The time-dependent attractor for beam equation with rotational inertia and structural damping

Xuan Wang; Wei Wang; Xuan Wang; Wei Wang

doi:10.3934/math.2025682

AIMS Mathematics

2025, Volume 10, Issue 7: 15206-15230. doi: 10.3934/math.2025682

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Research article Topical Sections

The time-dependent attractor for beam equation with rotational inertia and structural damping

Xuan Wang ,
Wei Wang ^,

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, China

Received: 13 March 2025 Revised: 30 May 2025 Accepted: 04 June 2025 Published: 01 July 2025
MSC : 35B40, 35B41, 37L15, 37L30

In this paper, we discussed the asymptotic behavior of the solutions to the beam equation with rotational inertia and structural damping:
$ \varepsilon(t)(1+(-\Delta) ^{\alpha})\partial^{2}_tu+\Delta^2 u+\gamma(-\Delta)^{\theta}\partial_tu+f(u) = g(x), $
where $ \varepsilon(t) $ was a decreasing bounded function. We found a more optimized subcritical exponent $ p^{*} = \frac{N+2\theta}{N-4} $ depending on $ \theta. $ Additionally, we showed that when the growth exponent $ p $ of nonlinear terms $ f(u) $ was within the range $ 1\leqslant p < p^{*}, $ the well-posedness and regularity were established. Finally, within the theory of process on time-dependent spaces, we investigated the existence of the time-dependent attractor by using the contraction function method and more detailed estimates in the time-dependent space $ \mathcal{H}_{t}^{\alpha}. $ The results refined and extended the model and the work in literature [Longtime behavior for an extensible beam equation with rotational inertia and structural nonlinear damping, J. Math. Anal. Appl., 496 (2021), 124785.] from general energy space to time-dependent space in some sense.
- rotational inertia,
- regularity,
- time-dependent global attractors,
- structural damping
Citation: Xuan Wang, Wei Wang. The time-dependent attractor for beam equation with rotational inertia and structural damping[J]. AIMS Mathematics, 2025, 10(7): 15206-15230. doi: 10.3934/math.2025682

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Abstract

In this paper, we discussed the asymptotic behavior of the solutions to the beam equation with rotational inertia and structural damping:

$ \varepsilon(t)(1+(-\Delta) ^{\alpha})\partial^{2}_tu+\Delta^2 u+\gamma(-\Delta)^{\theta}\partial_tu+f(u) = g(x), $

where $ \varepsilon(t) $ was a decreasing bounded function. We found a more optimized subcritical exponent $ p^{*} = \frac{N+2\theta}{N-4} $ depending on $ \theta. $ Additionally, we showed that when the growth exponent $ p $ of nonlinear terms $ f(u) $ was within the range $ 1\leqslant p < p^{*}, $ the well-posedness and regularity were established. Finally, within the theory of process on time-dependent spaces, we investigated the existence of the time-dependent attractor by using the contraction function method and more detailed estimates in the time-dependent space $ \mathcal{H}_{t}^{\alpha}. $ The results refined and extended the model and the work in literature [Longtime behavior for an extensible beam equation with rotational inertia and structural nonlinear damping, J. Math. Anal. Appl., 496 (2021), 124785.] from general energy space to time-dependent space in some sense.

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