DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2009.2.xx DYNAMICAL SYSTEMS SERIES S Volume 2, Number 1, March 2009 pp. 1–XXX NON-AUTONOMOUS ATTRACTORS FOR INTEGRO-DIFFERENTIAL EVOLUTION EQUATIONS T. Caraballo Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico Facultad de Matem´ aticas, Universidad de Sevilla Apartado de Correos 1160, 41080-Sevilla, Spain P. E. Kloeden J.W. Goethe-Universit¨ at , FB Mathematik, Postfach 11 19 32 D-60054 Frankfurt a.M., Germany Abstract. We show that infinite-dimensional integro-differential equations which involve an integral of the solution over the time interval since start- ing can be formulated as non-autonomous delay differential equations with an infinite delay. Moreover, when conditions guaranteeing uniqueness of solutions do not hold, they generate a non-autonomous (possibly) multi-valued dynami- cal system (MNDS). The pullback attractors here are defined with respect to a universe of subsets of the state space with sub-exponetial growth, rather than restricted to bounded sets. The theory of non-autonomous pullback attractors is extended to such MNDS in a general setting and then applied to the original integro-differential equations. Examples based on the logistic equations with and without a diffusion term are considered. 1. Introduction. The main aim of this paper is to show that a wide class of integro-differential partial differential equations can be analyzed within the frame- work of non-autonomous dynamical systems, and the long-time behaviour of their solutions can be investigated with the help of the theory of pullback attractors. This theory is now well established as has been extensively developed over the last one and a half decades. Pullback attractors have proven to be appropriate concepts to describe the long–time behaviour of many dynamical systems arising in science, especially those exhibiting non-autonomity (see, e.g. Caraballo et al. [15], Cheban et al. [20], Chepyzhov and Vishik [21]), Chueshov [22], Crauel and Flandoli [23], Flandoli and Schmalfuß [25], Kloeden [28], Kloeden and Schmalfuß [29], Robinson [33], Schmalfuß [34], amongst many others). Integro-differential equations appear in various branches of science (e.g. in mod- elling the growth of parasite population, in Lotka-Volterra predator-prey systems, in reaction-diffusion models with memory, and their relevance is without doubt. In general, the models containing in their equations some kind of delay terms are now being studied extensively, since it is assumed that in many phenomena from reality, 2000 Mathematics Subject Classification. Primary: 34G25, 34K25, 35R10; Secondary: 34D45, 37C70, 47H20. Key words and phrases. Integro-differential equation, differential equation with infinite delay, set-valued process, set-valued non-autonomous dynamical system, pullback attractor. Partially supported by Ministerio de Educaci´ on y Ciencia (Spain), FEDER (European Commu- nity) under grant MTM2005-01412, and Consejer´ ıa de Innovaci´ on Ciencia y Empresa de la Junta de Andaluc´ ıa (Spain) under grant P07-FQM-02468. 1