zbMATH — the first resource for mathematics

Convergence to equilibrium for a parabolic-hyperbolic phase-field system with dynamical boundary condition. (English) Zbl 1154.35329
Summary: This paper is concerned with the well-posedness and the asymptotic behavior of solutions to the following parabolic-hyperbolic phase-field system
\[ \begin{cases}(\theta+\chi)_t -\Delta \theta = 0\\ \chi_{tt}+\chi_t -\Delta_{\chi}+ \varphi(\chi)-\theta = 0\end{cases}\tag{1} \] in \(\Omega \times (0,+\infty)\), subject to the Neumann boundary condition for \(\theta\) \[ \partial_{\nu}\theta = 0, \quad \text{on } \Gamma \times (0, +\infty)\tag{2} \] the dynamical boundary condition for \(\chi\)
\[ \partial_{\nu}\chi + \chi +\chi_t=0,\quad \text{on } \Gamma \times (0,+\infty)\tag{3} \] and the initial
\[ \theta(0)=\theta_0, \quad \chi(0)=\chi_0,\quad \chi_t(0)=\chi_1,\quad \text{in } \Omega,\tag{4} \] where \(\Omega\) is a bounded domain in \(\mathbb R^3\) with smooth boundary \(\Gamma , \nu \) is the outward normal direction to the boundary and \(\varphi\) is a real analytic function. In this paper we first establish the existence and uniqueness of a global strong solution to \((1)--(4)\). Then, we prove its convergence to an equilibrium as time goes to infinity and we provide an estimate of the convergence rate.

35G30 Boundary value problems for nonlinear higher-order PDEs
35K55 Nonlinear parabolic equations
35L70 Second-order nonlinear hyperbolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI
[1] Aizicovici, S.; Feireisl, E.; Issard-Roch, F., Long time convergence of solutions to a phase-field system, Math. methods appl. sci., 24, 277-287, (2001) · Zbl 0984.35026
[2] Aizicovici, S.; Petzeltová, H., Convergence of solutions of phase-field systems with a nonconstant latent heat, Dynam. systems appl., 14, 163-173, (2005) · Zbl 1076.35018
[3] Bates, P.; Zheng, S., Inertial manifolds and inertial sets for the phase-field equations, J. dynam. differential equations, 4, 375-398, (1992) · Zbl 0758.35040
[4] Brochet, D.; Hilhorst, D.; Chen, X., Finite-dimensional exponential attractor for the phase field model, Appl. anal., 49, 197-212, (1993) · Zbl 0790.35052
[5] Brokate, M.; Sprekels, J., Hysteresis and phase transitions, (1996), Springer New York · Zbl 0951.74002
[6] Caginalp, G., An analysis of a phase field model of a free boundary, Arch. ration. mech. anal., 92, 205-245, (1986) · Zbl 0608.35080
[7] Chill, R., On the łojasiewicz – simon gradient inequality, J. funct. anal., 201, 572-601, (2003) · Zbl 1036.26015
[8] Chill, R.; Jendoubi, M.A., Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear anal., 53, 1017-1039, (2003) · Zbl 1033.34066
[9] Chueshov, I.; Eller, M.; Lasiecka, I., On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. partial differential equations, 27, 1901-1951, (2002) · Zbl 1021.35020
[10] Elliott, C.; Zheng, S., Global existence and stability of solution to the phase field equations, (), 45-58
[11] Feireisl, E.; Issard-Roch, F.; Petzeltová, H., Long-time behaviour and convergence towards equilibria for conserved phase field model, Discrete contin. dyn. syst., 10, 230-252, (2004) · Zbl 1060.35018
[12] Feireisl, E.; Issard-Roch, F.; Petzeltová, H., A non-smooth version of the łojasiewicz – simon theorem with applications to non-local phase-field systems, J. differential equations, 199, 1-21, (2004) · Zbl 1062.35152
[13] Feireisl, E.; Simondon, F., Convergence for semilinear degenerate parabolic equations in several space dimensions, J. dynam. differential equations, 12, 647-673, (2000) · Zbl 0977.35069
[14] Galenko, P.; Jou, D., Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Phys. rev. E, 71, 046125(13), (2005)
[15] Grasselli, M.; Pata, V., Existence of a universal attractor for a parabolic – hyperbolic phase-field systems, Adv. math. sci. appl., 13, 443-459, (2003) · Zbl 1057.37068
[16] Grasselli, M.; Pata, V., Asymptotic behavior of a parabolic – hyperbolic system, Comm. pure appl. anal., 3, 849-881, (2004) · Zbl 1079.35022
[17] Grasselli, M.; Petzeltová, H.; Schimperna, G., Long time behavior of solutions to the Caginalp system with singular potential, Z. anal. anwendungen, 25, 51-72, (2006) · Zbl 1128.35021
[18] M. Grasselli, H. Petzeltová, G. Schimperna, Convergence to stationary solutions for a parabolic – hyperbolic phase-field system, Comm. Pure Appl. Anal., in press
[19] Greenberg, J.M., Elastic phase transitions: A new model, Phys. D, 108, 209-235, (1997) · Zbl 0963.74547
[20] Haraux, A.; Jendoubi, M.A., Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. var. partial differential equations, 9, 95-124, (1999) · Zbl 0939.35122
[21] Haraux, A., Systèmes dynamiques dissipatifs et applications, (1991), Masson Paris · Zbl 0726.58001
[22] Huang, S.Z.; Takáč, P., Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear anal., 46, 675-698, (2001) · Zbl 1002.35022
[23] Jendoubi, M.A., A simple unified approach to some convergence theorem of L. Simon, J. funct. anal., 153, 187-202, (1998) · Zbl 0895.35012
[24] 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
[25] Jiménez-Casas, A.; Rodríguez-Bernal, A., Asymptotic behaviour for a phase field model in higher order Sobolev spaces, Rev. mat. complut., 15, 213-248, (2002) · Zbl 1009.35011
[26] Kalantarov, V.K., On the minimal global attractor of a system of phase field equations, Kraev. zadachi mat. fiz. i smezh. voprosy teor. funktsii, Zap. nauchn. sem. leningrad. otdel. mat. inst. Steklov. (LOMI), J. math. sci., 70, 3, 1767-1777, (1994), (in Russian); translation in · Zbl 0835.35158
[27] Laurençot, Ph., Long-time behaviour for a model of phase-field type, Proc. roy. soc. Edinburgh sect. A, 126, 167-185, (1996) · Zbl 0851.35055
[28] Li, W., Long-time convergence of solution to phase-field system with Neumann boundary conditions, Chinese ann. math. ser. A, 26, 659-668, (2005), (in Chinese) · Zbl 1090.35037
[29] Lions, J.-L.; Magenes, E., Non-homogeneous boundary value problems and applications. vol. I, (1973), Springer New York · Zbl 0251.35001
[30] Łojasiewicz, S., Une propriété topologique des sous-ensembles analytiques réels, (), 87-89
[31] Łojasiewicz, S., Sur la geometrie semi- et sous-analytique, Ann. inst. Fourier (Grenoble), 43, 1575-1595, (1963) · Zbl 0803.32002
[32] Łojasiewicz, S., Ensembles semi-analytiques. notes, (1965), I.H.E.S. Bures-sur-Yvette
[33] Matano, H., Convergence of solutions of one-dimensional semilinear parabolic equations, J. math. Kyoto univ., 18, 221-227, (1978) · Zbl 0387.35008
[34] Mironescu, P.; Rădulescu, V., Nonlinear sturm – liouville type problems with a finite number of solutions, (), 54-67
[35] Nirenberg, L., Topics in nonlinear functional analysis, (1974), Courant Institute of Mathematical Science New York University, New York · Zbl 0286.47037
[36] Polačik, P.; Rybakowski, K.P., Nonconvergent bounded trajectories in semilinear heat equations, J. differential equations, 124, 472-494, (1996) · Zbl 0845.35054
[37] Polačik, P.; Simondon, F., Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains, J. differential equations, 186, 586-610, (2002) · Zbl 1024.35046
[38] Rybka, P.; Hoffmann, K.-H., Convergence of solutions to cahn – hilliard equation, Comm. partial differential equations, 24, 1055-1077, (1999) · Zbl 0936.35032
[39] Simon, L., Asymptotics for a class of non-linear evolution equation with applications to geometric problems, Ann. of math., 118, 525-571, (1983) · Zbl 0549.35071
[40] Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, (1988), Springer New York · Zbl 0662.35001
[41] Webb, G.F., Compactness of bounded trajectories of dynamical systems in infinite dimensional space, Proc. roy. soc. Edinburgh sect. A, 84, 19-34, (1979) · Zbl 0414.34042
[42] H. Wu, M. Grasselli, S. Zheng, Convergence to equilibrium for a parabolic – hyperbolic phase-field system with Neumann boundary conditions. Math. Models Methods Appl. Sci., in press · Zbl 1120.35024
[43] 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
[44] Wu, H.; Zheng, S., Convergence to equilibrium for the damped semilinear wave equation with critical exponent and dissipative boundary condition, Quart. appl. math., 64, 167-188, (2006) · Zbl 1120.35025
[45] Zeidler, E., Nonlinear functional analysis and its applications. I. fixed-point theorems, (1986), Springer New York
[46] Zelenyak, T.I., Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differ. uravn., 4, 34-45, (1968)
[47] Zhang, Z., Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Comm. pure appl. anal., 4, 683-693, (2005) · Zbl 1082.35033
[48] Zheng, S., Nonlinear evolution equations, Pitman ser. monogr. and surv. on pure and appl. math., vol. 133, (2004), Chapman & Hall/CRC Boca Raton, FL
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.