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Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions. (English) Zbl 1068.35018
The paper contains the proof of the result announced in [Gakuto Int. Series; Math. Sci. Appl. 20, 382–390 (2004; Zbl 1068.35052)]. The initial boundary value problem is considered \[ \begin{aligned} &\frac{\partial u}{\partial t}=\Delta\mu,\quad \mu=-\Delta u-u+u^3\quad t>0,\;x\in\Omega\\ &\frac{\partial \mu}{\partial \nu}=0,\quad \frac 1{\Gamma_s}\frac{\partial u}{\partial t}=\sigma_s\Delta_{\|}\,u-\frac{\partial u}{\partial \nu}+h_s-g_s u\quad t>0,\;x\in\partial\Omega\tag{1}\\ &u(x,0)=u_0(x), \quad x\in\Omega \end{aligned} \] where \(\Omega\) is a bounded domain from \(\mathbb R^n\), \(n\leq 3\), \(\Gamma_s>0,\;\sigma_s>0,\;g_s>0\), \(h_s\) are given constants, \(\Delta_{\|}\) is the tangential Laplacian operator, \(\nu\) is the outward normal to \(\partial\Omega\).
It is proved that \(\lim_{t\to\infty}\| u(\cdot,t)-\psi\|_{H^3(\Omega)}=0\) where \(\psi\) is an equilibrium to the problem (1) \[ \begin{aligned} & -\Delta \psi-\psi+\psi^3=\text{const}\quad\;x\in\Omega\\ &\sigma_s\Delta_{\|}\,\psi-\frac{\partial \psi}{\partial \nu}+h_s-g_s \psi=0\quad t>0,\;x\in\partial\Omega\\ &\int_{\Omega}\psi(x)\,dx=\int_{\Omega}u_0(x)\,dx. \end{aligned} \]

MSC:
35B40 Asymptotic behavior of solutions to PDEs
37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
74N20 Dynamics of phase boundaries in solids
35K35 Initial-boundary value problems for higher-order parabolic equations
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[1] Cahn, J.W.; Hilliard, E., Free energy of a nonuniform system, I, interfacial free energy, J. chem. phys., 28, 258-367, (1958)
[2] Dafermos, C., Asymptotic behavior of solutions of evolution equations, (), 103-124
[3] Elliott, C.M.; Zheng, S., On the cahn – hilliard equation, Arch. rational mech. anal., 96, 339-357, (1986) · Zbl 0624.35048
[4] Escher, J., Quasilinear parabolic systems with dynamical boundary conditions, Comm. partial differential equations, 18, 7&8, 1309-1364, (1993) · Zbl 0816.35059
[5] Escher, J., On quasilinear fully parabolic boundary value problems, Differential integral equations, 7, 5, 1325-1343, (1994) · Zbl 0815.35049
[6] Escher, J., On the qualitative behavior of some semilinear parabolic problems, Differential integral equations, 8, 247-267, (1995) · Zbl 0814.35053
[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, 351-370, (1995) · Zbl 0828.34007
[8] Henry, D., Geometric theory of semilinear parabolic equations, Lectures notes in mathematics, Vol. 840, (1981), Springer Berlin · Zbl 0456.35001
[9] Hintermann, T., Evolution equations with dynamic boundary conditions, Proc. roy. soc. Edinburgh (section A), 113, 43-65, (1989) · Zbl 0699.35045
[10] Huang, S.Z.; Tak\(ác̆\), P., Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear anal., 46, 675-698, (2001) · Zbl 1002.35022
[11] Jendoubi, M.A., A simple unified approach to some convergence theorem of L. Simon, J. functional anal., 153, 187-202, (1998) · Zbl 0895.35012
[12] Jendoubi, M.A., Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. differential equations, 144, 302-312, (1998) · Zbl 0912.35028
[13] Kenzler, R.; Eurich, F.; Maass, P.; Rinn, B.; Schropp, J.; Bohl, E.; Dieterich, W., Phase separation in confined geometriessolving the cahn – hilliard equation with generic boundary conditions, Comput. phys. comm., 133, 139-157, (2001) · Zbl 0985.65114
[14] S. Lojasiewicz, Une propri\(é\)t\(é\) topologique des sous-ensembles analytiques re\(é\)s, Colloque Internationaux du C.N.R.S. #117, Les equations aux deriv\(é\)es parielles, 1963, pp. 87-89.
[15] Lojasiewicz, S., Sur la geometrie semi- et sous-analytique, Ann. inst. Fourier (Grenoble), 43, 1575-1595, (1963) · Zbl 0803.32002
[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, 221-227, (1978) · Zbl 0387.35008
[18] Nicolaenko, B.; Scheurer, B.; Temam, R., Some global dynamical properties of a class of pattern formation equations, Comm. partial differential equations, 14, 245-297, (1989) · Zbl 0691.35019
[19] Nirenberg, L., Topics in nonlinear functional analysis, (1974), Courant Institute of Mathematical Science New York · Zbl 0286.47037
[20] Novick-Cohen, A.; Peletier, A., Steady states of the one-dimensional cahn – hilliard equation, Proc. roy. soc. Edinburgh (section A), 123, 6, 1071-1098, (1993) · Zbl 0818.35127
[21] Novick-Cohen, A.; Segel, L.A., Nonlinear aspects of the cahn – hilliard equation, Physica D, 10, 277-298, (1984)
[22] Polac̆ik, P.; Pybakowski, K.P., Nonconvergent bounded trajectories in semilinear heat equations, J. differential equations, 124, 472-494, (1996) · Zbl 0845.35054
[23] Polac̆ik, P.; Simondon, F., Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains, J. differential equations, 186, 586-610, (2002) · Zbl 1024.35046
[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, 1-21, (2003)
[25] Racke, R.; Zheng, S., The cahn – hilliard equation with dynamical boundary conditions, Adv. differential equations, 8, 1, 83-110, (2003) · Zbl 1035.35050
[26] Rybka, P.; Hoffmann, K.H., Convergence of solutions to cahn – hilliard equation, Comm. partial differential equations, 24, 5&6, 1055-1077, (1999) · Zbl 0936.35032
[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, 525-571, (1983) · Zbl 0549.35071
[29] Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, Applied mathematical science, Vol. 68, (1988), Springer New York · Zbl 0662.35001
[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. · Zbl 0232.35053
[31] Zheng, S., Asymptotic behavior of solutions to the cahn – hilliard equation, Appl. anal., 23, 165-184, (1986) · Zbl 0582.34070
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