# American Institute of Mathematical Sciences

2019, 26: 36-53. doi: 10.3934/era.2019.26.004

## On higher-order anisotropic perturbed Caginalp phase field systems

 Faculté des Sciences et Techniques, Université Marien Ngouabi, B.P 69, Brazzaville, Congo

* Corresponding author: Clesh Deseskel Elion Ekohela

Received  March 2019 Revised  June 2019 Published  July 2019

Our aim in this paper is to study the existence and uniqueness of solution for hyperbolic relaxations of higher-order anisotropic Caginalp phase field systems with homogeous Dirichlet boundary conditions with regular potentials.

Citation: Clesh Deseskel Elion Ekohela, Daniel Moukoko. On higher-order anisotropic perturbed Caginalp phase field systems. Electronic Research Announcements, 2019, 26: 36-53. doi: 10.3934/era.2019.26.004
##### References:
 [1] S. Agmon, Lectures on Elliptic Boundary Value Problems, princeton, 1965.  Google Scholar [2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I. Com. on Pure and Appl. Math., 12 (1959), 623–727. doi: 10.1002/cpa.3160120405.  Google Scholar [3] S. Aizicovici and E. Feireisl, Long-time stabilization of solutions to a phase-field model with memory, Journal of Evolution Equations, 1 (2001), 69–84. doi: 10.1007/PL00001365.  Google Scholar [4] S. Aizicovici, E. Feireisl and F. Issard-Roch, Long-time convergence of solutions to a phase-field system, Mathematical Methods in the Applied Sciences, 24 (2001), 277–287. doi: 10.1002/mma.215.  Google Scholar [5] D. Brochet, D. Hilhorst, A. Novick-Cohen, et al., Maximal attractor and inertial sets for a conserved phase field model, Advances in Differential Equations, 1 (1996), 547–578.  Google Scholar [6] G. Caginalp, An analysis of a phase field model of a free boundary, Archive for Rational Mechanics and Analysis, 92 (1986), 205–245. doi: 10.1007/BF00254827.  Google Scholar [7] G. Caginalp, Conserved-phase field system, Implications for Kinetic Undercooling. Physical, Review B, 38 (1988), 789. Google Scholar [8] G. Caginalp, The dynamics of a conserved phase field system: Stefan-like, hele-shaw, and cahn-hilliard models as asymptotic limits, IMA Journal of Applied Mathematics, 44 (1990), 77–94. doi: 10.1093/imamat/44.1.77.  Google Scholar [9] G. Caginalp and E. Esenturk, Anisotropic phase field equations of arbitrary order, Discrete and Continuous Dynamical Systems-S, 4 (2011), 311–350. doi: 10.3934/dcdss.2011.4.311.  Google Scholar [10] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system, i. interfacial free energy, The Journal of Chemical Physics, 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.  Google Scholar [11] P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Zeitschrift Für Angewandte Mathematik und Physik (ZAMP), 19 (1968), 614-627.  doi: 10.1007/BF01594969.  Google Scholar [12] X. Chen, G. Caginalp and E. Esenturk, Interface conditions for a phase field model with anisotropic and non-local interactions, Archive for Rational Mechanics and Analysis, 202 (2011), 349–372. doi: 10.1007/s00205-011-0429-8.  Google Scholar [13] L. Cherfils and A. Miranville, On the caginalp system with dynamic boundary conditions and singular potentials, Applications of Mathematics, 54 (2009), 89–115. doi: 10.1007/s10492-009-0008-6.  Google Scholar [14] G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. Ⅰ. Macroscopic limits, Journal of Statistical Physics, 87 (1997), 37–61. doi: 10.1007/BF02181479.  Google Scholar [15] G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions Ⅱ: Interface motion, SIAM Journal on Applied Mathematics, 58 (1998), 1707–1729. doi: 10.1137/S0036139996313046.  Google Scholar [16] M. Grasselli and H. Wu, Well-posedness and long-time behavior for the modified phase-field crystal equation, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2743–2783. doi: 10.1142/S0218202514500365.  Google Scholar [17] R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Physica D: Nonlinear Phenomena, 63 (1993), 410-423.  doi: 10.1016/0167-2789(93)90120-P.  Google Scholar [18] C. Laurence, A. Miranville and S. Peng, Higher-order models in phase separation, Journal of Applied Analysis and Computation, 7 (2017), 39–56.  Google Scholar [19] A. Miranville, Some mathematical models in phase transition, Discrete and Continuous Dynamical Systems-S, 7 (2014), 271–306. doi: 10.3934/dcdss.2014.7.271.  Google Scholar [20] A. Miranville, Higher-order anisotropic caginalp phase-field systems, Mediterranean Journal of Mathematics, 13 (2016), 4519–4535. doi: 10.1007/s00009-016-0760-2.  Google Scholar [21] A. Miranville, On higher-order anisotropic conservative caginalp phase-field systems, Applied Mathematics and Optimization, 77 (2018), 297–314. doi: 10.1007/s00245-016-9375-z.  Google Scholar [22] A. Miranville and R. Quintanilla, A Caginalp phase field system based on type Ⅲ heat conduction with two temperatures, Quarterly of Applied Mathematics, 74 (2016), 375–398. doi: 10.1090/qam/1430.  Google Scholar [23] A. J. Ntsokongo, On higher-order anisotropic caginalp phase-field systems with polynomial nonlinear terms, J. Appl. Anal. Comput, 7 (2017), 992–1012.  Google Scholar [24] R. Quintanilla, A well-posed problem for the dual-phase-lag heat conduction, Journal of Thermal Stresses, 31 (2008), 260-269.   Google Scholar [25] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

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##### References:
 [1] S. Agmon, Lectures on Elliptic Boundary Value Problems, princeton, 1965.  Google Scholar [2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I. Com. on Pure and Appl. Math., 12 (1959), 623–727. doi: 10.1002/cpa.3160120405.  Google Scholar [3] S. Aizicovici and E. Feireisl, Long-time stabilization of solutions to a phase-field model with memory, Journal of Evolution Equations, 1 (2001), 69–84. doi: 10.1007/PL00001365.  Google Scholar [4] S. Aizicovici, E. Feireisl and F. Issard-Roch, Long-time convergence of solutions to a phase-field system, Mathematical Methods in the Applied Sciences, 24 (2001), 277–287. doi: 10.1002/mma.215.  Google Scholar [5] D. Brochet, D. Hilhorst, A. Novick-Cohen, et al., Maximal attractor and inertial sets for a conserved phase field model, Advances in Differential Equations, 1 (1996), 547–578.  Google Scholar [6] G. Caginalp, An analysis of a phase field model of a free boundary, Archive for Rational Mechanics and Analysis, 92 (1986), 205–245. doi: 10.1007/BF00254827.  Google Scholar [7] G. Caginalp, Conserved-phase field system, Implications for Kinetic Undercooling. Physical, Review B, 38 (1988), 789. Google Scholar [8] G. Caginalp, The dynamics of a conserved phase field system: Stefan-like, hele-shaw, and cahn-hilliard models as asymptotic limits, IMA Journal of Applied Mathematics, 44 (1990), 77–94. doi: 10.1093/imamat/44.1.77.  Google Scholar [9] G. Caginalp and E. Esenturk, Anisotropic phase field equations of arbitrary order, Discrete and Continuous Dynamical Systems-S, 4 (2011), 311–350. doi: 10.3934/dcdss.2011.4.311.  Google Scholar [10] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system, i. interfacial free energy, The Journal of Chemical Physics, 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.  Google Scholar [11] P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Zeitschrift Für Angewandte Mathematik und Physik (ZAMP), 19 (1968), 614-627.  doi: 10.1007/BF01594969.  Google Scholar [12] X. Chen, G. Caginalp and E. Esenturk, Interface conditions for a phase field model with anisotropic and non-local interactions, Archive for Rational Mechanics and Analysis, 202 (2011), 349–372. doi: 10.1007/s00205-011-0429-8.  Google Scholar [13] L. Cherfils and A. Miranville, On the caginalp system with dynamic boundary conditions and singular potentials, Applications of Mathematics, 54 (2009), 89–115. doi: 10.1007/s10492-009-0008-6.  Google Scholar [14] G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. Ⅰ. Macroscopic limits, Journal of Statistical Physics, 87 (1997), 37–61. doi: 10.1007/BF02181479.  Google Scholar [15] G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions Ⅱ: Interface motion, SIAM Journal on Applied Mathematics, 58 (1998), 1707–1729. doi: 10.1137/S0036139996313046.  Google Scholar [16] M. Grasselli and H. Wu, Well-posedness and long-time behavior for the modified phase-field crystal equation, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2743–2783. doi: 10.1142/S0218202514500365.  Google Scholar [17] R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Physica D: Nonlinear Phenomena, 63 (1993), 410-423.  doi: 10.1016/0167-2789(93)90120-P.  Google Scholar [18] C. Laurence, A. Miranville and S. Peng, Higher-order models in phase separation, Journal of Applied Analysis and Computation, 7 (2017), 39–56.  Google Scholar [19] A. Miranville, Some mathematical models in phase transition, Discrete and Continuous Dynamical Systems-S, 7 (2014), 271–306. doi: 10.3934/dcdss.2014.7.271.  Google Scholar [20] A. Miranville, Higher-order anisotropic caginalp phase-field systems, Mediterranean Journal of Mathematics, 13 (2016), 4519–4535. doi: 10.1007/s00009-016-0760-2.  Google Scholar [21] A. Miranville, On higher-order anisotropic conservative caginalp phase-field systems, Applied Mathematics and Optimization, 77 (2018), 297–314. doi: 10.1007/s00245-016-9375-z.  Google Scholar [22] A. Miranville and R. Quintanilla, A Caginalp phase field system based on type Ⅲ heat conduction with two temperatures, Quarterly of Applied Mathematics, 74 (2016), 375–398. doi: 10.1090/qam/1430.  Google Scholar [23] A. J. Ntsokongo, On higher-order anisotropic caginalp phase-field systems with polynomial nonlinear terms, J. Appl. Anal. Comput, 7 (2017), 992–1012.  Google Scholar [24] R. Quintanilla, A well-posed problem for the dual-phase-lag heat conduction, Journal of Thermal Stresses, 31 (2008), 260-269.   Google Scholar [25] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar
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