AIMS Mathematics, 2016, 1(1): 24-42. doi: 10.3934/Math.2016.1.24

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

On the Caginalp phase-field system based on the Cattaneo law with nonlinear coupling

1 Universit´e des Sciences et techniques de Masuku, Franceville, Gabon
2 Universit´e de Poitiers, Laboratoire de Math´ematiques et Applications, UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie - T´el´eport 2, F-86962 Chasseneuil Futuroscope Cedex, France

We focus in this paper on a Caginalp phase-field system based on the Cattaneo law with nonlinear coupling. We start our analysis by establishing existence, uniqueness and regularity based on Moser’s iterations. We finish with the study of the spatial behavior of the solutions in a semi-infinite cylinder, assuming the existence of such solutions.
  Figure/Table
  Supplementary
  Article Metrics

References

1. G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Ration. Mech. Anal., 92 (1986): 205-245.

2. S. Aizicovici, E. Feireisl, Long-time stabilization of solutions to a phase-field model with memory. J. Evol. Equ., 1 (2001), 69-84.

3. S. Aizicovici, E. Feireisl, Long-time convergence of solutions to a phase-field system, Math. Methods Appl. Sci., 24 (2001), 277-287.

4. D. Brochet, X. Chen, and D. Hilhorst, Finite dimensional exponential attractors for the phase-field model, Appl. Anal., 49 (1993), 197-212.

5. M. Brokate, J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996.

6. L. Cherfils, A. Miranville, Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl.,(2007), 17107-129.

7. L. Cherfils, A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.

8. R. Chill, E. Fasangov´a, and J. Pr¨uss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462.

9. C.I. Christov, P.M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 94 (2005), 154-301.

10. J.N. Flavin, R.J. Knops, and L.E. Payne, Decay estimates for the constrained elastic cylinder of variable cross-section, Quart. Appl. Math., 47 (1989), 325-350.

11. S. Gatti, A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in: Di erential Equations: Inverse and Direct Problems (Proceedings of the workshop “Evolution Equations: Inverse and Direct Problems ”, Cortona, June 21-25, 2004), in A. Favini, A. Lorenzi (Eds), A Series of Lecture Notes in Pure and Applied Mathematics, 251 (2006), 149-170.

12. C. Giorgi, M. Grasselli, and V. Pata, Uniform attractors for a phase-field model with memory and quadratic nonlinearity, Indiana Univ. Math. J., 48 (1999), 1395-1446.

13. M. Grasseli, 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), 1475-1509.

14. M. Grasselli, On the large time behavior of a phase-field system with memory, Asymptot. Anal., 56 (2008), 229-249.

15. M. Grasselli, V. Pata, Robust exponential attractors for a phase-field system with memory J. Evol. Equ., 5 (2005), 465-483.

16. M. Grasselli, H. Petzeltov´a, and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potentials, Z. Anal. Anwend., 25 (2006), 51-73.

17. M. Grasselli, H. Wu, and S. Zheng, Asymptotic behavior of a non-isothermal Ginzburg-Landau model, Quart. Appl. Math., 66 (2008), 743-770.

18. A.E. Green, P.M. Naghdi, A new thermoviscous theory for fluids, J. Non-Newtonian Fluid Mech., 56 (1995), 289-306.

19. A.E. Green, P.M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Roy. Soc. Lond. A., 432 (1991), 171-194.

20. A.E. Green, P.M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal. Stresses, 15 (1992), 253-264.

21. J. Jiang, Convergence to equilibrium for a parabolic-hyperbolic phase-field model with Cattaneo heat flux law, J. Math. Anal. Appl., 341 (2008), 149-169.

22. J. Jiang, Convergence to equilibrium for a fully hyperbolic phase field model with Cattaneo heat flux law, Math. Methods Appl. Sci., 32 (2009), 1156-1182.

23. Ph. Laurenc¸ot, Long-time behaviour for a model of phase-field type, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 167-185.

24. A. Miranville, R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894.

25. A. Miranville, R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal. TMA., 71 (2009), 2278-2290.

26. A. Miranville, R. Quintanilla, A Caginalp phase-field system with a nonlinear coupling. Nonlinear Anal.: Real World Applications, 11 (2010), 2849-2861.

27. A. Miranville, S. Zelik, Robust exponential attractors for singularly perturbed phase-field type equations, Electron. J. Diff. Equ., (2002), 1-28.

28. A. Miranville, S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in: C.M. Dafermos, M. Pokorny (Eds.) Handbook of Differential Equations, Evolutionary Partial Differential Equations. Elsevier, Amsterdam, 2008.

29. A. Novick-Cohen, A phase field system with memory: Global existence, J. Int. Equ. Appl. 14 (2002), 73-107.

30. R. Quintanilla, On existence in thermoelasticity without energy dissipation, J. Thermal. Stresses, 25 (2002), 195-202.

31. R. Quintanilla, End e ects in thermoelasticity, Math. Methods Appl. Sci.. 24 (2001), 93-102.

32. R. Quintanilla, R. Racke, Stability in thermoelasticity of type III, Discrete Contin. Dyn. Syst. B, 3 (2003), 383-400.

33. R. Quintanilla, Phragm´en-Lindel¨of alternative for linear equations of the anti-plane shear dynamic problem in viscoelasticity, Dynam. Contin. Discrete Impuls. Systems, 2 (1996), 423-435.

34. R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, second edition, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997.

35. Z. Zhang, Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Comm. Pure Appl. Anal., 4 (2005), 683-693.

Copyright Info: © 2016, Alain Miranville, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved