Global attractor for a non-autonomous integro-differential equation in materials with memory

• Published in: Nonlinear analysis 2010-07, Vol.73 (1), p.183-201
• Main Authors: Caraballo, T., Garrido-Atienza, M.J., Schmalfuß, B., Valero, J.
• Format: Article
• Language: English
• Subjects: Asymptotic behavior; Delayed reaction–diffusion equations; Exact sciences and technology; Global analysis, analysis on manifolds; Integro-differential equations with memory; Mathematical analysis; Mathematics; Multivalued dynamical systems; Non-autonomous (pullback) attractors; Ordinary differential equations; Partial differential equations; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2010.03.012

The long-time behavior of an integro-differential parabolic equation of diffusion type with memory terms, expressed by convolution integrals involving infinite delays and by a forcing term with bounded delay, is investigated in this paper. The assumptions imposed on the coefficients are weak in the sense that uniqueness of solutions of the corresponding initial value problems cannot be guaranteed. Then, it is proved that the model generates a multivalued non-autonomous dynamical system which possesses a pullback attractor. First, the analysis is carried out with an abstract parabolic equation. Then, the theory is applied to the particular integro-differential equation which is the objective of this paper. The general results obtained in the paper are also valid for other types of parabolic equations with memory.