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November  2013, 33(11&12): 5347-5377. doi: 10.3934/dcds.2013.33.5347

Global existence via a multivalued operator for an Allen-Cahn-Gurtin equation

Michel Pierre 1, and  Morgan Pierre 2,

1. 

ENS Cachan Bretagne, IRMAR, EUB, Campus de Ker Lann, 35170 Bruz
2. 

Université de Poitiers, Laboratoire de Mathématiques, et Applications UMR CNRS 7348, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil, France

Received  March 2012 Revised  December 2012 Published  May 2013

The main goal of this paper is to prove existence of global solutions in time for an Allen-Cahn-Gurtin model of pseudo-parabolic type. Local solutions were known to ``blow up" in some sense in finite time. It is proved that the equation is actually governed by a monotone-like operator. It turns out to be multivalued and measure-valued. The measures are singular with respect to the Lebesgue measure. This operator allows to extend the local solutions globally in time and to fully solve the evolution problem. The asymptotic behavior is also analyzed.
Keywords: singular measures., Pseudo-parabolic, capacity, global existence, monotone operator, Allen-Cahn, blow-up.
Mathematics Subject Classification: Primary: 35K85, 35D99; Secondary: 65M6.
Citation: Michel Pierre, Morgan Pierre. Global existence via a multivalued operator for an Allen-Cahn-Gurtin equation. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5347-5377. doi: 10.3934/dcds.2013.33.5347
References:
[1]

H. Abels and M. Wilke, Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 67 (2007), 3176-3193. doi: 10.1016/j.na.2006.10.002.  Google Scholar

[2]

H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland Publishing Co., Amsterdam, 1973.  Google Scholar

[3]

H. Brezis and F. Browder, Sur une propriété des espaces de Sobolev, C. R. Acad. Sci. Paris, 287 (1978), 113-115.  Google Scholar

[4]

T. Cazenave and A. Haraux, "Introduction aux Problèmes D'évolution Semi-Linéaires," Mathématiques & Applications (Paris), 1, Ellipses, Paris, 1990.  Google Scholar

[5]

L. Cherfils and A. Miranville, Finite dimensional attractors for a model of Allen-Cahn equation based on a microforce balance, C. R. Acad. Sci. Paris, Sér. I, Math., 329 (1999), 1109-1114. doi: 10.1016/S0764-4442(00)88483-9.  Google Scholar

[6]

L. Cherfils and Mo. Pierre, Non-global existence for an Allen-Cahn-Gurtin equation with logarithmic free energy, J. Evol. Equ., 8 (2008), 727-748. doi: 10.1007/s00028-008-0412-5.  Google Scholar

[7]

E. DiBenedetto and Mi. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J., 30 (1981), 821-854. doi: 10.1512/iumj.1981.30.30062.  Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 2001.  Google Scholar

[9]

M. Grun-Rehomme, Caractérisation du sous-différentiel d'intégrandes convexes dans les espaces de Sobolev, J. Math. Pures et Appl., 56 (1977), 149-156.  Google Scholar

[10]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[11]

A. Henrot et Mi. Pierre, "Variation et Optimisation de Formes: Une Analyse Géométrique," Mathématiques & Applications 48, Springer, 2005.  Google Scholar

[12]

J.-L. Lions, "Quelques Méthodes de Résolution de Problèmes aux Limites non Linéaires," Dunod., 1969.  Google Scholar

[13]

A. Miranville, A model of Cahn-Hilliard equation based on a microforce balance, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1247-1252. doi: 10.1016/S0764-4442(99)80448-0.  Google Scholar

[14]

V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," Springer Series in Computational Mathematics, 25, Springer-Verlag, Berlin, 2006.  Google Scholar

show all references

References:
[1]

H. Abels and M. Wilke, Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 67 (2007), 3176-3193. doi: 10.1016/j.na.2006.10.002.  Google Scholar

[2]

H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland Publishing Co., Amsterdam, 1973.  Google Scholar

[3]

H. Brezis and F. Browder, Sur une propriété des espaces de Sobolev, C. R. Acad. Sci. Paris, 287 (1978), 113-115.  Google Scholar

[4]

T. Cazenave and A. Haraux, "Introduction aux Problèmes D'évolution Semi-Linéaires," Mathématiques & Applications (Paris), 1, Ellipses, Paris, 1990.  Google Scholar

[5]

L. Cherfils and A. Miranville, Finite dimensional attractors for a model of Allen-Cahn equation based on a microforce balance, C. R. Acad. Sci. Paris, Sér. I, Math., 329 (1999), 1109-1114. doi: 10.1016/S0764-4442(00)88483-9.  Google Scholar

[6]

L. Cherfils and Mo. Pierre, Non-global existence for an Allen-Cahn-Gurtin equation with logarithmic free energy, J. Evol. Equ., 8 (2008), 727-748. doi: 10.1007/s00028-008-0412-5.  Google Scholar

[7]

E. DiBenedetto and Mi. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J., 30 (1981), 821-854. doi: 10.1512/iumj.1981.30.30062.  Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 2001.  Google Scholar

[9]

M. Grun-Rehomme, Caractérisation du sous-différentiel d'intégrandes convexes dans les espaces de Sobolev, J. Math. Pures et Appl., 56 (1977), 149-156.  Google Scholar

[10]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[11]

A. Henrot et Mi. Pierre, "Variation et Optimisation de Formes: Une Analyse Géométrique," Mathématiques & Applications 48, Springer, 2005.  Google Scholar

[12]

J.-L. Lions, "Quelques Méthodes de Résolution de Problèmes aux Limites non Linéaires," Dunod., 1969.  Google Scholar

[13]

A. Miranville, A model of Cahn-Hilliard equation based on a microforce balance, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1247-1252. doi: 10.1016/S0764-4442(99)80448-0.  Google Scholar

[14]

V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," Springer Series in Computational Mathematics, 25, Springer-Verlag, Berlin, 2006.  Google Scholar

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