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

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... hus 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. 2000 Mathematics Subject Classification. Primary: 35B41, 35K55, 37L30; Secondary: 80A22. Key words and phrases. Phase-field systems, dynamic boundary conditions, global attractors, exponential attractors. 1 2 CIPRIAN G. GAL AND MAURIZIO GRASSELLI The mathematical literature regarding system (1.1)-(1.2) is rather vast. In particular, well-posedness results can be found in [53, 16] (see also [14, 52] ). The analysis of the dissipative dynamical system generated by equations like (1.1)-(1.2) has been carried out in a number of papers (see [4, 5, 6, 31, 35, 36, 37, 39] ), proving theorems about existence of global and/or exponential attractors. Lately, the asymptotic behavior of single solutions has been investigated by means of the Lojasiewicz-Simon inequality (see [11, 32] where singular potentials are considered, cf. also [57] ). System (1.1)-(1.2) can be viewed as a singular perturbation of the celebrated Cahn-Hilliard equation that accounts for phase separation dynamics (see [8] , cf. also [43] and references therein). In fact, if we formally set ε = 0 in equation (1.2), then we can easily deduce the (viscous) Cahn-Hilliard equation (see [44] )