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Symmetries and blow-up phenomena for a Dirichlet problem with a large parameter
Solvability and asymptotic analysis of a generalization of the Caginalp phase field system
1. | Dipartimento di Matematica ``F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy |
2. | Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, I-27100 Pavia |
References:
[1] |
V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,'' Noordhoff, Leyden, 1976. |
[2] |
H. Brezis, "Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,'' North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973. |
[3] |
G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.
doi: 10.1007/BF00254827. |
[4] |
P. Colli, G. Gilardi and M. Grasselli, Global smooth solution to the standard phase field model with memory, Adv. Differential Equations, 2 (1997), 453-486. |
[5] |
P. Colli, G. Gilardi and M. Grasselli, Well-posedness of the weak formulation for the phase-field model with memory, Adv. Differential Equations, 2 (1997), 487-508. |
[6] |
G. Duvaut, Résolution d'un problème de Stefan (fusion d'un bloc de glace à zéro degré), C. R. Acad. Sci. Paris S閞. A-B, 276 (1973), A1461-A1463. |
[7] |
M. Frémond, "Non-smooth Thermomechanics,'' Springer-Verlag, Berlin, 2002. |
[8] |
A.E. Green and P.M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Roy. Soc. Lond. A, 432 (1991), 171-194.
doi: 10.1098/rspa.1991.0012. |
[9] |
A.E. Green and P.M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253-264.
doi: 10.1080/01495739208946136. |
[10] |
A.E. Green and P.M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208.
doi: 10.1007/BF00044969. |
[11] |
A.E. Green and P.M. Naghdi, A new thermoviscous theory for fluids, J. Non-Newtonian Fluid Mech., 56 (1995), 289-306.
doi: 10.1016/0377-0257(94)01288-S. |
[12] |
O.A. Ladyženskaja, V.A. Solonnikov, and N.N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'' Trans. Amer. Math. Soc. 23, Amer. Math. Soc., Providence, RI, 1968. |
[13] |
J.L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'' Dunod Gauthier-Villars, Paris, 1969. |
[14] |
A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal., 71 (2009), 2278-2290.
doi: 10.1016/j.na.2009.01.061. |
[15] |
A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894.
doi: 10.1080/00036810903042182. |
[16] |
A. Miranville and R. Quintanilla, A Caginalp phase-field system with a nonlinear coupling, Nonlinear Anal. Real World Appl., 11 (2010), 2849-2861.
doi: 10.1016/j.nonrwa.2009.10.008. |
[17] |
A. Miranville and R. Quintanilla, A type III phase-field system with a logarithmic potential, Appl. Math. Lett., 24 (2011), 1003-1008.
doi: 10.1016/j.aml.2011.01.016. |
[18] |
J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
show all references
References:
[1] |
V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,'' Noordhoff, Leyden, 1976. |
[2] |
H. Brezis, "Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,'' North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973. |
[3] |
G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.
doi: 10.1007/BF00254827. |
[4] |
P. Colli, G. Gilardi and M. Grasselli, Global smooth solution to the standard phase field model with memory, Adv. Differential Equations, 2 (1997), 453-486. |
[5] |
P. Colli, G. Gilardi and M. Grasselli, Well-posedness of the weak formulation for the phase-field model with memory, Adv. Differential Equations, 2 (1997), 487-508. |
[6] |
G. Duvaut, Résolution d'un problème de Stefan (fusion d'un bloc de glace à zéro degré), C. R. Acad. Sci. Paris S閞. A-B, 276 (1973), A1461-A1463. |
[7] |
M. Frémond, "Non-smooth Thermomechanics,'' Springer-Verlag, Berlin, 2002. |
[8] |
A.E. Green and P.M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Roy. Soc. Lond. A, 432 (1991), 171-194.
doi: 10.1098/rspa.1991.0012. |
[9] |
A.E. Green and P.M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253-264.
doi: 10.1080/01495739208946136. |
[10] |
A.E. Green and P.M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208.
doi: 10.1007/BF00044969. |
[11] |
A.E. Green and P.M. Naghdi, A new thermoviscous theory for fluids, J. Non-Newtonian Fluid Mech., 56 (1995), 289-306.
doi: 10.1016/0377-0257(94)01288-S. |
[12] |
O.A. Ladyženskaja, V.A. Solonnikov, and N.N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'' Trans. Amer. Math. Soc. 23, Amer. Math. Soc., Providence, RI, 1968. |
[13] |
J.L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'' Dunod Gauthier-Villars, Paris, 1969. |
[14] |
A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal., 71 (2009), 2278-2290.
doi: 10.1016/j.na.2009.01.061. |
[15] |
A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894.
doi: 10.1080/00036810903042182. |
[16] |
A. Miranville and R. Quintanilla, A Caginalp phase-field system with a nonlinear coupling, Nonlinear Anal. Real World Appl., 11 (2010), 2849-2861.
doi: 10.1016/j.nonrwa.2009.10.008. |
[17] |
A. Miranville and R. Quintanilla, A type III phase-field system with a logarithmic potential, Appl. Math. Lett., 24 (2011), 1003-1008.
doi: 10.1016/j.aml.2011.01.016. |
[18] |
J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
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