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

摘要

In this paper we study the global existence and blow-up of solutions to the fourth order equations u_(tt) + u_t + Δ~2u - αΔu -n/∑/i=1 ?/?x_i(θ_i(u_(x_i))) = f (u), x ∈ Ω, t > 0, where α ≥ 0. Under 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 → ∞.