no. 4
Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 4, pp. 565-582.

In this paper we give a detailed study of the global attractors for parabolic equations governed by the p-Laplacian in a heterogeneous medium. Not only the existence but also the infinite dimensionality of the global attractors is presented by showing that their ε-Kolmogorov entropy behaves as a polynomial of the variable 1/ϵ as ε tends to zero, which is not observed for non-degenerate parabolic equations. The upper and lower bounds for the Kolmogorov ε-entropy of infinite-dimensional attractors are also obtained.

@article{AIHPC_2011__28_4_565_0,
     author = {Efendiev, Messoud A. and \^Otani, Mitsuharu},
     title = {Infinite-dimensional attractors for parabolic equations with {\protect\emph{p}-Laplacian} in heterogeneous medium},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {565--582},
     publisher = {Elsevier},
     volume = {28},
     number = {4},
     year = {2011},
     doi = {10.1016/j.anihpc.2011.03.006},
     zbl = {1242.35159},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.03.006/}
}
TY  - JOUR
AU  - Efendiev, Messoud A.
AU  - Ôtani, Mitsuharu
TI  - Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2011
SP  - 565
EP  - 582
VL  - 28
IS  - 4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2011.03.006/
DO  - 10.1016/j.anihpc.2011.03.006
LA  - en
ID  - AIHPC_2011__28_4_565_0
ER  - 
%0 Journal Article
%A Efendiev, Messoud A.
%A Ôtani, Mitsuharu
%T Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium
%J Annales de l'I.H.P. Analyse non linéaire
%D 2011
%P 565-582
%V 28
%N 4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2011.03.006/
%R 10.1016/j.anihpc.2011.03.006
%G en
%F AIHPC_2011__28_4_565_0
Efendiev, Messoud A.; Ôtani, Mitsuharu. Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 4, pp. 565-582. doi : 10.1016/j.anihpc.2011.03.006. http://www.numdam.org/articles/10.1016/j.anihpc.2011.03.006/

[1] H. Brézis, Opérateurs Maximaux Monotone et Semi-Groupes de Contractions dans les Espaces Hilbert, North-Holland Math. Stud. vol. 5 (1973) | Zbl

[2] E. Dibenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York (1993) | Zbl

[3] M.A. Efendiev, M. Ôtani, Infinite-dimensional attractors for evolution equations with p-Laplacian and their Kolmogorov entropy, Differential Integral Equations 20 (2007), 1201-1209 | Zbl

[4] M.A. Efendiev, S. Zelik, Upper and lower bounds for the Kolmogorov entropy of the attractor for reaction–diffusion equation in an unbounded domain, J. Dynam. Differential Equations 14 (2002), 369-403 | Zbl

[5] M.A. Efendiev, S. Zelik, Finite and infinite dimensional attractors for porous media equations, Proc. Lond. Math. Soc. 96 (2008), 51-77 | Zbl

[6] E. Feireisl, Ph. Laurencot, F. Simondon, Global attractors for degenerate parabolic equations on unbounded domains, J. Differential Equations 129 no. 2 (1996), 239-261 | Zbl

[7] A.N. Kolmogorov, V.M. Tikhomirov, ℰ-entropy and ε-capacity of sets in functional space, Amer. Math. Soc. Transl. Ser. 2 vol. 17 (1961), 277-364

[8] M. Nakao, N. Aris, On global attractor for nonlinear parabolic equations of m-Laplacian type, J. Math. Anal. Appl. 331 (2007), 793-809 | Zbl

[9] M. Ôtani, Non-monotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations 46 no. 12 (1982), 268-299 | Zbl

[10] M. Ôtani, L ∞ -energy method and its applications, Nonlinear Partial Differential Equations and Their Applications, GAKUTO Internat. Ser. Math. Sci. Appl. vol. 20, Gakkotosho, Tokyo (2004), 505-516 | Zbl

[11] M. Ôtani, L ∞ -energy method — Basic tools and usage, Vasile Staicu (ed.), Differential Equations, Chaos and Variational Problems, Progr. Nonlinear Differential Equations Appl. vol. 75, Birkhäuser (2007), 357-376

[12] S. Takeuchi, T. Yokota, Global attractors for a class of degenerate diffusion equations, Electron. J. Differential Equations 2003 no. 76 (2003), 1-13 | EuDML | Zbl

[13] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. vol. 68, Springer-Verlag, New York (1997) | Zbl

Cité par Sources :