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On the Cahn-Hilliard equation. (English) Zbl 0624.35048
This paper is concerned with the Cahn-Hilliard equation which arises in the study of phase separation in cooling of binary solutions such as alloys, glasses and polymer mixtures. The initial boundary value problem for the Cahn-Hilliard equation is studied. Global existence and finite time blow up results are obtained. Error bounds for the finite element Galerkin approximation are proved.
When the solution globally exists, the asymptotic behavior of solution and the multiplicity of stationary solutions have been considered by the second author [Appl. Anal. 23, 165-184 (1986; Zbl 0613.35042)].

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35K35 Initial-boundary value problems for higher-order parabolic equations
35A35 Theoretical approximation in context of PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
Citations:
Zbl 0613.35042
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References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York (1975).
[2] J. W. Cahn & J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial fre
[3] J. Carr, M. E. Gurtin & M. Slemrod, Structural phase transitions on a finite interval, Ar · Zbl 0564.76075
[4] D. S. Cohen & J. D. Murray, A generalized diffusion model for growth and dispersal in a popu · Zbl 0474.92013
[5] J. Douglas, Jr., T. Dupont &am
[6] M. Hazewinkel, J. F. Kaashoek & B. Leynse, Pattern formation for a one dimensional evolution equation based on Thom’s river basin model, Report # 8519/B, Econometric Institute, Erasmus University (1985).
[7] S. Klainerman & G. Ponce, Global small amplitude solutions to nonlinear evolution equations, · Zbl 0509.35009
[8] J. L. Lions & E. Magenes, Non-homogeneous boundary value problems and applications, Vol. II, Springer-Verlag (1972).
[9] A. Novick-Cohen, Energy methods for the Cahn-Hilliard equation, IMA Preprint # 157, (1985). · Zbl 0685.35050
[10] A. Novick-Cohen & L. A. Segel, Nonlinear aspects of the Cahn-Hi
[11] G. I. Sivashinsky, On cellular instability in the solidification of a dilu
[12] V. Thomée, Some convergence results for Galerkin methods for parabolic boundary value problems, in Mathematical Aspects of Finite Element Methods in Partial Differential Equations, ed. C. de Boor, Academic Press (1974), p. 55–84.
[13] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, L. N. M. # 1054, Springer-Verlag, Berlin (1984).
[14] L. Wahlbin, On maximum norm error estimates for Galerkin approximations to one dimensional second order parabolic boundary value proble · Zbl 0295.65053
[15] M · Zbl 0232.35060
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