Asymptotic behavior of strongly damped nonlinear beam equations

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.