A New Approach to Equations with Memory

Abstract

In this work, we present a novel approach to the mathematical analysis of equations with memory based on the notion of a state, namely, the initial configuration of the system which can be unambiguously determined by the knowledge of the future dynamics. As a model, we discuss the abstract version of an equation arising from linear viscoelasticity. It is worth mentioning that our approach goes back to the heuristic derivation of the state framework, devised by L.Deseri, M.Fabrizio and M.J.Golden in "The concept of minimal state in viscoelasticity: new free energies and applications to PDEs", Arch. Ration. Mech. Anal., vol. 181 (2006) pp.43-96. Starting from their physical motivations, we develop a suitable functional formulation which, as far as we know, is completely new.

39 pages, no figure

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• Coleman-Gurtin type equations with dynamic boundary conditions
• Flow decomposition for heat equations with memory
• A dynamic model for viscoelasticity in domains with time-dependent cracks
• Observability for heat equations with time-dependent analytic memory
• Dynamic crack growth in viscoelastic materials with memory
• Markovian Embedding of Nonlinear Memory via Spectral Representation