Long time behavior of a singular perturbation of the viscous Cahn–Hilliard–Gurtin equation |
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Authors: | Ahmed Bonfoh Maurizio Grasselli Alain Miranville |
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Institution: | 1. Université de Lomé, FDS, Département de Mathématiques, BP: 1515, Lomé, Togo;2. Dipartimento di Matematica ‘F. Brioschi’, Politecnico di Milano, Via E. Bonardi 9, 20133 Milano, Italy;3. Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 6086 ‐ SP2MI, Boulevard Marie et Pierre Curie ‐ Téléport 2, 86962 Chasseneuil Futuroscope Cedex, France |
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Abstract: | We consider a singular perturbation of the generalized viscous Cahn–Hilliard equation based on constitutive equations introduced by Gurtin. This equation rules the order parameter ρ, which represents the density of atoms, and it is given on a n‐rectangle (n?3) with periodic boundary conditions. We prove the existence of a family of exponential attractors that is robust with respect to the perturbation parameter ε>0, as ε goes to 0. In a similar spirit, we analyze the stability of the global attractor. If n=1, 2, then we also construct a family of inertial manifolds that is continuous with respect to ε. These results improve and generalize the ones contained in some previous papers. Finally, we establish the convergence of any trajectory to a single equilibrium via a suitable version of the ?ojasiewicz–Simon inequality, provided that the potential is real analytic. Copyright © 2007 John Wiley & Sons, Ltd. |
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Keywords: | generalized Cahn– Hilliard equations singular perturbation exponential attractors inertial manifolds convergence to stationary states |
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