April  1999, 5(2): 375-390. doi: 10.3934/dcds.1999.5.375

Uniqueness and long-time behavior for the conserved phase-field system with memory

1. 

Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, I-27100 Pavia, Italy

2. 

Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy

3. 

Institut Elie Cartan, Université de Nancy 1, B.P. 239, 54506 Vandœuvre les Nancy Cedex, France

4. 

Department of Mathematics, Technion-IIT, Haifa 32000, Israel

Received  June 1998 Revised  November 1998 Published  January 1999

This paper is concerned with a conserved phase-field model with memory. We include memory by replacing the standard Fourier heat law with a constitutive assumption of Gurtin-Pipkin type, and the system is conservative in the sense that the initial mass of the order parameter as well as the energy are preserved during the evolution. A Cauchy-Neumann problem is investigated for this model which couples a Volterra integro-differential equation with fourth order dynamics for the phase field. A sharp uniqueness theorem is proven by demonstrating continuous dependence for a suitably weak formulation. With regard to the long-time behavior, the limit points of the trajectories are completely characterized.
Citation: Pierluigi Colli, Gianni Gilardi, Philippe Laurençot, Amy Novick-Cohen. Uniqueness and long-time behavior for the conserved phase-field system with memory. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 375-390. doi: 10.3934/dcds.1999.5.375
[1]

Ke Xu, M. Gregory Forest, Xiaofeng Yang. Shearing the I-N phase transition of liquid crystalline polymers: Long-time memory of defect initial data. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 457-473. doi: 10.3934/dcdsb.2011.15.457

[2]

Norbert Požár, Giang Thi Thu Vu. Long-time behavior of the one-phase Stefan problem in periodic and random media. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 991-1010. doi: 10.3934/dcdss.2018058

[3]

Chang Zhang, Fang Li, Jinqiao Duan. Long-time behavior of a class of nonlocal partial differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 749-763. doi: 10.3934/dcdsb.2018041

[4]

Hongtao Li, Shan Ma, Chengkui Zhong. Long-time behavior for a class of degenerate parabolic equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2873-2892. doi: 10.3934/dcds.2014.34.2873

[5]

Shan Ma, Chunyou Sun. Long-time behavior for a class of weighted equations with degeneracy. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1889-1902. doi: 10.3934/dcds.2020098

[6]

Yang Liu. Long-time behavior of a class of viscoelastic plate equations. Electronic Research Archive, 2020, 28 (1) : 311-326. doi: 10.3934/era.2020018

[7]

Elena Bonetti, Pierluigi Colli, Mauro Fabrizio, Gianni Gilardi. Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1001-1026. doi: 10.3934/dcdsb.2006.6.1001

[8]

Eduard Feireisl, Françoise Issard-Roch, Hana Petzeltová. Long-time behaviour and convergence towards equilibria for a conserved phase field model. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 239-252. doi: 10.3934/dcds.2004.10.239

[9]

Elena Bonetti, Elisabetta Rocca. Global existence and long-time behaviour for a singular integro-differential phase-field system. Communications on Pure & Applied Analysis, 2007, 6 (2) : 367-387. doi: 10.3934/cpaa.2007.6.367

[10]

Xianpeng Hu, Hao Wu. Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3437-3461. doi: 10.3934/dcds.2015.35.3437

[11]

Manuel Núñez. The long-time evolution of mean field magnetohydrodynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 465-478. doi: 10.3934/dcdsb.2004.4.465

[12]

Irena Lasiecka, To Fu Ma, Rodrigo Nunes Monteiro. Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1037-1072. doi: 10.3934/dcdsb.2018141

[13]

Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767

[14]

Lu Yang, Meihua Yang. Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2627-2650. doi: 10.3934/dcdsb.2017102

[15]

Jianping Wang, Mingxin Wang. Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020121

[16]

Zhenhua Zhang. Asymptotic behavior of solutions to the phase-field equations with neumann boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 683-693. doi: 10.3934/cpaa.2005.4.683

[17]

Giulio Schimperna, Antonio Segatti, Ulisse Stefanelli. Well-posedness and long-time behavior for a class of doubly nonlinear equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 15-38. doi: 10.3934/dcds.2007.18.15

[18]

Horst Osberger. Long-time behavior of a fully discrete Lagrangian scheme for a family of fourth order equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 405-434. doi: 10.3934/dcds.2017017

[19]

Joackim Bernier, Michel Mehrenberger. Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium. Kinetic & Related Models, 2020, 13 (1) : 129-168. doi: 10.3934/krm.2020005

[20]

Jean-Paul Chehab, Pierre Garnier, Youcef Mammeri. Long-time behavior of solutions of a BBM equation with generalized damping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1897-1915. doi: 10.3934/dcdsb.2015.20.1897

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (12)

[Back to Top]