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Optimal scalar products in the Moore-Gibson-Thompson equation

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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  • 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.

    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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