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

Lishan Liu (), Fenglong Sun () and Yonghong Wu ()
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Lishan Liu: Qufu Normal University
Fenglong Sun: Qufu Normal University
Yonghong Wu: Curtin University

Partial Differential Equations and Applications, 2020, vol. 1, issue 5, 1-18

Abstract: 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.

Keywords: Petrovsky equation; Memory term; Weak damping; Blow-up; Concavity method; 35L70; 65M60 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s42985-020-00031-1

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