Lower semicontinuity of attractors for non-autonomous dynamical systems

Ergodic Theory and Dynamical Systems

Volumn 29, Issue 6, 2009, Pages 1765-1780

  Carvalho, Alexandre N ^a   Langa, José A ^b   Robinson, James C ^c  

^a UNIVERSITY OF SÃO PAULO   (Brazil)

^b UNIVERSITY OF SEVILLE   (Spain)

^c UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 74349095065     PISSN: 01433857     EISSN: 14694417     Source Type: Journal    
DOI: 10.1017/S0143385708000850     Document Type: Article

References (30)

1. E. A. M. Abreu and A. N. Carvalho. Lower semicontinuity of attractors for parabolic problems with Dirichlet boundary conditons in varying domains. Mat. Contemp. 27 (2004), 37-51.
2. J. M. Arrieta and A. N. Carvalho. Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain. J. Differential Equations 199 (2004), 143-178.
3. P. Bates, K. Lu and C. Zeng. Existence and persistence of invariant manifolds for semiflows in Banach spaces. Mem. Amer. Math. Soc. 135(645) (1999).
4. A. Berger and S. Siegmund. Uniformly attracting solutions of nonautonomous differential equations. Nonlinear Anal. 68(12) (2008), 3789-3811.
5. S. M. Bruschi, A. N. Carvalho, J. W. Cholewa and T. Dłotko. Uniform exponential dichotomy and continuity of attractors for singularly perturbed damped wave equations. J. Dynam. Differential Equations 18 (2006), 767-814.
6. T. Caraballo, J. A. Langa and J. C. Robinson. Upper semicontinuity of attractors for small random perturbations of dynamical systems. Comm. Partial Differential Equations 23(9-10) (1998), 1557-1581.
7. T. Caraballo and J. A. Langa. On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10(4) (2003), 491-513.
8. V. L. Carbone, A. N. Carvalho and K. Schiabel-Silva. Continuity of attractors for parabolic problems with localized large diffusion. Nonlinear Anal. 68(3) (2008), 515-535.
9. A. N. Carvalho and S. Piskarev. A general approximation scheme for attractors of abstract parabolic problems. Numer. Funct. Anal. Optim. 27 (2006), 785-829.
10. A. N. Carvalho and J. A. Langa. Non-autonomous perturbation of autonomous semilinear differential equations: continuity of local stable and unstable manifolds. J. Differential Equations 233(2) (2007), 622-653.
11. A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez. Characterization of non-autonomous attractors of a perturbed gradient system. J. Differential Equations 236 (2007), 570-603.
12. H. Crauel, A. Debussche and F. Flandoli. Random attractors. J. Dynam. Differential Equations 9 (1995), 307-341.
13. D. N. Cheban, P. E. Kloeden and B. Schmalfuß. The relationship between pullback, forward and global attractors of nonautonomous dynamical systems. Nonlinear Dyn. Syst. Theory 2(2) (2002), 125-144.
14. V. V. Chepyzhov and M. I. Vishik. Attractors for Equations of Mathematical Physics (AMS Colloquium Publications, 49). American Mathematical Society, Providence, RI, 2002.
15. M. Efendiev, S. Zelik and A. Miranville. Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems. Proc. Roy. Soc. Edinburgh Sect. A 135(4) (2005), 703-730.
16. C. M. Elliott and I. N. Kostin. Lower semicontinuity of a non-hyperbolic attractor for the viscous Cahn- Hilliard equation. Nonlinearity 9 (1996), 687-702.
17. J. K. Hale, X. B. Lin and G. Raugel. Upper semicontinuity of attractors for approximation of semigroups and partial differential equations. J. Math. Comput. 50 (1988), 89-123.
18. J. K. Hale. Asymptotic Behavior of Dissipative Systems (Mathematical Surveys and Monographs, 25). American Mathematical Society, Providence, RI, 1988.
19. J. K. Hale and G. Raugel. Lower semicontinuity of attractors of gradient systems and applications. Ann. Mat. Pura Appl. 154(4) (1989), 281-326.
20. D. Henry. Geometric Theory of Semilinear Parabolic Equations (Lecture Notes in Mathematics, 840). Springer, Berlin, 1981.
21. P. E. Kloeden and B. Schmalfuß. Asymptotic behaviour of non-autonomous difference inclusions. Systems Control Lett. 33 (1998), 275-280.
22. I. N. Kostin. Lower semicontinuity of a non-hyperbolic attractor. J. London Math. Soc. 52 (1995), 568-582.
23. J. A. Langa, J. C. Robinson, A. Suárez and A. Vidal-López. The structure of attractors in non-autonomous perturbations of gradient-like systems. J. Differential Equations 234(2) (2007), 607-625.
24. J. A. Langa, J. C. Robinson and A. Suárez. Bifurcation from zero of a complete trajectory for nonautonomous logistic PDEs. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15(8) (2005), 2663-2669.
25. V. A. Pliss and G. R. Sell. Robustness of exponential dichotomies in infinite dimensional dynamical systems. J. Dynam. Differential Equations 11 (1999), 471-513.
26. V. A. Pliss and G. R. Sell. Perturbations of foliated bundles and evolutionary equations. Ann. Mat. Pura Appl. (4) 185(Suppl.) (2006), S325-S388.
27. J. C. Robinson. Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge University Press, Cambridge, 2001.
28. A. Rodríguez-Beral and A. Vidal-López. Extremal equilibria and asymptotic behavior of parabolic nonlinear reaction-diffusion equations. Nonlinear Elliptic and Parabolic Problems (Progress in Nonlinear Differential Equations and their Applications, 64). Birkhäuser, Basel, 2005, pp. 509-516.
29. A. M. Stuart and A. R. Humphries. Dynamical Systems and Numerical Analysis. Cambridge University Press, Cambridge, 1996.
30. R. Temam. Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York, 1988.