On the Hyperbolicity Properties of Inertial Manifolds of Reaction--Diffusion Equations

TL;DR

This paper investigates the conditions under which inertial manifolds for 3D reaction-diffusion equations are normally hyperbolic or absolutely normally hyperbolic, providing examples and theoretical insights.

Contribution

It constructs examples of reaction-diffusion systems that do not admit normally hyperbolic inertial manifolds and distinguishes classes based on hyperbolicity properties.

Findings

A specific coupled system with cubic nonlinearity lacks a normally hyperbolic inertial manifold.

An example differentiates systems that admit inertial manifolds from those that do not.

Discussion on the existence of absolutely normally hyperbolic inertial manifolds.

Abstract

For 3D reaction--diffusion equations, we study the problem of existence or nonexistence of an inertial manifold that is normally hyperbolic or absolutely normally hyperbolic. We present a system of two coupled equations with a cubic nonlinearity which does not admit a normally hyperbolic inertial manifold. An example separating the classes of such equations admitting an inertial manifold and a normally hyperbolic inertial manifold is constructed. Similar questions concerning absolutely normally hyperbolic inertial manifolds are discussed.