Global existence and asymptotic behavior for a mildly degenerate Kirchhoff wave equation with boundary damping
Author:
Qiong Zhang
Journal:
Quart. Appl. Math. 70 (2012), 253-267
MSC (2010):
Primary 35L80, 35B40
DOI:
https://doi.org/10.1090/S0033-569X-2012-01281-0
Published electronically:
February 3, 2012
MathSciNet review:
2953102
Full-text PDF Free Access
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Additional Information
Abstract: In this article a degenerate nonlinear dissipative wave equation of Kirchhoff type with nonlinear boundary damping is considered. We prove the existence, uniqueness and regularity of the global solution of the system when the initial data are small enough and the geometry of the domain satisfies suitable assumptions. We also obtain the polynomial decay property of the global solution.
References
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References
- A. Arosio and S. Garavaldi, On the mildly degenerate Kirchhoff string, Mathematical Methods in Applied Science, 14 (1991), 177-195. MR 1099324 (92c:35072)
- V. Barbu, I. Lasiecka, and M. A. Rammaha, On nonlinear wave equations with degenerate damping and source terms, Transactions of the American Mathematical Society, 357 (2005), 2571-2611. MR 2139519 (2006a:35203)
- E. H. de Brito, Decay estimates for the generalized damped extensible string and beam equation, Nonlinear Analysis, 8 (1984), 1489-1496. MR 769410 (86k:34056)
- G.F. Carrier, On the non-linear vibration problem of the elastic string, Quarterly of Applied Mathematics, 3 (1945), 157-165. MR 0012351 (7:13h)
- M. M. Cavalcanti, U V. N. Domingos Cavalcanti, J. S. Prates Filho, and J. A. Soriano, Existence and exponential decay for a Kirchhoff-Carrier model with viscosity, Journal of Mathematical Analysis and Applications, 226 (1998), 40-60. MR 1646453 (99g:35084)
- P. D’Ancona and S. Spagnolo, Nonlinear perturbations of the Kirchhoff equation, Communications on Pure and Applied Mathematics, 47 (1994), 1005-1029. MR 1283880 (95d:35105)
- M. Ghisi, Global solutions for dissipative Kirchhoff strings with non-Lipschitz nonlinear term, Journal of Differential Equations, 230 (2006), 128-139. MR 2270549 (2008a:35196)
- R. Izaguirrea, R. Fuentesb and M. M. Miranda, Existence of local solutions of the Kirchhoff-Carrier equation in Banach spaces, Nonlinear Analysis, 68 (2008), 3565-3580. MR 2401368 (2009d:35233)
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Additional Information
Qiong Zhang
Affiliation:
Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China
Email:
zhangqiong@bit.edu.cn, qiongzhg@gmail.com
Keywords:
Degenerate equation,
Kirchhoff equation,
boundary dissipation,
global existence,
polynomial decay.
Received by editor(s):
May 4, 2010
Published electronically:
February 3, 2012
Additional Notes:
The work is supported by the NSF of China (60504001, 60974033), SRF for ROCS, SEM, China (20080732041).
Article copyright:
© Copyright 2012
Brown University
The copyright for this article reverts to public domain 28 years after publication.