Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term |
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Authors: | Maurizio Grasselli,Hana Petzeltová ,Giulio Schimperna |
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Affiliation: | a Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Via Bonardi, 9, I-20133 Milano, Italy b Mathematical Institute AS CR, ?itná, 25, CZ-115 67 Praha, Czech Republic c Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Via Ferrata, 1, I-27100 Pavia, Italy |
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Abstract: | We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term χtt, χ being the order parameter, which is linearly coupled with an evolution equation for the (relative) temperature ?. The latter can be of hyperbolic type if the Cattaneo-Maxwell heat conduction law is assumed. The state variables and the chemical potential are subject to the homogeneous Neumann boundary conditions. We first provide conditions which ensure the well-posedness of the initial and boundary value problem. Then, we prove that the corresponding dynamical system is dissipative and possesses a global attractor. Moreover, assuming that the nonlinear potential is real analytic, we establish that each trajectory converges to a single steady state by using a suitable version of the ?ojasiewicz-Simon inequality. We also obtain an estimate of the decay rate to equilibrium. |
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Keywords: | 35B40 35B41 35R35 80A22 |
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