Opuscula Math. 42, no. 2 (2022), 239-255
https://doi.org/10.7494/OpMath.2022.42.2.239
Opuscula Mathematica
Exponential decay of solutions to a class of fourth-order nonlinear hyperbolic equations modeling the oscillations of suspension bridges
Abstract. This paper is concerned with a class of fourth-order nonlinear hyperbolic equations subject to free boundary conditions that can be used to describe the nonlinear dynamics of suspension bridges.
Keywords: fourth-order nonlinear hyperbolic equations, weak solutions, exponential decay, a family of potential wells.
Mathematics Subject Classification: 35L35, 35D30, 35B40.
- R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
- S. Baraket, V.D. Radulescu, Combined effects of concave-convex nonlinearities in a fourth-order problem with variable exponent, Adv. Nonlinear Stud. 16 (2016), 409-419.
- J.M.W. Brownjohn, Observations on non-linear dynamic characteristics of suspension bridges, Earthquake Engng. Struct. Dyn. 23 (1994), 1351-1367.
- A. Ferrero, F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges, Discrete Contin. Dyn. Syst. 35 (2015), 5879-5908.
- F. Gazzola, Nonlinearity in oscillating bridges, Electron. J. Differential Equations 211 (2013), 1642-1654.
- W. Lacarbonara, Nonlinear Structural Mechanics: Theory, Dynamical Phenomena and Modeling,
- A.C. Lazer, P.J. McKenna, Large scale oscillatory behaviour in loaded asymmetric systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 243-274.
- W. Lian, J. Wang, R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations 269 (2020), 4914-4959.
- W. Lian, R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal. 9 (2020), 612-632.
- Y. Liu, J. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal. 64 (2006), 2665-2687.
- P.J. McKenna, W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Ration. Mech. Anal. 98 (1987), 167-177.
- P.J. McKenna, W. Walter, Travelling waves in a suspension bridge, SIAM J. Appl. Math. 50 (1990), 703-715.
- A. Mohammed, V.D. Radulescu, A. Vitolo, Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness, Adv. Nonlinear Stud. 9 (2020), 39-64.
- L.E. Payne, D.H. Sattinger, Sadle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), 273-303.
- R.H. Plaut, F.M. Davis, Sudden lateral asymmetry and torsional oscillations of section models of suspension bridges, J. Sound Vib. 307 (2007), 894-905.
- V. Radulescu, Nonlinear Analysis - Theory and Methods, Springer, New York, 2019.
- D.H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30 (1968), 148-172.
- E. Ventsel, T. Krauthammer, Thin Plates and Shells: Theory, Analysis, and Applications, Marcel Dekker, New York, 2001.
- Y. Wang, Finite time blow-up and global solutions for fourth order damped wave equations, J. Math. Anal. Appl. 418 (2014), 713-733.
- R. Xu, W. Lian, Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math. 63 (2020), 321-356.
- R. Xu, J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal. 264 (2013), 2732-2763.
- R. Xu, X. Wang, Y. Yang, S. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys. 59 (2018), 061503.
- R. Xu, M. Zhang, S. Chen, Y. Yang, J. Shen, The initial-boundary value problems for a class of six order nonlinear wave equation, Discrete Contin. Dyn. Syst. 37 (2017), 5631-5649.
- Yang Liu
- Northwest Minzu University, College of Mathematics and Computer Science, No. 1, Northwest New Village, Lanzhou 730030, P.R. China
- Sichuan University, College of Mathematics, No. 24, South Section 1, Yihuan Road, Chengdu 610065, P.R. China
- Chao Yang (corresponding author)
- Harbin Engineering University, College of Mathematical Sciences, No. 145, Nantong Street, Harbin 150001, P.R. China
- Communicated by Runzhang Xu.
- Received: 2021-06-11.
- Revised: 2021-10-14.
- Accepted: 2021-10-15.
- Published online: 2022-02-25.
