# American Institute of Mathematical Sciences

January  2012, 32(1): 167-190. doi: 10.3934/dcds.2012.32.167

## Asymptotic behavior of solutions to a one-dimensional full model for phase transitions with microscopic movements

 1 Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China 2 Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, 100088

Received  July 2010 Revised  November 2010 Published  September 2011

This paper is devoted to the study of long-time behavior of the solutions to a one-dimensional full model for the first order phase transitions. Our system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature $\theta$, which is coupled with an evolution equation for the phase change parameter $f$ with a third-order nonlinearity $G_2'(f)$ in place of the customarily constant latent heat. The main novelty of this paper is that we perform an argument to establish Lemma 3.1 which enables us to obtain uniform estimates of the global solutions with respect to time. Asymptotic behavior of the solutions as time goes to infinity and the compactness of the orbit are obtained. Furthermore, we investigate the dynamics of the system and prove the existence of global attractors.
Citation: Jie Jiang, Boling Guo. Asymptotic behavior of solutions to a one-dimensional full model for phase transitions with microscopic movements. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 167-190. doi: 10.3934/dcds.2012.32.167
##### References:
 [1] V. Berti, M. Fabrizio and C. Giorgi, Well-posedness for solid-liquid phase transitions with a fourth-order nonlinearity, Phys. D, 236 (2007), 13-21. doi: 10.1016/j.physd.2007.07.009. [2] E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Existence and boundedness of solutions for a singular phase field system, J. Differential Equations, 246 (2009), 3260-3295. [3] G. Bonfanti, M. Frémond and F. Luterotti, Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl., 10 (2000), 1-24. [4] G. Bonfanti, M. Frémond and F. Luterotti, Local solutions to the full model of phase transitions with dissipation, Adv. Math. Sci. Appl., 11 (2001), 791-810. [5] M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002. [6] E. Feireisl, Mathematical theory of compressible, viscous, and heat conducting fluids, Comput. Math. Appl., 53 (2007), 461-490. doi: 10.1016/j.camwa.2006.02.042. [7] E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369. doi: 10.1002/mma.1089. [8] M. Fabrizio, Ginzburg-Landau equations and first and second order phase transitions, Internat. J. Engrg. Sci., 44 (2006), 529-539. doi: 10.1016/j.ijengsci.2006.02.006. [9] M. Fabrizio, Ice-water and liquid-vapor phase transitions by a Ginzburg-Landau model, J. Math. Phys., 49 (2008), 102902. doi: 10.1063/1.2992478. [10] P. Germain, "Cours de Méchanique des Milieux Continus. Tome I: Théorie Générale," Masson er Cie, Éditeurs, Paris 1973. [11] B. Guo and P. Zhu, Global existence of smooth solution to nonlinear thermoviscoelastic system with clamped boundary conditions in solid-like materials, Comm. Math. Phys., 203 (1999), 365-383. doi: 10.1007/s002200050617. [12] J. Jiang and Y. Zhang, Counting the set of equilibria for a one-dimensional full model for phase transitions with microscopic movements, Q. Appl. Math., to appear. [13] Ph. Laurençot, G. Schimperna and U. Stefanelli, Global existence of a strong solution to the one-dimensional full model for irreversible phase transitions, J. Math. Anal. Appl., 271 (2002), 426-442. doi: 10.1016/S0022-247X(02)00127-0. [14] F. Luterotti and U. Stefanelli, Existence result for the one-dimensional full model of phase transitions, Z. Anal. Anwendungen, 21 (2002), 335-350. [15] F. Luterotti and U. Stefanelli, Errata and addendum to: "Existence result for the one-dimensional full model of phase transitions", [Z. Anal. Anwendungen, 21 (2002), 335-350], Z. Anal. Anwendungen, 22 (2003), 239-240. [16] F. Luterotti, G. Schimperna and U. Stefanelli, Existence results for a phase transition model based on microscopic movements, Differential Equations: Inverse and Direct Problems, (2006), 245-263. [17] F. Luterotti, G. Schimperna and U. Stefanelli, Existence result for a nonlinear model related to irreversible phase changes, Math. Models Methods Appl. Sci., 11 (2001), 809-825. doi: 10.1142/S0218202501001112. [18] R. Racke and S. Zheng, Global existence and asymptotic behavior in nonlinear thermoviscoelasticity, J. Diff. Equ., 134 (1997), 46-67. doi: 10.1006/jdeq.1996.3216. [19] E. Rocca and R. Rossi, Global existence of strong solutions to the one-dimensional full model for phase transition in thermoviscoelastic materials, Applications of Mathematics, 53 (2008), 485-520. [20] C. Shang, Asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with constant temperature boundary conditions, Asymptot. Anal., 55 (2007), 229-251. [21] C. Shang, Global attractor for the Ginzburg-Landau thermoviscoelastic systems with hinged boundary conditions, J. Math. Anal. Appl., 343 (2008), 1-21. doi: 10.1016/j.jmaa.2008.01.043. [22] W. Shen and S. Zheng, On the coupled Cahn-Hilliard equations, Comm. Partial Differential Equations, 18 (1993), 701-727. [23] W. Shen and S. Zheng, Maximal attractor for the coupled Cahn-Hilliard equations, Nonlinear Anal., 49 (2002), 21-34. doi: 10.1016/S0362-546X(00)00246-7. [24] J. Sprekels and S. Zheng, Maximal attractor for the system of a Landau-Ginzburg theory for structural phase transitions in shape memory alloys, Phys. D, 121 (1998), 252-262. doi: 10.1016/S0167-2789(98)00167-5. [25] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, 68, 1988. [26] H. Wu and S. Zheng, Global attractor for the 1-D thin film equation, Asympt. Anal., 51, 101-111. [27] S. Zheng, "Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems," Pitman Series Monographs and Surveys in Pure and Applied Mathematics, 76, 1995.

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##### References:
 [1] V. Berti, M. Fabrizio and C. Giorgi, Well-posedness for solid-liquid phase transitions with a fourth-order nonlinearity, Phys. D, 236 (2007), 13-21. doi: 10.1016/j.physd.2007.07.009. [2] E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Existence and boundedness of solutions for a singular phase field system, J. Differential Equations, 246 (2009), 3260-3295. [3] G. Bonfanti, M. Frémond and F. Luterotti, Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl., 10 (2000), 1-24. [4] G. Bonfanti, M. Frémond and F. Luterotti, Local solutions to the full model of phase transitions with dissipation, Adv. Math. Sci. Appl., 11 (2001), 791-810. [5] M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002. [6] E. Feireisl, Mathematical theory of compressible, viscous, and heat conducting fluids, Comput. Math. Appl., 53 (2007), 461-490. doi: 10.1016/j.camwa.2006.02.042. [7] E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369. doi: 10.1002/mma.1089. [8] M. Fabrizio, Ginzburg-Landau equations and first and second order phase transitions, Internat. J. Engrg. Sci., 44 (2006), 529-539. doi: 10.1016/j.ijengsci.2006.02.006. [9] M. Fabrizio, Ice-water and liquid-vapor phase transitions by a Ginzburg-Landau model, J. Math. Phys., 49 (2008), 102902. doi: 10.1063/1.2992478. [10] P. Germain, "Cours de Méchanique des Milieux Continus. Tome I: Théorie Générale," Masson er Cie, Éditeurs, Paris 1973. [11] B. Guo and P. Zhu, Global existence of smooth solution to nonlinear thermoviscoelastic system with clamped boundary conditions in solid-like materials, Comm. Math. Phys., 203 (1999), 365-383. doi: 10.1007/s002200050617. [12] J. Jiang and Y. Zhang, Counting the set of equilibria for a one-dimensional full model for phase transitions with microscopic movements, Q. Appl. Math., to appear. [13] Ph. Laurençot, G. Schimperna and U. Stefanelli, Global existence of a strong solution to the one-dimensional full model for irreversible phase transitions, J. Math. Anal. Appl., 271 (2002), 426-442. doi: 10.1016/S0022-247X(02)00127-0. [14] F. Luterotti and U. Stefanelli, Existence result for the one-dimensional full model of phase transitions, Z. Anal. Anwendungen, 21 (2002), 335-350. [15] F. Luterotti and U. Stefanelli, Errata and addendum to: "Existence result for the one-dimensional full model of phase transitions", [Z. Anal. Anwendungen, 21 (2002), 335-350], Z. Anal. Anwendungen, 22 (2003), 239-240. [16] F. Luterotti, G. Schimperna and U. Stefanelli, Existence results for a phase transition model based on microscopic movements, Differential Equations: Inverse and Direct Problems, (2006), 245-263. [17] F. Luterotti, G. Schimperna and U. Stefanelli, Existence result for a nonlinear model related to irreversible phase changes, Math. Models Methods Appl. Sci., 11 (2001), 809-825. doi: 10.1142/S0218202501001112. [18] R. Racke and S. Zheng, Global existence and asymptotic behavior in nonlinear thermoviscoelasticity, J. Diff. Equ., 134 (1997), 46-67. doi: 10.1006/jdeq.1996.3216. [19] E. Rocca and R. Rossi, Global existence of strong solutions to the one-dimensional full model for phase transition in thermoviscoelastic materials, Applications of Mathematics, 53 (2008), 485-520. [20] C. Shang, Asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with constant temperature boundary conditions, Asymptot. Anal., 55 (2007), 229-251. [21] C. Shang, Global attractor for the Ginzburg-Landau thermoviscoelastic systems with hinged boundary conditions, J. Math. Anal. Appl., 343 (2008), 1-21. doi: 10.1016/j.jmaa.2008.01.043. [22] W. Shen and S. Zheng, On the coupled Cahn-Hilliard equations, Comm. Partial Differential Equations, 18 (1993), 701-727. [23] W. Shen and S. Zheng, Maximal attractor for the coupled Cahn-Hilliard equations, Nonlinear Anal., 49 (2002), 21-34. doi: 10.1016/S0362-546X(00)00246-7. [24] J. Sprekels and S. Zheng, Maximal attractor for the system of a Landau-Ginzburg theory for structural phase transitions in shape memory alloys, Phys. D, 121 (1998), 252-262. doi: 10.1016/S0167-2789(98)00167-5. [25] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, 68, 1988. [26] H. Wu and S. Zheng, Global attractor for the 1-D thin film equation, Asympt. Anal., 51, 101-111. [27] S. Zheng, "Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems," Pitman Series Monographs and Surveys in Pure and Applied Mathematics, 76, 1995.
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