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Finite time blow-up for a nonlinear viscoelastic Petrovsky equation with high initial energy

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Abstract

In this paper, we study the initial boundary value problem for a Petrovsky type equation with a memory term, a linear weak damping and superlinear source. Finite time blow-up results have been obtained for the case in which the initial energy \(E(0)\le M\), where M is a positive constant. By utilizing Levine’s classical concavity method, we give a new blow-up criterion which includes the case of \(E(0)>M\) and derive an explicit upper bound for the blow-up time. By using the Fountain Theorem, we show that the problem with arbitrary positive initial energy always admits weak solutions blowing up in finite time.

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Acknowledgements

The authors were supported financially by the National Natural Science Foundation of China (11871302) and Natural Science Foundation of Shandong Province of China (ZR2019BA029, ZR2017MA036). The support from the Australian Research council for the research is also acknowledged.

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Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, Shandong, People’s Republic of China

    Lishan Liu & Fenglong Sun

  2. Department of Mathematics and Statistics, Curtin University, Perth, WA, 6845, Australia

    Lishan Liu & Yonghong Wu

Authors
  1. Lishan Liu
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  2. Fenglong Sun
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  3. Yonghong Wu
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All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Correspondence to Lishan Liu or Fenglong Sun.

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Additional information

This article is part of the topical collection dedicated to Prof. Dajun Guo for his 85th birthday, edited by Yihong Du, Zhaoli Liu, Xingbin Pan, and Zhitao Zhang.

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Liu, L., Sun, F. & Wu, Y. Finite time blow-up for a nonlinear viscoelastic Petrovsky equation with high initial energy. SN Partial Differ. Equ. Appl. 1, 31 (2020). https://doi.org/10.1007/s42985-020-00031-1

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