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A note on parabolic equation with nonlinear dynamical boundary condition. (English) Zbl 1185.35131
The authors consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition related to the so-called Wentzell boundary condition. Existence and uniqueness of global solutions are proved which imply also the existence of a global attractor. Further, a suitable Łojasiewicz-Simon type inequality is derived in order to show the convergence of the global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms \(f,g\) are real analytic. Moreover, an estimate for the convergence rate is provided.

MSC:
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B41 Attractors
35B45 A priori estimates in context of PDEs
35K58 Semilinear parabolic equations
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[1] Wu, H., Convergence to equilibrium for the semilinear parabolic equation with dynamic boundary condition, Adv. math. sci. appl., 17, 1, 67-88, (2007) · Zbl 1145.35406
[2] Taira, K., ()
[3] Favini, A.; Goldstein, G.R.; Goldstein, J.A.; Romanelli, S., The heat equation with generalized Wentzell boundary condition, J. evol. equ., 2, 1-19, (2002) · Zbl 1043.35062
[4] Favini, A.; Goldstein, G.R.; Goldstein, J.A.; Romanelli, S., Nonlinear boundary conditions for nonlinear second order differential operators on \(C [0, 1]\), Arch. math. (basel), 76, 391-400, (2001) · Zbl 0990.34054
[5] Favini, A.; Goldstein, G.R.; Goldstein, J.A.; Romanelli, S., The heat equation with nonlinear general Wentzell boundary condition, Adv. differential equations, 11, 481-510, (2006) · Zbl 1149.35051
[6] 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, 139-157, (2001) · Zbl 0985.65114
[7] Gal, C.; Grasselli, M., The non-isothermal allen – cahn equation with dynamical boundary condition, Discrete contin. dyn. syst., 22, 4, 1009-1040, (2008) · Zbl 1160.35353
[8] Gal, C.; Grasselli, M., On the asymptotic behavior of the cagnalp system with dynamical boundary conditions, Comm. pure anal. appl., 8, 2, 689-710, (2009) · Zbl 1171.35337
[9] Gatti, S.; Miranville, A., Asymptotic behavior of a phase-field system with dynamic boundary conditions, (), 149-170 · Zbl 1123.35310
[10] Cherfils, L.; Miranville, A., On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. math., 54, 2, 89-115, (2009) · Zbl 1212.35012
[11] C. Gal, M. Grasselli, A. Miranville, Non-isothermal Allen-Cahn equations with coupled dynamic boundary conditions, preprint, 2008 · Zbl 1178.35076
[12] Gal, C., Global well-posedness for the non-isothermal cahn – hilliard with dynamical boundary conditions, Adv. differential equations, 12, 11, 1241-1274, (2007) · Zbl 1162.35386
[13] Gal, C., Well-posedness and long time behavior of the non-isothermal viscous cahn – hilliard equation with dynamic boundary conditions, Dyn. partial differ. equ., 5, 1, 39-67, (2008) · Zbl 1162.35043
[14] Gal, C.; Grasselli, M.; Miranville, A., Robust exponential attractors for singularly perturbed phase-field equations with dynamic boundary conditions, Nonlinear differential equations appl., 15, 4-5, 535-556, (2008) · Zbl 1180.35119
[15] Gal, C.; Miranville, A., Uniform global attractors for non-isothermal viscous and non-viscous cahn – hilliard equations with dynamic boundary conditions, Nonlinear anal. RWA, 10, 3, 1738-1766, (2009) · Zbl 1183.35052
[16] Miranville, A.; Zelik, S., Exponential attractors for the cahn – hilliard equation with dynamical boundary conditions, Math. methods appl. sci., 28, 709-735, (2005) · Zbl 1068.35020
[17] Racke, R.; Zheng, S., The cahn – hilliard equation with dynamical boundary conditions, Adv. difference equ., 8, 1, 83-110, (2003) · Zbl 1035.35050
[18] Haraux, A., Systèmes dynamiques dissipatifs et applications, (1991), Masson Paris · Zbl 0726.58001
[19] Polac˘ik, P.; Simondon, F., Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains, J. differential equations, 186, 586-610, (2002) · Zbl 1024.35046
[20] Simon, L., Asymptotics for a class of nonlinear evolution equation with applications to geometric problems, Ann. math., 118, 525-571, (1983) · Zbl 0549.35071
[21] Wu, H.; Zheng, S., Convergence to equibrium for the cahn – hilliard equation with dynamic boundary conditions, J. differential equations, 204, 511-531, (2004) · Zbl 1068.35018
[22] Chill, R.; Fasangova, E.; Pruess, J., Convergence to steady states of solutions of the cahn – hilliard and Caginalp equations with dynamic boundary conditions, Math. nachr., 13, 1448-1462, (2006) · Zbl 1107.35058
[23] Wu, H.; Zheng, S., Convergence to equilibrium for the damped semilinear wave equation with critical exponent and dissipative boundary condition, Quart. appl. math., 64, 1, 167-188, (2006) · Zbl 1120.35025
[24] Gal, C.; Wu, H., Asymptotic behavior of a cahn – hilliard equation with Wentzell boundary conditions and mass conversation, Discrete contin. dyn. syst., 22, 4, 1041-1063, (2008) · Zbl 1158.35052
[25] Haraux, A.; Jendoubi, M.A., Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptot. anal., 26, 21-36, (2001) · Zbl 0993.35017
[26] Zheng, S., ()
[27] Wu, H., Long-time behavior for a nonlinear plate equation with thermal memory, J. math. anal. appl., 348, 650-670, (2008) · Zbl 1152.35076
[28] Wu, H., Convergence to equilibrium for a cahn – hilliard model with the Wentzell boundary condition, Asymptot. anal., 54, 71-92, (2007) · Zbl 1139.34028
[29] Wu, H.; Grasselli, M.; Zheng, S., Convergence to equilibrium for a parabolic – hyperbolic phase-field system with Neumann boundary conditions, Math. models methods appl. sci., 17, 1, 1-29, (2007)
[30] Temam, R., ()
[31] Haraux, A.; Jendoubi, M.A., Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. var., 9, 95-124, (1999) · Zbl 0939.35122
[32] C. Gal, G.R. Goldstein, J.A. Goldstein, S. Romanelli, Fredholm alternative, semilinear elliptic problems and Wentzell boundary conditions, preprint, 2008
[33] Zeidler, E., Nonlinear functional analysis and its applications, I, (1986), Springer New York, Berlin, Heidelberg
[34] P. Mironescu, V. Radulescu, Nonlinear Sturm-Liouville type problems with finite number of solutions, Matarom, Laboratoire d’Analyse Numerique, Universite Pierre et Marie Curie (Paris VI), 3, 1993, pp. 54-67
[35] S. Łojasiewicz, 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
[36] Jendoubi, M.A., A simple unified approach to some convergence theorem of L. Simon, J. funct. anal., 153, 187-202, (1998) · Zbl 0895.35012
[37] Grasselli, M.; Wu, H.; Zheng, S., Asymptotic behavior of a non-isothermal ginzburg – landau model, Quart. appl. math., 66, 743-770, (2008) · Zbl 1171.35018
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