Optimal scalar products in the Moore-Gibson-Thompson equation

Marta Pellicer; Joan Solà-Morales

doi:10.3934/eect.2019011

Article Contents

2019, Volume 8, Issue 1: 203-220. Doi: 10.3934/eect.2019011

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Optimal scalar products in the Moore-Gibson-Thompson equation

Marta Pellicer^1, , and
Joan Solà-Morales^2,

1.
Dpt. d'Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, EPS-P4, Campus de Montilivi, 17071 Girona, Catalunya, Spain
2.
Dpt. de Matemàtiques, Universitat Politècnica de Catalunya, ETSEIB-UPC, Av. Diagonal 647, 08028 Barcelona, Catalunya, Spain

* Corresponding author: martap@imae.udg.edu
* Corresponding author: martap@imae.udg.edu

Received: May 31, 2017

Revised: August 31, 2017

Early access: December 2018

Published: January 2019

Both authors are part of the Catalan research groups 2014 SGR 1083 and 2017 SGR 1392. J. Sol`a-Morales has been supported by the MINECO grants MTM2014-52402-C3-1-P and MTM2017-84214-C2-1-P (Spain). M. Pellicer has been supported by the MINECO grants MTM2014-52402- C3-3-P and MTM2017-84214-C2-2-P (Spain), and also by MPC UdG 2016/047 (U. of Girona, Catalonia).

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Abstract

We study the third order in time linear dissipative wave equation known as the Moore-Gibson-Thompson equation, that appears as the linearization of a the Jordan-Moore-Gibson-Thompson equation, an important model in nonlinear acoustics. The same equation also arises in viscoelasticity theory, as a model which is considered more realistic than the usual Kelvin-Voigt one for the linear deformations of a viscoelastic solid. In this context, it is known as the Standard Linear Viscoelastic model. We complete the description in [13] 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 optimal exponential decay estimates for the solutions as $ t\to\infty $, whether the operator is normal or not.

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Mathematics Subject Classification: Primary: 35L05, 35L35, 47D03, 35B40; Secondary: 35Q60, 35Q74.

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Figure 1. Plots of the eigenvalues of the operator $ A $ (circles) in the complex plane (in solid lines, the real and complex axes), showing different possibilities for $ \sigma_{max}(A) $. In all of them, the dashed line represents $ \textrm{Re} (\lambda) = -\frac{1}{2}\left( \frac{1}{\alpha}-\frac{1}{\beta}\right) $, which is the limit of the real parts of the nonreal eigenvalues, and the point marked as a square is $ -\frac{1}{\beta} $, which is the limit of the real ones. In panel (1a), we can see an example of the $ \alpha/\beta>1/3 $ case and, hence, $ \sigma_{max} = \textrm{Re}(\lambda^1_2) $, while in the others $ \alpha/\beta<1/3 $. In panel (1c) we can see the limit situation between cases represented in panels (1b) and (1d)

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