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Applicable Analysis
An International Journal
Volume 99, 2020 - Issue 2
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Articles

Asymptotic behavior for a past history viscoelastic problem with acoustic boundary conditions

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Pages 249-269 | Received 01 Jul 2017, Accepted 14 Jun 2018, Published online: 30 Jun 2018

References

  • Morse PM, Ingard KU. Theoretical acoustics. New York: McGraw-Hill; 1968.
  • Beale JT. Spectral properties of an acoustic boundary condition. Indiana Univ Math J. 1976;25(9):895–917. doi: 10.1512/iumj.1976.25.25071
  • Beale JT, Rosencrans SI. Acoustic boundary conditions. Bull Amer Math Soc. 1974;80:1276–1278. doi: 10.1090/S0002-9904-1974-13714-6
  • Avalos G, Lasiecka I, Rebarber R. Uniform decay properties of a model in structural acoustics. J Math Pures Appl. 2000;79(10):1057–1072. doi: 10.1016/S0021-7824(00)00173-2
  • Bucci F, Lasiecka I. Exponential decay rates for structural acoustic model with an over damping on the interface and boundary layer dissipation. Appl Anal. 2002;81(4):977–999. doi: 10.1080/0003681021000004555
  • Cousin AT, Frota CL, Larkin NA. Global solvability and asymptotic behavior of a hyperbolic problem with acoustic boundary conditions. Funkcial Ekvac. 2001;44(3):471–485.
  • Frota CL, Goldstein JA. Some nonlinear wave equations with acoustic boundary conditions. J Differ Equ. 2000;164(1):92–109. doi: 10.1006/jdeq.1999.3743
  • Lasiecka I. Boundary stabilization of a 3-dimensional structural acoustic model. J Math Pures Appl. 1999;78(2):203–232. doi: 10.1016/S0021-7824(01)80009-X
  • Mugnolo D. Abstract wave equations with acoustic boundary conditions. Math Nachr. 2006;279(3):299–318. doi: 10.1002/mana.200310362
  • Park JY, Ha TG. Well-posedness and uniform decay rates for the Klein-Gordon equation with damping term and acoustic boundary conditions. J Math Phys. 2009;50(1):013506. 18. doi: 10.1063/1.3040185
  • Boukhatem Y, Benabderrahmanne B. Existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions. Nonlin Anal. 2014;97:191–209. doi: 10.1016/j.na.2013.11.019
  • Park JY, Park SH. Decay rate estimates for wave equations of memory type with acoustic boundary conditions. Nonlinear Anal TMA. 2011;74(3):993–998. doi: 10.1016/j.na.2010.09.057
  • Abdallah C, Dorato P, Benitez-Read J, et al. Delayed positive feedback can stabilize oscillatory system. ACC. San Francisco; 1993. p. 3106–3107.
  • Datko R, Lagnese J, Polis MP. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J Control Optim. 1986;24:152–156. doi: 10.1137/0324007
  • Nicaise S, Pignotti C. Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J Control Optim. 2006;45:1561–1585. doi: 10.1137/060648891
  • Nicaise S, Pignotti C. Stabilization of the wave equation with boundary or internal distributed delay. Differ Int Equ. 2008;21:935–958.
  • Xu CQ, Yung SP, Li LK. Stabilization of the wave system with input delay in the boundary control. ESAIM: Control Optim Calc Var. 2006;12:770–785. doi: 10.1051/cocv:2006021
  • Chen M, Liu W, Zhou W. Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms. Adv Nonlin Anal. Online First. doi: 10.1515/anona-2016-0085.
  • Liu W, Chen M. Well-posedness and exponential decay for a porous thermoelastic system with second sound and a time-varying delay term in the internal feedback. Continuum Mech Thermodyn. 2017;29:731–746. https://doi.org/10.1007/s00161-017-0556-z.
  • Liu W, Zhu B, Li G, et al. General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term. Evol Equ Control Theory. 2017;6(2):239–260. doi: 10.3934/eect.2017013
  • Ning ZH, Shen CX, Zhao XP. Stabilization of the wave equation with variable coefficients and a delay in dissipative internal feedback. J Math Anal Appl. 2013;405:148–155. doi: 10.1016/j.jmaa.2013.01.072
  • Ning ZH, Shen CX, Zhao XP, et al. Nonlinear boundary stabilization of the wave equations with variable coefficients and time dependent delay. Appl Math Comput. 2014;232:511–520.
  • Wu J. Well-posedness for a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions. Nonlin Anal TMA. 2012;75(18):6562–6569. doi: 10.1016/j.na.2012.07.032
  • Yao PF. Observability inequality for exact controllability of wave equations with variable coefficients. SIAM J Control Optim. 1999;37:1568–1599. doi: 10.1137/S0363012997331482
  • Lasiecka I, Triggiani R, Yao PF. Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. J Math Anal Appl. 1999;235:13–57. doi: 10.1006/jmaa.1999.6348
  • Guesmia A. Asymptotic stability of abstract dissipative systems with infinite memory. J Math Appl. 2011;382:748–760.
  • Messaoudi SA, Apalara TA. Asymptotic stability of thermoelasticity type III with delay term and infinite memory. IMA J Math Control Inf. 2013;32:1–21. doi: 10.1093/imamci/dnt024.
  • Guesmia A. Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay. IMA J Math Control Inform. 2013;30:507–526. doi: 10.1093/imamci/dns039
  • Guesmia A, Messaoudi SA. A new approach to the stability of an abstract system in the presence of infinite history. J Math Anal Appl. 2014;416:212–228. doi: 10.1016/j.jmaa.2014.02.030
  • Messaoudi SA, Al-Gharabli M. A general stability result for a nonlinear wave equation with infinite memory. Appl Math Lett. 2013;26:1082–1086. doi: 10.1016/j.aml.2013.06.002
  • Liu W, Zhao W. Stabilization of a thermoelastic laminated beam with past history. Appl Math Optim. 1–31. doi: 10.1007/s00245-017-9460-y.
  • Li G, Wang D, Zhu B. Well-posedness and general decay of solution for a transmission problem with past history and delay. Electron J Differ Equ. 2016;2016(23):1–21.
  • Messaoudi SA, Mustafa MI. On the control of solutions of viscoelastic equations with boundary feedback. Nonlin Anal. 2010;72:3602–3611. doi: 10.1016/j.na.2009.12.040
  • Benaissa A, Benaissa A, Messaoudi SA. Global existence and energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks. J Math Phys. 2012;53:123514. doi: 10.1063/1.4765046
  • Dafermos CM. Asymptotic stability in viscoelasticity. Arch Ration Mech Anal. 1970;37:297–308. doi: 10.1007/BF00251609
  • Nicaise S, Pignotti C. Interior feedback stabilization of wave equations with time dependent delay. Electron J Differ Equ. 2011;2011(41):1–20.
  • Graber PJ, Said-Houari B. On the wave equation with semilinear porous acoustic boundary conditions. J. Differ Equ. 2012;252:4898–4941. doi: 10.1016/j.jde.2012.01.042
  • Kato T. Nonlinear semigroups and evolution equations. J Math Soc Japan. 1967;19:508–520. doi: 10.2969/jmsj/01940508
  • Arnold VI. Mathematical methods of classical mechanics. New York: Springer; 1989.
  • Cavalcanti MM, Cavalcanti VD, Lasiecka I. Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction. J Differ Equ. 2007;236:407–459. doi: 10.1016/j.jde.2007.02.004

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