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2007, Volume 19, Issue 4: 609-629. Doi: 10.3934/dcds.2007.19.609
This issue Previous Article Next Article An invariant set generated by the domain topology for parabolic semiflows with small diffusion

Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals

  • 1.

    Department of Applied Mathematics and Informations, Ryukoku University, Seta, Otsu, 520-2194

  • 2.

    Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu 520-2194

  • 3.

    Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, 520-2194

Received:  December 2006
Revised:  June 2007
Published online:  September 2007
Abstract / Introduction Related Papers Cited by
    Abstract
  • We investigate stationary solutions of the one-dimensional Cahn-Hilliard equation with the diffusion coefficient and the total mass of the density as two given parameters. We solve the equation completely in the whole parameter space by using the Jacobi elliptic functions and complete elliptic integrals. In addition to counting the stationary solutions, which was studied by Grinfeld and Novick-Cohen, we provide an exact expression of the solutions. We also illustrate global bifurcation diagrams together with the asymptotic behavior of the solutions as the diffusion coefficient vanishes.
    Mathematics Subject Classification: 34B15, 35C05, 35B32, 33E05.

    Citation:
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