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General decay properties of abstract linear viscoelasticity

  • Monica ContiEmail author
  • Vittorino Pata
Article

Abstract

We consider a linear Volterra integro-differential equation of hyperbolic type, which can be viewed as an abstract version of the equation
$$\begin{aligned} \partial _{tt} u(t)- \varDelta u(t) +\displaystyle \int \limits _0^t\mu (s)\varDelta u(t-s)\mathrm{d}s=0 \end{aligned}$$
describing the motion of linearly viscoelastic solids. We establish some decay results for the associated energy, under assumptions that do not involve differential inequalities for the convolution kernel \(\mu \).

Keywords

Volterra integro-differential equation Viscoelasticity Decay of the energy 

Mathematics Subject Classification

35B40 45K05 45M05 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Politecnico di Milano - Dipartimento di MatematicaMilanItaly

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