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

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.

References in corpus (3)

• A non-local reduction principle for cocycles in Hilbert spaces
• The Poincaré-Bendixson theory for certain compact semi-flows in Banach spaces
• Nonlinear semigroups for delay equations in Hilbert spaces, inertial manifolds and dimension estimates

Cited by in corpus (4)

• Hidden and unstable periodic orbits as a result of homoclinic bifurcations in the Suarez-Schopf delayed oscillator and the irregularity of ENSO
• The Poincaré-Bendixson theory for certain compact semi-flows in Banach spaces
• Nonlinear semigroups for delay equations in Hilbert spaces, inertial manifolds and dimension estimates
• Frequency theorem and inertial manifolds for neutral delay equations