Infinite-dimensional exponential attractors for nonlinear reaction-diffusion systems in unbounded domains and their approximation

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 460, Issue 2044, 2004, Pages 1107-1129

  Efendiev, M ^a   Miranville, A ^b   Zelik, S ^b  

^a UNIVERSITY OF STUTTGART   (Germany)

^b UNIVERSITÉ DE POITIERS   (France)

Author keywords

Exponential attractors; Global attractor; Kolmogorov's entropy; Reaction diffusion systems; Unbounded domains

Indexed keywords

ARTICLE; DIFFUSION; ENTROPY; MATHEMATICAL ANALYSIS; MOLECULAR INTERACTION; NONLINEAR SYSTEM; REACTION ANALYSIS; SOLUTION AND SOLUBILITY;

EID: 1642485066     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2003.1182     Document Type: Article

References (28)

1. Abergel, F. 1990 Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains. J. Diff. Eqns 83, 359-370.
2. Babin, A. & Nicolaenko, B. 1995 Exponential attractors of reaction-diffusion systems in an unbounded domain. J. Dynam. Diff. Eqns 7, 567-590.
3. Babin, A. & Vishik, M. 1990 Attractors of partial differential evolution equations in an unbounded domain. Proc. R. Soc. Edinb. A 116, 221-243.
4. Babin, A. & Vishik, M. 1992 Attractors of evolutionary equations. Amsterdam: North-Holland.
5. Chepyzhov, V. & Vishik, M. 1998 Kolmogorov's ε-entropy for the attractor of reaction-diffusion equations. Mat. Sb. 189, 81-110.
6. Chepyzhov, V. & Vishik, M. 2002 Attractors for equations of mathematical physics. Providence, RI: American Mathematical Society.
7. Collet, P. & Eckmann, J. 1999 Extensive properties of the complex Ginzburg-Landau equation. Commun. Math. Phys. 200, 699-722.
8. Collet, P. & Eckmann, J. 2000 Topological entropy and ε-entropy for damped hyperbolic equations. Annls Inst. H. Poincaré Analyse Non Linéaire 1, 715-752.
9. Eden, A., Foias, C., Nicolaenko, B. & Temam, R. 1994 Exponential attractors for dissipative evolution equations. Wiley.
10. n with a strong nonlinearity. Discrete Contin. Dynam. Syst. 15, 399-424.
11. Efendiev, M. & Miranville, A. 2003 On the dimension of the global attractor for dissipative reaction-diffusion systems. Appl. Math. Lett. 16, 351-355.
12. Efendiev, M. & Zelik, S. 2001 The attractor for a nonlinear reaction-diffusion system in the unbounded domain. Commun. Pure Appl. Math. 54, 625-688.
13. Efendiev, M. & Zelik, S. 2002 Upper and lower bounds for the Kolmogorov entropy of the attractor for a reaction-diffusion equation in an unbounded domain. J. Dynam. Diff. Eqns 14, 369-403.
14. 3. C. R. Acad. Sci. Paris Sér. I 330, 713-718.
15. Efendiev, M., Miranville, A. & Zelik, S. 2002 Global and exponential attractors for nonlinear reaction-diffusion systems in unbounded domains. Université de Poitiers Preprints, vol. 163, pp. 1-45.
16. Efendiev, M., Miranville, A. & Zelik, S. 2003 Infinite dimensional attractors for a non-autonomous reaction-diffusion system. Math. Nachr. 248-249, 72-96.
17. Hale, J. 1987 Asymptotic behavior of dissipative systems. In Mathematical Surveys and Monographs, vol. 25. Providence, RI: American Mathematical Society.
18. Kolmogorov, A. & Tikhomirov, V. 1993 ε-entropy and ε-capacity of sets in functional spaces. In Selected works of A. N. Kolmogorov, vol. 3. Dordrecht: Kluwer.
19. Ladyzhenskaya, O. 1991 Attractors for semigroups of evolution equations. Cambridge University Press.
20. Ladyzhenskaya, O., Solonnikov, O. & Uraltseva, N. 1967 Linear and quasilinear equations of parabolic type. Providence, RI: American Mathematical Society.
21. n. J. Diff. Eqns 132, 87-106.
22. Mielke, A. & Schneider, G. 1995 Attractors for modulation equations on unbounded domains: existence and comparison. Nonlinearity 8, 743-768.
23. Temam, R. 1988 Infinite-dimensional dynamical systems in mechanics and physics. Springer.
24. n and the estimation of its ε-entropy. Math. Notes 65, 941-943.
25. Zelik, S. 2000 Spatial and dynamical chaos generated by reaction-diffusion systems in unbounded domains. Dynamik, Analysis, effiziente Simulation und Ergodentheorie, Freie Universität Berlin, preprint 38/00, pp. 1-60.
26. Zelik, S. 2001a The attractor for a nonlinear hyperbolic equation in an unbounded domain. Discrete Contin. Dynam. Syst. A 7, 593-641.
27. Zelik, S. 2001b The attractor for a nonlinear reaction-diffusion system in an unbounded domain and Kolmogorov's epsilon-entropy. Math. Nachr. 232, 129-179.
28. Zelik, S. 2003 Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity Commun. Pure Appl. Math. 56, 584-637.