# STABILIZATION FOR THE VISCOELASTIC KIRCHHOFF TYPE EQUATION WITH A NONLINEAR SOURCE

Kim, Daewook

• Accepted : 2016.01.30
• Published : 2016.01.30
• 19 2

#### Abstract

In this paper, we study the viscoelastic Kirchhoff type equation with a nonlinear source $$u^{{\prime}{\prime}}-M(x,t,{\parallel}{\bigtriangledown}u(t){\parallel}^2){\bigtriangleup}u+{\int}_0^th(t-{\tau})div[a(x){\bigtriangledown}u({\tau})]d{\tau}+{\mid}u{\mid}^{\gamma}u=0$$. Under the smallness condition with respect to Kirchhoff coefficient and the relaxation function and other assumptions, we prove the uniform decay rate of the Kirchhoff type energy.

#### Keywords

viscoelastic Kirchhoff type equation;energy decay rate;energy functional;smallness condition

#### References

1. F. Li, Z. Zhao and Y. Chen, Global existence and uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation, J Nonlinear Analysis: Real World Applications, 12 (2011), 1759-1773. https://doi.org/10.1016/j.nonrwa.2010.11.009
2. F. Li and Z. Zhao, Uniform energy decay rates for nonlinear viscoelastic wave equation with nonlocal boundary damping, Nonlinear Analysis: Real World Applications, 74 (2011), 3468-3477.
3. C. F. Carrier, On the vibration problem of elastic string, J. Appl. Math., 3 (1945), 151-165.
4. R. W. Dickey, The initial value problem for a nonlinear semi-infinite string, Proc. Roy. Soc. Edinburgh Vol. 82 (1978), 19-26. https://doi.org/10.1017/S0308210500011008
5. S. Y. Lee and C. D. Mote, Vibration control of an axially moving string by boundary control, ASME J. Dyna. Syst., Meas., Control, 118 (1996), 66-74. https://doi.org/10.1115/1.2801153
6. Y. Li, D. Aron and C. D. Rahn, Adaptive vibration isolation for axially moving strings: Theory and experiment, Automatica, 38 (1996), 379-390.
7. J. L. Lions, On some question on boundary value problem of mathematical physics, 1, in: G.M. de La Penha, L. A. Medeiros (Eds.), Contemporary Developments of Continuum Mechanics and Partial Differential Equations, North-Holland, Amsterdam, 1978.
8. M. Aassila and D. Kaya, On Local Solutions of a Mildly Degenerate Hyperbolic Equation, Journal of Mathematical Analysis and Applications, 238 (1999), 418-428. https://doi.org/10.1006/jmaa.1999.6517
9. F. Pellicano and F. Vestroni, Complex dynamics of high-speed axially moving systems, Journal of Sound and Vibration, 258 (2002), 31-44. https://doi.org/10.1006/jsvi.2002.5070
10. G. Kirchhoff, Vorlesungen uber Mechanik, Teubner, Leipzig, 1983.
11. G. Kirchhoff, Asymptotic behavior of a nonlinear Kirchhoff type equation with spring boundary conditions, Computers and Mathematics with Applications 62 (2011), 3004-3014. https://doi.org/10.1016/j.camwa.2011.08.011
12. G. Kirchhoff, Stabilization for the Kirchhoff type equation from an axially moving heterogeneous string modeling with boundary feedback control, Nonlinear Analysis: Theory, Methods and Applications 75 (2012), 3598-3617.
13. J. Limaco, H. R. Clark, and L. A. Medeiros, Vibrations of elastic string with nonhomogeneous material, Journal of Mathematical Analysis and Applications 344 (2008), 806-820. https://doi.org/10.1016/j.jmaa.2008.02.051

#### Acknowledgement

Supported by : National Research Foundation of Korea (NRF)