# American Institute of Mathematical Sciences

March  2009, 8(2): 689-710. doi: 10.3934/cpaa.2009.8.689

## On the asymptotic behavior of the Caginalp system with dynamic boundary conditions

 1 Department of Mathematics, University of Missouri, Columbia, MO, 65211, United States 2 Dipartimento di Matematica "F. Brioschi", Politecnico di Milano, I-20133 Milano

Received  March 2008 Revised  September 2008 Published  December 2008

We consider a phase-field system of Caginalp type on a three-dimensional bounded domain. The order parameter $\psi$ fulfills a dynamic boundary condition, while the (relative) temperature $\theta$ is subject to a boundary condition of Dirichlet, Neumann, Robin or Wentzell type. The corresponding class of initial and boundary value problems has already been studied by the authors, proving well-posedness results and the existence of global as well as exponential attractors. Here we intend to show first that the previous analysis can be redone for larger phase-spaces, provided that the bulk potential has a fourth-order growth at most whereas the boundary potential has an arbitrary polynomial growth. Moreover, assuming the potentials to be real analytic, we demonstrate that each trajectory converges to a single equilibrium by means of a Łojasiewicz-Simon type inequality. We also obtain a convergence rate estimate.
Citation: Ciprian G. Gal, M. Grasselli. On the asymptotic behavior of the Caginalp system with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 689-710. doi: 10.3934/cpaa.2009.8.689
 [1] 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 [2] 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 [3] 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 [4] 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 [5] A. M. Micheletti, Angela Pistoia. Multiple eigenvalues of the Laplace-Beltrami operator and deformation of the Riemannian metric. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 709-720. doi: 10.3934/dcds.1998.4.709 [6] Micol Amar, Roberto Gianni. Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1739-1756. doi: 10.3934/dcdsb.2018078 [7] 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 [8] 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 [9] 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 [10] 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 [11] 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 [12] Andrea Bonito, Roland Glowinski. On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2115-2126. doi: 10.3934/cpaa.2014.13.2115 [13] 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 [14] 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 [15] 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 [16] 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 [17] 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 [18] 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 [19] Kei Matsuura, Mitsuharu Otani. Exponential attractors for a quasilinear parabolic equation. Conference Publications, 2007, 2007 (Special) : 713-720. doi: 10.3934/proc.2007.2007.713 [20] 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

2018 Impact Factor: 0.925

## Metrics

• PDF downloads (9)
• HTML views (0)
• Cited by (15)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]