×

Existence of strong solutions and global attractors for the suspension bridge equations. (English) Zbl 1125.34019

The authors treat a nonlinear ordinary second-order differential equation by establishing some a priori bounds and then proving the convergence of the Faedo-Galerkin approximation method.
Unfortunately, the equation does not describe the deflection of a suspension bridge, whose equation is completely different. In addition, the language is poor and full of mistakes, and the references ignore the huge, classical, literature of the problem.

MSC:

34B45 Boundary value problems on graphs and networks for ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
35B41 Attractors
74H45 Vibrations in dynamical problems in solid mechanics
34D05 Asymptotic properties of solutions to ordinary differential equations
35Q72 Other PDE from mechanics (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lazer, A. C.; McKenna, P. J., Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32, 4, 537-578 (1990) · Zbl 0725.73057
[2] Humphreys, L. D., Numerical mountain pass solutions of a suspension bridge equation, Nonlinear Anal.(TMA), 28, 11, 1811-1826 (1997) · Zbl 0877.35126
[3] Lazer, A. C.; McKenna, P. J., Large scale oscillatory behavior in loaded asymmetric systems, Nonlinear. Anal. I. H.P., 4, 243-274 (1987) · Zbl 0633.34037
[4] McKenna, P. J.; Walter, W., Nonlinear oscillation in a suspension bridge, Non. Anal., 39, 731-743 (2000)
[5] Choi, Q. H.; Jung, T., A nonlinear suspension bridge equation with nonconstant load, Nonlinear. Anal., 35, 649-668 (1999) · Zbl 0934.74031
[6] Y. An, On the Suspension Bridge Equations and the Relevant Problems, Doctoral Thesis, 2001; Y. An, On the Suspension Bridge Equations and the Relevant Problems, Doctoral Thesis, 2001
[7] Ma, Q. F.; Wang, S. H.; Zhong, C. K., Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J., 51, 6, 1541-1559 (2002) · Zbl 1028.37047
[8] Temam, R., Infinite Dimensional Dynamical System in Mechanics and Physics (1997), Springer-Verlag · Zbl 0871.35001
[9] Ma, Q. Z.; Zhong, C. K., Existence of strong global attractors for hyperbolic equation with linear memory, Appl. Math. Comput., 157, 745-758 (2004) · Zbl 1068.45018
[10] Ball, J. M., Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42, 61-90 (1973) · Zbl 0254.73042
[11] Ma, Q. Z.; Zhong, C. K., Existence of global attractors for the suspension bridge equations, J. Sichuan Univ., 43, 2 (2006) · Zbl 1111.35089
[12] Ma, Q. Z.; Zhong, C. K., Global attractors of strong solutions for nonclassical diffusion equation, J. Lanzhou Univ., 40, 5, 7-9 (2004) · Zbl 1090.35145
[13] Zhong, C. K.; Yang, M. H.; Sun, C. Y., The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations., 223, 2, 367-399 (2006) · Zbl 1101.35022
[14] Hale, J. K., Asymptotic Behavior of Dissipative Systems (1988), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0642.58013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.