On the initial–boundary problem for fourth order wave equations with damping, strain and source terms

In this paper we study the global existence and blow-up of solutions to the fourth order equations u t t + u t + Δ 2 u − α Δ u − ∑ i = 1 n ∂ ∂ x i ( θ i ( u x i ) ) = f ( u ) , x ∈ Ω , t > 0 , where α ≥ 0 . appropriate assumptions on the initial data and parameters in the above equation we establish two results on blow-up of solutions with arbitrary initial energy, − ∞ < E ( 0 ) < + ∞ . Also, by using a potential well we show the global existence of solutions for the fourth order wave equation with some θ i ( s ) and f ( s ) . Especially, it is proved that the energy decays exponentially as t → ∞ .