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