Skip to main content Accessibility help

Autonomous and non-autonomous unbounded attractors under perturbations

  • Alexandre N. Carvalho (a1) and Juliana F. S. Pimentel (a2)


Pullback attractors with forwards unbounded behaviour are to be found in the literature, but not much is known about pullback attractors with each and every section being unbounded. In this paper, we introduce the concept of unbounded pullback attractor, for which the sections are not required to be compact. These objects are addressed in this paper in the context of a class of non-autonomous semilinear parabolic equations. The nonlinearities are assumed to be non-dissipative and, in addition, defined in such a way that the equation possesses unbounded solutions as the initial time goes to -∞, for each elapsed time. Distinct regimes for the non-autonomous term are taken into account. Namely, we address the small non-autonomous perturbation and the asymptotically autonomous cases.



Hide All
1Arrieta, J. and Carvalho, A.. Abstract parabolic problems with critical nonlinearities and applications to Navier Stokes and Heat Equations. Trans. Amer. Math. Soc. 352 (2000), 285310.
2Arrieta, J., Carvalho, A. and Rodríguez-Bernal, A.. Parabolic problems with nonlinear boundary conditions and critical nonlinearities. J. Differ. Equ. 156 (1999), 376406.
3Arrieta, J., Carvalho, A. and Rodríguez-Bernal, A.. Attractors for parabolic problems with nonlinear boundary condition. Uniform bounds. Comm. Partial Differ. Equ. 25 (2000), 137.
4Ben-Gal, N.. Grow-Up Solutions and Heteroclinics to Infinity for Scalar Parabolic PDEs. PhD Thesis, Division of Applied Mathematics, Brown University, 2010.
5Bortolan, M., Caraballo, T., Carvalho, A. and Langa, J.. Skew product semiflows and Morse decomposition. J. Differ. Equ. 255 (2013), 24362462.
6Bortolan, M., Carvalho, A. and Langa, J.. Structure of attractors for skew product semiflows. J. Differ. Equ. 257 (2014), 490522.
7Bortolan, M., Carvalho, A., Langa, J. and Raugel, G.. Non-autonomous perturbations of Morse-Smale semigroups: stability of the phase diagram. Preprint.
8Carvalho, A., Langa, J. and Robinson, J.. Structure and bifurcation of pullback attractors in a non-autonomous Chafee-Infante equation. Proc. Amer. Math. Soc. 140 (2011), 23572373.
9Carvalho, A., Langa, J. and Robinson, J.. Attractors for infinite-dimensional non-autonomous dynamical systems. In Applied mathematical sciences (ed. Antman, S.S., Holmes, P. and Sreenivasan, K.), vol.182 (New York: Springer-Verlag, 2013).
10Chepyzhov, V. V.. On unbounded invariant sets and attractors of some semilinear equations and systems of parabolic type. Russian Math. Surveys 42 (1987), 167168.
11Chepyzhov, V. V.. Unbounded attractors of some parabolic systems of differential equations and estimates on their dimension. Soviet Math. Dokl. 38 (1989), 449.
12Chepyzhov, V. V. and Goritskii, A. Yu.. Unbounded attractors of evolution equations, Properties of Global Attractors of Partial Differential Equations. Adv. Soviet Math. 10, Amer. Math. Soc., Providence, RI (1992), 85128.
13Hell, J.. Conley index at infinity. Topol. Methods Nonlinear Anal. 42 (2013), 137167.
14Henry, D.. Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840 (Berlin-New York: Springer-Verlag, 1981).
15Langa, J., Robinson, J., Rodríguez-Bernal, A., Suárez, A. and Vidal-López, A.. Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations. Discrete Contin. Dyn. Syst. 18 (2007), 483497.
16Miklavčič, M.. A sharp condition for existence of an inertial manifold. J. Dynam. Differ. Equ. 3 (1991), 437456.
17Pimentel, J. and Rocha, C.. A permutation related to non-compact global attractors for slowly non-dissipative systems. J. Dynam. Differ. Equ. 28 (2016), 128.


MSC classification

Autonomous and non-autonomous unbounded attractors under perturbations

  • Alexandre N. Carvalho (a1) and Juliana F. S. Pimentel (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed