Rocky Mountain Journal of Mathematics

Asymptotic behavior of strongly damped nonlinear beam equations

Ahmed Y. Abdallah

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Abstract

For $\xi \in [1/2,3/4] $, the existence of the global attractor for the evolutionary equation corresponding to the following strongly damped nonlinear beam equation $(1+\beta A^{1/2}) u_{tt}+\delta A^{1/2}u_{t}+\alpha Au+g(\Vert u \Vert _{\xi -1/4}^{2}) A^{1/2}u=f$, $t>0$, has been studied in $\mathcal {D}_{H}(A^{\xi }) \times \mathcal {D}_{H}(A^{\xi -1/4})$. Such an equation is related to a nonlinear beam equation as well as Timoshenko's equation.

The main difficulty of our work comes from the terms $\beta A^{1/2}u_{tt}$ and $g(\Vert u \Vert _{\xi -1/4}^{2}) A^{1/2}u$, representing the rotational inertia of the beam and the tension within the beam due to its extensibility, respectively. We overcome the difficulty of introducing the solution, bounded absorbing set, and $\kappa $-contracting property by carefully using the fractional power theory and suitable time-uniform a priori estimates.

Article information

Source
Rocky Mountain J. Math., Volume 46, Number 4 (2016), 1071-1088.

Dates
First available in Project Euclid: 19 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1476884576

Digital Object Identifier
doi:10.1216/RMJ-2016-46-4-1071

Mathematical Reviews number (MathSciNet)
MR3563175

Zentralblatt MATH identifier
1362.35166

Subjects
Primary: 35B41: Attractors 35B45: A priori estimates

Keywords
Beam equation bounded dissipative $\kappa $-contracting global attractor

Citation

Abdallah, Ahmed Y. Asymptotic behavior of strongly damped nonlinear beam equations. Rocky Mountain J. Math. 46 (2016), no. 4, 1071--1088. doi:10.1216/RMJ-2016-46-4-1071. https://projecteuclid.org/euclid.rmjm/1476884576


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