On the Relative Decay of Unbounded Semigroups on the Domain of the Generator

Автор(и)

  • Grigory M. Sklyar Faculty of Computer Science and Information Technology, West Pomeranian University of Technology in Szczecin, Zolnierska 49, 71-210 Szczecin, Poland
  • Piotr Polak Faculty of Computer Science and Information Technology, West Pomeranian University of Technology in Szczecin, Zolnierska 49, 71-210 Szczecin, Poland
  • Bartosz Wasilewski Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
    University of Szczecin, Doctoral School, Mickiewicza 16, 70-383 Szczecin, Poland

DOI:

https://doi.org/10.15407/mag20.01.094

Ключові слова:

C0-півгрупи, асимптотична поведiнка

Анотація

Дослiджується асимптотика $C_0$-пiвгрупи в областi визначення генератора. Зокрема, ми аналiзуємо поведiнку $||T(t)(A-\lambda I)^{-1}||$, коли час прямує до нескiнченностi. Нашi результати розширюють деякi наявнi результати на випадок, коли перетин спектра генератора з уявною вiссю є непорожнiм. Наведено також конструктивний приклад класу необмежених $C_0$-напiвгруп з чисто уявним точковим спектром, для яких наша теорема може бути застосована.

Mathematical Subject Classification 2020: 47D06

Посилання

J. van Neerven, The asymptotic behaviour of semigroups of linear operators, Birkhäuser, Basel, 1996. https://doi.org/10.1007/978-3-0348-9206-3

K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.

G.M. Sklyar and V. Shirman, On asymptotic stability of linear differential equation in Banach spaces, (Russian) Teoria Funk., Funkt. Anal. Prilozh. 37 (1982), 127--132.

Y. Lyubich and V. Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math. 88 (1988), 37--42. https://doi.org/10.4064/sm-88-1-37-42

W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Am. Math. Soc. 306 (1988), 837--852. https://doi.org/10.1090/S0002-9947-1988-0933321-3

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann. 347 (2010). 455--478. https://doi.org/10.1007/s00208-009-0439-0

C.J.K. Batty, A. Borichev, and Y. Tomilov, $Lp$-tauberian theorems and $Lp$-rates for energy decay, J. Funct. Anal 270 (2016) 1153--1201. https://doi.org/10.1016/j.jfa.2015.12.003

F. Huang, Strong Asymptotic Stability of Linear Dynamical Systems in Banach Spaces, J. Differ. Equ. 104, (1993), 307--324. https://doi.org/10.1006/jdeq.1993.1074

C.J.K Batty and S. Yeates, Weighted and local stability of semigroups of operators, Math. Proc. Cambridge Philos. Soc.129 (2000), 85--98. https://doi.org/10.1017/S0305004199004363

J. Woźniak and M. Firkowski, Optimal decay ratio of damped slowly rotating Timoshenko beams, ZAMM Z. Angew. Math. Mech. 99 (2019), https://doi.org/10.1002/zamm.201800222

C.J.K. Batty, Asymptotic behaviour of semigroups of operators, Banach Center Publ. 30 (1994), 35--52. https://doi.org/10.4064/-30-1-35-52

C.J.K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ. 8, 2008, 765--780. https://doi.org/10.1007/s00028-008-0424-1

G.M. Sklyar and P. Polak, On asymptotic estimation of a discrete type $C_0$-semigroups on dense sets: Application to neutral type systems, Appl. Math. Optim. 75 (2017), 175--192. https://doi.org/10.1007/s00245-015-9327-z

G.M. Sklyar, On the decay of bounded semigroup on the domain of its generator, Vietnam J. Math. 43 (2015), 207--213. https://doi.org/10.1007/s10013-014-0093-z

V. Phóng, Theorems of Katznelson-Tzafriri type for semigroups of operators, J. Func. Anal. 103 (1992), 74--84. https://doi.org/10.1016/0022-1236(92)90135-6

V. Phóng, Semigroups with nonquasianalytic growth, Studia Mathematica 104 (1993), 229--241. https://doi.org/10.4064/sm-104-3-229-241

Yu.I. Lyubich, V.I Matsaev, and G.M. Fel'dman On representations with a separable spectrum, Funct. Anal. Its Appl. 7 (1973) 129--136. https://doi.org/10.1007/BF01078885

G.Q. Xu and S.P. Yung, The expansion of a semigroup and a Riesz basis criterion, J. Differ. Equ. 210 (2005), 1--24. https://doi.org/10.1016/j.jde.2004.09.015

H. Zwart, Riesz basis for strongly continuous groups, J. Differ. Equ. 249 (2010), 2397--2408. https://doi.org/10.1016/j.jde.2010.07.020

G.M. Sklyar and P. Polak, Notes on the asymptotic properties of some class of unbounded strongly continuous semigroups, J. Math. Phys. Anal. Geom. 15 (2019), 412--424. https://doi.org/10.15407/mag15.03.412

G.M. Sklyar, On the maximal asymptotics for linear differential equations in Banach spaces, Taiwan. J. Math. 14 (2010), 2203--2217. https://doi.org/10.11650/twjm/1500406070

G.M. Sklyar. Lack of maximal asymptotics for linear differential equations in Banach spaces. Dokl. Math. 81 (2010), 265--267. https://doi.org/10.1134/S1064562410020286

G.M. Sklyar, V. Marchenko, and P. Polak, Sharp polynomial bounds for certain $C_0$-groups generated by operators with non-basis family of eigenvectors, J. Funct. Anal. 280 (2021) https://doi.org/10.1016/j.jfa.2020.108864

G.M. Sklyar and V. Marchenko, Hardy inequality and the construction of infinitesimal operators with non-basis family of eigenvectors, J. Func. Anal. 272 (2017), 1017--1043. https://doi.org/10.1016/j.jfa.2016.11.001

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Sklyar, G. M.; Polak, P.; Wasilewski, B. On the Relative Decay of Unbounded Semigroups on the Domain of the Generator. Журн. мат. фіз. анал. геом. 2024, 20, 94–111.

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Том 20 № 1 (2024)

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