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Energy decay rate for the Kirchhoff type wave equation with acoustic boundary

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In this paper, we study uniform exponential stabilization of the vibrations of the Kirchhoff 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.

Abstract

In this paper, we study uniform exponential stabilization of the vibrations of the Kirchhoff 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.

참고문헌 (13)

  1. G. C. Gorain, "Exponential eneragy decay estimates for the solutions of n-dimensional Kirchho type wave equation", Applied Mathematics and Computation, 117, pp.235~242 (2006)
  2. M. A. Horn, "Exact controllability and uniform stabilization of the Kirchho plate equation with boundary feedback acting via bending moments", J. Math. Anal. Appl, 167, pp.557~581 (1992)
  3. G. Kirchho, "Vorlesungen ubear Mathematische Physik" (1977), Mechanik(Teubner)
  4. J. Y. Park, "Decay rate estimates for wave equations of memory type with acoustic boundary conditions", NONLINEAR ANALYSIS-THEORY METHODS APPLICATIONS, 74(3), pp.993~998 (2011), PERGAMON-ELSEVIER SCIENCE LTD
  5. J. T. Beal, "Acoustic boundary conditions", Bull. Amer. Math. Soc, 80, pp.1276~1278 (1974)
  6. C. L. Frota, "Some nonlinear wave equations with acoustic boundary conditions", J. Di er. Equ, 164, pp.92~109 (2000)
  7. H. Harrison, "Plane and circular motion of a string", J. Acoust. Soc. Am, 20, pp.874~875 (1948)
  8. A. T. Cousin, "On a system of Klein-Gordon type equations with acoustic boundary conditions", J. Math. Anal. Appl, 293, pp.293~309 (2004)
  9. C. L. Frota, "Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains", Progr. Nonlinear Di erential Equations Appl, 66, pp.297~312 (2005)
  10. Y. H. Kang, "Stabilization of Kirchho type wave equation with locally distributed damping", Applied Mathematics Letters, 22(5), pp.719~722 (2009)
  11. J. Y. Park, "Some nonlinear wave equations with nonlinear memory source term and acoustic boundary conditions", NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION, 27, pp.889~903 (2006), TAYLOR & FRANCIS INC
  12. J. Y. Park, "Well-posedness and uniform decay rates for the Klein- Gordon equation with damping term and acoustic boundary conditions", J. Math. Phys, 50 (2009)[DOI]
  13. A. Vicente, "Wave equations with acoustic", memory boundary conditions, 2009, pp.27(3), 29~39

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