Partial Inertial Manifolds for infinite-dimensional dynamical systems: Example for P.D.E.s with a state-dependent delay

TL;DR

This paper introduces the concept of Partial Inertial Manifolds to analyze the long-term behavior of dissipative differential equations, especially when classical inertial manifolds are absent or unknown, demonstrated through an example involving PDEs with state-dependent delay.

Contribution

It presents a new notion of Partial Inertial Manifolds that extends the analysis of dissipative systems beyond classical inertial manifold existence.

Findings

Partial Inertial Manifolds can exist where classical ones do not

The concept is demonstrated with an example involving PDEs with delay

Provides a new tool for studying long-term dynamics of complex systems

Abstract

We propose a new notion of Partial Inertial Manifold to study the long-time asymptotic behavior of dissipative differential equations. As shown on an example, such manifolds may exist in the cases when the classical Inertial manifold does not exist (or not known to exist).