A generalization of the Caginalp phase-field system based on the Cattaneo law. (English) Zbl 1167.35304

Summary: We consider in this article a generalization of the Caginalp model for phase transitions based on the Maxwell-Cattaneo law, instead of the classical Fourier law. In particular, we prove the existence and uniqueness of solutions. We finally study the spatial behavior of the solutions in a semi-infinite cylinder.


35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35L70 Second-order nonlinear hyperbolic equations
35K55 Nonlinear parabolic equations
80A22 Stefan problems, phase changes, etc.
Full Text: DOI


[1] Caginalp, G., An analysis of a phase field model of a free boundary, Arch. ration. mech. anal., 92, 205-245, (1986) · Zbl 0608.35080
[2] Aizicovici, S.; Feireisl, E., Long-time stabilization of solutions to a phase-field model with memory, J. evol. equ., 1, 69-84, (2001) · Zbl 0973.35037
[3] 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
[4] Brochet, D.; Chen, X.; Hilhorst, D., Finite dimensional exponential attractors 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] Cherfils, L.; Miranville, A., Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. math. sci. appl., 17, 107-129, (2007) · Zbl 1145.35042
[7] L. Cherfils, A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math. (in press) · Zbl 1212.35012
[8] Chill, R.; Fašangovà, E.; Prüss, J., Convergence to steady states of solutions of the cahn – hilliard equation with dynamic boundary conditions, Math. nachr., 279, 1448-1462, (2006) · Zbl 1107.35058
[9] Gatti, S.; Miranville, A., Asymptotic behavior of a phase-field system with dynamic boundary conditions, (), 149-170 · Zbl 1123.35310
[10] Grasselli, M.; Miranville, A.; Pata, V.; Zelik, S., Well-posedness and long time behavior of a parabolic – hyperbolic phase-field system with singular potentials, Math. nachr., 280, 1475-1509, (2007) · Zbl 1133.35017
[11] Miranville, A.; Zelik, S., Robust exponential attractors for singularly perturbed phase-field type equations, Electron. J. differential equations, 2002, 1-28, (2002) · Zbl 1004.35024
[12] Zhang, Z., Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Commun. pure appl. anal., 4, 683-693, (2005) · Zbl 1082.35033
[13] Gal, C.G.; Grasselli, M., The nonisothermal allen – cahn equation with dynamic boundary conditions, Discrete contin. dyn. systems A, 22, 1009-1040, (2008) · Zbl 1160.35353
[14] Christov, C.I.; Jordan, P.M., Heat conduction paradox involving second-sound propagation in moving media, Phys. rev. lett., 94, 154301, (2005)
[15] Grasselli, M.; Pata, V., Existence of a universal attractor for a fully hyperbolic phase-field system, J. evol. equ., 4, 27-51, (2004) · Zbl 1063.35038
[16] Jiang, J., Convergence to equilibrium for a parabolic – hyperbolic phase-field model with Cattaneo heat flux law, J. math. anal. appl., 341, 149-169, (2008) · Zbl 1139.35019
[17] J. Jiang, Convergence to equilibrium for a fully hyperbolic phase field model with Cattaneo heat flux law, Math. Methods Appl. Sci. (in press) · Zbl 1180.35107
[18] Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, () · Zbl 0871.35001
[19] Flavin, J.N.; Knops, R.J.; Payne, L.E., Decay estimates for the constrained elastic cylinder of variable cross-section, Quart. appl. math., 47, 325-350, (1989) · Zbl 0706.73015
[20] Quintanilla, R., End effects in thermoelasticity, Math. methods appl. sci., 24, 93-102, (2001) · Zbl 0989.35028
[21] Quintanilla, R., Damping of end effects in a thermoelastic theory, Appl. math. lett., 14, 137-141, (2001) · Zbl 0971.74037
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.