## Rocky Mountain Journal of Mathematics

### Asymptotic behavior of strongly damped nonlinear beam equations

Ahmed Y. Abdallah

#### 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

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

#### 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

#### References

• M. Coti Zelati, Global and exponential attractors for the singularly perturbed extensible beam, Discr. Cont. Dynam. Syst. 25 (2009), 1041–1060.
• H. Gao, Global dynamics of a nonlinear beam equation with strong structural damping, J. Partial Diff. Eqs. 12 (1999), 324–336.
• J.M. Ghidaglia and A. Marzocchi, Long time behavior of strongly damped wave equations, global attractors and their dimensions, SIAM J. Math. Anal. 22 (1991), 879–895.
• C. Giorgi, M.G. Naso, V. Pata and M. Potomkin, Global attractors for the extensible thermoelastic beam system, J. Diff. Eqs. 9 (2009), 3496–3517.
• C. Giorgi, V. Pata and E. Vuk, On the extensible viscoelastic beam, Nonlinearity 21 (2008), 713–733.
• M. Grasselli, V. Pata and G. Prouse, Longtime behavior of a viscoelastic Timoshenko beam, Discr. Cont. Dynam. Syst. 10 (2004), 337–348.
• J. Hale, Asymptotic behavior of dissipative systems, Math. Surv. Mono. 25, American Mathematical Society, Providence, RI, 1988.
• T.F. Ma and V. Narciso, Global attractor for a model of extensible beam with nonlinear damping and source terms, Nonlin. Anal. 73 (2010), 3402–3412.
• A. Pazy, Semigroups of linear operators and applications to partial differential equations, Appl. Math. Sci. 44, Springer-Verlag, New York, 1983.
• G. Perla Menzala and E. Zuazua, Timoshenko's beam equation as limit of a nonlinear one-dimensional von Kármán system, Proc. Roy. Soc. Edinb. 130 (2000), 855–875.
• R. Racke and C. Shang, Global attractors for nonlinear beam equations, Proc. Roy. Soc. Edinb. 142 (2012), 1087–1107.
• G.R. Sell and Y. You, Dynamics of evolutionary equations, Appl. Math. Sci. 143, Springer-Verlag, New York, 2002.
• D. Sevcovic, Existence and limiting behavior for damped nonlinear evolution equations with nonlocal terms, Comm. Math. Univ. Carolin. 31 (1990), 283–293.
• M. Tucsnak, Semi-internal stabilization for a nonlinear Bernoulli-Euler equation, Math. Meth. Appl. Sci. 19 (1996), 897–907.