June  2007, 6(2): 367-387. doi: 10.3934/cpaa.2007.6.367

Global existence and long-time behaviour for a singular integro-differential phase-field system

1. 

Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, via Ferrata, 1, 27100, Pavia, Italy

2. 

Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, via Saldini, 50, 20133, Milano, Italy

Received  February 2006 Revised  August 2006 Published  March 2007

This paper deals with a singular integro-differential PDE system describing phase transitions in terms of nonlinear evolution equations for micromotions and for the entropy. The model is derived from a non-convex free energy functional, possibly accounting for thermal memory effects. After recovering a global existence result for a related initial and boundary value problem, the long-time behaviour of the solutions is investigated. In particular, it is proved that the elements of the $\omega$-limit set (i.e. the cluster points) of the solution trajectories solve the steady state system which is naturally associated to the evolution problem.
Citation: 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
[1]

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

[2]

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

[3]

Pavel Krejčí, Jürgen Sprekels. Long time behaviour of a singular phase transition model. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1119-1135. doi: 10.3934/dcds.2006.15.1119

[4]

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

[5]

Pierluigi Colli, Danielle Hilhorst, Françoise Issard-Roch, Giulio Schimperna. Long time convergence for a class of variational phase-field models. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 63-81. doi: 10.3934/dcds.2009.25.63

[6]

Ahmed Bonfoh, Cyril D. Enyi. Large time behavior of a conserved phase-field system. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1077-1105. doi: 10.3934/cpaa.2016.15.1077

[7]

Sergiu Aizicovici, Hana Petzeltová. Convergence to equilibria of solutions to a conserved Phase-Field system with memory. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 1-16. doi: 10.3934/dcdss.2009.2.1

[8]

M. Grasselli, Hana Petzeltová, Giulio Schimperna. Convergence to stationary solutions for a parabolic-hyperbolic phase-field system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 827-838. doi: 10.3934/cpaa.2006.5.827

[9]

Stig-Olof Londen, Hana Petzeltová. Convergence of solutions of a non-local phase-field system. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 653-670. doi: 10.3934/dcdss.2011.4.653

[10]

Federico Mario Vegni. Dissipativity of a conserved phase-field system with memory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 949-968. doi: 10.3934/dcds.2003.9.949

[11]

Eduardo Cuesta. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Conference Publications, 2007, 2007 (Special) : 277-285. doi: 10.3934/proc.2007.2007.277

[12]

Honghu Liu. Phase transitions of a phase field model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 883-894. doi: 10.3934/dcdsb.2011.16.883

[13]

Monica Conti, Stefania Gatti, Alain Miranville. A singular cahn-hilliard-oono phase-field system with hereditary memory. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3033-3054. doi: 10.3934/dcds.2018132

[14]

Maurizio Grasselli, Alain Miranville, Giulio Schimperna. The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 67-98. doi: 10.3934/dcds.2010.28.67

[15]

Narcisse Batangouna, Morgan Pierre. Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system. Communications on Pure & Applied Analysis, 2018, 17 (1) : 1-19. doi: 10.3934/cpaa.2018001

[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]

Yihong Du, Yoshio Yamada. On the long-time limit of positive solutions to the degenerate logistic equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 123-132. doi: 10.3934/dcds.2009.25.123

[18]

Maurizio Grasselli, Giulio Schimperna. Nonlocal phase-field systems with general potentials. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5089-5106. doi: 10.3934/dcds.2013.33.5089

[19]

José Luiz Boldrini, Gabriela Planas. A tridimensional phase-field model with convection for phase change of an alloy. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 429-450. doi: 10.3934/dcds.2005.13.429

[20]

Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]