December  2008, 22(4): 1009-1040. doi: 10.3934/dcds.2008.22.1009

The non-isothermal Allen-Cahn equation with dynamic boundary conditions

1. 

Department of Mathematics, University of Missouri, Columbia, MO 65211, United States

2. 

Dipartimento di Matematica, Politecnico di Milano, 20133 Milano, Italy

Received  March 2007 Revised  September 2007 Published  September 2008

We consider a model of nonisothermal phase transitions taking place in a bounded spatial region. The order parameter $\psi$ is governed by an Allen-Cahn type equation which is coupled with the equation for the temperature $\theta$. The former is subject to a dynamic boundary condition recently proposed by some physicists to account for interactions with the walls. The latter is endowed with a boundary condition which can be a standard one (Dirichlet, Neumann or Robin) or a dynamic one of Wentzell type. We thus formulate a class of initial and boundary value problems whose local existence and uniqueness is proven by means of a fixed point argument. The local solution becomes global owing to suitable a priori estimates. Then we analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. In particular, we demonstrate the existence of the global attractor as well as of an exponential attractor.
Citation: 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
[1]

S. Gatti, M. Grasselli, V. Pata, M. Squassina. Robust exponential attractors for a family of nonconserved phase-field systems with memory. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 1019-1029. doi: 10.3934/dcds.2005.12.1019

[2]

Maurizio Grasselli, Hao Wu. Robust exponential attractors for the modified phase-field crystal equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2539-2564. doi: 10.3934/dcds.2015.35.2539

[3]

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

[4]

Gianluca Mola. Global attractors for a three-dimensional conserved phase-field system with memory. Communications on Pure & Applied Analysis, 2008, 7 (2) : 317-353. doi: 10.3934/cpaa.2008.7.317

[5]

Monica Conti, Stefania Gatti, Alain Miranville. Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 485-505. doi: 10.3934/dcdss.2012.5.485

[6]

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

[7]

Alain Miranville, Elisabetta Rocca, Giulio Schimperna, Antonio Segatti. The Penrose-Fife phase-field model with coupled dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4259-4290. doi: 10.3934/dcds.2014.34.4259

[8]

Alain Miranville, Costică Moroşanu. Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 537-556. doi: 10.3934/dcdss.2016011

[9]

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

[10]

Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113

[11]

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

[12]

Nobuyuki Kenmochi, Noriaki Yamazaki. Global attractor of the multivalued semigroup associated with a phase-field model of grain boundary motion with constraint. Conference Publications, 2011, 2011 (Special) : 824-833. doi: 10.3934/proc.2011.2011.824

[13]

Ciprian G. Gal. Robust exponential attractors for a conserved Cahn-Hilliard model with singularly perturbed boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (4) : 819-836. doi: 10.3934/cpaa.2008.7.819

[14]

Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041

[15]

Mei-Qin Zhan. Global attractors for phase-lock equations in superconductivity. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 243-256. doi: 10.3934/dcdsb.2002.2.243

[16]

Ahmed Y. Abdallah. Exponential attractors for second order lattice dynamical systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 803-813. doi: 10.3934/cpaa.2009.8.803

[17]

Xiaoying Han. Exponential attractors for lattice dynamical systems in weighted spaces. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 445-467. doi: 10.3934/dcds.2011.31.445

[18]

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

[19]

David Cheban. Global attractors of nonautonomous quasihomogeneous dynamical systems. Conference Publications, 2001, 2001 (Special) : 96-101. doi: 10.3934/proc.2001.2001.96

[20]

Alessia Berti, Valeria Berti, Ivana Bochicchio. Global and exponential attractors for a Ginzburg-Landau model of superfluidity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 247-271. doi: 10.3934/dcdss.2011.4.247

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (34)

Other articles
by authors

[Back to Top]