### 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.

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.

#### Keywords

Generalized Cahn-Hilliard equations; singular perturbations; inertial manifolds

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