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A Class of Fourth Order Damped Wave Equations with Arbitrary Positive Initial Energy

Part of: Partial differential equations Hyperbolic equations and systems

Published online by Cambridge University Press:  14 September 2018

Yang Liu ,
Jia Mu  and
Yujuan Jiao
Yang Liu
Affiliation:
College of Mathematics, Sichuan University, Chengdu 610065, People's Republic of China (liuyangnufn@163.com) College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou 730124, People's Republic of China
Jia Mu
Affiliation:
College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou 730124, People's Republic of China
Yujuan Jiao
Affiliation:
College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou 730124, People's Republic of China
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Abstract

In this paper, we study the initial boundary value problem for a class of fourth order damped wave equations with arbitrary positive initial energy. In the framework of the energy method, we further exploit the properties of the Nehari functional. Finally, the global existence and finite time blow-up of solutions are obtained.

Keywords

fourth order damped wave equationsglobal existencefinite time blow-uparbitrary positive initial energy

MSC classification

Primary: 35A01: Existence problems
Secondary: 35B44: Blow-up 35L35: Initial-boundary value problems for higher-order hyperbolic equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 165 - 178
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Brownjohn, J. M. W., Observations on non-linear dynamic characteristics of suspension bridges, Earthq. Eng. Struct. D 23(12) (1994), 1351–1367.Google Scholar
2.Cavalcanti, M. M. and Domingos Cavalcanti, V. N., Existence and asymptotic stability for evolution problem on manifolds with damping and source terms, J. Math. Anal. Appl. 291(1) (2004), 109–127.Google Scholar
3.Cavalcanti, M. M., Domingos Cavalcanti, V. N. and Martinez, P., Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations 203(1) (2004), 119–158.Google Scholar
4.Esquivel-Avila, J. A., A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations, Nonlinear Anal. 52(4) (2003), 1111–1127.Google Scholar
5.Ferrero, A. and Gazzola, F., A partially hinged rectangular plate as a model for suspension bridges, Discrete Contin. Dyn. Syst. Ser. A 35(12) (2015), 5879–5908.Google Scholar
6.Gazzola, F., Nonlinearity in oscillating bridges, Electron. J. Differential Equations 211(3) (2013), 1642–1654.Google Scholar
7.Gazzola, F. and Squassina, M., Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. Henri Poincaré Probab. Stat. 23 (2006), 185–207.Google Scholar
8.Ikehata, R., Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal. 27(10) (1996), 1165–1175.Google Scholar
9.Lacarbonara, W., Nonlinear structural mechanics (Springer, 2013).Google Scholar
10.Levine, H. A., Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu tt = −Au+𝒻(u), Trans. Amer. Math. Soc. 192 (1974), 1–21.Google Scholar
11.Levine, H. A., Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1974), 138–146.Google Scholar
12.Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires (Dunod, 1969).Google Scholar
13.Liu, Y. and Zhao, J., On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal. 64(12) (2006), 2665–2687.Google Scholar
14.Liu, Y., Xu, R. and Yu, T., Global existence, nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations, Nonlinear Anal. 68(11) (2008), 3332–3348.Google Scholar
15.Nakao, M. and Ono, K., Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z. 214 (1993), 325–342.Google Scholar
16.Ono, K., Global existence, asymptotic behaviour, and global non-existence of solutions for damped non-linear wave equations of Kirchhoff type in the whole space, Math. Methods Appl. Sci. 23(6) (2000), 535–560.Google Scholar
17.Payne, L. E. and Sattinger, D. H., Sadle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22(3–4) (1975), 273–303.Google Scholar
18.Plaut, R. H. and Davis, F. M., Sudden lateral asymmetry and torsional oscillations of section models of suspension bridges, J. Sound Vib. 307(3) (2007), 894–905.Google Scholar
19.Sattinger, D. H., On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal. 30(2) (1968), 148–172.Google Scholar
20.Tsutsumi, M., On solutions of semilinear differential equations in a Hilbert space, Math. Japan. 17 (1972), 173–193.Google Scholar
21.Vitillaro, E., Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149(2) (1999), 155–182.Google Scholar
22.Wang, Y., Finite time blow-up and global solutions for fourth order damped wave equations, J. Math. Anal. Appl. 418(2) (2014), 713–733.Google Scholar
23.Xu, R., Asymptotic behavior and blow up of solutions for semilinear parabolic equations at critical energy level, Math. Comput. Simulation 80(4) (2009), 808–813.Google Scholar