Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions

Journal of Differential Equations

Volumn 204, Issue 2, 2004, Pages 511-531

  Wu, Hao ^a   Zheng, Songmu ^a  

^a FUDAN UNIVERSITY   (China)

Author keywords

Cahn Hilliard equation; Dynamic boundary condition; Simon Lojasiewicz inequality

Indexed keywords

EID: 4344654674     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jde.2004.05.004     Document Type: Article

References (31)

1. Cahn J.W. Hilliard E. Free energy of a nonuniform system, I, Interfacial free energy J. Chem. Phys. 28 1958 258-367
2. Dafermos C. Asymptotic behavior of solutions of evolution equations in: Crandall M.G. (Ed.), Nonlinear Evolution Equations 1977 103-124 Academic Press New York
3. Elliott C.M. Zheng S. On the Cahn-Hilliard equation Arch. Rational Mech. Anal. 96 1986 339-357
4. Escher J. Quasilinear parabolic systems with dynamical boundary conditions Comm. Partial Differential Equations 18 7 & 8 1993 1309-1364
5. Escher J. On quasilinear fully parabolic boundary value problems Differential Integral Equations 7 5 1994 1325-1343
6. Escher J. On the qualitative behavior of some semilinear parabolic problems Differential Integral Equations 8 1995 247-267
7. Grinfeld M. Novick-Cohen A. Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments Proc. Roy. Soc. Edinburgh (Section A) 125 2 1995 351-370
8. Henry D. Geometric Theory of Semilinear Parabolic Equations Lectures Notes in Mathematics Vol. 840 1981 Springer Berlin
9. Hintermann T. Evolution equations with dynamic boundary conditions Proc. Roy. Soc. Edinburgh (Section A) 113 1989 43-65
10. Huang S.Z. Takác P. Convergence in gradient-like systems which are asymptotically autonomous and analytic Nonlinear Anal. 46 2001 675-698
11. Jendoubi M.A. A simple unified approach to some convergence theorem of L. Simon J. Functional Anal. 153 1998 187-202
12. Jendoubi M.A. Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity J. Differential Equations 144 1998 302-312
13. Kenzler R. Eurich F. Maass P. Rinn B. Schropp J. Bohl E. Dieterich W. Phase separation in confined geometries: solving the Cahn-Hilliard equation with generic boundary conditions Comput. Phys. Comm. 133 2001 139-157
14. S. Lojasiewicz, Une propriété topologique des sous-ensembles analytiques reés, Colloque Internationaux du C.N.R.S. #117, Les equations aux derivées parielles, 1963, pp. 87-89
15. Lojasiewicz S. Sur la geometrie semi- et sous-analytique Ann. Inst. Fourier (Grenoble) 43 1963 1575-1595
16. Lojasiewicz S. Ensemble semi-analytic 1965 IHES Bures-sur-Yvette
17. Matano H. Convergence of solutions of one-dimensional semilinear parabolic equations J. Math. Kyoto Univ. 18-2 1978 221-227
18. Nicolaenko B. Scheurer B. Temam R. Some global dynamical properties of a class of pattern formation equations Comm. Partial Differential Equations 14 1989 245-297
19. Nirenberg L. Topics in Nonlinear Functional Analysis 1974 Courant Institute of Mathematical Science New York
20. Novick-Cohen A. Peletier A. Steady states of the one-dimensional Cahn-Hilliard equation Proc. Roy. Soc. Edinburgh (Section A) 123 6 1993 1071-1098
21. Novick-Cohen A. Segel L.A. Nonlinear aspects of the Cahn-Hilliard equation Physica D 10 1984 277-298
22. Polacik P. Pybakowski K.P. Nonconvergent bounded trajectories in semilinear heat equations J. Differential Equations 124 1996 472-494
23. Polacik P. Simondon F. Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains J. Differential Equations 186 2002 586-610
24. Prüss J. Racke R. Zheng S. Maximal Regularity and Asymptotic Behavior of Solutions for the Cahn-Hilliard Equation with Dynamic Boundary Condition Konstanzer Schrift. Math. Inform. 189 2003 1-21
25. Racke R. Zheng S. The Cahn-Hilliard equation with dynamical boundary conditions Adv. Differential Equations 8 1 2003 83-110
26. Rybka P. Hoffmann K.H. Convergence of solutions to Cahn-Hilliard equation Comm. Partial Differential Equations 24 5 & 6 1999 1055-1077
27. Sell G. You Y. Dynamics of Evolutionary Equations 2001 Springer New York
28. Simon L. Asymptotics for a class of nonlinear evolution equation with applications to geometric problems Ann. Math. 118 1983 525-571
29. Temam R. Infinite-dimensional dynamical systems in mechanics and physics Applied Mathematical Science Vol. 68 1988 Springer New York
30. T.I. Zelenyak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differentsial'nye Uravneniya (1968) 17-22
31. Zheng S. Asymptotic behavior of solutions to the Cahn-Hilliard equation Appl. Anal. 23 1986 165-184