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A new approach to equations with memory M. Fabrizio, C. Giorgi, V. Pata

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A new approach to equations with memory M. Fabrizio, C. Giorgi, V. Pata

Abstract

We discuss a novel approach to the mathematical analysis of equations with memory, based on the notion of a 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

+

→ R

+

such that

G(0) = lim

s→0

G(s) > lim

s→∞

G(s) > 0 we consider an abstract version of the evolution equation

∂

tt

u(x, t) − ∆ h

G(0)u(x, t) + Z

∞

0

G

0

(s)u(x, t − s)ds i

= 0 arising from linear viscoelasticity.

1

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