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

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.