April  2014, 7(2): 271-306. doi: 10.3934/dcdss.2014.7.271

Some mathematical models in phase transition

1. 

Laboratoire de Mathématiques et Applications, Université de Poitiers, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex

Received  March 2013 Revised  July 2013 Published  September 2013

Our aim in these notes is to discuss the well-posedness and the asymptotic behavior, in terms of finite-dimensional attractors, of models in phase transition. In particular, we focus on the Caginalp phase field model.
Citation: Alain Miranville. Some mathematical models in phase transition. Discrete and Continuous Dynamical Systems - S, 2014, 7 (2) : 271-306. doi: 10.3934/dcdss.2014.7.271
References:
[1]

S. Aizicovici, E. Feireisl and F. Issard-Roch, Long-time convergence of solutions to a phase-field system, Math. Methods Appl. Sci., 24 (2001), 277-287. doi: 10.1002/mma.215.

[2]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Commun. Partial Diff. Eqns., 4 (1979), 827-868. doi: 10.1080/03605307908820113.

[3]

S. M. Allen and J. W. Cahn, A microscopic theory of antiphase boundary motion and its application to antiphase domain coarsening, J. Chem. Phys., 28 (1957), 258-267. doi: 10.1016/0001-6160(79)90196-2.

[4]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[5]

P. W. Bates and S. Zheng, Inertial manifolds and inertial sets for the phase-field equations, J. Dyn. Diff. Eqns., 4 (1992), 375-398. doi: 10.1007/BF01049391.

[6]

D. Brochet, X. Chen and D. Hilhorst, Finite dimensional exponential attractors for the phase-field model, Appl. Anal., 49 (1993), 197-212. doi: 10.1080/00036819108840173.

[7]

D. Brochet and D. Hilhorst, Universal attractor and inertial sets for the phase field model, Appl. Math. Letters, 4 (1991), 59-62. doi: 10.1016/0893-9659(91)90076-8.

[8]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Applied Mathematical Sciences, 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.

[9]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 2 (1958), 258-267. doi: 10.1063/1.1744102.

[10]

G. Caginalp, The role of microscopic anisotropy in the macroscopic behavior of a phase boundary, Ann. Physics, 172 (1986), 136-155. doi: 10.1016/0003-4916(86)90022-9.

[11]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[12]

G. Caginalp, Stefan and Hele-shaw type models as asymptotic limits of the phase-field equations, Phys. Review A, 39 (1989), 5887-5896. doi: 10.1103/PhysRevA.39.5887.

[13]

G. Caginalp and X. Chen, Phase field equations in the singular limit of sharp interface problems, in "On the Evolution of Phase Boundaries" (ed. M. Gurtin), IMA Vol. Math. Appl., 43, Springer, New York, (1992), 1-27. doi: 10.1007/978-1-4613-9211-8_1.

[14]

G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, European J. Appl. Math., 9 (1998), 417-445. doi: 10.1017/S0956792598003520.

[15]

G. Caginalp, X. Chen and C. Eck, Numerical tests of a phase field model with second order accuracy, SIAM J. Appl. Math., 68 (2008), 1518-1534. doi: 10.1137/070680965.

[16]

G. Caginalp and E. A. Socolovsky, Efficient computation of a sharp interface spreading via phase field methods, Appl. Math. Letters, 2 (1989), 117-120. doi: 10.1016/0893-9659(89)90002-5.

[17]

B. Chalmers, "Principles of Solidification," R. E. Krieger Publishing, Huntington, New York, 1977. doi: 10.1007/978-1-4684-1854-5_5.

[18]

X. Chen, G. Caginalp and C. Eck, A rapidly converging phase field model, Discrete Cont. Dyn. Systems, 15 (2006), 1107-1034.

[19]

L. Cherfils and A. Miranville, Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 17 (2007), 107-129.

[20]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115. doi: 10.1007/s10492-009-0008-6.

[21]

R. Chill, E. Fašangovà and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462. doi: 10.1002/mana.200410431.

[22]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Review Letters, 94 (2005), 154301. doi: 10.1103/PhysRevLett.94.154301.

[23]

C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations, in "Free Boundary Value Problems" (Oberwolfach, 1989), Internat. Ser. Numer. Math., Vol. 95, Birkhäuser, Basel, (1990), 46-58.

[24]

H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Review Letters, 79 (1997), 893-896. doi: 10.1103/PhysRevLett.79.893.

[25]

H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Letters, 42 (1998), 49-54.

[26]

C. G. Gal, A Cahn-Hilliard model in bounded domains with permeable walls, Math. Methods Appl. Sci., 29 (2006), 2009-2036. doi: 10.1002/mma.757.

[27]

C. G. Gal and M. Grasselli, On the asymptotic behavior of the Caginalp system with dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 689-710. doi: 10.3934/cpaa.2009.8.689.

[28]

C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions, in "Nonlinear Phenomena with Energy Dissipation: Mathematical Analysis, Modeling and Simulation," Gakuto Int. Ser. Math. Sci. Appl., Vol. {29}, Gakkōtosho, Tokyo, (2008), 117-139.

[29]

S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in "Differential Equations: Inverse and Direct Problems" (eds. A. Favini and A. Lorenzi), Lecture Notes in Pure and Applied Mathematics, Vol. 251, Chapman & Hall, Boca Raton, FL, (2006), 149-170. doi: 10.1201/9781420011135.ch9.

[30]

J. W. Gibbs, "Collected Works," Yale University Press, New Haven, 1948.

[31]

G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881.

[32]

M. Grasselli, A. Miranville, V. Pata and S. Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., 280 (2007), 1475-1509. doi: 10.1002/mana.200510560.

[33]

A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Royal Society London A, 432 (1991), 171-194. doi: 10.1098/rspa.1991.0012.

[34]

M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.

[35]

S. I. Hariharan and G. W. Young, Comparison of asymptotic solutions of a phase-field model to a sharp-interface model, SIAM J. Appl. Math., 62 (2001), 244-263. doi: 10.1137/S0036139900374908.

[36]

B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity, 12 (1999), 1263-1275. doi: 10.1088/0951-7715/12/5/303.

[37]

B. R. Hunt, T. Sauer and J. A. Yorke, Prevalence: A translation-invariant "almost every'' for infinite-dimensional spaces, Bull. Amer. Math. Soc., 27 (1992), 217-238. doi: 10.1090/S0273-0979-1992-00328-2.

[38]

M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal., 153 (1998), 187-202. doi: 10.1006/jfan.1997.3174.

[39]

G. Lamé and B. P. Clapeyron, Mémoire sur la solidification par refroidissement d'un globe solide, Ann. Chem. Phys., 47 (1831), 250-256.

[40]

L. D. Landau and E. M. Lifschitz, "Statistical Physics (Part 1)," Third edition, Pergamon, New York, 1980.

[41]

S. Łojasiewicz, "Ensembles Semi-Analytiques," IHES, Bures-sur-Yvette, 1965.

[42]

A. M. Meirmanov, The classical solution of a multidimensional Stefan problem for quasilinear parabolic equations (in Russian), Mat. Sb. (N.S.), 112 (1980), 170-192.

[43]

A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal. TMA, 71 (2009), 2278-2290. doi: 10.1016/j.na.2009.01.061.

[44]

A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894. doi: 10.1080/00036810903042182.

[45]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations. Vol. IV" (eds. C. M. Dafermos and M. Pokorny), Elsevier/North Holland, Amsterdam, (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0.

[46]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Cont. Dyn. Systems, 28 (2010), 275-310. doi: 10.3934/dcds.2010.28.275.

[47]

O. A. Oleinik, A method of solution of the general Stefan problem, Soviet Math. Dokl., 1 (1960), 1350-1353.

[48]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43 (1990), 44-62. doi: 10.1016/0167-2789(90)90015-H.

[49]

O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a "thermodynamically consistent'' phase-field model, Physica D, 69 (1993), 107-113. doi: 10.1016/0167-2789(93)90183-2.

[50]

J. C. Robinson, Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces, Nonlinearity, 22 (2009), 711-728. doi: 10.1088/0951-7715/22/4/001.

[51]

L. Rubinstein, On the solution of Stefan's problem, (in Russian) Izvestia Akad. Nauk SSSR, 11 (1947), 37-54.

[52]

S. I. Serdyukov, N. M. Voskresenskii, V. K. Bel'nov and I. I. Karpov, Extended irreversible thermodynamics and generalization of the dual-phase-lag model in heat transfer, J. Non-Equilib. Thermodyn., 28 (2003), 1-13. doi: 10.1515/JNETDY.2003.013.

[53]

L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. Math., 118 (1983), 525-571. doi: 10.2307/2006981.

[54]

J. Sprekels and S. Zheng, Global smooth solutions to a thermodynamically consistent model of phase-field type in higher space dimensions, J. Math. Anal. Appl., 176 (1993), 200-223. doi: 10.1006/jmaa.1993.1209.

[55]

J. Stefan, Uber einige Probleme der Theorie der Warmeleitung, S.-B. Wien Akad. Mat. Natur., 98 (1889), 173-484.

[56]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997.

[57]

A. Visintin, Introduction to Stefan-type problems, in "Handbook of Differential Equations: Evolutionary Equations. Vol. IV" (eds. C. M. Dafermos and M. Pokorny), Elsevier/North Holland, Amsterdam, (2008), 377-484. doi: 10.1016/S1874-5717(08)00008-X.

[58]

S. Zelik, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 24 (2000), 1-25.

[59]

Z. Zhang, Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Commun. Pure Appl. Anal., 4 (2005), 683-693. doi: 10.3934/cpaa.2005.4.683.

[60]

S. Zheng, Global existence for a thermodynamically consistent model of phase field type, Diff. Integral Eqns., 5 (1992), 241-253.

show all references

References:
[1]

S. Aizicovici, E. Feireisl and F. Issard-Roch, Long-time convergence of solutions to a phase-field system, Math. Methods Appl. Sci., 24 (2001), 277-287. doi: 10.1002/mma.215.

[2]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Commun. Partial Diff. Eqns., 4 (1979), 827-868. doi: 10.1080/03605307908820113.

[3]

S. M. Allen and J. W. Cahn, A microscopic theory of antiphase boundary motion and its application to antiphase domain coarsening, J. Chem. Phys., 28 (1957), 258-267. doi: 10.1016/0001-6160(79)90196-2.

[4]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[5]

P. W. Bates and S. Zheng, Inertial manifolds and inertial sets for the phase-field equations, J. Dyn. Diff. Eqns., 4 (1992), 375-398. doi: 10.1007/BF01049391.

[6]

D. Brochet, X. Chen and D. Hilhorst, Finite dimensional exponential attractors for the phase-field model, Appl. Anal., 49 (1993), 197-212. doi: 10.1080/00036819108840173.

[7]

D. Brochet and D. Hilhorst, Universal attractor and inertial sets for the phase field model, Appl. Math. Letters, 4 (1991), 59-62. doi: 10.1016/0893-9659(91)90076-8.

[8]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Applied Mathematical Sciences, 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.

[9]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 2 (1958), 258-267. doi: 10.1063/1.1744102.

[10]

G. Caginalp, The role of microscopic anisotropy in the macroscopic behavior of a phase boundary, Ann. Physics, 172 (1986), 136-155. doi: 10.1016/0003-4916(86)90022-9.

[11]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[12]

G. Caginalp, Stefan and Hele-shaw type models as asymptotic limits of the phase-field equations, Phys. Review A, 39 (1989), 5887-5896. doi: 10.1103/PhysRevA.39.5887.

[13]

G. Caginalp and X. Chen, Phase field equations in the singular limit of sharp interface problems, in "On the Evolution of Phase Boundaries" (ed. M. Gurtin), IMA Vol. Math. Appl., 43, Springer, New York, (1992), 1-27. doi: 10.1007/978-1-4613-9211-8_1.

[14]

G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, European J. Appl. Math., 9 (1998), 417-445. doi: 10.1017/S0956792598003520.

[15]

G. Caginalp, X. Chen and C. Eck, Numerical tests of a phase field model with second order accuracy, SIAM J. Appl. Math., 68 (2008), 1518-1534. doi: 10.1137/070680965.

[16]

G. Caginalp and E. A. Socolovsky, Efficient computation of a sharp interface spreading via phase field methods, Appl. Math. Letters, 2 (1989), 117-120. doi: 10.1016/0893-9659(89)90002-5.

[17]

B. Chalmers, "Principles of Solidification," R. E. Krieger Publishing, Huntington, New York, 1977. doi: 10.1007/978-1-4684-1854-5_5.

[18]

X. Chen, G. Caginalp and C. Eck, A rapidly converging phase field model, Discrete Cont. Dyn. Systems, 15 (2006), 1107-1034.

[19]

L. Cherfils and A. Miranville, Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 17 (2007), 107-129.

[20]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115. doi: 10.1007/s10492-009-0008-6.

[21]

R. Chill, E. Fašangovà and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462. doi: 10.1002/mana.200410431.

[22]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Review Letters, 94 (2005), 154301. doi: 10.1103/PhysRevLett.94.154301.

[23]

C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations, in "Free Boundary Value Problems" (Oberwolfach, 1989), Internat. Ser. Numer. Math., Vol. 95, Birkhäuser, Basel, (1990), 46-58.

[24]

H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Review Letters, 79 (1997), 893-896. doi: 10.1103/PhysRevLett.79.893.

[25]

H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Letters, 42 (1998), 49-54.

[26]

C. G. Gal, A Cahn-Hilliard model in bounded domains with permeable walls, Math. Methods Appl. Sci., 29 (2006), 2009-2036. doi: 10.1002/mma.757.

[27]

C. G. Gal and M. Grasselli, On the asymptotic behavior of the Caginalp system with dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 689-710. doi: 10.3934/cpaa.2009.8.689.

[28]

C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions, in "Nonlinear Phenomena with Energy Dissipation: Mathematical Analysis, Modeling and Simulation," Gakuto Int. Ser. Math. Sci. Appl., Vol. {29}, Gakkōtosho, Tokyo, (2008), 117-139.

[29]

S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in "Differential Equations: Inverse and Direct Problems" (eds. A. Favini and A. Lorenzi), Lecture Notes in Pure and Applied Mathematics, Vol. 251, Chapman & Hall, Boca Raton, FL, (2006), 149-170. doi: 10.1201/9781420011135.ch9.

[30]

J. W. Gibbs, "Collected Works," Yale University Press, New Haven, 1948.

[31]

G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881.

[32]

M. Grasselli, A. Miranville, V. Pata and S. Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., 280 (2007), 1475-1509. doi: 10.1002/mana.200510560.

[33]

A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Royal Society London A, 432 (1991), 171-194. doi: 10.1098/rspa.1991.0012.

[34]

M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.

[35]

S. I. Hariharan and G. W. Young, Comparison of asymptotic solutions of a phase-field model to a sharp-interface model, SIAM J. Appl. Math., 62 (2001), 244-263. doi: 10.1137/S0036139900374908.

[36]

B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity, 12 (1999), 1263-1275. doi: 10.1088/0951-7715/12/5/303.

[37]

B. R. Hunt, T. Sauer and J. A. Yorke, Prevalence: A translation-invariant "almost every'' for infinite-dimensional spaces, Bull. Amer. Math. Soc., 27 (1992), 217-238. doi: 10.1090/S0273-0979-1992-00328-2.

[38]

M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal., 153 (1998), 187-202. doi: 10.1006/jfan.1997.3174.

[39]

G. Lamé and B. P. Clapeyron, Mémoire sur la solidification par refroidissement d'un globe solide, Ann. Chem. Phys., 47 (1831), 250-256.

[40]

L. D. Landau and E. M. Lifschitz, "Statistical Physics (Part 1)," Third edition, Pergamon, New York, 1980.

[41]

S. Łojasiewicz, "Ensembles Semi-Analytiques," IHES, Bures-sur-Yvette, 1965.

[42]

A. M. Meirmanov, The classical solution of a multidimensional Stefan problem for quasilinear parabolic equations (in Russian), Mat. Sb. (N.S.), 112 (1980), 170-192.

[43]

A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal. TMA, 71 (2009), 2278-2290. doi: 10.1016/j.na.2009.01.061.

[44]

A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894. doi: 10.1080/00036810903042182.

[45]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations. Vol. IV" (eds. C. M. Dafermos and M. Pokorny), Elsevier/North Holland, Amsterdam, (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0.

[46]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Cont. Dyn. Systems, 28 (2010), 275-310. doi: 10.3934/dcds.2010.28.275.

[47]

O. A. Oleinik, A method of solution of the general Stefan problem, Soviet Math. Dokl., 1 (1960), 1350-1353.

[48]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43 (1990), 44-62. doi: 10.1016/0167-2789(90)90015-H.

[49]

O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a "thermodynamically consistent'' phase-field model, Physica D, 69 (1993), 107-113. doi: 10.1016/0167-2789(93)90183-2.

[50]

J. C. Robinson, Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces, Nonlinearity, 22 (2009), 711-728. doi: 10.1088/0951-7715/22/4/001.

[51]

L. Rubinstein, On the solution of Stefan's problem, (in Russian) Izvestia Akad. Nauk SSSR, 11 (1947), 37-54.

[52]

S. I. Serdyukov, N. M. Voskresenskii, V. K. Bel'nov and I. I. Karpov, Extended irreversible thermodynamics and generalization of the dual-phase-lag model in heat transfer, J. Non-Equilib. Thermodyn., 28 (2003), 1-13. doi: 10.1515/JNETDY.2003.013.

[53]

L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. Math., 118 (1983), 525-571. doi: 10.2307/2006981.

[54]

J. Sprekels and S. Zheng, Global smooth solutions to a thermodynamically consistent model of phase-field type in higher space dimensions, J. Math. Anal. Appl., 176 (1993), 200-223. doi: 10.1006/jmaa.1993.1209.

[55]

J. Stefan, Uber einige Probleme der Theorie der Warmeleitung, S.-B. Wien Akad. Mat. Natur., 98 (1889), 173-484.

[56]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997.

[57]

A. Visintin, Introduction to Stefan-type problems, in "Handbook of Differential Equations: Evolutionary Equations. Vol. IV" (eds. C. M. Dafermos and M. Pokorny), Elsevier/North Holland, Amsterdam, (2008), 377-484. doi: 10.1016/S1874-5717(08)00008-X.

[58]

S. Zelik, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 24 (2000), 1-25.

[59]

Z. Zhang, Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Commun. Pure Appl. Anal., 4 (2005), 683-693. doi: 10.3934/cpaa.2005.4.683.

[60]

S. Zheng, Global existence for a thermodynamically consistent model of phase field type, Diff. Integral Eqns., 5 (1992), 241-253.

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