DCDS
Non--autonomous and random attractors for delay random semilinear equations without uniqueness
Tomás Caraballo M. J. Garrido-Atienza B. Schmalfuss José Valero
Discrete & Continuous Dynamical Systems - A 2008, 21(2): 415-443 doi: 10.3934/dcds.2008.21.415
We first prove the existence and uniqueness of pullback and random attractors for abstract multi-valued non-autonomous and random dynamical systems. The standard assumption of compactness of these systems can be replaced by the assumption of asymptotic compactness. Then, we apply the abstract theory to handle a random reaction-diffusion equation with memory or delay terms which can be considered on the complete past defined by $\mathbb{R}^{-}$. In particular, we do not assume the uniqueness of solutions of these equations.
keywords: multi-valued non-autonomous and random dynamical systems pullback and random attractors delay differential equations. Cocycles
DCDS
Existence of exponentially attracting stationary solutions for delay evolution equations
Tomás Caraballo M. J. Garrido-Atienza B. Schmalfuss
Discrete & Continuous Dynamical Systems - A 2007, 18(2&3): 271-293 doi: 10.3934/dcds.2007.18.271
We consider the exponential stability of semilinear stochastic evolution equations with delays when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution exponentially stable, for which we use a general random fixed point theorem for general cocycles. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly, by means of the theory of random dynamical systems and their conjugation properties.
keywords: stationary solutions exponential stability. random dynamical systems delay equations cocycles
DCDS-B
Stabilization of stationary solutions of evolution equations by noise
Tomás Caraballo P.E. Kloeden B. Schmalfuss
Discrete & Continuous Dynamical Systems - B 2006, 6(6): 1199-1212 doi: 10.3934/dcdsb.2006.6.1199
We investigate the existence, uniqueness and exponential stability of non-constant stationary solutions of stochastic semilinear evolution equations. Our main result shows, in particular, that noise can have a stabilization effect on deterministic equations. Moreover, we do not require any commutative condition on the noise terms.
keywords: stationary solutions stabilization. random dynamical systems exponential stability

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