American Institute of Mathematical Sciences

Advanced
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 and 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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[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.

[11]

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

[12]

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

[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.

[14]

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

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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[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.

[11]

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

[12]

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

[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.

[14]

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

[1]

Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051
[2]

Wenjun Liu, Jiangyong Yu, Gang Li. Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4337-4366. doi: 10.3934/dcdss.2021121
[3]

Xiaoli Zhu, Fuyi Li, Ting Rong. Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2465-2485. doi: 10.3934/cpaa.2015.14.2465
[4]

Yinghua Li, Shijin Ding, Mingxia Huang. Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1507-1523. doi: 10.3934/dcdsb.2016009
[5]

Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2291-2300. doi: 10.3934/dcdsb.2017096
[6]

Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042
[7]

Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058
[8]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[9]

Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535
[10]

Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3633-3651. doi: 10.3934/dcdsb.2018308
[11]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[12]

Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086
[13]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072
[14]

Hirokazu Ninomiya, Masaharu Taniguchi. Global stability of traveling curved fronts in the Allen-Cahn equations. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 819-832. doi: 10.3934/dcds.2006.15.819
[15]

Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823
[16]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations and Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517
[17]

Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661
[18]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[19]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations and Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012
[20]

Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy