ASYMPTIOTIC BEHAVIOR FOR THE VISCOELASTIC KIRCHHOFF TYPE EQUATION WITH AN INTERNAL TIME-VARYING DELAY TERM

Title & Authors
ASYMPTIOTIC BEHAVIOR FOR THE VISCOELASTIC KIRCHHOFF TYPE EQUATION WITH AN INTERNAL TIME-VARYING DELAY TERM
Kim, Daewook;

Abstract
In this paper, we study the viscoelastic Kirchhoff type equation with the following nonlinear source and time-varying delay $\small{u_{tt}-M(x,t,{\parallel}{\nabla}u(t){\parallel}^2){\Delta}u+{\int_{0}^{t}}h(t-{\tau})div[a(x){\nabla}u({\tau})}$$\small{]}$d{\tau}\\+{\parallel}u{\parallel}^{\gamma}u+{\mu}_1u_t(x,t)+{\mu}_2u_t(x,t-s(t))
Keywords
viscoelastic Kirchhoff type equation;internal time-varying delay;energy decay rate;energy functional;smallness condition;
Language
English
Cited by
References
1.
S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations 41 (2011), 1-20.

2.
W. Liu, General decay rate estimate for the energy of a weak viscoelastic equation with an internal time-varying delay term, Taiwanese journal of mathematics 17 (2013), 2101-2115.

3.
W. Liu, Stabilization for the viscoelastic Kirchhoff type equation with nonlinear source, East Asian Math. J. 32 (2016), 117-128.

4.
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.

5.
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.

6.
C. F. Carrier, On the vibration problem of elastic string, J. Appl. Math., 3 (1945), 151-165.

7.
R. W. Dickey, The initial value problem for a nonlinear semi-infinite string, Proc. Roy. Soc. Edinburgh Vol. 82 (1978), 19-26.

8.
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.

9.
Y. Li, D. Aron and C. D. Rahn, Adaptive vibration isolation for axially moving strings: Theory and experiment, Automatica, 38 (1996), 379-390.

10.
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.

11.
M. Aassila and D. Kaya, On Local Solutions of a Mildly Degenerate Hyperbolic Equation, Journal of Mathematical Analysis and Applications, 238 (1999), 418-428.

12.
F. Pellicano and F. Vestroni, Complex dynamics of high-speed axially moving systems, Journal of Sound and Vibration, 258 (2002), 31-44.

13.
G. Kirchhoff, Vorlesungen uber Mechanik, Teubner, Leipzig, 1983.

14.
G. Kirchhoff, Asymptotic behavior of a nonlinear Kirchhoff type equation with spring boundary conditions, Computers and Mathematics with Applications 62 (2011), 3004-3014.

15.
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.

16.
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.