American Institute of Mathematical Sciences

Advanced
November  2013, 12(6): 2811-2827. doi: 10.3934/cpaa.2013.12.2811

Well-posedness and long time behavior of an Allen-Cahn type equation

Haydi Israel 1,

1. 

UMR 6086 CNRS. Laboratoire de Mathématiques et Applications - Université de Poitiers, SP2MI - Boulevard Marie et Pierre Curie - Téléport 2, BP30179 - 86962 Futuroscope Chasseneuil Cedex, France

Received  August 2011 Revised  January 2012 Published  May 2013

The aim of this article is to study the existence and uniqueness of solutions for an equation of Allen-Cahn type and to prove the existence of the finite-dimensional global attractor as well as the existence of exponential attractors.
Keywords: global attractor, exponential attractor., Allen-Cahn equation, continuity.
Mathematics Subject Classification: 37L30, 35B45, 35A05, 35B4.
Citation: Haydi Israel. Well-posedness and long time behavior of an Allen-Cahn type equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2811-2827. doi: 10.3934/cpaa.2013.12.2811
References:
[1]

A. Bonfoh and A. Miranville, On Cahn-Hilliard-Gurtin equations,, in, 47 (2001), 3455.  doi: 10.1016/S0362-546X(01)00463-1.  Google Scholar

[2]

M. Carrive, A. Miranville, A. Piétrus and J. M. Rakotoson, Weakly coupled dynamical systems and applications,, Asymptotic Analysis, 30 (2002), 161.   Google Scholar

[3]

A. Eden, C. Foias, B. Nicolaenko and R. and Temam, "Exponential Attractors for Dissipative Evolution Equations,", Masson, (1994).   Google Scholar

[4]

M.Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system,, Math. Nachr., 272 (2004), 11.   Google Scholar

[5]

G. Karali, and A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution,, J. Differential Equations, 235 (2007), 418.  doi: 10.1016/j.jde.2006.12.021.  Google Scholar

[6]

G. Karali and T. Ricciardi, On the convergence of a fourth order evolution equation to the Allen-Cahn equation,, Nonlinear Anal., 72 (2010), 4271.  doi: 10.1016/j.na.2010.02.003.  Google Scholar

[7]

A. Katsoulakis and G. Vlachos, From microscopic interactions to macroscopic laws of cluster evolution,, Phys. Rev. Lett., 84 (2000), 1511.  doi: 10.1103/PhysRevLett.84.1511.  Google Scholar

[8]

S. Mikhailov, M. Hildebrand and G. Ertl, Nonequilibrium nanostructures in condensed reactive systems,, in, 567 (2001), 252.  doi: 10.1007/3-540-44698-2_16.  Google Scholar

[9]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103.  doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[10]

A. Miranville, Some generalizations of the Cahn-Hilliard equation,, Asymptot. Anal., 22 (2000), 235.   Google Scholar

[11]

A. Miranville, Long-time behavior of some models of Cahn-Hilliard equations in deformable continua,, Nonlinear Anal. Real World Appl., 2 (2001), 273.  doi: 10.1016/S0362-546X(00)00104-8.  Google Scholar

[12]

C. Robinson, "Infinite-dimensional Dynamical Systems,'', Cambridge Universtity Press, (2001).   Google Scholar

[13]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,'', Springer-Verlag, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

show all references

References:
[1]

A. Bonfoh and A. Miranville, On Cahn-Hilliard-Gurtin equations,, in, 47 (2001), 3455.  doi: 10.1016/S0362-546X(01)00463-1.  Google Scholar

[2]

M. Carrive, A. Miranville, A. Piétrus and J. M. Rakotoson, Weakly coupled dynamical systems and applications,, Asymptotic Analysis, 30 (2002), 161.   Google Scholar

[3]

A. Eden, C. Foias, B. Nicolaenko and R. and Temam, "Exponential Attractors for Dissipative Evolution Equations,", Masson, (1994).   Google Scholar

[4]

M.Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system,, Math. Nachr., 272 (2004), 11.   Google Scholar

[5]

G. Karali, and A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution,, J. Differential Equations, 235 (2007), 418.  doi: 10.1016/j.jde.2006.12.021.  Google Scholar

[6]

G. Karali and T. Ricciardi, On the convergence of a fourth order evolution equation to the Allen-Cahn equation,, Nonlinear Anal., 72 (2010), 4271.  doi: 10.1016/j.na.2010.02.003.  Google Scholar

[7]

A. Katsoulakis and G. Vlachos, From microscopic interactions to macroscopic laws of cluster evolution,, Phys. Rev. Lett., 84 (2000), 1511.  doi: 10.1103/PhysRevLett.84.1511.  Google Scholar

[8]

S. Mikhailov, M. Hildebrand and G. Ertl, Nonequilibrium nanostructures in condensed reactive systems,, in, 567 (2001), 252.  doi: 10.1007/3-540-44698-2_16.  Google Scholar

[9]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103.  doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[10]

A. Miranville, Some generalizations of the Cahn-Hilliard equation,, Asymptot. Anal., 22 (2000), 235.   Google Scholar

[11]

A. Miranville, Long-time behavior of some models of Cahn-Hilliard equations in deformable continua,, Nonlinear Anal. Real World Appl., 2 (2001), 273.  doi: 10.1016/S0362-546X(00)00104-8.  Google Scholar

[12]

C. Robinson, "Infinite-dimensional Dynamical Systems,'', Cambridge Universtity Press, (2001).   Google Scholar

[13]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,'', Springer-Verlag, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[1]

Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703
[2]

Luyi Ma, Hong-Tao Niu, Zhi-Cheng Wang. Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111
[3]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[4]

Hirokazu Ninomiya, Masaharu Taniguchi. Global stability of traveling curved fronts in the Allen-Cahn equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 819-832. doi: 10.3934/dcds.2006.15.819
[5]

Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015
[6]

Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679
[7]

Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577
[8]

Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319
[9]

Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009
[10]

Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024
[11]

Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407
[12]

Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4169-4190. doi: 10.3934/dcdsb.2019077
[13]

Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717
[14]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012
[15]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517
[16]

Fang Li, Kimie Nakashima. Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1391-1420. doi: 10.3934/dcds.2012.32.1391
[17]

Takeshi Ohtsuka, Ken Shirakawa, Noriaki Yamazaki. Optimal control problem for Allen-Cahn type equation associated with total variation energy. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 159-181. doi: 10.3934/dcdss.2012.5.159
[18]

Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823
[19]

Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154
[20]

Giorgio Fusco. Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1807-1841. doi: 10.3934/cpaa.2017088

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences