zbMATH — the first resource for mathematics

Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy. (English) Zbl 1121.35018
Summary: We investigate the asymptotic behavior of the nonlinear Cahn-Hilliard equation with a logarithmic free energy and similar singular free energies. We prove an existence and uniqueness result with the help of monotone operator methods, which differs from the known proofs based on approximation by smooth potentials. Moreover, we apply the Lojasiewicz-Simon inequality to show that each solution converges to a steady state as time tends to infinity.

MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K55 Nonlinear parabolic equations 35Q99 Partial differential equations of mathematical physics and other areas of application 47H05 Monotone operators and generalizations 47J35 Nonlinear evolution equations 80A22 Stefan problems, phase changes, etc.
Full Text:
References:
 [1] Bonfoh, A., A fourth-order parabolic equation with a logarithmic nonlinearity, Bull. austral. math. soc., 69, 35-48, (2004) · Zbl 1052.35074 [2] Brézis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans LES espaces de Hilbert, (), Notas de Matemática (50) · Zbl 0252.47055 [3] Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system. I. interfacial energy, J. chem. phys., 28, 2, 258-267, (2005) [4] Cazenave, T.; Haraux, A., (), Translated from the 1990 French original by Yvan Martel and revised by the authors [5] Chill, R., On the łojasiewicz – simon gradient inequality, J. funct. anal., 201, 2, 572-601, (2003) · Zbl 1036.26015 [6] Chill, R.; Fašangová, E.; Prüss, J., Convergence to steady states of solutions of the cahn – hilliard equation with dynamic boundary conditions, Math. nachr., 279, 13-14, 1448-1462, (2006) · Zbl 1107.35058 [7] Debussche, A.; Dettori, L., On the cahn – hilliard equation with a logarithmic free energy, Nonlinear anal., 24, 10, 1491-1514, (1995) · Zbl 0831.35088 [8] Dupaix, C., A singularly perturbed phase field model with a logarithmic nonlinearity: upper-semicontinuity of the attractor, Nonlinear anal., 41, 725-744, (2000), (5-6, Ser. A: Theory Methods) · Zbl 0960.35005 [9] C.M. Elliot, S. Luckhaus, A generalized equation for phase separation of a multi-component mixture with interfacial free energy, 1991. preprint SFB 256 Bonn No. 195 [10] Elliott, C.M.; Zheng, S., On the cahn – hilliard equation, Arch. ration. mech. anal., 96, 339-357, (1986) · Zbl 0624.35048 [11] Elliott, C.M.; Garcke, H., On the cahn – hilliard equation with degenerate mobility, SIAM J. math. anal., 27, 2, 404-423, (1996) · Zbl 0856.35071 [12] Garcke, H., On a cahn – hilliard model for phase separation with elastic misfit, Ann. inst. H. Poincaré anal. non linéaire, 22, 165-185, (2005) · Zbl 1072.35081 [13] Giga, Y., Analyticity of the semigroup generated by the Stokes operator in $$L_r$$ spaces, Math. Z., 178, 297-329, (1981) · Zbl 0473.35064 [14] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (), Reprint of the 1998 edition · Zbl 0691.35001 [15] Hoffmann, K.-H.; Rybka, P., Convergence of solutions to cahn – hilliard equation, Comm. partial differential equations, 24, 1055-1077, (1999) · Zbl 0936.35032 [16] Kenmochi, N.; Niezgódka, M.; Pawlow, I., Subdifferential operator approach to the cahn – hilliard equation with constraint, J. differential equations, 117, 320-354, (1995) · Zbl 0823.35073 [17] Miranville, A.; Zelik, S., Robust exponential attractors for cahn – hilliard type equations with singular potentials, Math. methods appl. sci., 27, 5, 545-582, (2004) · Zbl 1050.35113 [18] Prüss, J.; Racke, R.; Zheng, S., Maximal regularity and asymptotic behaviour of solutions for the cahn – hilliard equation with dynamic boundary conditions, Ann. mat. pura appl., 185, 4, 627-648, (2006) · Zbl 1232.35081 [19] Prüss, J.; Wilke, M., Maximal $$L_p$$-regularity and long-time behaviour of the non-isothermal cahn – hilliard equation with dynamic boundary conditions, (), 209-236 · Zbl 1109.35060 [20] Showalter, R.E., () [21] Wu, H.; Zheng, S., Convergence to equilibrium for the cahn – hilliard equation with dynamic boundary conditions, J. differential equations, 204, 511-531, (2004) · Zbl 1068.35018 [22] Zeidler, E., Nonlinear functional analysis and its applications. II/A, (1990), Springer-Verlag New York, Linear monotone operators, Translated from the German by the author and Leo F. Boron
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.