# American Institute of Mathematical Sciences

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July  2017, 22(5): 1779-1800. doi: 10.3934/dcdsb.2017106

## Attractors for a random evolution equation with infinite memory: Theoretical results

 1 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain 2 Institut für Mathematik, Institut für Stochastik, Ernst Abbe Platz 2,07737-Jena, Germany 3 Universidad Miguel Hernandez de Elche, Centro de Investigación Operativa, Avda. Universidad s/n, 03202-Elche (Alicante), Spain

* Corresponding author

Received  April 2016 Revised  June 2016 Published  March 2017

Fund Project: This work has been partially supported by FEDER and Spanish Ministerio de Economĺa y Competitividad, project MTM2015-63723-P, and by Junta de Andalucĺa under Proyecto de Excelencia P12-FQM-1492.

The long-time behavior of solutions (more precisely, the existence of random pullback attractors) for an integro-differential parabolic equation of diffusion type with memory terms, more particularly with terms containing both finite and infinite delays, as well as some kind of randomness, is analyzed in this paper. We impose general assumptions not ensuring uniqueness of solutions, which implies that the theory of multivalued dynamical system has to be used. Furthermore, the emphasis is put on the existence of random pullback attractors by exploiting the techniques of the theory of multivalued nonautonomous/random dynamical systems.

Citation: Tomás Caraballo, María J. Garrido-Atienza, Björn Schmalfuss, José Valero. Attractors for a random evolution equation with infinite memory: Theoretical results. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1779-1800. doi: 10.3934/dcdsb.2017106
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