# ENERGY DECAY RATE FOR THE KELVIN-VOIGT TYPE WAVE EQUATION WITH BALAKRISHNAN-TAYLOR DAMPING AND ACOUSTIC BOUNDARY

Kang, Yong Han

• Accepted : 2016.03.17
• Published : 2016.05.31
• 19 4

#### Abstract

In this paper, we study exponential stabilization of the vibrations of the Kelvin-Voigt type wave equation with Balakrishnan-Taylor damping and 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 Kang [3]. Energy decay rate are obtained by the exponential stability of solutions by using multiplier technique.

#### Keywords

Kelvin-Voigt type;Energy decay;Balakrishnan-Taylor damping;Acoustic boundary;Stabilization;Lyapunov functional

#### References

1. 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
2. A.V. Balakishnan and L.W. Taylor, Distributed parameter nonlinear damping models for flight structures, Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989.
3. A. Vicente, Wave equations with acoustic/memory boundary conditions, Bol. Soc. Parana. Mat.27 (2009), no. 3, 29-39, Springer-Verlag, New York, 1972.
4. A. Zarai and N.-E. Tatar, Global existence and polynominal decay for a problem Balakrishnan-Taylor damping, Archivum Mathematicum(BRNO) 46 (2010), 157-176.
5. 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
6. 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.
7. 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.
8. G. Kirchhoff, Vorlesungen ubear Mathematische Physik, Mechanik(Teubner) 1977.
9. H. Harrison, Plane and circular motion of a string, J. Acoust. Soc. Am.20 (1948), 874-875.
10. 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
11. 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
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. https://doi.org/10.1063/1.3040185
13. 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
14. 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
15. R.W. Bass and D. Zes, Spillover, nonlinearity and exible structures, The Fourth NASA Workship on Computational Control of Flexible Aerospace Systems, NASA Conference Publication 10065 (L.W. Taylor, ed.), 1991, 1-14.
16. Y.H. Kang, Energy decay rate for the Kirchhoff type wave equation with acoustic boundary condition, East Asian Mathematical Journal 28 (2012), no. 3, 339-345. https://doi.org/10.7858/eamj.2012.28.3.339
17. Y.H. Kang, Energy decay rates for the Kelvin-Voigt type wave equation with acoustic boundary condition, J. KSIAM. 16 (2012), no. 2, 85-91.

#### Cited by

1. A global nonexistence of solutions for a quasilinear viscoelastic wave equation with acoustic boundary conditions vol.2018, pp.1, 2018, https://doi.org/10.1186/s13661-018-1057-0