Long time behavior of a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation

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.