Abstract

A phase-field model based on the Gurtin-Pipkin heat flux law is considered. The resulting system has been investigated by Colli and Lauren\c cot who proved existence and uniqueness results, when the time relaxation coefficient is strictly positive. The aim of this paper is the study of the asymptotic behaviour of such a solution, as the time relaxation goes to zero, and of the related limit problem

Author information

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Giovanna Bonfanti

Dip. di Elettronica per l'Automazione, Università di Brescia, Via Branze 38, 25123 Brescia, Italy

Fabio Luterotti

Dip. di Elettronica per l'Automazione, Università di Brescia, Via Branze 38, 25123 Brescia, Italy

References

  1. S. Aizicovici, V. Barbu: Existence and asymptotic results for a system of integro-partial differential equations, NoDEA Nonlinear Differential Equations Appl. 3 (1996) 1-18.https://doi.org/10.1007/BF01194214Google Scholar
  2. C. Baiocchi: Sulle equazioni differenziali astratte lineari del primo e del secondo ordine in spazi di Hilbert, Ann. Mat. Pura Appl. 76(4) (1967) 233-304.https://doi.org/10.1007/BF02412236Google Scholar
  3. V. Barbu: Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.https://doi.org/10.1007/978-94-010-1537-0Google Scholar
  4. G. Bonfanti, F. Luterotti: Global solution to a phase-field model with memory and quadratic nonlinearity, Adv. Math. Sci. Appl., to appear.Google Scholar
  5. G. Bonfanti, F. Luterotti: Regularity and convergence results for a phase-field model with memory, Math. Meth. Appl. Sci. 21 (1998) 1085-1105.https://doi.org/10.1002/(SICI)1099-1476(199808)21:12<1085::AID-MMA986>3.0.CO;2-6Google Scholar
  6. H. Brézis: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Math. Studies 5, North-Holland, Amsterdam, 1973.Google Scholar
  7. G. Caginalp: An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal. 92 (1986) 205-245.https://doi.org/10.1007/BF00254827Google Scholar
  8. L. Chiusano, P. Colli: Positive kernels and zero time relaxation in a phase-field model with memory, Comm. Appl. Anal. 2 (1998) 531-550.Google Scholar
  9. P. Colli, G. Gilardi, M. Grasselli: Global smooth solution to the standard phase-field model with memory, Advances Differential Equations 2 (1997) 453-486.https://doi.org/10.57262/ade/1366742252Google Scholar
  10. P. Colli, G. Gilardi, M. Grasselli: Well-posedness of the weak formulation for the phase-field model with memory, Advances Differential Equations 2 (1997) 487-508.https://doi.org/10.57262/ade/1366742253Google Scholar
  11. P. Colli, G. Gilardi, M. Grasselli: Convergence of phase field to phase relaxation models with memory, Ann. Univ. Ferrara Sez. VII (N.S.) Sc. Mat. 41 (1995) 1-14.https://doi.org/10.1007/BF02825252Google Scholar
  12. P. Colli, G. Gilardi, M. Grasselli: Asymptotic analysis of a phase field model with memory for vanishing time relaxation, Hiroshima Math. J., to appear.Google Scholar
  13. G. J. Fix: Phase field models for free boundary problems, In: A. Fasano, M. Primicerio (eds.): Free Boundary Problems: Theory and Applications II, Pitman Res. Notes Math. Ser. 79, Longman, London (1983) 580-589.Google Scholar
  14. M. Frémond, A. Visintin: Dissipation dans le changement de phase. Surfusion. Changement de phase irréversible, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 301 (1985) 1265-1268.Google Scholar
  15. G. Gentili, C. Giorgi: Thermodynamic properties and stability for the heat flux equation with linear memory, Quart. Appl. Math. 51 (1993) 343-362.https://doi.org/10.1090/qam/1218373Google Scholar
  16. M. E. Gurtin, A. C. Pipkin: A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968) 113-126.https://doi.org/10.1007/BF00281373Google Scholar
  17. O. Penrose, P. C. Fife: Thermodynamically consistent models of phase field type for the kinetics of phase transitions, Physica D 43 (1990) 44-62.https://doi.org/10.1016/0167-2789(90)90015-HGoogle Scholar

Suggested citation

G. Bonfanti, F. Luterotti. “Asymptotic Analysis to a Phase-Field Model with a Nonsmooth Memory Kernel.” Journal of Convex Analysis 6 (1999), No. 1, 41–57. https://doi.org/10.68381/jca06004

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Published by Heldermann Verlag, 1999. Rights now held by Banach Press LLC.