ABSTRACT
Inertial manifolds are positively invariant, finite dimensional, exponentially attracting Lipschitz manifolds. This chapter gives the precise definition of this notion for a continuous semiflow defined on a Banach space, and shows how an inertial manifold can be constructed when continuous semiflow satisfies some natural conditions on its geometrical structure. It considers the cone invariance property, and one of several versions of the strong squeezing property. The cone invariance and decay properties describe a sort of dichotomy principle, whereby either the difference of two motions can never leave a certain cone (cone invariance property), or, if it does, the distance between the motions decays exponentially (decay property). The chapter also discusses in detail a model of the Chafee-Infante reaction-diffusion equations in one dimension of space, and its hyperbolic perturbation.
