On the regularized Moore-Gibson-Thompson equation

Filippo Dell'Oro; Lorenzo Liverani; Vittorino Pata

doi:10.3934/dcdss.2023025

Article Contents

2023, Volume 16, Issue 9: 2326-2338. Doi: 10.3934/dcdss.2023025

This issue Previous Article Optimal control of a nonconserved phase field model of Caginalp type with thermal memory and double obstacle potential Next Article A phase field approach for direct calculation of interfacial free energy of solid-gas and solid-liquid interfaces in Lennard-Jones systems

On the regularized Moore-Gibson-Thompson equation

Politecnico di Milano, 20133 Milano, Italy

^*Corresponding author: Vittorino Pata
^*Corresponding author: Vittorino Pata

In memory of Professor Gunduz Caginalp

Received: October 2022

Early access: February 2023

Published: September 2023

Abstract / Introduction Full Text(HTML) Figure(2) Related Papers Cited by

Abstract

We study the regularized MGT equation

$ u_{ttt} + \alpha u_{tt} +\beta Au_t +\gamma A u +\delta A u_{tt} = 0 $

where $ A $ is a strictly positive unbounded operator and $ \alpha, \beta, \gamma, \delta>0 $. The effect of the regularizing term $ \delta A u_{tt} $ translates into having an analytic semigroup $ S(t) = e^{t{{\mathbb A}}} $ of solutions. Moreover, the asymptotic properties of the semigroup are ruled by the stability number

$ \varkappa = \beta - \frac{\gamma}{\alpha +\delta \lambda_0} $

which, contrary to the case of the standard MGT equation, depends also on the minimum $ \lambda_0>0 $ of the spectrum of $ A $.

Keywords:

Mathematics Subject Classification: Primary: 35B35, 35G05, 35Q79, 47D06; Secondary: 76N15.

Citation:

Full Text(HTML)

Figure 1. Spectrum of $ {{\mathbb A}} $ (in red) for $ \delta = 0.3 $ (left) and $ \delta = 0.1 $ (right)

Download: Full-size image PowerPoint slide

Figure 2. Spectrum of $ {{\mathbb A}} $ (in red) for $ \delta = 0.01 $ (left) and $ \delta = 0 $ (right)

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena, 3 (1949), 83-101.
[2]	C. Cattaneo, Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée, C. R. Acad. Sci. Paris, 247 (1958), 431-433.
[3]	W. Chester, Resonant oscillations in closed tubes, J. Fluid Mech., 18 (1964), 44-64.
[4]	B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208. doi: 10.1007/BF01596912.
[5]	M. Conti, F. Dell'Oro, L. Liverani and V. Pata, Spectral analysis and stability of the Moore-Gibson-Thompson-Fourier model, J. Dynam. Differential Equations, (in press).
[6]	M. Conti, V. Pata and R. Quintanilla, Thermoelasticity of Moore-Gibson-Thompson type with history dependence in the temperature, Asymptot. Anal., 120 (2020), 1-21. doi: 10.3233/ASY-191576.
[7]	F. Dell'Oro and V. Pata, On the Moore-Gibson-Thompson equation and its relation to linear viscoelasticity, Appl. Math. Optim., 76 (2017), 641-655. doi: 10.1007/s00245-016-9365-1.
[8]	F. Dell'Oro and V. Pata, A hierarchy of heat conduction laws, Discrete Contin. Dyn. Syst. Ser. S, (in press).
[9]	K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.
[10]	L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394. doi: 10.1090/S0002-9947-1978-0461206-1.
[11]	P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2189-2205. doi: 10.3934/dcdsb.2014.19.2189.
[12]	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.
[13]	B. Kaltenbacher, I. Lasiecka and M. K. Pospieszalska, Well-posedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 1250035, 34 pp. doi: 10.1142/S0218202512500352.
[14]	I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems, Cambridge University Press, Cambridge, 2000.
[15]	Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, 1999.
[16]	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.
[17]	F. K. Moore and W. E. Gibson, Propagation of weak disturbances in a gas subject to relaxation effects, J. Aerospace Sci., 27 (1960), 117-127. doi: 10.2514/8.8418.
[18]	R. Nittka, Three-Line Proofs, Ulmer Seminare Volume 16, Ulm University, 2011.
[19]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[20]	J. Prüss, On the spectrum of $\text{C}_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.1090/S0002-9947-1984-0743749-9.
[21]	G. G. Stokes, An examination of the possible effect of the radiation of heat on the propagation of sound, Philos. Mag. Series 4, 1 (1851), 305-317.
[22]	P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, 1972.