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

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Frequency theorem for parabolic equations and its relation to inertial manifolds theory. / Anikushin, Mikhail.

в: Journal of Mathematical Analysis and Applications, Том 505, № 1, 125454, 01.01.2022.

Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование

BibTeX

@article{a069ae9ace4349908e3e22b9a9ea6d60,

title = "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.",

keywords = "Frequency theorem, Inertial manifolds, Lyapunov functionals, Parabolic equations",

author = "Mikhail Anikushin",

note = "Publisher Copyright: {\textcopyright} 2021 Elsevier Inc.",

year = "2022",

month = jan,

day = "1",

doi = "10.1016/j.jmaa.2021.125454",

language = "English",

volume = "505",

journal = "Journal of Mathematical Analysis and Applications",

issn = "0022-247X",

publisher = "Elsevier",

number = "1",

}

RIS

TY - JOUR

T1 - Frequency theorem for parabolic equations and its relation to inertial manifolds theory

AU - Anikushin, Mikhail

PY - 2022/1/1

Y1 - 2022/1/1

N2 - 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.

AB - 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.

KW - Frequency theorem

KW - Inertial manifolds

KW - Lyapunov functionals

KW - Parabolic equations

UR - http://www.scopus.com/inward/record.url?scp=85124184022&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/ef04aa2c-2068-307a-ac9a-c53cca4696fa/

U2 - 10.1016/j.jmaa.2021.125454

DO - 10.1016/j.jmaa.2021.125454

M3 - Article

AN - SCOPUS:85124184022

VL - 505

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

M1 - 125454

ER -

ID: 95166346