Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
Hostname: page-component-54cdcc668b-pb9vg Total loading time: 3.608 Render date: 2021-03-09T02:02:32.649Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }
EnglishFrançais
Proceedings of the Edinburgh Mathematical Society Proceedings of the Edinburgh Mathematical Society

Article contents

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

Part of: Hyperbolic equations and systems Partial differential equations

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
Corresponding
E-mail address:
liuyangnufn@163.com
Get access
Rights & Permissions[Opens in a new window]

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 equations global existence finite time blow-up arbitrary 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 

Access options

Get access to the full version of this content by using one of the access options below.
If you should have access and can't see this content please contact technical support.

References

1.Brownjohn, J. M. W., Observations on non-linear dynamic characteristics of suspension bridges, Earthq. Eng. Struct. D 23(12) (1994), 1351–1367.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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), 535–560.3.0.CO;2-H>CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
19.Sattinger, D. H., On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal. 30(2) (1968), 148–172.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 60 *
View data table for this chart

* Views captured on Cambridge Core between 14th September 2018 - 9th March 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

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

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

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

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *