Non-autonomous attractors for integro-differential evolution equations
Tomás Caraballo - Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain (email)
Abstract: 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, differential equation with
infinite delay, set-valued process, set-valued non-autonomous
dynamical system, pullback attractor.
Received: May 2008; Revised: July 2008; Published: January 2009.