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 and 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.

[2]

F. Alabau-Boussouira, P. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, Journal of Functional Analysis, 254 (2008), 1342-1372. doi: 10.1016/j.jfa.2007.09.012.

[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, Ser., 347 (2009), 867-872. doi: 10.1016/j.crma.2009.05.011.

[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), 1473-1517. doi: 10.1016/j.jde.2009.12.005.

[5]

F. Alabau-Boussouira, On some recent advances on stabilization for hyperbolic equations, CIME Foundation Subseries, Control of Partial Differential Equations, Springer Verlag, 2048 (2012), 1-100. doi: 10.1007/978-3-642-27893-8_1.

[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), 113-147. doi: 10.1016/j.jde.2016.03.006.

[7]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Analysis, 68 (2008), 177-193. doi: 10.1016/j.na.2006.10.040.

[8]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, Cham/Heidelberg/New York, 2015. doi: 10.1007/978-3-319-22903-4.

[9]

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

[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), viii+183 pp. doi: 10.1090/memo/0912.

[11]

D. G. Crighton, Model equations of nonlinear acoustics, Ann. Rev. Fluid Mech., 11 (1979), 11-33. doi: 10.1146/annurev.fl.11.010179.000303.

[12]

B. Coleman, M. Fabrizio and D. Owen, On the thermodynamics of second sound in dielectric crystals, Arch. Rat. Mech. Anal, 80 (1982), 135-158. doi: 10.1007/BF00250739.

[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), 4188-4222. doi: 10.1016/j.jde.2016.06.025.

[14]

W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Cont. Mechanics Thermodynamics, 5 (1993), 3-50. doi: 10.1007/BF01135371.

[15]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal., 81 (2002), 1245-1264. doi: 10.1080/0003681021000035588.

[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), 1193-1229. doi: 10.1016/j.jde.2012.10.016.

[17]

H. Fattorini, The Cauchy Problem, Addison Wesley, 1983.

[18]

X. Han and M. Wang, General decay rates of energy for the second order evolutions equations with memory, Acta Applicanda, Math., 110 (2010), 195-207. doi: 10.1007/s10440-008-9397-x.

[19]

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

[20]

P. Jordan, Second-Sound phenomena in inviscidm thermally relaxing gases, Discrete and Continuous Dynamical Systems, Series B, 19 (2014), 2189-2205. doi: 10.3934/dcdsb.2014.19.2189.

[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, 2009), Dresden, Germany, 9 September 2009. see also J. Acoust. Soc. Amer., 124 (2008), 2491-2491. doi: 10.1121/1.4782790.

[22]

P. M. Jordan, Private Communication, May 2015.

[23]

B. Kaltenbacher, Mathematics of nonlinear acoustics, Evolution Equations and Control Theory, 4 (2015), 447-491. doi: 10.3934/eect.2015.4.447.

[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), 971-988.

[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), 1250035, 34 pp. doi: 10.1142/S0218202512500352.

[26]

V. P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust., 16 (1971), 467-470.

[27]

I. Lasiecka, Global solvability of Moore-Gibson-Thompson equation with memory raising in nonlinear acoustics, Journal of Evolution Equations, to appear 2016.

[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), 031504, 18pp. doi: 10.1063/1.4793988.

[29]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation, Differential and Integral Equations, 6 (1993), 507-533.

[30]

I. Lasiecka and X. Wang, General decay rate of Moore-Gibson-Thompson equation with memory -Part II, Journal Differential Equations, 259 (2015), 7610-7635. doi: 10.1016/j.jde.2015.08.052.

[31]

I. Lasiecka and X. Wang, Moore Gibson Thompson equation with memory, part I, ZAMP, 67 (2016), 23pp. doi: 10.1007/s00033-015-0597-8.

[32]

G. Lebon and A. Cloot, Propagation of ultrasonic sound waves in dissipative dilute gases and extended irreversible thermodynamics, Wave Motion, 11 (1989), 23-32. doi: 10.1016/0165-2125(89)90010-3.

[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), 1896-1929. doi: 10.1002/mma.1576.

[34]

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

[35]

S. A. Messaoudi, General decay of solutions of a viscoelastic equation, JMAA, 341 (2008), 1457-1467. doi: 10.1016/j.jmaa.2007.11.048.

[36]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271. doi: 10.1007/s00245-011-9138-9.

[37]

S. Messaoudi and M. Mustafa, General stability result for viscoelastic wave equations, Journal Mathematical Pgysics, 53 (2012), 053702, 14pp. doi: 10.1063/1.4711830.

[38]

F. Moore and W. Gibson, Propagation of weak disturbances in a gas subject to relaxing effects, Journal Aero/Space SCi., 27 (1960), 117-127.

[39]

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

[40]

J. Pruss, Decay properties for the solutions of a partial differential equations with memory, Archiv der Mathematik, 92 (2009), 158-173. doi: 10.1007/s00013-008-2936-x.

[41]

J. Pruss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87. Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[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), 117-127.

[43]

M. Renardy, Mathematical Analysis of Viscoelastic Flows, CBMS-NSF Conference Series in Applied Mathematics, 73, SIAM, 2000. doi: 10.1137/1.9780898719413.

[44]

J. Rivera and A. Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic materials, Quart. Appl. Math., 59, (2001), 557-578.

[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, New York-London, 1977.

[46]

P. Stokes, An examination of the possible effect of the radiation of heat on the propagation of sound, Philosophical Magazine Series, 4 (1851), 3015-3317.

show all references

References:
[1]

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

[2]

F. Alabau-Boussouira, P. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, Journal of Functional Analysis, 254 (2008), 1342-1372. doi: 10.1016/j.jfa.2007.09.012.

[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, Ser., 347 (2009), 867-872. doi: 10.1016/j.crma.2009.05.011.

[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), 1473-1517. doi: 10.1016/j.jde.2009.12.005.

[5]

F. Alabau-Boussouira, On some recent advances on stabilization for hyperbolic equations, CIME Foundation Subseries, Control of Partial Differential Equations, Springer Verlag, 2048 (2012), 1-100. doi: 10.1007/978-3-642-27893-8_1.

[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), 113-147. doi: 10.1016/j.jde.2016.03.006.

[7]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Analysis, 68 (2008), 177-193. doi: 10.1016/j.na.2006.10.040.

[8]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, Cham/Heidelberg/New York, 2015. doi: 10.1007/978-3-319-22903-4.

[9]

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

[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), viii+183 pp. doi: 10.1090/memo/0912.

[11]

D. G. Crighton, Model equations of nonlinear acoustics, Ann. Rev. Fluid Mech., 11 (1979), 11-33. doi: 10.1146/annurev.fl.11.010179.000303.

[12]

B. Coleman, M. Fabrizio and D. Owen, On the thermodynamics of second sound in dielectric crystals, Arch. Rat. Mech. Anal, 80 (1982), 135-158. doi: 10.1007/BF00250739.

[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), 4188-4222. doi: 10.1016/j.jde.2016.06.025.

[14]

W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Cont. Mechanics Thermodynamics, 5 (1993), 3-50. doi: 10.1007/BF01135371.

[15]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal., 81 (2002), 1245-1264. doi: 10.1080/0003681021000035588.

[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), 1193-1229. doi: 10.1016/j.jde.2012.10.016.

[17]

H. Fattorini, The Cauchy Problem, Addison Wesley, 1983.

[18]

X. Han and M. Wang, General decay rates of energy for the second order evolutions equations with memory, Acta Applicanda, Math., 110 (2010), 195-207. doi: 10.1007/s10440-008-9397-x.

[19]

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

[20]

P. Jordan, Second-Sound phenomena in inviscidm thermally relaxing gases, Discrete and Continuous Dynamical Systems, Series B, 19 (2014), 2189-2205. doi: 10.3934/dcdsb.2014.19.2189.

[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, 2009), Dresden, Germany, 9 September 2009. see also J. Acoust. Soc. Amer., 124 (2008), 2491-2491. doi: 10.1121/1.4782790.

[22]

P. M. Jordan, Private Communication, May 2015.

[23]

B. Kaltenbacher, Mathematics of nonlinear acoustics, Evolution Equations and Control Theory, 4 (2015), 447-491. doi: 10.3934/eect.2015.4.447.

[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), 971-988.

[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), 1250035, 34 pp. doi: 10.1142/S0218202512500352.

[26]

V. P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust., 16 (1971), 467-470.

[27]

I. Lasiecka, Global solvability of Moore-Gibson-Thompson equation with memory raising in nonlinear acoustics, Journal of Evolution Equations, to appear 2016.

[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), 031504, 18pp. doi: 10.1063/1.4793988.

[29]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation, Differential and Integral Equations, 6 (1993), 507-533.

[30]

I. Lasiecka and X. Wang, General decay rate of Moore-Gibson-Thompson equation with memory -Part II, Journal Differential Equations, 259 (2015), 7610-7635. doi: 10.1016/j.jde.2015.08.052.

[31]

I. Lasiecka and X. Wang, Moore Gibson Thompson equation with memory, part I, ZAMP, 67 (2016), 23pp. doi: 10.1007/s00033-015-0597-8.

[32]

G. Lebon and A. Cloot, Propagation of ultrasonic sound waves in dissipative dilute gases and extended irreversible thermodynamics, Wave Motion, 11 (1989), 23-32. doi: 10.1016/0165-2125(89)90010-3.

[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), 1896-1929. doi: 10.1002/mma.1576.

[34]

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

[35]

S. A. Messaoudi, General decay of solutions of a viscoelastic equation, JMAA, 341 (2008), 1457-1467. doi: 10.1016/j.jmaa.2007.11.048.

[36]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271. doi: 10.1007/s00245-011-9138-9.

[37]

S. Messaoudi and M. Mustafa, General stability result for viscoelastic wave equations, Journal Mathematical Pgysics, 53 (2012), 053702, 14pp. doi: 10.1063/1.4711830.

[38]

F. Moore and W. Gibson, Propagation of weak disturbances in a gas subject to relaxing effects, Journal Aero/Space SCi., 27 (1960), 117-127.

[39]

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

[40]

J. Pruss, Decay properties for the solutions of a partial differential equations with memory, Archiv der Mathematik, 92 (2009), 158-173. doi: 10.1007/s00013-008-2936-x.

[41]

J. Pruss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87. Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[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), 117-127.

[43]

M. Renardy, Mathematical Analysis of Viscoelastic Flows, CBMS-NSF Conference Series in Applied Mathematics, 73, SIAM, 2000. doi: 10.1137/1.9780898719413.

[44]

J. Rivera and A. Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic materials, Quart. Appl. Math., 59, (2001), 557-578.

[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, New York-London, 1977.

[46]

P. Stokes, An examination of the possible effect of the radiation of heat on the propagation of sound, Philosophical Magazine Series, 4 (1851), 3015-3317.

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