# American Institute of Mathematical Sciences

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
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