Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
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ENERGY DECAY RATES FOR THE KELVIN-VOIGT TYPE WAVE EQUATION WITH ACOUSTIC BOUNDARY

  • Seo, Young-Il (National Fisheries Research and Development Institute) ;
  • Kang, Yong-Han (Institute of Liberal Education, Catholic University of Daegu)
  • Received : 2012.02.11
  • Accepted : 2012.06.15
  • Published : 2012.06.25
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Abstract

In this paper, we study uniform exponential stabilization of the vibrations of the Kelvin-Voigt type wave equation with acoustic boundary in a bounded domain in $R^n$. To stabilize the systems, we incorporate separately, the internal material damping in the model as like Gannesh C. Gorain [1]. Energy decay rates are obtained by the exponential stability of solutions by using multiplier technique.

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References

  1. G.C. Gorain, Exponential eneragy decay estimates for the solutions of n-dimensional Kirchhoff type wave equation, Applied Mathematics and Computation 117, 235-242 (2006).
  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, 557-581 (1992). 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(3), 993-998 (2011). 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, 1276-1278 (1974). 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, 92-109 (2000). https://doi.org/10.1006/jdeq.1999.3743
  7. H. Harrison, Plane and circular motion of a string, J. Acoust. Soc. Am. 20, 874-875 (1948).
  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, 293-309 (2004). 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, 297-312 (2005).
  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(5), 719-722 (2009). 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, 889-903 (2006). 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(3), 29-39 (2009). Springer-Verlag, New York, 1972.

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