Geodesic
Inertial invariant manifolds of a nonlinear semigroup of operators in a Hilbert space
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the All-Russian Scientific Conference «Differential Equations and Their Applications» dedicated to the 85th anniversary of Professor M. T. Terekhin. Ryazan State University named for S. A. Yesenin, Ryazan, May 17-18, 2019. Part 2, Tome 186 (2020), pp. 57-66.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we examine the existence and analyze properties of inertial manifolds of a nonlinear semigroup of operators in a Hilbert space. This questions were studied in a general setting that allows generalizing results of the well-known works of K. Foias, J. Sell, and R. Temam. Our reasoning is based on the scheme of proofs of similar assertions proposed earlier by S. Sternberg and F. Hartman for ordinary autonomous differential equations.
Keywords: inertial invariant manifold, Hilbert space, operator. 
@article{INTO_2020_186_a7,
     author = {A. N. Kulikov},
     title = {Inertial invariant manifolds of a nonlinear semigroup of operators in a {Hilbert} space},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {57--66},
     publisher = {mathdoc},
     volume = {186},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_186_a7/}
}
TY  - JOUR
AU  - A. N. Kulikov
TI  - Inertial invariant manifolds of a nonlinear semigroup of operators in a Hilbert space
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2020
SP  - 57
EP  - 66
VL  - 186
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2020_186_a7/
LA  - ru
ID  - INTO_2020_186_a7
ER  - 
%0 Journal Article
%A A. N. Kulikov
%T Inertial invariant manifolds of a nonlinear semigroup of operators in a Hilbert space
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2020
%P 57-66
%V 186
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2020_186_a7/
%G ru
%F INTO_2020_186_a7
A. N. Kulikov. Inertial invariant manifolds of a nonlinear semigroup of operators in a Hilbert space. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the All-Russian Scientific Conference «Differential Equations and Their Applications» dedicated to the 85th anniversary of Professor M. T. Terekhin. Ryazan State University named for S. A. Yesenin, Ryazan, May 17-18, 2019. Part 2, Tome 186 (2020), pp. 57-66. http://geodesic.mathdoc.fr/item/INTO_2020_186_a7/

[2] Gokhberg I. I., Krein M. G., Vvedenie v teoriyu nesamosopryazhennykh operatorov, Nauka, M., 1965

[3] Kolesov A. Yu., Kulikov A. N., Rozov N. Kh., “Invariantnye tory odnogo klassa tochechnykh otobrazhenii: printsip koltsa”, Differ. uravn., 39:5 (2003), 584–601 | MR | Zbl

[4] Kolesov A. Yu., Kulikov A. N., Rozov N. Kh., “Invariantnye tory odnogo klassa tochechnykh otobrazhenii: sokhranenie invariantnogo tora pri vozmuscheniyakh”, Differ. uravn., 39:6 (2003), 738–753 | MR | Zbl

[5] Krein S. G., Lineinye differentsialnye operatory v banakhovom prostranstve, Nauka, M., 1977

[6] Kulikov A. N., “Integralnye mnogoobraziya giperbolicheskikh uravnenii v sluchae, blizkom k kriticheskomu, odnoi pary chisto mnimykh kornei”, Vestn. YarGU., 13 (1975), 94–117

[7] Kulikov A. N., “O gladkikh invariantnykh mnogoobraziyakh polugruppy nelineinykh operatorov v banakhovom prostranstve”, Issledovaniya po ustoichivosti i teorii kolebanii, Yaroslavl, 1976, 114–129

[8] Kulikov A. N., “Odno zamechanie o svoistvakh dvumernykh invariantnykh mnogoobrazii”, Issledovaniya po ustoichivosti i teorii kolebanii, Yaroslavl, 1978, 78–80

[9] Kulikov A. N., “K voprosu o edinstvennosti invariantnogo mnogoobraziya v kriticheskom sluchae”, Issledovaniya po ustoichivosti i teorii kolebanii, Yaroslavl, 1979, 81–85

[10] Kulikov A. N., “Bifurkatsii invariantnykh torov”, Issledovaniya po ustoichivosti i teorii kolebanii, Yaroslavl, 1983, 112–117

[11] Kulikov A. N., Inertsialnye mnogoobraziya nelineinykh avtonomnykh differentsialnykh uravnenii v gilbertovom prostranstve, Preprint No 85 In-ta prikl. mat. im. M. V. Keldysha, M., 1991

[12] Lyapunov A. M., Obschaya zadacha ob ustoichivosti dvizheniya, Merkurii, M., 2000 | MR

[13] Marsden Dzh., Mak-Kraken M., Bifurkatsii rozhdeniya tsikla i ee prilozheniya, Mir, M., 1980

[14] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1965

[15] Sobolevskii P. E., “Ob uravneniyakh parabolicheskogo tipa v banakhovom prostranstve”, Tr. Mosk. mat. o-va., 10 (1961), 297–350

[16] Khartman F., Obyknovennye differentsialnye uravneniya, Mir, M., 1970

[17] Khille E., Fillips R., Obyknovennye differentsialnye uravneniya, Mir, M., 1970

[18] Yakubov S. Ya., “Razreshimost zadachi Koshi dlya abstraktnykh giperbolicheskikh uravnenii vtorogo poryadka i ikh prilozheniya”, Tr. Mosk. mat. o-va., 23 (1970), 37–60 | Zbl

[19] Foias G., Sell G. R., Temam R., “Inertial manifolds for nonlinear evolutionary equations”, J. Differ. Equ., 73 (1988), 309–388 | DOI | MR

[20] Segal I., “Nonlinear semigroups”, Ann. Math., 78:2 (1963), 339–364 | DOI | MR | Zbl

[21] Slemrod M., “Asymptotic behavior of $C_0$ semigroups as determined by the spectrum of the generator”, Indiana Univ. Math. J., 25 (1977), 783–792 | DOI | MR

[22] Sternberg S., “Local contractions ant theorem of Poincaré”, Am. J. Math., 79 (1957), 809–824 | DOI | MR | Zbl

[23] Zabcsyk J., “A note on $C_0$ semigroups”, Bull. Acad. Pol. Sci. Sér. Sci. Math. Astron. Phys., 27:8 (1975), 895–898 | MR