Open Access
2016 Asymptotic behavior of strongly damped nonlinear beam equations
Ahmed Y. Abdallah
Rocky Mountain J. Math. 46(4): 1071-1088 (2016). DOI: 10.1216/RMJ-2016-46-4-1071


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.


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Ahmed Y. Abdallah. "Asymptotic behavior of strongly damped nonlinear beam equations." Rocky Mountain J. Math. 46 (4) 1071 - 1088, 2016.


Published: 2016
First available in Project Euclid: 19 October 2016

zbMATH: 1362.35166
MathSciNet: MR3563175
Digital Object Identifier: 10.1216/RMJ-2016-46-4-1071

Primary: 35B41 , 35B45

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

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

Vol.46 • No. 4 • 2016
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