Sufficient and Necessary Criteria for Existence of Pullback Attractors for Non-compact Random Dynamical Systems

By Bixiang Wang. 2012

460 citations recorded by the source corpus.

We study pullback attractors of non-autonomous non-compact dynamical systems generated by differential equations with non-autonomous deterministic as well as stochastic forcing terms. We first introduce the concepts of pullback attractors and asymptotic compactness for such systems. We then prove a sufficient and necessary condition for existence of pullback attractors. We also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors. For random systems containing periodic deterministic forcing terms, we show the pullback attractors are also periodic. As an application of the abstract theory, we prove the existence of a unique pullback attractor for Reaction-Diffusion equations on $\R^n$ with both deterministic and random external terms. Since Sobolev embeddings are not compact on unbounded domains, the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions.

Subfields: Computational Theory and Mathematics, Computer Networks and Communications, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/f88201a4a095d5225aa30136cb28e4437e58becf

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Citations in the corpus

Exponential Stability of Non-Autonomous Stochastic Delay Lattice Systems with Multiplicative Noise (2015, methodology): “We recall some concepts on pullback attractors for non-autonomous random dynamical systems from[34].”
Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise (2015, methodology): “Motivated by the ideas of [11, 20, 26, 29], in this article, we first give some conditions for bounding the fractal dimension of a random invariant set for a nonautonomous random dynamical system originated from Debussche [11].”
Existence, Stability and Bifurcation of Random Complete and Periodic Solutions of Stochastic Parabolic Equations (2013, background): “Similarly, the structures of random attractors of stochastic equations are fully described by random complete solutions as demonstrated in [51].”