On the asymptotic behavior of semilinear wave equations with degenerate dissipation and source terms

We investigate the asymptotic behavior of weak solutions to the semilinear non autonomous wave equation u_tt - Δu + u_t|u_t|^p-1 = V(t)u|u|^p-1 + f(.,t), where V(t) is a positive time dependent potential satisfying V(t) = O((1 + t)^-λ) as t → +∞ and f_t decays to 0 as t → +∞. We show that for 0 ≤ λ ≤ p there are initial values such that the energy norm of the corresponding solutions grows at least polynomially as t → +∞, while if λ > p the energy norm remains uniformly bounded for any choice of initial values; moreover, in certain cases there is an absorbing ball for the orbits.