We consider a model of nonisothermal phase transitions taking place in a bounded spatial region. The order parameter is governed by an Allen-Cahn type equation which is coupled with the equation for the temperature. 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.

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

GRASSELLI, MAURIZIO
2008-01-01

Abstract

We consider a model of nonisothermal phase transitions taking place in a bounded spatial region. The order parameter is governed by an Allen-Cahn type equation which is coupled with the equation for the temperature. 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/528713
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