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- W2003629394 abstract "We study the stability of attractors under non-autonomous perturbations that are uniformly small in time. While in general the pullback attractors for the non-autonomous problems converge towards the autonomous attractor only in the Hausdorff semi-distance (upper semicontinuity), the assumption that the autonomous attractor has a ‘gradient-like’ structure (the union of the unstable manifolds of a finite number of hyperbolic equilibria) implies convergence (i.e. also lower semicontinuity) provided that the local unstable manifolds perturb continuously. We go further when the underlying autonomous system is itself gradient-like, and show that all trajectories converge to one of the hyperbolic trajectories as t → ∞ . In finite-dimensional systems, in which we can reverse time and apply similar arguments to deduce that all bounded orbits converge to a hyperbolic trajectory as t → − ∞ , this implies that the ‘gradient-like’ structure of the attractor is also preserved under small non-autonomous perturbations: the pullback attractor is given as the union of the unstable manifolds of a finite number of hyperbolic trajectories." @default.
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- W2003629394 date "2007-03-01" @default.
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- W2003629394 title "The stability of attractors for non-autonomous perturbations of gradient-like systems" @default.
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- W2003629394 doi "https://doi.org/10.1016/j.jde.2006.11.016" @default.
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