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Blow-Up Phenomena for a Class of Extensible Beam Equations

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Abstract

In this paper, we investigate the initial boundary value problem of the following nonlinear extensible beam equation with nonlinear damping term

$$\begin{aligned} u_{t t}+\Delta ^2 u-M\left( \Vert \nabla u\Vert ^2\right) \Delta u-\Delta u_t+\left| u_t\right| ^{r-1} u_t=|u|^{p-1} u \end{aligned}$$

which was considered by Yang et al. (Adv Nonlinear Stud 22:436–468, 2022). We consider the problem with the nonlinear damping and establish the finite time blow-up of the solution for the initial data at arbitrary high energy level, including the estimate lower and upper bounds of the blow-up time. The result provides some affirmative answer to the open problems given in Yang et al. (2022).

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Acknowledgements

The authors would like to thank the referees for the careful reading of this paper and for the valuable suggestions to improve the presentation and the style of the paper. This paper is supported by the Innovative Funds Plan of Henan University of Technology (No. 2020ZKCJ09) and National Natural Science Foundation of China (No. 11801145).

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Authors and Affiliations

  1. School of Science, Henan University of Technology, Zhengzhou, 450001, China

    Gongwei Liu, Mengyun Yin & Suxia Xia

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  1. Gongwei Liu
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  2. Mengyun Yin
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  3. Suxia Xia
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Correspondence to Gongwei Liu.

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Liu, G., Yin, M. & Xia, S. Blow-Up Phenomena for a Class of Extensible Beam Equations. Mediterr. J. Math. 20, 266 (2023). https://doi.org/10.1007/s00009-023-02469-0

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