Geodesic
Rates of convergence of approximate attractors for parabolic equations
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 37, Tome 336 (2006), pp. 67-111 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We estimate rates of convergence of global attractors of approximations to the global attractor of a semilinear parabolic equation. We consider a general equation for which all fixed points are hyperbolic and the Chafee–Infante equation having a nonhyperbolic fixed point. The results are applied to an implicit discretization of a parabolic equation. 
@article{ZNSL_2006_336_a4,
     author = {V. S. Kolezhuk and S. Yu. Pilyugin},
     title = {Rates of convergence of approximate attractors for parabolic equations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {67--111},
     year = {2006},
     volume = {336},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_336_a4/}
}
TY  - JOUR
AU  - V. S. Kolezhuk
AU  - S. Yu. Pilyugin
TI  - Rates of convergence of approximate attractors for parabolic equations
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2006
SP  - 67
EP  - 111
VL  - 336
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2006_336_a4/
LA  - en
ID  - ZNSL_2006_336_a4
ER  - 
%0 Journal Article
%A V. S. Kolezhuk
%A S. Yu. Pilyugin
%T Rates of convergence of approximate attractors for parabolic equations
%J Zapiski Nauchnykh Seminarov POMI
%D 2006
%P 67-111
%V 336
%U http://geodesic.mathdoc.fr/item/ZNSL_2006_336_a4/
%G en
%F ZNSL_2006_336_a4
V. S. Kolezhuk; S. Yu. Pilyugin. Rates of convergence of approximate attractors for parabolic equations. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 37, Tome 336 (2006), pp. 67-111. http://geodesic.mathdoc.fr/item/ZNSL_2006_336_a4/

[1] A. V. Babin, M. I. Vishik, Attractors of Evolution Equations, Studies in Math. and its Appl., 25, North-Holland, Amsterdam, 1992 | MR | Zbl

[2] G. R. Sell, Y. You, Dynamics of Evolutionary Equations, Appl. Math. Sci., 143, Springer-Verlag, New York, Inc., 2002 | MR | Zbl

[3] D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, New York, 1981 | MR | Zbl

[4] J. K. Hale, X.-B. Lin, G. Raugel, “Upper semicontinuity of attractors for approximations of semigroups and partial differential equations”, Math. in Comput., 50 (1988), 89–123 | DOI | MR | Zbl

[5] J. K. Hale, G. Raugel, “Lower semicontinuity of attractors of gradient systems and applications”, Ann. Mat. Pura Appl., 154 (1989), 281–326 | DOI | MR | Zbl

[6] I. N. Kostin, S. Yu. Pilyugin, “Uniform exponential stability of attractors for perturbed evolution equations”, Dokl. Ros. Akad. Nauk, 369 (1999), 449–450 | MR | Zbl

[7] I. N. Kostin, S. Yu. Pilyugin, “Uniform exponential stability of attractors for a family of systems”, J. Evol. Equat. (to appear)

[8] P. Brunovský, S.-N. Chow, “Generic properties of stationary state solutions of reaction-diffusion equations”, J. Differential Equations, 53 (1984), 1–23 | DOI | MR | Zbl

[9] N. Chafee, E. F. Infante, “A bifurcation problem for a nonlinear partial differential equation of parabolic type”, Applicable Anal., 4 (1974), 17–37 | DOI | MR | Zbl

[10] I. N. Kostin, “Rate of attraction to a non-hyperbolic attractor”, Asymptot. Anal., 16 (1998), 203–222 | MR | Zbl

[11] A. Al-Nayef, P. Diamond, P. Kloeden, V. Kozyakin, A. Pokrovskii, “Bi-shadowing and delay equations”, Dynamics and Stability of Systems, 11 (1996), 121–134 | MR | Zbl

[12] D. B. Henry, “Some infinite dimensional Morse-Smale systems defined by parabolic partial differential equations”, J. Differential Equations, 53 (1985), 401–458 | MR

[13] Z. Nitecki, Differentiable Dynamics, MIT Press, 1971 | MR

[14] J. K. Hale, L. T. Magalhães, W. M. Oliva, An Introduction to Infinite Dimensional Dynamical Systems – Geometric Theory, Appl. Math. Sci., 47, Springer-Verlag, Berlin, New York, 1984 | MR | Zbl

[15] K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Math. and its Appl., 501, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000 | MR | Zbl

[16] J. W. Robbin, “A structural stability theorem”, Annals of Math., 94 (1971), 447–493 | DOI | MR | Zbl

[17] S. Yu. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Mathematics, 1706, Springer-Verlag, Berlin, New York, 1999 | MR | Zbl

[18] C. Pugh, M. Shub, “The $\Omega$-stability theorem for flows”, Inventiones Mathematicae, 11 (1970), 150–158 | DOI | MR | Zbl

[19] C. Robinson, “Structural stability of vector fields”, Ann. Math., 99 (1974), 154–175 | DOI | MR | Zbl

[20] Kolezhuk V. S., “Tipichnye svoistva diskretizatsii parabolicheskikh uravnenii”, Nelineinye dinamicheskie sistemy, Vyp. 5, Izdatelstvo Sankt-Peterburgskogo universiteta, SPb., 2003, 43–58

[21] W.-J. Beyn, S. Yu. Pilyugin, “Attractors of reaction diffusion systems on infinite lattices”, J. Dynamics Differential Equations, 15 (2003), 485–515 | DOI | MR | Zbl

[22] S. Larsson, Nonsmooth data error estimates with application to the study of the long-time behavior of finite element solutions of semilinear parabolic equations, Preprint 1992-36, Dept. Math. Chalmers Univ. Techn. Göteborg Univ., 1992