Bulletin of the Australian Mathematical Society
- Bulletin of the Australian Mathematical Society / Volume 69 / Issue 01 / February 2004, pp 35-48
- Copyright © Australian Mathematical Society 2004
- DOI: http://dx.doi.org/10.1017/S0004972700034249 (About DOI), Published online: 17 April 2009
Research Article
A fourth-order parabolic equation with a logarithmic nonlinearlity
Ahmed Bonfoha1
a1 University Kuala Lumpur, Malaysia France Institute, Sec. 14, Jln Teras Jernang, 43650 Bdr Baru Bangi, Selangor D.E., Malaysia
We consider some generalisations of the Cahn—Hilliard equation based on constitutive equations derived by M. Gurtin in (1996) with a logarithmic free energy. Compared to the classical Cahn—Hilliard equation (see [4, 5]), these models take into account the work of internal microforces and the anisotropy of the material. We obtain the existence and uniqueness of solutions results and then prove the existence of finite dimensional attractors.
(Received May 07 2003)
- [1] Barrett, J. and Blowey, F., ‘Finite element approximation of the Cahn—Hilliard equation with concentration dependent mobility’, Math. Comp. 68 1996, 487–517. [OpenURL Query Data] [CrossRef] [Google Scholar]
- [2] Bonfoh, A., ‘Some Cahn—Hilliard—Gurtin models with a logarithmic potential’, Appl. Math. Lett. (to appear). [OpenURL Query Data] [Google Scholar]
- [3] Bonfoh, A. and Miranville, A., ‘On Cahn—Hilliard—Gurtin equations’, (Proceedings of the 3rd World Congress of Nonlinear Analysts), Nonlin. Anal. 47 2001, 3455–3466. [OpenURL Query Data] [CrossRef] [Google Scholar]
- [4] Cahn, J.W., ‘On spinodal decomposition’, Acta Metall. 9 1961, 795–801. [OpenURL Query Data] [CrossRef] [Google Scholar]
- [5] Cahn, J.W. and Hilliard, J.E., ‘Free energy of a non-uniform system I. Interfacial free energy’, J. Chem. Phys. 2 1958, 258–267. [OpenURL Query Data] [CrossRef] [Google Scholar]
- [6] Cherfils, L. and Miranville, A., ‘Generalized Cahn—Hilliard equations with a logarithmic free energy’, Rev. Real Acad. Cienc. Exactas Fís. Nat. (Esp.) 94 2000, 19–32. [OpenURL Query Data] [Google Scholar]
- [7] Debussche, A. and Dettori, L., ‘On the Cahn—Hilliard equation with a logarithmic free energy’, Nonlinear Anal. 24 1995, 1491–1514. [OpenURL Query Data] [CrossRef] [MathSciNet] [Google Scholar]
- [8] Eden, A., Foias, C., Nicolaenko, B. and Temam, R., Exponential attractors for dissipative evolution equations (Masson, Paris, 1994). [Google Scholar]
- [9] Elliott, C.M. and Garcke, H., ‘On the Cahn—Hilliard equation with degenerate mobility’, SIAM J. Math. Anal. 27 1996, 404–423. [OpenURL Query Data] [CrossRef] [MathSciNet] [Google Scholar]
- [10] Gurtin, M., ‘Generalized Ginzburg—Landau and Cahn—Hilliard equations based on a microforce balance’, Phys. D 92 1996, 178–192. [OpenURL Query Data] [CrossRef] [MathSciNet] [Google Scholar]
- [11] Lions, J.L., Quelques méthodes de résolutions des problèmes aux limites non linéaires (Dunod, Paris, 1969). [Google Scholar]
- [12] Miranville, A., ‘Some generalizations of the Cahn—Hilliard equation’, Asymptotic Anal. 22 2000, 235–259. [OpenURL Query Data] [Google Scholar]
- [13] Temam, R., Infinite dimensional dynamical systems in mechanics and physics, (2nd Edition) (Springer-Verlag, Berlin, Heidelberg, New York, 1997). [Google Scholar]
