Global and exponential attractors for a Caginalp type phase-field problem

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1. [1] Babin A., Nicolaenko B., Exponential attractors of reaction-diffusion systems in an unbounded domain, J. Dynam. Differential Equations, 1995, 7(4), 567–590 http://dx.doi.org/10.1007/BF02218725 Zbl0846.35061
2. [2] Bates P.W., Zheng S.M., Inertial manifolds and inertial sets for the phase-field equations, J. Dynam. Differential Equations, 1992, 4(2), 375–398 http://dx.doi.org/10.1007/BF01049391 Zbl0758.35040
3. [3] Brezis H., Analyse Fonctionnelle, Collect. Math. Appl. Maitrise, Masson, Paris, 1983 Zbl0511.46001
4. [4] Brochet D., Hilhorst D., Universal attractor and inertial sets for the phase field model, Appl. Math. Lett., 1991, 4(6), 59–62, 1991 http://dx.doi.org/10.1016/0893-9659(91)90076-8 Zbl0773.35028
5. [5] Brochet D., Hilhorst D., Chen X., Finite-dimensional exponential attractor for the phase field model, Appl. Anal., 1993, 49(3–4), 197–212 http://dx.doi.org/10.1080/00036819108840173 Zbl0790.35052
6. [6] Brochet D., Hilhorst D., Novick-Cohen A., Finite-dimensional exponential attractor for a model for order-disorder and phase separation, Appl. Math. Lett., 1994, 7(3), 83–87 http://dx.doi.org/10.1016/0893-9659(94)90118-X Zbl0803.35076
7. [7] Caginalp G., An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 1986, 92(3), 205–245 http://dx.doi.org/10.1007/BF00254827 Zbl0608.35080
8. [8] Caginalp G., The role of microscopic anisotropy in the macroscopic behavior of a phase boundary, Ann. Physics, 1986, 172(1), 136–155 http://dx.doi.org/10.1016/0003-4916(86)90022-9 
9. [9] Cherfils L., Miranville A., Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 2007, 17(1), 107–129 Zbl1145.35042
10. [10] Cherfils L., Miranville A., On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 2009, 54(2), 89–115 http://dx.doi.org/10.1007/s10492-009-0008-6 Zbl1212.35012
11. [11] Christov C.I., Jordan P.M., Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 2005, 94(15), #154301 http://dx.doi.org/10.1103/PhysRevLett.94.154301 
12. [12] Conti M., Gatti S., Miranville A., Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser., 2012, S5(3), 485–505 Zbl1244.35067
13. [13] Eden A., Foias C., Nicolaenko B., Temam R., Exponential Attractors for Dissipative Evolution Equations, RAM Res. Appl. Math., 37, Masson/John Wiley & Sons, Paris/Chichester, 1994 
14. [14] Fabrie P., Galusinski C., Exponential attractors for a partially dissipative reaction system, Asymptotic Anal., 1996, 12(4), 329–354 Zbl1028.35026
15. [15] Flavin J.N., Knops R.J., Payne L.E., Decay estimates for the constrained elastic cylinder of variable cross section, Quart. Appl. Math., 1989, 47(2), 325–350 Zbl0706.73015
16. [16] Gurtin M.E., Chen P.J., On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 1968, 19(4), 614–627 http://dx.doi.org/10.1007/BF01594969 Zbl0159.15103
17. [17] Landau L.D., Lifshitz E.M., Statistical Physics I, 3rd ed., Butterworth-Heinemann, Oxford, 1980 Zbl0080.19702
18. [18] Lions J.-L., Magenes E., Problèmes aux Limites non Homogènes et Applications, 2, Travaux et Recherches Mathématiques, 18, Dunod, Paris, 1968 Zbl0197.06701
19. [19] Miranville A., Exponential attractors for a class of evolution equations by a decomposition method, C. R. Acad. Sci. Paris Sér. I Math., 1999, 328(2), 145–150 http://dx.doi.org/10.1016/S0764-4442(99)80153-0 Zbl1101.35334
20. [20] Miranville A., Some models of Cahn-Hilliard equations in nonisotropic media, M2AN Math. Model. Numer. Anal., 2000, 34(3), 539–554 http://dx.doi.org/10.1051/m2an:2000155 Zbl0965.35170
21. [21] Miranville A., Some mathematical models in phase transition, Ravello, 2009 (lecture notes) 
22. [22] Miranville A., Quintanilla R., A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal., 2009, 71(5–6), 2278–2290 http://dx.doi.org/10.1016/j.na.2009.01.061 Zbl1167.35304
23. [23] Miranville A., Quintanilla R., Some generalizations of the Caginalp phase-field system, Appl. Anal., 2009, 88(6), 897–894 http://dx.doi.org/10.1080/00036810903042182 Zbl1178.35194
24. [24] Miranville A., Quintanilla R., A Caginalp phase-field system with a nonlinear coupling, Nonlinear Anal. Real World Appl., 2010, 11(4), 2849–2861 http://dx.doi.org/10.1016/j.nonrwa.2009.10.008 Zbl1197.35082
25. [25] Miranville A., Zelik S., Attractors for dissipative partial differential equations in bounded and unbounded domains, In: Handbook of Differential Equations: Evolutionary Equations, IV, Handb. Diff. Equ., Elsevier/North-Holland, Amsterdam, 2008, 103–200 http://dx.doi.org/10.1016/S1874-5717(08)00003-0 
26. [26] Quintanilla R., Spatial stability for the quasi-static problem of thermoelasticity, J. Elasticity, 2004, 76(2), 93–105 http://dx.doi.org/10.1007/s10659-004-3334-7 Zbl1060.74020
27. [27] Temam R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Appl. Math. Sci., 68, Springer, New York, 1997 Zbl0871.35001