Optimal scalar products in the Standard Linear Viscoelastic Model

TL;DR

This paper analyzes the spectral properties of the Standard Linear Viscoelastic Model, revealing conditions under which the generator becomes normal and deriving improved decay estimates for solutions over time.

Contribution

It provides a detailed spectral analysis of the model's generator, identifying parameter conditions for normality and establishing sharper decay estimates for solutions.

Findings

Generator can be made normal with a new scalar product

Complete orthogonal eigenfunctions are constructed

Sharper decay estimates for solutions are obtained

Abstract

We study the third order in time linear dissipative wave equation known as the Standard Linear Viscoelastic Model, that appears also as the linearization of the so-called Moore-Gibson-Thompson equation in Nonlinear Acoustics. We complete the description in a paper by R. Marchand et al. (2012) of the spectrum of the generator of the corresponding group of operators and show that, apart from some exceptional values of the parameters, this generator can be made to be a normal operator with a new scalar product, with a complete set of orthogonal eigenfunctions. Using this property we also obtain sharper decay estimates for the solutions as time tends to infinity, both when the operator is normal or not.