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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 126, Issue 1
  • January 1996, pp. 167-185

Long-time behaviour for a model of phase-field type

  • Ph. Laurençot (a1)
  • DOI: http://dx.doi.org/10.1017/S0308210500030663
  • Published online: 14 November 2011
Abstract

In this paper, we study a model of phase-field type for the kinetics of phase transitions which was considered by Halperin, Hohenberg and Ma and which includes the phase-field equations. We study the well-posedness of the corresponding initial boundary value problem in an open bounded subset in space dimension lower than or equal to 3 and prove that, under suitable conditions, the long-time behaviour of the solutions to this problem is described by a maximal attractor.

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3H. Amann . Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In Function Spaces, Differential Operators and Nonlinear Analysis, eds H. Triebel and H. J. Schmeisser Teubner-Texte Math. 133, 9126 (Stuttgart: Teubner, 1993).

4P. W. Bates and S. Zheng . Inertial manifolds and inertial sets for phase-field equations. J. Dynamics Differential Equations 4 (1992), 375–97.

5D. Brochet , X. Chen and D. Hilhorst . Finite dimensional exponential attractor for the phase field model. Appl. Anal. 49 (1993), 197212.

6D. Brochet and D. Hilhorst . Universal attractor and inertial sets for the phase field model. Appl. Math. Lett. 4(1991), 5962.

7G. Caginalp . An analysis of a phase-field model of a free boundary. Arch. Rational Mech. Anal. 92 (1986), 205–45.

11B. I. Halperin , P. C. Hohenberg and S. K. Ma . Renormalization-group methods for critical dynamics: I. Recursion relations and effects of energy conservation. Phys. Rev. B 10 (1974), 139–53.

13O. Penrose and P. C. Fife . Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Phys. D 43 (1990), 4462.

14R. Temam . Infinite-dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68 (New York: Springer, 1988).

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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