December  2016, 5(4): 661-676. doi: 10.3934/eect.2016024

On long time behavior of Moore-Gibson-Thompson equation with molecular relaxation

1. 

State University of Londrina, Department of Mathematics, Londrina, Paraná, Brazil

2. 

University of Memphis, Department of Mathematical Sciences, 373 Dunn Hall, Memphis, TN 38152

3. 

State University of Maringá , Department of Mathematics, Maringá, Paraná, Brazil

Received  August 2016 Revised  September 2016 Published  October 2016

A third order in time nonlinear equation is considered. This particular model is motivated by High Frequency Ultra Sound (HFU) technology which accounts for thermal and molecular relaxation. The resulting equations give rise to a quasilinear-like evolution with a potentially degenerate damping [23]. The purpose of this paper is twofold: (1) to provide a brief review of recent results in the area of long time behavior of solutions to of MGT equation, (2) to provide recent results pertaining to decay of energy associated with the model accounting for molecular relaxation which is locally distributed.
Citation: Arthur Henrique Caixeta, Irena Lasiecka, Valéria Neves Domingos Cavalcanti. On long time behavior of Moore-Gibson-Thompson equation with molecular relaxation. Evolution Equations & Control Theory, 2016, 5 (4) : 661-676. doi: 10.3934/eect.2016024
References:
[1]

S. Adhikari, Structural Dynamic Analysis with Generalized Damping Models: Analysis,, Wiley-ISTE, (2013). doi: 10.1002/9781118572023. Google Scholar

[2]

F. Alabau-Boussouira, P. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory,, Journal of Functional Analysis, 254 (2008), 1342. doi: 10.1016/j.jfa.2007.09.012. Google Scholar

[3]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations,, C. R. Acad. Sci. Paris, 347 (2009), 867. doi: 10.1016/j.crma.2009.05.011. Google Scholar

[4]

F. Alabau-Boussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems,, Journal of Differential Equations, 248 (2010), 1473. doi: 10.1016/j.jde.2009.12.005. Google Scholar

[5]

F. Alabau-Boussouira, On some recent advances on stabilization for hyperbolic equations,, CIME Foundation Subseries, 2048 (2012), 1. doi: 10.1007/978-3-642-27893-8_1. Google Scholar

[6]

A. H. Caixeta, V. N. Domingos Cavalcanti and I. Lasiecka, Global attractors for a third order in time nonlinear dynamics,, Journal of Differential Equations, 261 (2016), 113. doi: 10.1016/j.jde.2016.03.006. Google Scholar

[7]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems,, Nonlinear Analysis, 68 (2008), 177. doi: 10.1016/j.na.2006.10.040. Google Scholar

[8]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems,, Springer, (2015). doi: 10.1007/978-3-319-22903-4. Google Scholar

[9]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar

[10]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Memoirs of the American Mathematical Society, 195 (2008). doi: 10.1090/memo/0912. Google Scholar

[11]

D. G. Crighton, Model equations of nonlinear acoustics,, Ann. Rev. Fluid Mech., 11 (1979), 11. doi: 10.1146/annurev.fl.11.010179.000303. Google Scholar

[12]

B. Coleman, M. Fabrizio and D. Owen, On the thermodynamics of second sound in dielectric crystals,, Arch. Rat. Mech. Anal, 80 (1982), 135. doi: 10.1007/BF00250739. Google Scholar

[13]

F. Dell'Oro, I. Lasiecka and V. Pata, The Moore-Gibson-Thompson equation with memory in the critical case,, Journal Differential Equations, 261 (2016), 4188. doi: 10.1016/j.jde.2016.06.025. Google Scholar

[14]

W. Dreyer and H. Struchtrup, Heat pulse experiments revisited,, Cont. Mechanics Thermodynamics, 5 (1993), 3. doi: 10.1007/BF01135371. Google Scholar

[15]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory,, Appl. Anal., 81 (2002), 1245. doi: 10.1080/0003681021000035588. Google Scholar

[16]

P. Geredelli, I. Lasiecka and J. Webster, Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer,, J. Differential Equations, 254 (2013), 1193. doi: 10.1016/j.jde.2012.10.016. Google Scholar

[17]

H. Fattorini, The Cauchy Problem,, Addison Wesley, (1983). Google Scholar

[18]

X. Han and M. Wang, General decay rates of energy for the second order evolutions equations with memory,, Acta Applicanda, 110 (2010), 195. doi: 10.1007/s10440-008-9397-x. Google Scholar

[19]

W. J. Hrusa, J. A. Nohel and M. Renardy, Mathematical Problems in Viscoelasticity,, Pitman Monographs and Surveys in Pure and Applied Mathematics, 35 (1987). Google Scholar

[20]

P. Jordan, Second-Sound phenomena in inviscidm thermally relaxing gases,, Discrete and Continuous Dynamical Systems, 19 (2014), 2189. doi: 10.3934/dcdsb.2014.19.2189. Google Scholar

[21]

P. M. Jordan, Nonlinear acoustic phenomena in viscous thermally relaxing fluids: Shock bifurcation and the emergence of diffusive solitons,, in The 9th Int. Conf. on Theoretical and Computational Acoustics (ICTCA, 124 (2008), 2491. doi: 10.1121/1.4782790. Google Scholar

[22]

P. M. Jordan, Private Communication,, May 2015., (2015). Google Scholar

[23]

B. Kaltenbacher, Mathematics of nonlinear acoustics,, Evolution Equations and Control Theory, 4 (2015), 447. doi: 10.3934/eect.2015.4.447. Google Scholar

[24]

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. Google Scholar

[25]

B. Kaltenbacher, I. Lasiecka and M. 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). doi: 10.1142/S0218202512500352. Google Scholar

[26]

V. P. Kuznetsov, Equations of nonlinear acoustics,, Sov. Phys. Acoust., 16 (1971), 467. Google Scholar

[27]

I. Lasiecka, Global solvability of Moore-Gibson-Thompson equation with memory raising in nonlinear acoustics,, Journal of Evolution Equations, (2016). Google Scholar

[28]

I. Lasiecka, S. A. Messaoudi and M. I. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory,, Journal of Mathematical Physics, 54 (2013). doi: 10.1063/1.4793988. Google Scholar

[29]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation,, Differential and Integral Equations, 6 (1993), 507. Google Scholar

[30]

I. Lasiecka and X. Wang, General decay rate of Moore-Gibson-Thompson equation with memory -Part II,, Journal Differential Equations, 259 (2015), 7610. doi: 10.1016/j.jde.2015.08.052. Google Scholar

[31]

I. Lasiecka and X. Wang, Moore Gibson Thompson equation with memory, part I,, ZAMP, 67 (2016). doi: 10.1007/s00033-015-0597-8. Google Scholar

[32]

G. Lebon and A. Cloot, Propagation of ultrasonic sound waves in dissipative dilute gases and extended irreversible thermodynamics,, Wave Motion, 11 (1989), 23. doi: 10.1016/0165-2125(89)90010-3. Google Scholar

[33]

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. doi: 10.1002/mma.1576. Google Scholar

[34]

I. Muller and T. Ruggeri, Extended Thermodynamics,, vol 37, (1993). doi: 10.1007/978-1-4684-0447-0. Google Scholar

[35]

S. A. Messaoudi, General decay of solutions of a viscoelastic equation,, JMAA, 341 (2008), 1457. doi: 10.1016/j.jmaa.2007.11.048. Google Scholar

[36]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation,, Appl. Math. Optim., 64 (2011), 257. doi: 10.1007/s00245-011-9138-9. Google Scholar

[37]

S. Messaoudi and M. Mustafa, General stability result for viscoelastic wave equations,, Journal Mathematical Pgysics, 53 (2012). doi: 10.1063/1.4711830. Google Scholar

[38]

F. Moore and W. Gibson, Propagation of weak disturbances in a gas subject to relaxing effects,, Journal Aero/Space SCi., 27 (1960), 117. Google Scholar

[39]

K. Naugolnykh and L. Ostrovsky, Nonlinear Wave Processes in Acoustics,, Cambridge University Press, (1998). Google Scholar

[40]

J. Pruss, Decay properties for the solutions of a partial differential equations with memory,, Archiv der Mathematik, 92 (2009), 158. doi: 10.1007/s00013-008-2936-x. Google Scholar

[41]

J. Pruss, Evolutionary Integral Equations and Applications,, Monographs in Mathematics, (1993). doi: 10.1007/978-3-0348-8570-6. Google Scholar

[42]

F. K. Moore and W. E. Gibson, Propagation of weak disturbances in a gas subject to relaxation effects,, J. Aerospace Sci. Tech., 27 (1960), 117. Google Scholar

[43]

M. Renardy, Mathematical Analysis of Viscoelastic Flows,, CBMS-NSF Conference Series in Applied Mathematics, (2000). doi: 10.1137/1.9780898719413. Google Scholar

[44]

J. Rivera and A. Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic materials,, Quart. Appl. Math., 59 (2001), 557. Google Scholar

[45]

O. V. Rudenko and S. I. Soluyan, Theoretical Foundations of Nonlinear Acoustics,, Translated from the Russian by Robert T. Beyer. Studies in Soviet Science. Consultants Bureau, (1977). Google Scholar

[46]

P. Stokes, An examination of the possible effect of the radiation of heat on the propagation of sound,, Philosophical Magazine Series, 4 (1851), 3015. Google Scholar

show all references

References:
[1]

S. Adhikari, Structural Dynamic Analysis with Generalized Damping Models: Analysis,, Wiley-ISTE, (2013). doi: 10.1002/9781118572023. Google Scholar

[2]

F. Alabau-Boussouira, P. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory,, Journal of Functional Analysis, 254 (2008), 1342. doi: 10.1016/j.jfa.2007.09.012. Google Scholar

[3]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations,, C. R. Acad. Sci. Paris, 347 (2009), 867. doi: 10.1016/j.crma.2009.05.011. Google Scholar

[4]

F. Alabau-Boussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems,, Journal of Differential Equations, 248 (2010), 1473. doi: 10.1016/j.jde.2009.12.005. Google Scholar

[5]

F. Alabau-Boussouira, On some recent advances on stabilization for hyperbolic equations,, CIME Foundation Subseries, 2048 (2012), 1. doi: 10.1007/978-3-642-27893-8_1. Google Scholar

[6]

A. H. Caixeta, V. N. Domingos Cavalcanti and I. Lasiecka, Global attractors for a third order in time nonlinear dynamics,, Journal of Differential Equations, 261 (2016), 113. doi: 10.1016/j.jde.2016.03.006. Google Scholar

[7]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems,, Nonlinear Analysis, 68 (2008), 177. doi: 10.1016/j.na.2006.10.040. Google Scholar

[8]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems,, Springer, (2015). doi: 10.1007/978-3-319-22903-4. Google Scholar

[9]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar

[10]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Memoirs of the American Mathematical Society, 195 (2008). doi: 10.1090/memo/0912. Google Scholar

[11]

D. G. Crighton, Model equations of nonlinear acoustics,, Ann. Rev. Fluid Mech., 11 (1979), 11. doi: 10.1146/annurev.fl.11.010179.000303. Google Scholar

[12]

B. Coleman, M. Fabrizio and D. Owen, On the thermodynamics of second sound in dielectric crystals,, Arch. Rat. Mech. Anal, 80 (1982), 135. doi: 10.1007/BF00250739. Google Scholar

[13]

F. Dell'Oro, I. Lasiecka and V. Pata, The Moore-Gibson-Thompson equation with memory in the critical case,, Journal Differential Equations, 261 (2016), 4188. doi: 10.1016/j.jde.2016.06.025. Google Scholar

[14]

W. Dreyer and H. Struchtrup, Heat pulse experiments revisited,, Cont. Mechanics Thermodynamics, 5 (1993), 3. doi: 10.1007/BF01135371. Google Scholar

[15]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory,, Appl. Anal., 81 (2002), 1245. doi: 10.1080/0003681021000035588. Google Scholar

[16]

P. Geredelli, I. Lasiecka and J. Webster, Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer,, J. Differential Equations, 254 (2013), 1193. doi: 10.1016/j.jde.2012.10.016. Google Scholar

[17]

H. Fattorini, The Cauchy Problem,, Addison Wesley, (1983). Google Scholar

[18]

X. Han and M. Wang, General decay rates of energy for the second order evolutions equations with memory,, Acta Applicanda, 110 (2010), 195. doi: 10.1007/s10440-008-9397-x. Google Scholar

[19]

W. J. Hrusa, J. A. Nohel and M. Renardy, Mathematical Problems in Viscoelasticity,, Pitman Monographs and Surveys in Pure and Applied Mathematics, 35 (1987). Google Scholar

[20]

P. Jordan, Second-Sound phenomena in inviscidm thermally relaxing gases,, Discrete and Continuous Dynamical Systems, 19 (2014), 2189. doi: 10.3934/dcdsb.2014.19.2189. Google Scholar

[21]

P. M. Jordan, Nonlinear acoustic phenomena in viscous thermally relaxing fluids: Shock bifurcation and the emergence of diffusive solitons,, in The 9th Int. Conf. on Theoretical and Computational Acoustics (ICTCA, 124 (2008), 2491. doi: 10.1121/1.4782790. Google Scholar

[22]

P. M. Jordan, Private Communication,, May 2015., (2015). Google Scholar

[23]

B. Kaltenbacher, Mathematics of nonlinear acoustics,, Evolution Equations and Control Theory, 4 (2015), 447. doi: 10.3934/eect.2015.4.447. Google Scholar

[24]

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. Google Scholar

[25]

B. Kaltenbacher, I. Lasiecka and M. 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). doi: 10.1142/S0218202512500352. Google Scholar

[26]

V. P. Kuznetsov, Equations of nonlinear acoustics,, Sov. Phys. Acoust., 16 (1971), 467. Google Scholar

[27]

I. Lasiecka, Global solvability of Moore-Gibson-Thompson equation with memory raising in nonlinear acoustics,, Journal of Evolution Equations, (2016). Google Scholar

[28]

I. Lasiecka, S. A. Messaoudi and M. I. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory,, Journal of Mathematical Physics, 54 (2013). doi: 10.1063/1.4793988. Google Scholar

[29]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation,, Differential and Integral Equations, 6 (1993), 507. Google Scholar

[30]

I. Lasiecka and X. Wang, General decay rate of Moore-Gibson-Thompson equation with memory -Part II,, Journal Differential Equations, 259 (2015), 7610. doi: 10.1016/j.jde.2015.08.052. Google Scholar

[31]

I. Lasiecka and X. Wang, Moore Gibson Thompson equation with memory, part I,, ZAMP, 67 (2016). doi: 10.1007/s00033-015-0597-8. Google Scholar

[32]

G. Lebon and A. Cloot, Propagation of ultrasonic sound waves in dissipative dilute gases and extended irreversible thermodynamics,, Wave Motion, 11 (1989), 23. doi: 10.1016/0165-2125(89)90010-3. Google Scholar

[33]

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. doi: 10.1002/mma.1576. Google Scholar

[34]

I. Muller and T. Ruggeri, Extended Thermodynamics,, vol 37, (1993). doi: 10.1007/978-1-4684-0447-0. Google Scholar

[35]

S. A. Messaoudi, General decay of solutions of a viscoelastic equation,, JMAA, 341 (2008), 1457. doi: 10.1016/j.jmaa.2007.11.048. Google Scholar

[36]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation,, Appl. Math. Optim., 64 (2011), 257. doi: 10.1007/s00245-011-9138-9. Google Scholar

[37]

S. Messaoudi and M. Mustafa, General stability result for viscoelastic wave equations,, Journal Mathematical Pgysics, 53 (2012). doi: 10.1063/1.4711830. Google Scholar

[38]

F. Moore and W. Gibson, Propagation of weak disturbances in a gas subject to relaxing effects,, Journal Aero/Space SCi., 27 (1960), 117. Google Scholar

[39]

K. Naugolnykh and L. Ostrovsky, Nonlinear Wave Processes in Acoustics,, Cambridge University Press, (1998). Google Scholar

[40]

J. Pruss, Decay properties for the solutions of a partial differential equations with memory,, Archiv der Mathematik, 92 (2009), 158. doi: 10.1007/s00013-008-2936-x. Google Scholar

[41]

J. Pruss, Evolutionary Integral Equations and Applications,, Monographs in Mathematics, (1993). doi: 10.1007/978-3-0348-8570-6. Google Scholar

[42]

F. K. Moore and W. E. Gibson, Propagation of weak disturbances in a gas subject to relaxation effects,, J. Aerospace Sci. Tech., 27 (1960), 117. Google Scholar

[43]

M. Renardy, Mathematical Analysis of Viscoelastic Flows,, CBMS-NSF Conference Series in Applied Mathematics, (2000). doi: 10.1137/1.9780898719413. Google Scholar

[44]

J. Rivera and A. Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic materials,, Quart. Appl. Math., 59 (2001), 557. Google Scholar

[45]

O. V. Rudenko and S. I. Soluyan, Theoretical Foundations of Nonlinear Acoustics,, Translated from the Russian by Robert T. Beyer. Studies in Soviet Science. Consultants Bureau, (1977). Google Scholar

[46]

P. Stokes, An examination of the possible effect of the radiation of heat on the propagation of sound,, Philosophical Magazine Series, 4 (1851), 3015. Google Scholar

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