Blow-up of solution for a nonlinear Petrovsky type equation with memory

In this paper, we consider the nonlinear Petrovsky type equation utt+Δ2u−∫0tg(t−s)Δ2u(t,s)ds+|ut|m−2ut=|u|p−2uwith initial conditions and Dirichlet boundary conditions. Under suitable conditions of the initial data and the relaxation function, we prove that the solution with upper bounded initial energy blows up in finite time. Moreover, for the linear damping case, we show that the solution blows up in finite time by different method for nonpositive initial energy.