Upper semicontinuous dependence of pullback attractors on time scales. (English) Zbl 1103.34045
Dynamic equations on time scales are intrinsically nonautonomous. This means that their solutions fail to satisfy a semigroup property, even if the corresponding vector field does not depend explicitly on time (unless the time scale has constant graininess reducing the problem to an autonomous one). Therefore, only nonautonomous counterparts of established concepts in the theory of dynamical systems (attractors, invariant manifolds, Lyapunov functions, etc.) are appropriate to tackle such general dynamic equations on inhomogeneous time scales.
The present paper deals with pullback attractors – a natural and flexible nonautonomous attractor notion – of dynamic equations.
In particular, the paper contains an existence criterion for pullback attractors under a pullback dissipativity condition (the existence of a pullback absorbing set). As main results, however, the following is shown: (1) Pullback dissipativity is robust under small variations of the time scale. (2) Pullback attractors of dynamic equations depend upper semi-continuously on the time scale. Here, the set of all time scales is endowed with the topology generated by the Hausdorff distance.
It is worth to point out that this is one of the few papers dealing with the behavior of dynamic equations under variation of the underlying time scale, which (in contrast to classical perturbation results) affects not only the right-hand side, but also the differentiation operator.
The present paper deals with pullback attractors – a natural and flexible nonautonomous attractor notion – of dynamic equations.
In particular, the paper contains an existence criterion for pullback attractors under a pullback dissipativity condition (the existence of a pullback absorbing set). As main results, however, the following is shown: (1) Pullback dissipativity is robust under small variations of the time scale. (2) Pullback attractors of dynamic equations depend upper semi-continuously on the time scale. Here, the set of all time scales is endowed with the topology generated by the Hausdorff distance.
It is worth to point out that this is one of the few papers dealing with the behavior of dynamic equations under variation of the underlying time scale, which (in contrast to classical perturbation results) affects not only the right-hand side, but also the differentiation operator.
Reviewer: Christian Pötzsche (Minneapolis)
MSC:
| 34D45 | Attractors of solutions to ordinary differential equations |
| 39A14 | Partial difference equations |
| 39A13 | Difference equations, scaling (\(q\)-differences) |
| 37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
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