STABILIZATION FOR THE VISCOELASTIC KIRCHHOFF TYPE EQUATION WITH A NONLINEAR SOURCE

Title & Authors
STABILIZATION FOR THE VISCOELASTIC KIRCHHOFF TYPE EQUATION WITH A NONLINEAR SOURCE
Kim, Daewook;

Abstract
In this paper, we study the viscoelastic Kirchhoff type equation with a nonlinear source $\small{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})}$$\small{]}$d{\tau}+{\mid}u{\mid}^{\gamma}u
Keywords
viscoelastic Kirchhoff type equation;energy decay rate;energy functional;smallness condition;
Language
English
Cited by
1.
EXPONENTIAL STABILITY FOR THE GENERALIZED KIRCHHOFF TYPE EQUATION IN THE PRESENCE OF PAST AND FINITE HISTORY,;

East Asian mathematical journal, 2016. vol.32. 5, pp.659-675
2.
ASYMPTIOTIC BEHAVIOR FOR THE VISCOELASTIC KIRCHHOFF TYPE EQUATION WITH AN INTERNAL TIME-VARYING DELAY TERM,;

East Asian mathematical journal, 2016. vol.32. 3, pp.399-412
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.

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.

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.

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.

9.
F. Pellicano and F. Vestroni, Complex dynamics of high-speed axially moving systems, Journal of Sound and Vibration, 258 (2002), 31-44.

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.

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.