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On periodic solutions of a special type of the beam equation. (English) Zbl 0665.35015
The paper deals with the existence of time-periodic solutions to the following system of equations: \[ u_{tt}-cv_{tt}+u_{xxxx}+\alpha u_ t-\beta (\int^{\pi}_{0}u^ 2_ x(s,.)ds)u_{xx}=f^{(1)}, \]
\[ -cu_{tt}+\gamma v_{tt}+\delta v_{xxxx}+{\tilde \alpha}v_ t- {\tilde \beta}v_{xx}=f^{(2)}, \] where all constants are assumed to be positive and \(\gamma -c^ 2>0.\)
The main part of the article is an existence theorem in the case of time- periodic outer forces, which is proved by means of an a priori estimate. The system is reformulated in the weak sense, then truncated according to the Fourier method and transformed into a system of ordinary differential equations. Making use of Brouwer’s theorem, time-periodic solutions of the truncated system are found.
Finally it is proved that these truncated solutions tend to solutions of the original system, that remain time-periodic.
Reviewer: J.Řeháček
MSC:
35G20 Nonlinear higher-order PDEs
35B10 Periodic solutions to PDEs
74H45 Vibrations in dynamical problems in solid mechanics
35B45 A priori estimates in context of PDEs
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References:
[1] N. G. de Andrade: On a nonlinear Systems of Partial Differential Equations. Journal of Math. Anal. and Appl. 91 (1983), 119-130. · Zbl 0523.35026
[2] J. M. Ball: Initial Boundary Value Problems for an Extensible Beam. Journal of Math. Anal. and Appl. 42(1973), 61-90. · Zbl 0254.73042
[3] J. Kurzweil: Ordinary Differential Equations. Elsevier, Amsterdam, 1986. · Zbl 0667.34002
[4] S. P. Timošenko D. H. Young W. Weaver: Vibrations Problems in Engineering. New York 1974.
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