Lower and upper bounds for the blow-up time to a viscoelastic Petrovsky wave equation with variable sources and memory term

Under Dirichlet boundary conditions, we consider here a new type of viscoelastic Petrovsky wave equation involving variable sources and memory term utt+Δ2u−∫0tg(t−s)Δ2u(x,s)ds+|ut|m(.)−2ut=|u|p(.)−2u.We discuss the blow-up in finite time with arbitrary positive initial energy and suitable large initial values if p(.) and the relaxation function g satisfies some conditions. Employing a different method for higher bounded positive initial energy, not only finite time blow-up for solutions proved but also the lower and upper bounds for blowing up time are gotten.