# 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 and Control Theory, 2019, 8 (1) : 203-220. doi: 10.3934/eect.2019011
##### References:
 [1] M. S. Alves, C. Buriol, M. V. Ferreira, J. E. Muñoz Rivera, M. Sepúlveda and O. Vera, Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect, J. Math. Anal. Appl., 399 (2013), 472-479.  doi: 10.1016/j.jmaa.2012.10.019. [2] B. de Andrade and C. Lizama, Existence of asymptotically almost periodic solutions for damped wave equations, J. Math. Anal. Appl., 382 (2011), 761-771.  doi: 10.1016/j.jmaa.2011.04.078. [3] S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-25.  doi: 10.2140/pjm.1989.136.15. [4] J. A. Conejero, C. Lizama and F. Ródenas, Chaotic behaviour of the solutions of the Moore-Gibson-Thompson equation, Applied Mathematics and Information Sciences, 9 (2015), 2233-2238. [5] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in a Hilbert Space, American Mathematical Society, 1991. [6] G. C. Gorain, Stabilization for the vibrations modeled by the standard linear model of viscoelasticity, Proc. Indian Acad. Sci. (Math. Sci.), 120 (2010), 495-506.  doi: 10.1007/s12044-010-0038-8. [7] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. [8] B. Kaltenbacher, I. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control Cybernet, 40 (2011), 971-988. [9] V. K. Kalantarov and Y. Yilmaz, Decay and growth estimates for solutions of second-order and third-order differential-operator equations, Nonlinear Anal., 89 (2013), 1-7.  doi: 10.1016/j.na.2013.04.016. [10] I. Lasiecka and R. Triggiani, Control theory for partial differential equations: Continuous and approximation theories. I. Abstract parabolic systems, in Encyclopedia of Mathematics and its Applications, 74 (2000), xxii+644+I4pp. Cambridge University Press, Cambridge. [11] I. Lasiecka and X. Wang, Moore-Gibson-Thompson equation with memory, part Ⅱ: General decay of energy, J. Differential Equations, 259 (2015), 7610-7635.  doi: 10.1016/j.jde.2015.08.052. [12] C. R. da Luz, R. Ikehata and R. C. Charo, Asymptotic behavior for abstract evolution differential equations of second order, J. Differential Equations, 259 (2015), 5017-5039.  doi: 10.1016/j.jde.2015.06.012. [13] R. Marchand, T. McDevitt and R. Triggiani, An abstract semigroup approach to the third-order Moore-Gibson-Thompson partial differential equation arising in high-intensity ultrasound: structural decomposition, spectral analysis, exponential stability, Math. Methods Appl. Sci., 35 (2012), 1896-1929.  doi: 10.1002/mma.1576. [14] M. Pellicer and J. Solà-Morales, Analysis of a viscoelastic spring-mass model, J. Math. Anal. Appl., 294 (2004), 687-698.  doi: 10.1016/j.jmaa.2004.03.008. [15] M. Pellicer and J. Solà-Morales, Optimal decay rates and the selfadjoint property in overdamped systems, J. Differential Equations, 246 (2009), 2813-2828.  doi: 10.1016/j.jde.2009.01.010. [16] M. Pellicer and B. Said-Houari, Wellposedness and decay rates for the Cauchy problem of the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Applied Mathematics & Optimization, 2017, 1-32, http://arxiv.org/abs/1603.04270. doi: 10.1007/s00245-017-9471-8. [17] P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, 1972.

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##### References:
 [1] M. S. Alves, C. Buriol, M. V. Ferreira, J. E. Muñoz Rivera, M. Sepúlveda and O. Vera, Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect, J. Math. Anal. Appl., 399 (2013), 472-479.  doi: 10.1016/j.jmaa.2012.10.019. [2] B. de Andrade and C. Lizama, Existence of asymptotically almost periodic solutions for damped wave equations, J. Math. Anal. Appl., 382 (2011), 761-771.  doi: 10.1016/j.jmaa.2011.04.078. [3] S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-25.  doi: 10.2140/pjm.1989.136.15. [4] J. A. Conejero, C. Lizama and F. Ródenas, Chaotic behaviour of the solutions of the Moore-Gibson-Thompson equation, Applied Mathematics and Information Sciences, 9 (2015), 2233-2238. [5] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in a Hilbert Space, American Mathematical Society, 1991. [6] G. C. Gorain, Stabilization for the vibrations modeled by the standard linear model of viscoelasticity, Proc. Indian Acad. Sci. (Math. Sci.), 120 (2010), 495-506.  doi: 10.1007/s12044-010-0038-8. [7] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. [8] B. Kaltenbacher, I. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control Cybernet, 40 (2011), 971-988. [9] V. K. Kalantarov and Y. Yilmaz, Decay and growth estimates for solutions of second-order and third-order differential-operator equations, Nonlinear Anal., 89 (2013), 1-7.  doi: 10.1016/j.na.2013.04.016. [10] I. Lasiecka and R. Triggiani, Control theory for partial differential equations: Continuous and approximation theories. I. Abstract parabolic systems, in Encyclopedia of Mathematics and its Applications, 74 (2000), xxii+644+I4pp. Cambridge University Press, Cambridge. [11] I. Lasiecka and X. Wang, Moore-Gibson-Thompson equation with memory, part Ⅱ: General decay of energy, J. Differential Equations, 259 (2015), 7610-7635.  doi: 10.1016/j.jde.2015.08.052. [12] C. R. da Luz, R. Ikehata and R. C. Charo, Asymptotic behavior for abstract evolution differential equations of second order, J. Differential Equations, 259 (2015), 5017-5039.  doi: 10.1016/j.jde.2015.06.012. [13] R. Marchand, T. McDevitt and R. Triggiani, An abstract semigroup approach to the third-order Moore-Gibson-Thompson partial differential equation arising in high-intensity ultrasound: structural decomposition, spectral analysis, exponential stability, Math. Methods Appl. Sci., 35 (2012), 1896-1929.  doi: 10.1002/mma.1576. [14] M. Pellicer and J. Solà-Morales, Analysis of a viscoelastic spring-mass model, J. Math. Anal. Appl., 294 (2004), 687-698.  doi: 10.1016/j.jmaa.2004.03.008. [15] M. Pellicer and J. Solà-Morales, Optimal decay rates and the selfadjoint property in overdamped systems, J. Differential Equations, 246 (2009), 2813-2828.  doi: 10.1016/j.jde.2009.01.010. [16] M. Pellicer and B. Said-Houari, Wellposedness and decay rates for the Cauchy problem of the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Applied Mathematics & Optimization, 2017, 1-32, http://arxiv.org/abs/1603.04270. doi: 10.1007/s00245-017-9471-8. [17] P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, 1972.
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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