Abstract
We study a Penrose-Fife phase transition model coupled with homogeneous Neumann boundary conditions. Improving previous results, we show that the initial value problem for this model admits a unique solution under weak conditions on the initial data. Moreover, we prove asymptotic regularization properties of weak solutions.
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Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I., Comm. Pure Appl. Math. 12, 623–727 (1959)
Alikakos N.D.: L p bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations. 4, 827–868 (1979)
Baiocchi C.: Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert (Italian). Ann. Mat. Pura Appl. 4(76), 233–304 (1967)
V.Barbu, “Nonlinear Semigroups and Differential Equations in Banach Spaces”, Noordhoff, Leyden, 1976.
Barbu V., Colli P., Gilardi G., Grasselli M.: Existence, uniqueness, and longtime behavior for a nonlinear Volterra integrodifferential equation. Differential Integral Equations. 13, 1233–1262 (2000)
Bonforte M., Vázquez J.L.: Positivity, local smoothing, and Harnack inequalities for very-fast diffusion equations. Adv.Math. 2, 529–578 (2010)
H.Brezis, “Opérateurs Maximaux Monotones et Sémi-groupes de Contractions dans les Espaces de Hilbert”, North-Holland Math. Studies 5, North-Holland, Amsterdam, 1973.
Brezis H.: Integrales convexes dans les espaces de Sobolev. Israel J. Math. 13, 9–23 (1972)
Brezis H., Strauss W.A.: Semi-linear second-order elliptic equations in L 1, J.Math. Soc. Japan. 25, 565–590 (1973)
Colli P., Gilardi G., Rocca E., Schimperna G.: On a Penrose-Fife phase-field model with nonhomogeneous Neumann boundary conditions for the temperature. Differential Integral Equations. 17, 511–534 (2004)
Colli P., Laurençot Ph.: Weak solutions to the Penrose-Fife phase field model for a class of admissible heat flux laws. Phys.D: 111, 311–334 (1998)
P.Colli, Ph.Laurençot, and J.Sprekels, Global solution to the Penrose-Fife phase field model with special heat flux laws, Variations of domain and free-boundary problems in solid mechanics (Paris, 1997), 181–188, Solid Mech. Appl., 66, Kluwer Acad. Publ., Dordrecht, 1999.
Damlamian A., Kenmochi N.: Evolution equations generated by subdifferentials in the dual space of H 1(Ω), Discrete Contin. Dynam. Systems. 5, 269–278 (1999)
Feireisl E., Schimperna G.: Large time behavior of solutions to Penrose-Fife phase change models. Math. Methods Appl. Sci. 28, 2117–2132 (2005)
Gilardi G., Marson A.: On a Penrose-Fife type system with Dirichlet boundary conditions for the temperature. Math. Methods Appl. Sci. 26, 1303–1325 (2003)
Grun-Rehomme M.: Caractérisation du sous-différentiel d’intégrandes convexes dans les espaces de Sobolev (French). J.Math. Pures Appl. 9(56), 149–156 (1977)
Ito A., Kenmochi N., Kubo M.: Non-isothermal phase transition models with Neumann boundary conditions. Nonlinear Anal. 53, 977–996 (2003)
Kenmochi N.: Neumann problems for a class of nonlinear degenerate parabolic equations. Differential Integral Equations. 3, 253–273 (1990)
N.Kenmochi and M.Niezgódka, Systems of nonlinear parabolic equations for phase change problems, Adv. Math. Sci. Appl., 3 (1993/94), 89–117.
O.A.Ladyzhenskaya, V.A.Solonnikov, and N.N. Ural’ceva, “Linear and Quasi-linear Equations of Parabolic Type”, Transl. Math. Monogr., Amer. Math. Soc., Providence, RI, 1968.
Laurençot Ph.: Solutions to a Penrose-Fife model of phase-field type. J. Math. Anal. Appl. 185, 262–274 (1994)
Laurençot Ph.: Weak solutions to a Penrose-Fife model with Fourier law for the temperature. J. Math. Anal. Appl. 219, 331–343 (1998)
Miranville A., Zelik S.: Robust exponential attractors for Cahn-Hilliard type equations with singular potentials. Math. Methods Appl. Sci. 27, 545–582 (2004)
Nochetto R.H., Savaré G.: Nonlinear evolution governed by accretive operators in Banach spaces: error control and applications. Math. Models Methods Appl. Sci. 16, 439–477 (2006)
Penrose O., Fife P.C.: Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Phys. D. 43, 44–62 (1990)
Penrose O., Fife P.C.: On the relation between the standard phase-field model and a “thermodynamically consistent” phase-field model. Phys. D. 69, 107–113 (1993)
J.Prüss and M.Wilke, On conserved Penrose-Fife type models, ArXiv:1002.0928, to appear in Progress in Nonlinear Differential Equations and Their Applications.
Rocca E., Schimperna G.: Universal attractor for some singular phase transition systems. Phys.D. 192, 279–307 (2004)
Schimperna G.: Global and exponential attractors for the Penrose-Fife system. Math. Models Methods Appl. Sci. 19, 969–991 (2009)
R.E.Showalter, “Monotone Operators in Banach space and Nonlinear Partial Differential Equations”. Mathematical Surveys and Monographs, 49. American Mathematical Society, Providence, RI, 1997
Simon J.: Compact sets in the space L p(0,T;B). Ann. Mat. Pura Appl. (4), 146 65–96 (1987)
Sprekels J., Zheng S.: Global smooth solutions to a thermodynamically consistent model of phase-field type in higher space dimensions. J.Math. Anal. Appl. 176, 200–223 (1993)
Vázquez J.L.: “Smoothing and decay estimates for nonlinear diffusion equations”. Oxford University Press, Oxford (2006)
Zheng S.: Global existence for a thermodynamically consistent model of phase field type. Differential Integral Equations. 5, 241–253 (1992)
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Schimperna, G., Segatti, A. & Zelik, S. Asymptotic uniform boundedness of energy solutions to the Penrose-Fife model. J. Evol. Equ. 12, 863–890 (2012). https://doi.org/10.1007/s00028-012-0159-x
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DOI: https://doi.org/10.1007/s00028-012-0159-x