# American Institute of Mathematical Sciences

March  2019, 8(1): 203-220. doi: 10.3934/eect.2019011

## Optimal scalar products in the Moore-Gibson-Thompson equation

 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

Received  June 2017 Revised  September 2017 Published  March 2019 Early access  January 2019

Fund Project: 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).

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.

Citation: Marta Pellicer, Joan Solà-Morales. Optimal scalar products in the Moore-Gibson-Thompson equation. Evolution Equations & Control Theory, 2019, 8 (1) : 203-220. doi: 10.3934/eect.2019011
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##### References:
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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