Skip to main content
    • Aa
    • Aa

Lower semicontinuity of attractors for non-autonomous dynamical systems


This paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[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), 143178.

[4]A. Berger and S. Siegmund . Uniformly attracting solutions of nonautonomous differential equations. Nonlinear Anal. 68(12) (2008), 37893811.

[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), 767814.

[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), 15571581.

[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), 515535.

[9]A. N. Carvalho and S. Piskarev . A general approximation scheme for attractors of abstract parabolic problems. Numer. Funct. Anal. Optim. 27 (2006), 785829.

[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), 622653.

[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), 570603.

[12]H. Crauel , A. Debussche and F. Flandoli . Random attractors. J. Dynam. Differential Equations 9 (1995), 307341.

[16]C. M. Elliott and I. N. Kostin . Lower semicontinuity of a non-hyperbolic attractor for the viscous Cahn-Hilliard equation. Nonlinearity 9 (1996), 687702.

[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), 89123.

[19]J. K. Hale and G. Raugel . Lower semicontinuity of attractors of gradient systems and applications. Ann. Mat. Pura Appl. 154(4) (1989), 281326.

[21]P. E. Kloeden and B. Schmalfuß . Asymptotic behaviour of non-autonomous difference inclusions. Systems Control Lett. 33 (1998), 275280.

[22]I. N. Kostin . Lower semicontinuity of a non-hyperbolic attractor. J. London Math. Soc. 52 (1995), 568582.

[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), 607625.

[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), 26632669.

[25]V. A. Pliss and G. R. Sell . Robustness of exponential dichotomies in infinite dimensional dynamical systems. J. Dynam. Differential Equations 11 (1999), 471513.

[26]V. A. Pliss and G. R. Sell . Perturbations of foliated bundles and evolutionary equations. Ann. Mat. Pura Appl. (4) 185(Suppl.) (2006), S325S388.

[28]A. Rodríguez-Bernal 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. 509516.

[30]R. Temam . Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York, 1988.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *