Geodesic
Dichotomy property of solutions of quasilinear equations in problems on inertial manifolds
Sbornik. Mathematics, Tome 196 (2005) no. 4, pp. 485-511 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Exponential dichotomy properties are studied for non-autonomous quasilinear partial differential equations that can be written as an ordinary differential equation $du/dt+Au=F(u,t)$ in a Hilbert space $H$. It is assumed that the non-linear function $F(u,t)$ is essentially subordinated to the linear operator $A$; namely, the gap property from the theory of inertial manifolds must hold. Integral manifolds $M_+$ and $M_-$ attracting at an exponential rate an arbitrary solution of this equation as $t\to+\infty$ and $t\to-\infty$, respectively, are constructed. The general results established are applied to the study of the dichotomy properties of solutions of a one-dimensional reaction-diffusion system and of a dissipative hyperbolic equation of sine-Gordon type. 
@article{SM_2005_196_4_a1,
     author = {A. Yu. Goritskii and V. V. Chepyzhov},
     title = {Dichotomy property of solutions of quasilinear equations in problems on inertial manifolds},
     journal = {Sbornik. Mathematics},
     pages = {485--511},
     year = {2005},
     volume = {196},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2005_196_4_a1/}
}
TY  - JOUR
AU  - A. Yu. Goritskii
AU  - V. V. Chepyzhov
TI  - Dichotomy property of solutions of quasilinear equations in problems on inertial manifolds
JO  - Sbornik. Mathematics
PY  - 2005
SP  - 485
EP  - 511
VL  - 196
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_2005_196_4_a1/
LA  - en
ID  - SM_2005_196_4_a1
ER  - 
%0 Journal Article
%A A. Yu. Goritskii
%A V. V. Chepyzhov
%T Dichotomy property of solutions of quasilinear equations in problems on inertial manifolds
%J Sbornik. Mathematics
%D 2005
%P 485-511
%V 196
%N 4
%U http://geodesic.mathdoc.fr/item/SM_2005_196_4_a1/
%G en
%F SM_2005_196_4_a1
A. Yu. Goritskii; V. V. Chepyzhov. Dichotomy property of solutions of quasilinear equations in problems on inertial manifolds. Sbornik. Mathematics, Tome 196 (2005) no. 4, pp. 485-511. http://geodesic.mathdoc.fr/item/SM_2005_196_4_a1/

[1] Daletskii Yu. L., Krein M. G., Ustoichivost reshenii differentsialnykh uravnenii v banakhovom prostranstve, Nauka, M., 1970 | MR

[2] Iosida K., Funktsionalnyi analiz, Mir, M., 1967 | MR

[3] Massera Kh., Sheffer Kh., Lineinye differentsialnye uravneniya i funktsionalnye prostranstva, Mir, M., 1970 | MR | Zbl

[4] Foias C., Sell G. R., Temam R., “Inertial manifolds for nonlinear evolutionary equations”, J. Differential Equations, 73:2 (1988), 309–353 | DOI | MR | Zbl

[5] Mallet-Paret J., Sell G. R., “Inertial manifolds for reaction-diffusion equations in higher space dimensions”, J. Amer. Math. Soc., 4:4 (1988), 805–866 | DOI | MR

[6] Miklavčič M., “A sharp condition for existence of an inertial manifolds”, J. Dynam. Differential Equations, 3:3 (1991), 437–456 | DOI | MR | Zbl

[7] Mora X., “Finite-dimensional attracting invariant manifolds for damped semilinear wave equations”, Res. Notes Math., 155 (1987), 172–183 | MR | Zbl

[8] Romanov A. V., “Tochnye otsenki razmernosti integralnykh mnogoobrazii dlya nelineinykh parabolicheskikh uravnenii”, Izv. RAN. Ser. matem., 57:4 (1993), 36–54 | MR | Zbl

[9] Khenri D., Geometricheskaya teoriya polulineinykh parabolicheskikh uravnenii, Mir, M., 1977 | MR

[10] Chepyzhov V. V., Goritsky A. Yu., “Global integral manifolds with exponential tracking for nonautonomous equations”, Russian J. Math. Phys., 5:1 (1997), 9–28 | MR | Zbl

[11] Chepyzhov V. V., Goritsky A. Yu., “Explicit construction of integral manifolds with exponential tracking”, Appl. Anal., 71:1–4 (1999), 237–252 | DOI | MR | Zbl

[12] Chueshov I. D., Vvedenie v teoriyu beskonechnomernykh dinamicheskikh sistem, Akta, Kharkov, 1999 | MR | Zbl

[13] Hirsh M., Differential topology, Springer-Verlag, New York, 1976

[14] Temam R., Infinite-dimensional dynamical systems in mechanics and physics, Appl. Math., 68, Springer-Verlag, New York, 1988 | MR | Zbl

[15] Babin A. V., Vishik M. I., Attraktory evolyutsionnykh uravnenii, Nauka, M., 1989 | MR | Zbl

[16] Hale J. K., Asymptotic behaviour of dissipative systems, Math. Surveys Monogr., 25, Amer. Math. Soc., Providence, RI, 1988 | MR | Zbl

[17] Chepyzhov V. V., Vishik M. I., Attractors for equations of mathematical physics, Amer. Math. Soc. Colloq. Publ., 49, Amer. Math. Soc., Providence, RI, 2002 | MR | Zbl

[18] Chepyzhov, V. V., Vishik M. I., “A Hausdorff dimension estimate for kernel sections of non-autonomous evolution equations”, Indiana Univ. Math. J., 42:3 (1993), 1057–1076 | DOI | MR | Zbl