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Well-posedness for a phase-field transition system endowed with a polynomial nonlinearity and a general class of nonlinear dynamic boundary conditions

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Abstract

This work is devoted to the study of a phase-field transition system of Caginalp type endowed with a general polynomial nonlinearity and a general class of nonlinear and nonhomogeneous dynamic boundary conditions (in both unknown functions). The existence, uniqueness and regularity of solutions are established. Here we extend several results proved by some authors, including the already studied boundary conditions, which makes the present mathematical model capable of revealing the complexity of a wide class of physical phenomena (for instance, phase change in Ω at the boundary of Ω).

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References

  1. Allen S., Cahn J. W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metallurgica 27, 1084–1095 (1979)

    Article  Google Scholar 

  2. Benincasa T., Favini A., Moroşanu C.: A product formula approach to a non-homogeneous boundary optimal control problem governed by nonlinear phase-field transition system. Part I: A phase-field model. J. Optim. Theory Appl. 148, 14–30 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bird R. B., Stewart W. E., Lightfoot E. N.: Transport Phenomena. John Wiley, New York (2002)

    Google Scholar 

  4. Boldrini J. L., Caretta B. M. C., Fernández-Cara E.: Analysis of a twophase field model for the solidification of an alloy. J. Math. Anal. Appl. 357, 25–44 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caginalp G., Chen X.: Convergence of the phase field model to its sharp interface limits. European J. Appl. Math. 9, 417–445 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Calatroni L., Colli P.: Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions. Nonlinear Anal. 79, 12–27 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cârjă O., Miranville A., Moroşanu C.: On the existence, uniqueness and regularity of solutions to the phase-field system with a general regular potential and a general class of nonlinear and non-homogeneous boundary conditions. Nonlinear Anal. 113, 190–208 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carl S., Le V. K., Motreanu D.: Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications. Springer Monogr. Math., Springer, New York (2007)

    Google Scholar 

  9. Cavaterra C., Gal C., Grasselli M., Miranville A.: Phase-field systems with nonlinear coupling and dynamic boundary conditions. Nonlinear Anal. 72, 2375–2399 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Conti M., Gatti S., Miranville A.: Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions. Discrete Contin. Dyn. Syst. Ser. S 5, 485–505 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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. 95, Birkhäuser, Basel, 1990, 46–58.

  12. Fonseca I., Gangbo W.: Degree Theory in Analysis and Applications. Clarendon, Oxford (1995)

    MATH  Google Scholar 

  13. Gal C., Grasselli M.: On the asymptotic behavior of the Caginalp system with dynamic boundary conditions. Commun. Pure Appl. Anal. 8, 689–710 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gatti S., Miranville A.: Asymptotic behavior of a phase-field system with dynamic boundary conditions. In: Differential Equations: Inverse and Direct Problems, Lecture Notes Pure Appl. Math. 251, pp. 149–170. Chapman & Hall/CRC, Boca Raton, FL (2006)

    Chapter  Google Scholar 

  15. O. A. Ladyzhenskaya, B. A. Solonnikov and N. N. Uraltzava, Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1995.

  16. Lions J.L.: Control of Distributed Singular Systems. Gauthier-Villars, Paris (1985)

    MATH  Google Scholar 

  17. A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field transition system with non-homogeneous Cauchy-Neumann and nonlinear dynamic boundary conditions. Appl. Math. Modeling, doi:10.1016/j.apm.2015.04.039.

  18. Miranville A., Zelik S.: Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions. Math. Methods Appl. Sci. 28, 709–735 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. C. Moroşanu, Analysis and Optimal Control of Phase-Field Transition System: Fractional Steps Methods. Bentham Science Publishers, 2012; available at http://dx.doi.org/10.2174/97816080535061120101.

  20. Moroşanu C., Croitoru A.: Analysis of an iterative scheme of fractional steps type associated to the phase-field equation endowed with a general nonlinearity and Cauchy-Neumann boundary conditions. J. Math. Anal. Appl. 425, 1225–1239 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Moroşanu C., Motreanu D.: A generalized phase-field system. J. Math. Anal. Appl. 237, 515–540 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Moroşanu C., Motreanu D.: The phase field system with a general nonlinearity. Int. J. Differ. Equ. Appl. 1, 187–204 (2000)

    MathSciNet  MATH  Google Scholar 

  23. Pao C. V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992)

    MATH  Google Scholar 

  24. Penrose O., Fife P. C.: Thermodynamically consistent models of phase-field type for kinetics of phase transitions. Phys. D 43, 44–62 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  25. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. 2nd ed., Appl. Math. Sci., Springer, New York, 1997.

  26. Vaz C. L. D., Boldrini J. L.: A mathematical analysis of a nonisothermal Allen-Cahn type system. Math. Methods Appl.Sci. 35, 1392–1405 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Vaz C.L.D., Boldrini J.L.: A mathematical analysis of a nonisothermal Allen-Cahn type system: Error estimates. Math. Methods Appl. Sci. 35, 1406–1414 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. J. Zheng, J. Liu and H. Liu, State-constrained optimal control of phase-field equations with obstacle. Bound. Value Prob. 2013 (2013), doi: 10.1186/1687-2770-2013-234, 13 pages.

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  1. Faculty of Mathematics, “Al. I. Cuza” University, Bd. Carol I, No. 11, 700506, Iaşi, Romania

    Costică Moroşanu

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  1. Costică Moroşanu
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Moroşanu, C. Well-posedness for a phase-field transition system endowed with a polynomial nonlinearity and a general class of nonlinear dynamic boundary conditions. J. Fixed Point Theory Appl. 18, 225–250 (2016). https://doi.org/10.1007/s11784-015-0274-8

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