Optimal scalar products in the Standard Linear Viscoelastic Model

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.

17 pages, 1 figure

References in corpus (1)

• Wellposedness and decay rates for the Cauchy problem of the Moore-Gibson-Thompson equation arising in high intensity ultrasound

Cited by in corpus (7)

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• Determining kernels in linear viscoelasticity
• On the Jordan-Moore-Gibson-Thompson wave equation in hereditary fluids with quadratic gradient nonlinearity
• On the Cauchy problem for the standard linear solid model with heat conduction: Fourier versus Cattaneo
• Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary
• Vanishing relaxation time limit of the Jordan--Moore--Gibson--Thompson wave equation with Neumann and absorbing boundary conditions