Moore–Gibson–Thompson equation with memory, part I: exponential decay of energy

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• 作者：Irena Lasiecka ; Xiaojun Wang
• 关键词：Moore–Gibson–Thompson (MGT) equation ; High ; frequency ultrasound waves ; Memory ; Damping ; Multipliers ; Energy estimate ; Exponential decay
• 刊名：Zeitschrift f¨¹r angewandte Mathematik und Physik
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：67
• 期：2
• 全文大小：620 KB
• 参考文献：1.Adhikari S.: Structural Dynamic Analysis with Generalized Damping Models: Analysis, pp. 384. Wiley-ISTE, New York (2013)CrossRef
  2.Alabau-Boussouira F., Cannarsa P., Sforza D.: Decay estimates for second order evolution equations with memory. J Funct. Anal. 254, 1342–1372 (2008)MathSciNet CrossRef MATH
  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)MathSciNet CrossRef MATH
  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. J Differ. Equ. 248, 1473–1517 (2010)MathSciNet CrossRef MATH
  5.Alabau-Boussouira, F.: On some recent advances on stabilization for hyperbolic equations. In: Lecture Note in Mathematics 2048, CIME Foundation Subseries, Control of Partial Differential Equations, vol. 2048, pp. 1—100, Springer, Berlin (2012)
  6.Cavalcanti M.M., Cavalcanti A.D.D., Lasiecka I., Wang X.: Existence and sharp decay rate estimates for a von Karman system with long memory. Nonlinear Anal. Real World Appl. 22, 289–306 (2015)MathSciNet CrossRef MATH
  7.Cavalcanti M.M., Cavalcanti V.N.D., Martinez P.: General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal. 68(1), 177–193 (2008)MathSciNet CrossRef MATH
  8.Cavalcanti M.M., Oquendo H.P.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42(4), 1310–1324 (2003)MathSciNet CrossRef MATH
  9.Crighton D.G.: Model equations of nonlinear acoustics. Annu. Rev. Fluid Mech. 11, 1133 (1979)CrossRef MATH
  10.Dafermos C.M.: Asymptotic stability in viscoelasticity. Arch. Rat. Mech. Anal. 37, 297–308 (1970)MathSciNet CrossRef MATH
  11.Fabrizio M., Polidoro S.: Asymptotic decay for some differential systems with fading memory. Appl. Anal. 81(6), 1245–1264 (2002)MathSciNet CrossRef MATH
  12.Han X., Wang M.: General decay rates of energy for the second order evolutions equations with memory. Acta Appl. Math. 110, 195–207 (2010)MathSciNet CrossRef MATH
  13.Hrusa, W.J., Nohel, J.A., Renardy, M.: Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Longman (1987)
  14.Jordan P.M.: Nonlinear acoustic phenomena in viscous thermally relaxing fluids: Shock bifurcation and the emergence of diffusive solitons. J. Acoust. Soc. Am. 124, 2491 (2008)CrossRef
  15.Jordan P.M.: Private communication. March 2015
  16.Kaltenbacher, M.: Numerical Simulations of Mechatronic Sensors and Actuators, 2nd edn. Springer, Berlin, (2007)
  17.Kaltenbacher B., Lasiecka I., Marchand R.: well-posedness and exponential decay rates for the Moore–Gibson–Thompson equation arising in high intensity ultrasound. Control Cybern. 40(4), 971–988 (2011)MathSciNet MATH
  18.Kaltenbacher, B., Lasiecka, I., Pospieszalska, M.: 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(11) 1250035-1–1250035-34 (2012)
  19.Kuznetsov V.P.: Equations of nonlinear acoustics. Sov. Phys. Acoust. 16, 467–470 (1971)
  20.Lasiecka, I., Messaoudi, S.A., Mustafa, M.I.: Note on intrinsic decay rates for abstract wave equations with memory. J. Math. Phys. 54, 031504 (2013). doi:10.​1063/​1.​4793988
  21.Lasiecka I., Tataru D.: Uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation. Differ. Integr. Equ. 6, 507–533 (1993)MathSciNet MATH
  22.Lasiecka I., Wang X.: Intrinsic Decay Rate Estimates for Semilinear Abstract Second Order Equations with Memory. New Prospects in Direct, Inverse and Control Problems for Evolution Equations Springer INdAM Series 10, 271–303 (2014)MathSciNet
  23.Lasiecka, I., Wang, X.: Moore–Gibson–Thompson equation with memory, part II: general decay of energy. J. Differ. Equ. 259(12), 7610 -7635 (2015)
  24.Lebon G., Cloot A.: Propagation of ultrasonic sound waves in dissipative dilute gases and extended irreversible thermodynamics. Wave Motion 11, 23–32 (1989)CrossRef MATH
  25.Marchand R., McDevitt T., Triggiani R.: 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(15), 1896–1929 (2012)MathSciNet CrossRef MATH
  26.Messaoudi S.A.: General decay of solutions of a viscoelastic equation. JMAA 341, 1457–1467 (2008)MathSciNet MATH
  27.Messaoudi, S., Mustafa, M.: General stability result for viscoelastic wave equations. J. Math. Phys. 53, 053702-1–053702-14 (2012)
  28.Moore F.K., Gibson W.E.: Propagation of weak disturbances in a gas subject to relaxation effects. J. Aerospace Sci. Technol. 27, 117–127 (1960)CrossRef MATH
  29.Morrison, J.A.: Wave propagation in rods of Voigt material and visco-elastic materials with three parameter models. Q. Appl. Math., 14, 153–169 (1956–1957)
  30.Morawetz C.S., Ralston J.V., Strauss W.A.: Decay of solutions of the wave equation outside nontrapping obstacles. Commun. Pure Appl. Math. 30(4), 447508 (1977)MathSciNet CrossRef MATH
  31.Naugolnykh, K., Ostrovsky L.: Nonlinear wave processes in acoustics. Translated from the Russian. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, (1998)
  32.Pierce, A.D.: Acoustics, An Introduction to its Physical Principles and Applications. Acoustical Society of America, ASA Press (1989)
  33.Renardy, M.: Mathematical Analysis of Viscoelastic Flows, CBMS-NSF Conference Series in Applied Mathematics 73, SIAM (2000)
  34.Rivera J., Salvatierra A.: Asymptotic behaviour of the energy in partially viscoelastic materials. Q. Appl. Math. 59, 557–578 (2001)MathSciNet MATH
  35.Rudenko, O.V., Soluyan, S.I.: Theoretical foundations of nonlinear acoustics. Translated from the Russian by Robert T. Beyer. Studies in Soviet Science. Consultants Bureau, New York–London (1977). vii+274 pp
  36.Strauss, W.: Nonlinear invariant wave equations. Invariant wave equations (Proc. “Ettore Majorana” Internat. School of Math. Phys., Erice, 1977), pp. 197249, Lecture Notes in Phys., 73, Springer, Berlin-New York, 1978
  
• 作者单位：Irena Lasiecka (1) (2)
  Xiaojun Wang (1)
  
  1. Department of Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA
  2. IBS, Polish Academy of Sciences, Warsaw, Poland
  
• 刊物主题：Theoretical and Applied Mechanics; Mathematical Methods in Physics;
• 出版者：Springer Basel
• ISSN：1420-9039

文摘

We are interested in the Moore–Gibson–Thompson equation with memory $$\tau{u}_{ttt}+ \alpha u_{tt}+c^{2}\mathcal{A}u+b\mathcal{A}u_t -\int_0^{t}g(t-s)\mathcal{A} w(s){\rm {d}}s=0.$$This model arises in high-frequency ultrasound applications accounting for thermal flux and molecular relaxation times. According to revisited extended irreversible thermodynamics, thermal flux relaxation leads to the third-order derivative in time while molecular relaxation leads to non-local effects governed by memory terms. The resulting model is of hyperbolic type with viscous effects. We first classify the memory into three types. Then, we study how a memory term creates damping mechanism and how the memory causes energy decay even in the cases when the original memoryless system is unstable. Keywords Moore–Gibson–Thompson (MGT) equation High-frequency ultrasound waves Memory Damping Multipliers Energy estimate Exponential decay Mathematics Subject Classification 35Q70 35L05 74D99 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (36) References1.Adhikari S.: Structural Dynamic Analysis with Generalized Damping Models: Analysis, pp. 384. Wiley-ISTE, New York (2013)CrossRef2.Alabau-Boussouira F., Cannarsa P., Sforza D.: Decay estimates for second order evolution equations with memory. J Funct. Anal. 254, 1342–1372 (2008)MathSciNetCrossRefMATH3.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)MathSciNetCrossRefMATH4.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. J Differ. Equ. 248, 1473–1517 (2010)MathSciNetCrossRefMATH5.Alabau-Boussouira, F.: On some recent advances on stabilization for hyperbolic equations. In: Lecture Note in Mathematics 2048, CIME Foundation Subseries, Control of Partial Differential Equations, vol. 2048, pp. 1—100, Springer, Berlin (2012)6.Cavalcanti M.M., Cavalcanti A.D.D., Lasiecka I., Wang X.: Existence and sharp decay rate estimates for a von Karman system with long memory. Nonlinear Anal. Real World Appl. 22, 289–306 (2015)MathSciNetCrossRefMATH7.Cavalcanti M.M., Cavalcanti V.N.D., Martinez P.: General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal. 68(1), 177–193 (2008)MathSciNetCrossRefMATH8.Cavalcanti M.M., Oquendo H.P.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42(4), 1310–1324 (2003)MathSciNetCrossRefMATH9.Crighton D.G.: Model equations of nonlinear acoustics. Annu. Rev. Fluid Mech. 11, 1133 (1979)CrossRefMATH10.Dafermos C.M.: Asymptotic stability in viscoelasticity. Arch. Rat. Mech. Anal. 37, 297–308 (1970)MathSciNetCrossRefMATH11.Fabrizio M., Polidoro S.: Asymptotic decay for some differential systems with fading memory. Appl. Anal. 81(6), 1245–1264 (2002)MathSciNetCrossRefMATH12.Han X., Wang M.: General decay rates of energy for the second order evolutions equations with memory. Acta Appl. Math. 110, 195–207 (2010)MathSciNetCrossRefMATH13.Hrusa, W.J., Nohel, J.A., Renardy, M.: Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Longman (1987)14.Jordan P.M.: Nonlinear acoustic phenomena in viscous thermally relaxing fluids: Shock bifurcation and the emergence of diffusive solitons. J. Acoust. Soc. Am. 124, 2491 (2008)CrossRef15.Jordan P.M.: Private communication. March 201516.Kaltenbacher, M.: Numerical Simulations of Mechatronic Sensors and Actuators, 2nd edn. Springer, Berlin, (2007)17.Kaltenbacher B., Lasiecka I., Marchand R.: well-posedness and exponential decay rates for the Moore–Gibson–Thompson equation arising in high intensity ultrasound. Control Cybern. 40(4), 971–988 (2011)MathSciNetMATH18.Kaltenbacher, B., Lasiecka, I., Pospieszalska, M.: 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(11) 1250035-1–1250035-34 (2012)19.Kuznetsov V.P.: Equations of nonlinear acoustics. Sov. Phys. Acoust. 16, 467–470 (1971)20.Lasiecka, I., Messaoudi, S.A., Mustafa, M.I.: Note on intrinsic decay rates for abstract wave equations with memory. J. Math. Phys. 54, 031504 (2013). doi:10.​1063/​1.​4793988 21.Lasiecka I., Tataru D.: Uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation. Differ. Integr. Equ. 6, 507–533 (1993)MathSciNetMATH22.Lasiecka I., Wang X.: Intrinsic Decay Rate Estimates for Semilinear Abstract Second Order Equations with Memory. New Prospects in Direct, Inverse and Control Problems for Evolution Equations Springer INdAM Series 10, 271–303 (2014)MathSciNet23.Lasiecka, I., Wang, X.: Moore–Gibson–Thompson equation with memory, part II: general decay of energy. J. Differ. Equ. 259(12), 7610 -7635 (2015)24.Lebon G., Cloot A.: Propagation of ultrasonic sound waves in dissipative dilute gases and extended irreversible thermodynamics. Wave Motion 11, 23–32 (1989)CrossRefMATH25.Marchand R., McDevitt T., Triggiani R.: 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(15), 1896–1929 (2012)MathSciNetCrossRefMATH26.Messaoudi S.A.: General decay of solutions of a viscoelastic equation. JMAA 341, 1457–1467 (2008)MathSciNetMATH27.Messaoudi, S., Mustafa, M.: General stability result for viscoelastic wave equations. J. Math. Phys. 53, 053702-1–053702-14 (2012)28.Moore F.K., Gibson W.E.: Propagation of weak disturbances in a gas subject to relaxation effects. J. Aerospace Sci. Technol. 27, 117–127 (1960)CrossRefMATH29.Morrison, J.A.: Wave propagation in rods of Voigt material and visco-elastic materials with three parameter models. Q. Appl. Math., 14, 153–169 (1956–1957)30.Morawetz C.S., Ralston J.V., Strauss W.A.: Decay of solutions of the wave equation outside nontrapping obstacles. Commun. Pure Appl. Math. 30(4), 447508 (1977)MathSciNetCrossRefMATH31.Naugolnykh, K., Ostrovsky L.: Nonlinear wave processes in acoustics. Translated from the Russian. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, (1998)32.Pierce, A.D.: Acoustics, An Introduction to its Physical Principles and Applications. Acoustical Society of America, ASA Press (1989)33.Renardy, M.: Mathematical Analysis of Viscoelastic Flows, CBMS-NSF Conference Series in Applied Mathematics 73, SIAM (2000)34.Rivera J., Salvatierra A.: Asymptotic behaviour of the energy in partially viscoelastic materials. Q. Appl. Math. 59, 557–578 (2001)MathSciNetMATH35.Rudenko, O.V., Soluyan, S.I.: Theoretical foundations of nonlinear acoustics. Translated from the Russian by Robert T. Beyer. Studies in Soviet Science. Consultants Bureau, New York–London (1977). vii+274 pp36.Strauss, W.: Nonlinear invariant wave equations. Invariant wave equations (Proc. “Ettore Majorana” Internat. School of Math. Phys., Erice, 1977), pp. 197249, Lecture Notes in Phys., 73, Springer, Berlin-New York, 1978 About this Article Title Moore–Gibson–Thompson equation with memory, part I: exponential decay of energy Journal Zeitschrift für angewandte Mathematik und Physik 67:17 Online DateApril 2016 DOI 10.1007/s00033-015-0597-8 Print ISSN 0044-2275 Online ISSN 1420-9039 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Theoretical and Applied Mechanics Mathematical Methods in Physics Keywords 35Q70 35L05 74D99 Moore–Gibson–Thompson (MGT) equation High-frequency ultrasound waves Memory Damping Multipliers Energy estimate Exponential decay Industry Sectors Aerospace Engineering Oil, Gas & Geosciences Authors Irena Lasiecka (1) (2) Xiaojun Wang (1) Author Affiliations 1. Department of Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA 2. 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