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March  2009, 2(1): 17-36. doi: 10.3934/dcdss.2009.2.17

Non-autonomous attractors for integro-differential evolution equations

Tomás Caraballo 1, and  P.E. Kloeden 2,

1. 

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

FB Mathematik, Johann Wolfgang Goethe Universität, Postfach 11 19 32, D-60054 Frankfurt a.M.

Received  May 2008 Revised  July 2008 Published  January 2009

We show that infinite-dimensional integro-differential equations which involve an integral of the solution over the time interval since starting 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 dynamical 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.
Keywords: Integro-differential equation, set-valued process, differential equation with infinite delay, pullback attractor., set-valued non-autonomous dynamical system.
Mathematics Subject Classification: Primary: 34G25, 34K25, 35R10; Secondary: 34D45, 37C70, 47H2.
Citation: Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17
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