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

Author

Listed:
  • Lishan Liu

    (Qufu Normal University
    Curtin University)

  • Fenglong Sun

    (Qufu Normal University)

  • Yonghong Wu

    (Curtin University)

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 $$E(0)\le M$$ E ( 0 ) ≤ 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 $$E(0)>M$$ 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.

Suggested Citation

  • Lishan Liu & Fenglong Sun & Yonghong Wu, 2020. "Finite time blow-up for a nonlinear viscoelastic Petrovsky equation with high initial energy," Partial Differential Equations and Applications, Springer, vol. 1(5), pages 1-18, October.
    • Handle: RePEc:spr:pardea:v:1:y:2020:i:5:d:10.1007_s42985-020-00031-1
    DOI: 10.1007/s42985-020-00031-1
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    References listed on IDEAS

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    1. Li, Fushan & Gao, Qingyong, 2016. "Blow-up of solution for a nonlinear Petrovsky type equation with memory," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 383-392.
    2. Zhou, Jun, 2015. "Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 807-818.
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