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

GIORGI, Claudio;PATA, Vittorino
2010-01-01

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11379/39285
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