East Asian mathematical journal

  • 제28권3호
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  • Pages.339-345
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  • 2012
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  • 1226-6973(pISSN)
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  • 2287-2833(eISSN)
영남수학회 (The Youngnam Mathematical Society)
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ENERGY DECAY RATE FOR THE KIRCHHOFF TYPE WAVE EQUATION WITH ACOUSTIC BOUNDARY

  • Kang, Yong-Han (Institute of Liberal Education, Catholic University of Daegu)
  • 투고 : 2012.02.21
  • 심사 : 2012.03.15
  • 발행 : 2012.05.31
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⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we study uniform exponential stabilization of the vibrations of the Kirchho type wave equation with acoustic boundary in a bounded domain in $R^n$. To stabilize the system, we incorporate separately, the passive viscous damping in the model as like Gannesh C. Gorain [1]. Energy decay rate is obtained by the exponential stability of solutions by using multiplier technique.

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참고문헌

  1. G. C. Gorain, Exponential eneragy decay estimates for the solutions of n-dimensional Kirchhoff type wave equation, Applied Mathematics and Computation 117 (2006), 235-242.
  2. M. A. Horn, Exact controllability and uniform stabilization of the Kirchhoff plate equation with boundary feedback acting via bending moments, J. Math. Anal. Appl. 167 (1992), 557-581. https://doi.org/10.1016/0022-247X(92)90224-2
  3. G. Kirchhoff, Vorlesungen ubear Mathematische Physik, Mechanik(Teubner), 1977.
  4. J. Y. Park and S. H. Park, Decay rate estimates for wave equations of memory type with acoustic boundary conditions, Nonlinear Analysis: Theory, methods and Applications 74 (2011), no. 3, 993-998. https://doi.org/10.1016/j.na.2010.09.057
  5. J. T. Beal and S. I. Rosencrans, Acoustic boundary conditions, Bull. Amer. Math. Soc. 80 (1974), 1276-1278. https://doi.org/10.1090/S0002-9904-1974-13714-6
  6. C. L. Frota and J. A. Goldstein, Some nonlinear wave equations with acoustic boundary conditions, J. Differ. Equ. 164 (2000), 92-109. https://doi.org/10.1006/jdeq.1999.3743
  7. H. Harrison, Plane and circular motion of a string, J. Acoust. Soc. Am. 20 (1948), 874-875.
  8. A. T. Cousin, C. L. Frota and N. A. Larkin, On a system of Klein-Gordon type equations with acoustic boundary conditions, J. Math. Anal. Appl. 293 (2004), 293-309. https://doi.org/10.1016/j.jmaa.2004.01.007
  9. C. L. Frota and N. A. Larkin, Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains, Progr. Nonlinear Differential Equations Appl. 66 (2005), 297-312.
  10. Y. H. Kang, M. J. Lee and I. H. Jeong, Stabilization of Kirchhoff type wave equation with locally distributed damping, Applied Mathematics Letters. 22 (2009), no. 5, 719-722. https://doi.org/10.1016/j.aml.2008.08.009
  11. J. Y. Park and J. A. Kim, Some nonlinear wave equations with nonlinear memory source term and acoustic boundary conditions, Numer. Funct. Anal. Optim. 27 (2006), 889-903. https://doi.org/10.1080/01630560600884596
  12. J. Y. Park and T. G. Ha, Well-posedness and uniform decay rates for the Klein-Gordon equation with damping term and acoustic boundary conditions, J. Math. Phys. 50 (2009), Article No. 013506; doi:10.1063/1.3040185.
  13. A. Vicente, Wave equations with acoustic/memory boundary conditions, Bol. Soc. Parana. Mat. 27 (2009), no. 3, 29-39.

피인용 문헌

  1. General Energy Decay for a Viscoelastic Equation of Kirchhoff Type with Acoustic Boundary Conditions vol.14, pp.6, 2017, https://doi.org/10.1007/s00009-017-1038-z
  2. ENERGY DECAY RATES FOR THE KIRCHHOFF TYPE WAVE EQUATION WITH BALAKRISHNAN-TAYLOR DAMPING AND ACOUSTIC BOUNDARY vol.30, pp.3, 2014, https://doi.org/10.7858/eamj.2014.015
  3. ENERGY DECAY RATE FOR THE KELVIN-VOIGT TYPE WAVE EQUATION WITH BALAKRISHNAN-TAYLOR DAMPING AND ACOUSTIC BOUNDARY vol.32, pp.3, 2012, https://doi.org/10.7858/eamj.2016.026