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A New Approach to Equations with Memory

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Abstract

We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of 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 \({G : \mathbb{R}^+ \to \mathbb{R}^+}\) 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

$$\partial_{tt}{\varvec u}({\varvec x}, t) - \Delta\left[G(0){\varvec u}({\varvec x}, t) + \displaystyle\int_0^\infty G'(s){\varvec u}({\varvec x}, t - s){\rm d}s\right] = 0$$

arising from linear viscoelasticity.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, piazza di Porta San Donato 5, 40126, Bologna, Italy

    Mauro Fabrizio

  2. Dipartimento di Matematica, Università di Brescia, via Valotti 9, 25133, Brescia, Italy

    Claudio Giorgi

  3. Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, via Bonardi 9, 20133, Milan, Italy

    Vittorino Pata

Authors
  1. Mauro Fabrizio
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  2. Claudio Giorgi
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  3. Vittorino Pata
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Correspondence to Mauro Fabrizio.

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Communicated by C.M. Dafermos

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Fabrizio, M., Giorgi, C. & Pata, V. A New Approach to Equations with Memory. Arch Rational Mech Anal 198, 189–232 (2010). https://doi.org/10.1007/s00205-010-0300-3

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