Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Well-posedness and longtime behavior of the phase-field model with memory in a history space setting

Authors: Claudio Giorgi, Maurizio Grasselli and Vittorino Pata
Journal: Quart. Appl. Math. 59 (2001), 701-736
MSC: Primary 45K05; Secondary 35B30, 35B40
DOI: https://doi.org/10.1090/qam/1866554
MathSciNet review: MR1866554
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Abstract: A thermodynamically consistent phase-field model with memory, based on the linearized version of the Gurtin-Pipkin heat conduction law, is considered. The formulation of an initial and boundary value problem for the phase-field evolution system is framed in a history space setting. Namely, the summed past history of the temperature is regarded itself as a variable along with the temperature and the phase-field. Well-posedness results are discussed, as well as longtime behavior of solutions. Under suitable conditions, the existence of an absorbing set can be achieved.

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  • [1] S. Aizicovici and V. Barbu, Existence and asymptotic results for a system of integro-partial differential equations, NoDEA Nonlinear Differential Equations Appl. 3, 1-18 (1996) MR 1371092
  • [2] S. M. Allen and J. W. Cahn, A microscopic theory for antiphase motion and its application to antiphase domain coarsening, Acta Metall. 27, 1085-1095 (1979)
  • [3] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei, Noordhoff, Bucure§ti-Leyden, 1976 MR 0390843
  • [4] V. Barbu, A semigroup approach to an infinite delay equation in Hilbert space, in Abstract Cauchy Problems and Functional Differential Equations (F. Kappel and W. Schappacher, eds.), Pitman Res. Notes Math. Ser. 48, London, 1981 MR 617210
  • [5] G. Bonfanti and F. Luterotti, Global solution to a phase-field model with memory and quadratic nonlinearity, Adv. Math. Sci. Appl. 9, 523-538 (1999) MR 1690435
  • [6] G. Bonfanti and F. Luterotti, Regularity and convergence results for a phase-field model with memory, Math. Meth. Appl. Sci. 21, 1085-1105 (1998) MR 1637542
  • [7] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973
  • [8] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal. 96, 205-245 (1985) MR 816623
  • [9] J. W. Cahn and J. E. Hilliard, Free energy of a non-uniform system. I. Interfacial free enerqy, J. Chem. Phys. 28, 258-267 (1957)
  • [10] B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys. 18, 199-208 (1967) MR 0214334
  • [11] P. Colli and Ph. Laurençot, Existence and stabilization of solutions to the phase-field model with memory, J. Integral Equations Appl. 10, 169-194 (1998) MR 1646829
  • [12] P. Colli and Ph. Laurençot, Uniqueness of weak solutions to the phase-field model with memory, J. Math. Sci. Univ. Tokyo 5, 459-476 (1998) MR 1656061
  • [13] P. Colli, G. Gentili, and C. Giorgi, Nonlinear systems describing phase-transition models compatible with thermodynamics, Math. Models Methods. Appl. Sci. 9, 1015-1037 (1999) MR 1710273
  • [14] P. Colli, G. Gilardi, and M. Grasselli, Global smooth solution to the standard phase-field model with memory, Adv. Differential Equations 3, 453-486 (1997) MR 1441852
  • [15] P. Colli, G. Gilardi, and M. Grasselli, Well-posedness of the weak formation for the phase-field model with memory, Adv. Differential Equations 3, 487-508 (1997) MR 1441853
  • [16] P. Colli, G. Gilardi, and M. Grasselli, Convergence of phase field to phase relaxation models with memory, Ann. Univ. Ferrara Ser. VII (N.S.), Suppl. 41, 1-14 (1996) MR 1471011
  • [17] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, 297-308 (1970) MR 0281400
  • [18] M. Fabrizio and C. Giorgi, Sulla termodinamica dei materiali semplici, Boll. Un. Mat. Ital. B 5, 441-464 (1986)
  • [19] G. Gentili and C. Giorgi, Thermodynamic properties and stability for the heat flux equation with linear memory, Quart. Appl. Math. 51, 343-362 (1993) MR 1218373
  • [20] C. Giorgi, M. Grasselli, and V. Pata, Uniform attractors for a phase-field model with memory and quadratic nonlinearity, Indiana Univ. Math. J. 48, 1395-1445 (1999) MR 1757078
  • [21] C. Giorgi, A. Marzocchi, and V. Pata, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Differential Equations Appl. 5, 333-354 (1998) MR 1638908
  • [22] C. Giorgi, A. Marzocchi, and V. Pata, Uniform attractors for a non-autonomous semilinear heat equation with memory, Quart. Appl. Math. 58, 661-683 (2000) MR 1788423
  • [23] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31, 113-126 (1968) MR 1553521
  • [24] B. I. Halperin, P. C. Hoehenberg, and S. K. Ma, Renormalizaton-group methods for critical dynamics: I. Recursion relations and effects of energy conservation, Phys. Rev. B 10, 139-153 (1974)
  • [25] A. Haraux, Systémes dynamiques dissipatifs et applications, Recherches en Mathématiques Appliquées 17, Masson, Paris, 1991 MR 1084372
  • [26] D. D. Joseph and L. Preziosi, Heat waves, Rev. Modern Phys. 61, 41-73 (1989) MR 977943
  • [27] D. D. Joseph and L. Preziosi, Addendum to the paper ``Heat waves'' [Rev. Mod. Phys. 61 (1989), 41-73], Rev. Modern Phys. 62, 375-391 (1990) MR 1056235
  • [28] P. Laurençot, Long-time behaviour for a model of phase-field type, Proc. Roy. Soc. Edinburgh Sect. A 126, 167-185 (1996) MR 1378839
  • [29] J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I, Springer-Verlag, Berlin, 1972 MR 0350177
  • [30] V. Pata, G. Prouse, and M. I. Vishik, Traveling waves of dissipative non-autonomous hyperbolic equations in a strip, Adv. Differential Equations 3, 249-270 (1998) MR 1750416
  • [31] O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D 43, 44-62 (1990) MR 1060043
  • [32] J. Sprekels and S. Zheng, Maximal attractor for the system of a Landau-Ginzburg theory for structural phase transitions in shape memory alloys, Physica D 121, 252-262 (1998) MR 1645431
  • [33] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer-Verlag, New York, 1998 MR 953967
  • [34] A. A. Wheeler and G. B. McFadden, A $ \xi $-vector formulation of anisotropic phase-field models: 3D asymptotics, European J. Appl. Math. 7, 367-381 (1996) MR 1413370

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DOI: https://doi.org/10.1090/qam/1866554
Article copyright: © Copyright 2001 American Mathematical Society

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