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Long time behavior of a singular perturbation of the viscous Cahn–Hilliard–Gurtin equation
Author(s) -
Bonfoh Ahmed,
Grasselli Maurizio,
Miranville Alain
Publication year - 2008
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.938
Subject(s) - mathematics , attractor , rectangle , perturbation (astronomy) , singular perturbation , cahn–hilliard equation , inertial frame of reference , mathematical analysis , boundary (topology) , classical mechanics , geometry , partial differential equation , physics , quantum mechanics
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 Łojasiewicz–Simon inequality, provided that the potential is real analytic. Copyright © 2007 John Wiley & Sons, Ltd.