A New Approach to Equations with Memory

TL;DR

This paper introduces a novel functional approach to analyze equations with memory, particularly in viscoelasticity, by defining a state concept that captures the system's initial configuration from future dynamics.

Contribution

It develops a new functional framework for equations with memory, extending the state concept originally proposed for viscoelastic systems.

Findings

New functional formulation for equations with memory

Unambiguous state definition from future dynamics

Extension of state concept in viscoelasticity

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.