Frequency theorem for parabolic equations and its relation to inertial manifolds theory

TL;DR

This paper extends the Frequency Theorem to construct quadratic Lyapunov functionals for semilinear parabolic equations, linking spectral gap conditions with frequency inequalities to develop a more general inertial manifolds theory.

Contribution

It introduces a generalized Frequency Theorem for parabolic equations, connecting spectral gap conditions with frequency inequalities to advance inertial manifolds theory.

Findings

Frequency inequalities are related to spectral gap conditions.

Inertial manifolds can be constructed using the generalized Frequency Theorem.

The approach extends to some non-autonomous problems.

Abstract

We obtain a version of the Frequency Theorem (a theorem on solvability of certain operator inequalities), which allows to construct quadratic Lyapunov functionals for semilinear parabolic equations. We show that the well-known Spectral Gap Condition, which was used in the theory of inertial manifolds by C. Foias, R. Temam and G. R. Sell, is a particular case of some frequency inequality, which arises within the Frequency Theorem. In particular, this allows to construct inertial manifolds for semilinear parabolic equations (including also some non-autonomous problems) in the context of a more general geometric theory developed in our adjacent works. This theory is based on quadratic Lyapunov functionals and generalizes the frequency-domain approach used by R. A. Smith. We also discuss the optimality of frequency inequalities and its relationship with known old and recent results in the field.