1995), 575–586.

  8. 8.

    A. Novick-Cohen,On the viscous Cahn-Hiliard equation, In Material Instabilities in Continuum and Related Mathematical Problems, (edited by J. M. Ball), Oxford Univ. Press, Oxford, 1988.

  9. 9.

    A. Novick-Cohen and R. L. Pego,Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc. 324 (1991), 331–351.

  10. 10.

    S. Puri and K. Binder,Phenomenological theory for the formation of interfaces via the interdiffusion of layers, Phy. Review B 44 (1991), 9735–9738.

  11. 11.

    L. G. Reyna and M. J. Ward,Metastable internal layer dynamics for the viscous Cahn-Hilliard equation, Methods Appl. Anal. 2 (1995), 285–306.

  12. 12.

    S. Q. Wang and Q. Shi,Interdiffusion in binary polymer mixtures, Macromolecules 26 (1993), 1091–1096.

  13. 13.

    J. Yin,On the existence of nonnegative continuous solutions of the Cahn-Hilliard equation, J. Diff. Eqs. 97 (1992), 310–327.

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The first author was supported by Korea Research Foundation under grant KRF-2001-002-D00036.

S. M. Choo received his BS and MS from Seoul National University. He earned his Ph.D. at Seoul National University under the direction of S.K. Chung. He has been at Ulsan University since September, 2001. His research interests is numerical analysis.

S. K. Chung received his BS from Seoul National University and MS from Sogang University. He earned his Ph.D. at The University of Texas ar Arlington under the supervision of R. Kannan. He has been at Seoul National University since 1987. His research interests is numerical analysis.

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Choo, S.M., Chung, S.K. Asymtotic behaviour of the viscous Cahn-Hilliard equation. JAMC 11, 143–154 (2003). https://doi.org/10.1007/BF02935727

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AMS Mathematics Subject Classification

  • 35K55
  • 35K57
  • 35B05

Key words and Phrases

  • Viscous Cahn-Hilliard equation
  • regularity
  • decay property
  • comparison theorem