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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 |
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 [
References:
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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. |
show all references
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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