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 & 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.  doi: 10.1002/mma.215.  Google Scholar

[2]

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

[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.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[4]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, 25 (1992).   Google Scholar

[5]

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

[6]

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

[7]

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

[8]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Applied Mathematical Sciences, 121 (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

G. Caginalp and X. Chen, Phase field equations in the singular limit of sharp interface problems,, in, 43 (1992), 1.  doi: 10.1007/978-1-4613-9211-8_1.  Google Scholar

[14]

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

[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.  doi: 10.1137/070680965.  Google Scholar

[16]

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

[17]

B. Chalmers, "Principles of Solidification,", R. E. Krieger Publishing, (1977).  doi: 10.1007/978-1-4684-1854-5_5.  Google Scholar

[18]

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

[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.   Google Scholar

[20]

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

[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.  doi: 10.1002/mana.200410431.  Google Scholar

[22]

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

[23]

C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations,, in, (1990), 46.   Google Scholar

[24]

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

[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.   Google Scholar

[26]

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

[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.  doi: 10.3934/cpaa.2009.8.689.  Google Scholar

[28]

C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions,, in, (2008), 117.   Google Scholar

[29]

S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions,, in, (2006), 149.  doi: 10.1201/9781420011135.ch9.  Google Scholar

[30]

J. W. Gibbs, "Collected Works,", Yale University Press, (1948).   Google Scholar

[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.  doi: 10.3934/cpaa.2009.8.881.  Google Scholar

[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.  doi: 10.1002/mana.200510560.  Google Scholar

[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.  doi: 10.1098/rspa.1991.0012.  Google Scholar

[34]

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

[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.  doi: 10.1137/S0036139900374908.  Google Scholar

[36]

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

[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.  doi: 10.1090/S0273-0979-1992-00328-2.  Google Scholar

[38]

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

[39]

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

[40]

L. D. Landau and E. M. Lifschitz, "Statistical Physics (Part 1),", Third edition, (1980).   Google Scholar

[41]

S. Łojasiewicz, "Ensembles Semi-Analytiques,", IHES, (1965).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.na.2009.01.061.  Google Scholar

[44]

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

[45]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103.  doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[46]

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

[47]

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

[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.  doi: 10.1016/0167-2789(90)90015-H.  Google Scholar

[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.  doi: 10.1016/0167-2789(93)90183-2.  Google Scholar

[50]

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

[51]

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

[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.  doi: 10.1515/JNETDY.2003.013.  Google Scholar

[53]

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

[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.  doi: 10.1006/jmaa.1993.1209.  Google Scholar

[55]

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

[56]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Second edition, (1997).   Google Scholar

[57]

A. Visintin, Introduction to Stefan-type problems,, in, (2008), 377.  doi: 10.1016/S1874-5717(08)00008-X.  Google Scholar

[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.   Google Scholar

[59]

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

[60]

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

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.  doi: 10.1002/mma.215.  Google Scholar

[2]

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

[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.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[4]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, 25 (1992).   Google Scholar

[5]

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

[6]

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

[7]

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

[8]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Applied Mathematical Sciences, 121 (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

G. Caginalp and X. Chen, Phase field equations in the singular limit of sharp interface problems,, in, 43 (1992), 1.  doi: 10.1007/978-1-4613-9211-8_1.  Google Scholar

[14]

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

[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.  doi: 10.1137/070680965.  Google Scholar

[16]

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

[17]

B. Chalmers, "Principles of Solidification,", R. E. Krieger Publishing, (1977).  doi: 10.1007/978-1-4684-1854-5_5.  Google Scholar

[18]

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

[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.   Google Scholar

[20]

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

[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.  doi: 10.1002/mana.200410431.  Google Scholar

[22]

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

[23]

C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations,, in, (1990), 46.   Google Scholar

[24]

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

[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.   Google Scholar

[26]

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

[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.  doi: 10.3934/cpaa.2009.8.689.  Google Scholar

[28]

C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions,, in, (2008), 117.   Google Scholar

[29]

S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions,, in, (2006), 149.  doi: 10.1201/9781420011135.ch9.  Google Scholar

[30]

J. W. Gibbs, "Collected Works,", Yale University Press, (1948).   Google Scholar

[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.  doi: 10.3934/cpaa.2009.8.881.  Google Scholar

[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.  doi: 10.1002/mana.200510560.  Google Scholar

[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.  doi: 10.1098/rspa.1991.0012.  Google Scholar

[34]

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

[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.  doi: 10.1137/S0036139900374908.  Google Scholar

[36]

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

[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.  doi: 10.1090/S0273-0979-1992-00328-2.  Google Scholar

[38]

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

[39]

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

[40]

L. D. Landau and E. M. Lifschitz, "Statistical Physics (Part 1),", Third edition, (1980).   Google Scholar

[41]

S. Łojasiewicz, "Ensembles Semi-Analytiques,", IHES, (1965).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.na.2009.01.061.  Google Scholar

[44]

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

[45]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103.  doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[46]

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

[47]

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

[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.  doi: 10.1016/0167-2789(90)90015-H.  Google Scholar

[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.  doi: 10.1016/0167-2789(93)90183-2.  Google Scholar

[50]

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

[51]

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

[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.  doi: 10.1515/JNETDY.2003.013.  Google Scholar

[53]

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

[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.  doi: 10.1006/jmaa.1993.1209.  Google Scholar

[55]

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

[56]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Second edition, (1997).   Google Scholar

[57]

A. Visintin, Introduction to Stefan-type problems,, in, (2008), 377.  doi: 10.1016/S1874-5717(08)00008-X.  Google Scholar

[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.   Google Scholar

[59]

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

[60]

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

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