Advanced search
Publication Cover
Applicable Analysis
An International Journal
Volume 95, 2016 - Issue 5
Submit an article Journal homepage
378
Views
1
CrossRef citations to date
0
Altmetric
Articles

Decay estimates for the Cauchy problem for the damped extensible beam equation

Reinhard RackeDepartment of Mathematics and Statistics, University of Konstanz, Konstanz, 78457Germany.View further author information
&
Shuji YoshikawaDepartment of Engineering for Production and Environment, Graduate School of Science and Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama, Ehime, 790-8577Japan.Correspondence[email protected]
View further author information
Pages 1118-1136 | Received 13 Dec 2014, Accepted 16 May 2015, Published online: 08 Jun 2015

References

  • Woinowsky-Krieger S. The effect of axial force on the vibration of hinged bars. J. Appl. Mech. 1950;17:35–36.
  • Eisley JG. Nonlinear vibration of beams and rectangular plates. ZAMP. 1964;15:167–175.
  • Ball JM. Stability theory for an extensible beam. J. Differ. Equ. 1973;14:399–418.
  • Dickey RW. Free vibrations and dynamic buckling of the extensible beam. J. Math. Anal. Appl. 1970;29:443–454.
  • Ball JM. Initial-boundary value problems for an extensible beam. J. Math. Anal. Appl. 1973;42:61–90.
  • Nakao M. A difference inequality and its application to nonlinear evolution equations. J. Math. Soc. Japan. 1978;30:747–762.
  • Kawashima S, Nakao M, Ono K. On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term. J. Math. Soc. Japan. 1995;47:617–653.
  • Ono K. Asymptotic behavior of solutions for damped Kirchhoff equations in unbounded domain. Comm. Appl. Anal. 1999;3:101–114.
  • Brito EH. Decay estimates for the generalized damped extensible string and beam equations. Nonlinear Anal. Theory Methods Appl. 1984;8:1489–1496.
  • Biler P. Remark on the decay for damped string and beam equations. Nonlinear Anal. Theory Methods Appl. 1986;10:339–342.
  • Takeda H, Yoshikawa S. On the initial value problem of the semilinear beam equation with weak damping I: smoothing effect. J. Math. Anal. Appl. 2013;401:244–258.
  • Takeda H, Yoshikawa S. On the initial value problem of the semilinear beam equation with weak damping II: asymptotic profiles. J. Differ. Equ. 2012;253:3061–3080.
  • Liu K, Liu Z. Exponential decay of the energy of the Euler--Bernoulli beam with locally distributed Kelvin-Voigt damping. SIAM J. Control Opt. 1998;36:1086–1098.
  • Alves M, Muñoz Rivera J, Sepúlveda M, et al. The lack of exponential stability in certain transmission problems with localized Kelvin--Voigt dissipation. SIAM J. Appl. Math. 2014;74:345–365.
  • Hosono T, Ogawa T. Large time behavior and Lp--Lq estimate of solutions of 2-dimensional nonlinear damped wave equations. J. Differ. Equ. 2004;203:82–118.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Order Reprints Request Corporate Permissions

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

Request Academic Permissions

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.

Related research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.