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

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  • 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.
    Mathematics Subject Classification: Primary: 35B41, 35K55, 37L30; Secondary: 80A22.

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