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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

Ciprian G. Gal 1, and  M. Grasselli 2,

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.
Keywords: exponential attractors, Laplace-Beltrami operator, dynamic boundary conditions, Phase-field equation, global attractors, Łojasiewicz-Simon inequality, convergence to equilibrium.
Mathematics Subject Classification: Primary: 35B41, 35K55, 37L30; Secondary: 80A22.
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
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