Inertial manifolds for a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation

Abstract

We consider a singular perturbation of the generalized viscous Cahn-Hilliard equation based on constitutive equations introduced by M. E. Gurtin and we establish the existence of a family of inertial manifolds which is continuous with respect to the perturbation parameter $\varepsilon> 0$ as $\varepsilon$ goes to 0. In a recent paper, we proved a similar result for the singular perturbation of the standard viscous Cahn-Hilliard equation, applying a construction due to X. Mora and J. Sol\`a-Morales for equations involving linear self-adjoint operators only. Here we extend the result to the singularly perturbed Cahn-Hilliard-Gurtin equation which contains a non-self-adjoint operator. Our method can be applied to a larger class of nonlinear dynamical systems.