On a Caginalp phase-field system with a logarithmic nonlinearity

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• 作者：Charbel Wehbe
• 关键词：Caginalp phase ; field system ; Dirichlet boundary conditions ; well ; posedness ; long time behavior of solution ; global attractor ; exponential attractor ; Maxwell ; Cattaneo law ; logarithmic potential ; 35B40 ; 35B41 ; 35K51 ; 80A22 ; 80A20 ; 35Q53 ; 45K05 ; 35K55 ; 35G30 ; 92D50
• 刊名：Applications of Mathematics
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：60
• 期：4
• 页码：355-382
• 全文大小：242 KB
• 参考文献：[1] A. Babin, B. Nicolaenko: Exponential attractors for reaction-diffusion systems in an unbounded domain. J. Dyn. Differ. Equations 7 (1995), 567鈥?90.MATH MathSciNet CrossRef
  [2] D. Brochet, D. Hilhorst: Universal attractor and inertial sets for the phase field model. Appl. Math. Lett. 4 (1991), 59鈥?2.MATH MathSciNet CrossRef
  [3] D. Brochet, D. Hilhorst, X. Chen: Finite dimensional exponential attractor for the phase field model. Appl. Anal. 49 (1993), 197鈥?12.MATH MathSciNet CrossRef
  [4] D. Brochet, D. Hilhorst, A. Novick-Cohen: Finite-dimensional exponential attractor for a model for order-disorder and phase separation. Appl. Math. Lett. 7 (1994), 83鈥?7.MATH MathSciNet CrossRef
  [5] G. Caginalp: An analysis of a phase field model of a free boundary. Arch. Ration. Mech. Anal. 92 (1986), 205鈥?45.MATH MathSciNet CrossRef
  [6] G. Caginalp: The role of microscopic anisotropy in the macroscopic behavior of a phase boundary. Ann. Phys. 172 (1986), 136鈥?55.MATH MathSciNet CrossRef
  [7] L. Cherfils, A. Miranville: Some results on the asymptotic behavior of the Caginalp system with singular potentials. Adv. Math. Sci. Appl. 17 (2007), 107鈥?29.MATH MathSciNet
  [8] M. Conti, S. Gatti, A. Miranville: Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions. Discrete Contin. Dyn. Syst., Ser. S 5 (2012), 485鈥?05.MATH MathSciNet CrossRef
  [9] P. Fabrie, C. Galusinski: Exponential attractors for partially dissipative reaction system. Asymptotic Anal. 12 (1996), 329鈥?54.MATH MathSciNet
  [10] H. Gajewski, K. Zacharias: Global behaviour of a reaction-diffusion system modelling chemotaxis. Math. Nachr. 195 (1998), 77鈥?14.MATH MathSciNet CrossRef
  [11] M. Grasselli, A. Miranville, V. Pata, S. Zelik: Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials. Math. Nachr. 280 (2007), 1475鈥?509.MATH MathSciNet CrossRef
  [12] A. Kufner, O. John, S. Fu膷铆k: Function Spaces. Monographs and Textsbooks on Mechanics of Solids and Fluids. Mechanics: Analysis. Noordhoff International Publishing, Leyden, Academia, Prague, 1977.
  [13] L. D. Landau, E. M. Lifschitz: Course of Theoretical Physics. Vol. 1. Mechanics. Akademie, Berlin, 1981. (In German.)
  [14] A. Miranville: Exponential attractors for a class of evolution equations by a decomposition method. C. R. Acad. Sci., Paris, S茅r. I, Math. 328 (1999), 145鈥?50. (In English. Abridged French version.)MATH MathSciNet CrossRef
  [15] A. Miranville: On a phase-field model with a logarithmic nonlinearity. Appl. Math., Praha 57 (2012), 215鈥?29.MATH MathSciNet CrossRef
  [16] A. Miranville: Some mathematical models in phase transition. Discrete Contin. Dyn. Syst., Ser. S 7 (2014), 271鈥?06.MATH MathSciNet CrossRef
  [17] A. Miranville, R. Quintanilla: A generalization of the Caginalp phase-field system based on the Cattaneo law. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 2278鈥?290.MATH MathSciNet CrossRef
  [18] A. Miranville, R. Quintanilla: Some generalizations of the Caginalp phase-field system. Appl. Anal. 88 (2009), 877鈥?94.MATH MathSciNet CrossRef
  [19] A. Miranville, S. Zelik: Robust exponential attractors for Cahn-Hilliard type equations with singular potentials. Math. Methods Appl. Sci. 27 (2004), 545鈥?82.MATH MathSciNet CrossRef
  [20] A. Miranville, S. Zelik: Attractors for dissipative partial differential equations in bounded and unbounded domains. Handbook of Differential Equations: Evolutionary Equations. Vol. IV (C. M. Dafermos et al., eds.). Elsevier/North-Holland, Amsterdam, 2008, pp. 103鈥?00.
  [21] R. Temam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences 68, Springer, New York, 1997.MATH
  
• 作者单位：Charbel Wehbe (1)
  
  1. Zarazir station street, New Rawda, Beirut, Lebanon
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Applications of Mathematics
  Mechanics, Fluids and Thermodynamics
  Analysis
  Mathematical and Computational Physics
  Applied Mathematics and Computational Methods of Engineering
  Optimization
  
• 出版者：Springer Netherlands
• ISSN：1572-9109

文摘

We consider a phase field system based on the Maxwell Cattaneo heat conduction law, with a logarithmic nonlinearity, associated with Dirichlet boundary conditions. In particular, we prove, in one and two space dimensions, the existence of a solution which is strictly separated from the singularities of the nonlinear term and that the problem possesses a finite-dimensional global attractor in terms of exponential attractors. Keywords Caginalp phase-field system Dirichlet boundary conditions well-posedness long time behavior of solution global attractor exponential attractor Maxwell-Cattaneo law logarithmic potential