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Convergence to steady states of solutions of the Cahn-Hilliard and Caginalp equations with dynamic boundary conditions. (English) Zbl 1107.35058
Summary: We consider a solution of the Cahn-Hilliard equation or an associated Caginalp problem with dynamic boundary condition in the case of a general potential and prove that under some conditions on the potential it converges, as \(t\to \infty\), to a stationary solution. The main tool will be the Łojasiewicz-Simon inequality for the underlying energy functional.

MSC:
35K35 Initial-boundary value problems for higher-order parabolic equations
35B35 Stability in context of PDEs
34D05 Asymptotic properties of solutions to ordinary differential equations
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References:
[1] Bates, Phys. D 43 pp 335– (1990)
[2] Caginalp, Arch. Ration. Mech. Anal. 92 pp 205– (1986)
[3] Cahn, J. Chem. Phys. 28 pp 258– (1958)
[4] Cahn, J. Stat. Phys. 77 pp 183– (1994)
[5] Chill, J. Funct. Anal. 201 pp 572– (2003)
[6] , and , -Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type, Memoirs of the American Mathematical Society Vol. 166 (Amer. Math. Soc., Providence, RI, 2003).
[7] , and , Optimal Lp –Lq -regularity for vector-valued parabolic problems with inhomogeneous boundary data, submitted (2003).
[8] The Cahn–Hilliard model for the kinetics of phase transitions, in: Mathematical Models for Phase Change Problems, edited by J. F. Rodrigues, International Series of Numerical Mathematics Vol. 88 (Birkhäuser Verlag, Basel, 1989), pp. 35–73.
[9] Elliott, Arch. Ration. Mech. Anal. 96 pp 339– (1986)
[10] Escher, Math. Nachr. 257 pp 3– (2003)
[11] Hoffmann, Comm. Partial Differential Equations 24 pp 1055– (1999)
[12] Kenzler, Computer Phys. Comm. 133 pp 139– (2001)
[13] Ensembles semi-analytiques, I. H. E. S. Bures-sur-Yvette, preprint (1965).
[14] Novick–Cohen, Adv. Math. Sci. Appl. 8 pp 965– (1998)
[15] Prüss, Math. Bohem. 127 pp 311– (2002)
[16] Prüss, Conf. Semin. Mat. Univ. Bari 285 pp 1– (2003)
[17] , and , Global well-posedness and asymptotics for the Cahn–Hilliard equation with dynamic boundary conditions, Ann. Mat. Pura Appl. (4), to appear.
[18] Racke, Adv. Differential Equations 8 pp 83– (2003)
[19] Simon, Ann. of Math. (2) 118 pp 525– (1983)
[20] Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematics Sciences (Springer-Verlag, New York – Berlin – Heidelberg, 1988).
[21] Wu, J. Differential Equations 204 pp 511– (2004)
[22] Nonlinear Functional Analysis and Its Applications. I (Springer-Verlag, New York – Berlin – Heidelberg, 1990).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.