We discuss a novel approach to the mathematical analysis of equations with memory, based on the notion of a {\it state}. This is the initial configuration of the system at time $t=0$ which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function $G:\R^+\to\R^+$ such that $$G(0)=\lim_{s\to 0}G(s)>\lim_{s\to\infty}G(s)>0$$ we consider an abstract version of the evolution equation $$\partial_{tt} {\bm u}({\bm x},t)-\Delta\Big[G(0) {\bm u}({\bm x},t) +\int_0^\infty G'(s) {\bm u}({\bm x},t-s) {\rm d} s\Big]=0$$ arising from linear viscoelasticity.

### A new approach to equations with memory

#### Abstract

We discuss a novel approach to the mathematical analysis of equations with memory, based on the notion of a {\it state}. This is the initial configuration of the system at time $t=0$ which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function $G:\R^+\to\R^+$ such that $$G(0)=\lim_{s\to 0}G(s)>\lim_{s\to\infty}G(s)>0$$ we consider an abstract version of the evolution equation $$\partial_{tt} {\bm u}({\bm x},t)-\Delta\Big[G(0) {\bm u}({\bm x},t) +\int_0^\infty G'(s) {\bm u}({\bm x},t-s) {\rm d} s\Big]=0$$ arising from linear viscoelasticity.
##### Scheda breve Scheda completa Scheda completa (DC)
2010
File in questo prodotto:
File

accesso aperto

Tipologia: Altro materiale allegato
Licenza: DRM non definito
Dimensione 32.01 kB
FGP ARMA.pdf

accesso aperto

Tipologia: Full Text
Licenza: DRM non definito
Dimensione 437.06 kB
Abstract_FGP.pdf

accesso aperto

Tipologia: Abstract
Licenza: DRM non definito
Dimensione 41.03 kB
Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11379/39285