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June  2008, 21(2): 415-443. doi: 10.3934/dcds.2008.21.415

Non--autonomous and random attractors for delay random semilinear equations without uniqueness

Tomás Caraballo 1, , M. J. Garrido-Atienza 2, , B. Schmalfuss 3, and  José Valero 4,

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
2. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080–Sevilla, Spain
3. 

Institut für Mathematik Fakultät EIM, Universität Paderborn, Warburger Strasse 100, 33098 Paderborn
4. 

Centro de Investigación Operativa, Universidad Miguel Hernández, Avda Universidad s/n, 03202 Elche, Alicante

Received  April 2007 Revised  November 2007 Published  March 2008

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.
Mathematics Subject Classification: 60H15, 35K4.
Citation: Tomás Caraballo, M. J. Garrido-Atienza, B. Schmalfuss, José Valero. Non--autonomous and random attractors for delay random semilinear equations without uniqueness. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 415-443. doi: 10.3934/dcds.2008.21.415
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