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

Ahmed Bonfoh, Maurizio Grasselli, Alain Miranville

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.

Keywords


Generalized Cahn-Hilliard equations; singular perturbations; inertial manifolds

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