Skip to main content
SpringerLink
Log in

Global solvability of Moore–Gibson–Thompson equation with memory arising in nonlinear acoustics

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

A third-order in time nonlinear equation with memory term is considered. This particular model is motivated by high-frequency ultrasound technology which accounts for thermal and molecular relaxation. The resulting equations give rise to a quasilinear-like evolution with a potentially degenerate damping (Kaltenbacher in Evol Eqs Control Theory 4(4):447–491, 2015). Local and global (in time) existence of smooth solutions is studied. The main result of the paper states that with appropriate calibration of the memory kernel, solutions exist globally for sufficiently small and regular initial data. With exponentially decaying relaxation kernel said solutions exhibit exponential decay rates. The proof relies on the “barrier” method applied to a string of higher energy estimates, along with an abstract representation and the theory of viscoleastic evolutions developed in Pruss (Arch Math 92:158–173, 2009, Evolutionary integral equations and applications. Birkhauser, 2012).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Adhikari, Structural Dynamic Analysis with Generalized Damping Models: Analysis, Wiley-ISTE, (2013).

  2. Alabau F., Cannarsa P., Sforza D.: Decay estimates for second order evolution equations with memory. Journal of Functional Analysis 254, 1342–1372 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alabau-Boussouira F., Cannarsa P.: A general method for proving sharp energy decay rates for memory-dissipative evolution equations. C. R. Acad. Sci. Paris, Ser. I 347, 867–872 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alabau-Boussouira F.: 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, 1473–1517 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. F. Alabau-Boussouira, On some recent advances on stabilization for hyperbolic equations. Lecture Note in Mathematics 2048, CIME Foundation Subseries, Control of Partial Differential Equations, 1–100, Springer Verlag, 2048 (2012).

  6. M.M. Cavalcanti and V.N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Analysis. 68 # 1 177–193, (2008)

  7. D. G. Crighton, Model equations of nonlinear acoustics, Ann. Rev. Fluid Mech. 11, pp. 11–33, (1979)

  8. B. Coleman, M. Fabrizio, D. Owen. On the thermodynamics of second sound in dielectric crystals, Arch. Rat. Mech. Anal, 80, pp 135–158, (1982).

  9. F. Dell’Oro, I. Lasiecka and V. Pata, The Moore- Gibson-Thompson equation with memory in the critical case, Journal Differential Equations, 261, 4188–4222, (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dreyer W., Struchtrup H.: Heat pulse experiments revisited. Cont. Mechanics Thermodynamics, 5, 3–50 (1993)

    Article  MathSciNet  Google Scholar 

  11. M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal. 81 #6, 1245–1264, (2002)

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

  13. Han X., Wang M.: General decay rates of energy for the second order evolutions equations with memory. Acta Applicanda, Math., 110, 195–207 (2010)

    Article  MATH  Google Scholar 

  14. 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).

  15. P. Jordan, Second-Sound phenomena in inviscidm thermally relaxing gases, Discrete and Continuous Dynamical Systems, Series B, vol 19, 2189–2205, (2014).

  16. 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, 2491 (2009)

  17. P.M. Jordan. Private communication. May (2015).

  18. B. Kaltenbacher, Mathematics of nonlinear acoustics, Evolution Equations and Control Theory, vol 4, Nr 4, pp 447–491, (2015).

  19. 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, no. 4, 971–988, (2011).

  20. 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, no 11, (2012)

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

    Google Scholar 

  22. 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)

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

  24. I. Lasiecka and X. Wang, General decay rate of Moore-Gibson-Thompson equation with memory -Part II. Journal Differential Equations. 259, nr 12, 7610–7635, (2015).

  25. I. Lasiecka and X. Wang, Moore Gibson Thompson equation with memory, part I. ZAMP 67, 17, (2016)

  26. G. Lebon and A. Cloot, Propagation of ultrasonic sound waves in dissipative dilute gases and extended irreversible thermodynamics, Wave Motion 11, pp 23–32, (1989)

  27. 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 no. 15, 1896–1929, (2012)

  28. I. Muller and T. Ruggeri, Extended Thermodynamics, vol 37, Springer, (1993).

  29. Messaoudi S. A.: General decay of solutions of a viscoelastic equation. JMAA 341, 1457–1467 (2008)

    MathSciNet  MATH  Google Scholar 

  30. S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation. Appl. Math. Optim. 64, no 2, pp. 257–271, (2011)

  31. S. Messaoudi and M. Mustafa, General stability result for viscoelastic wave equation Journal Mathematical Pgysics. 53, (2012).

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

    Article  MATH  Google Scholar 

  33. K. Naugolnykh and L. Ostrovsky, Nonlinear wave processes in acoustics, Cambridge University Press, 1998.

  34. J. Pruss, Decay properties for the solutensofpartial differential equations with memory. Archiv der Mathematik, 92, pp 158–173, (2009).

  35. J. Pruss. Evolutionary Integral Equations and Applications, Birkhauser, (2012).

  36. Moore F.K., Gibson W.E.: Propagation of weak disturbances in a gas subject to relaxation effects. J. Aerospace Sci. Tech. 27, 117–127 (1960)

    Article  MATH  Google Scholar 

  37. M. Renardy, Mathematical Analysis of Viscoelastic Flows, CBMS-NSF Conference Series in Applied Mathematics 73, SIAM (2000)

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

    Article  MathSciNet  MATH  Google Scholar 

  39. 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).

  40. Professor Stokes. An examination of the possible effect of the radiation of heat on the propagation of sound, Philosophical Magazine Series 4, 1:4, 3015–317, (1851).

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA

    Irena Lasiecka

  2. IBS, Polish Academy of Sciences, Warsaw, Poland

    Irena Lasiecka

Authors
  1. Irena Lasiecka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Irena Lasiecka.

Additional information

Dedicated to Jan Prüss on the occasion of his retirement.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lasiecka, I. Global solvability of Moore–Gibson–Thompson equation with memory arising in nonlinear acoustics. J. Evol. Equ. 17, 411–441 (2017). https://doi.org/10.1007/s00028-016-0353-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-016-0353-3

Keywords

Search

Navigation