Inertial manifolds and foliations for asymptotically compact cocycles in Banach spaces

2020/12/07 by Mikhail Anikushin, Anikushin, Mikhail
Computer Science · Engineering · Mathematics · #35B42 #37L15 #37L25 #37L45 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Stability and Controllability of Differential Equations #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2012.03821

openalex publication_date 2020/12/07 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We study asymptotically compact nonautonomous dynamical systems given by abstract cocycles in Banach spaces. Our main assumptions are given by a squeezing property in a quadratic cone field (given by a family of indefinite quadratic Lyapunov-like functionals) and asymptotic compactness. Under such conditions it is possible to reconstruct foliations as in the theory of normally hyperbolic manifolds. Our approach allows to unify many "practical" theories of local and nonlocal invariant manifolds, such as stable/unstable/center manifolds and inertial manifolds, and, moreover, it often leads to optimal conditions for their existence. We give applications for semilinear parabolic equations and neutral delay equations, where the Frequency Theorem is used to construct constant cone fields.

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