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

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

Ciprian G. Gal 1, and  Maurizio Grasselli 2,

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.
Keywords: global attractors, exponential attractors, Phase-field systems, dynamic boundary conditions.
Mathematics Subject Classification: Primary: 35B41, 35K55, 37L30; Secondary: 80A22.
Citation: Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009
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