Boundary Stabilization of a More General Kirchhoff-Type Beam Equation ()
Abstract
Simultaneously, considering the viscous effect of material, damping of medium, geometrical nonlinearity, physical nonlinearity, we set up a more general equation of beam subjected to axial force and external load. We prove the existence and uniqueness of global solutions under non-linear boundary conditions which the model is added one damping mechanism at l end. What is more, we also prove the exponential decay property of the energy of above mentioned system.
Share and Cite:
Zhang, J. and Wang, D. (2012) Boundary Stabilization of a More General Kirchhoff-Type Beam Equation.
International Journal of Modern Nonlinear Theory and Application,
1, 97-101. doi:
10.4236/ijmnta.2012.13014.
Conflicts of Interest
The authors declare no conflicts of interest.
References
|
[1]
|
S. Woinowsky-Krieger, “The Effect of Axial Force on the Vibration of Hinged Bars,” Journal of applied Mechanics, Vol. 17, 1950, pp. 35-36.
|
|
[2]
|
J. M. Ball, “Initial-Boundary Value Problems for an Extensible Beam,” Journal of Mathematical Analysis and Applications, Vol. 42, No. 1, 1973, pp. 61-90.
doi:10.1016/0022-247X(73)90121-2
|
|
[3]
|
L. A. Mederios, “On a New Class of Nonlinear Wave Equations,” Journal of Mathematical Analysis and Applications, Vol. 69, No. 1, 1979, pp. 252-262.
|
|
[4]
|
M. Tucsnak, “Semi-Internal Stabilization for a Nonlinear Euler-Bernoulli Equation,” Mathematical Methods in the Applied Sciences, Vol. 19, No. 11, 1996, pp. 897-907.
doi:10.1002/(SICI)1099-1476(19960725)19:11<897::AID-MMA801>3.0.CO;2-#
|
|
[5]
|
S. Kouemou Patcheu, “On a Global Solution and Asymptotic Behavior for the Generalized Damped Extensible Beam Equation,” Journal of Differential Equations, Vol. 135, No. 2, 1997, pp. 299-314.
doi:10.1006/jdeq.1996.3231
|
|
[6]
|
F. M. To, “Boundary Stabilization for a Non-Linear Beam on Elastic Bearings,” Mathematical Methods in the Applied Sciences, Vol. 24, No. 8, 2001, pp. 583-594.
doi:10.1002/mma.230
|
|
[7]
|
J. M. Ball, “Stability Theory for an Extensible Beam,” Journal of Differential Equations, Vol. 14, No. 3, 1973, pp. 61-90. doi:10.1016/0022-0396(73)90056-9
|