We investigate the asymptotic behavior of weak solutions to the semilinear non autonomous wave equation utt - Δu + ut|ut|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 ft 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.
On the asymptotic behavior of semilinear wave equations with degenerate dissipation and source terms
Georgiev V.;
1998-01-01
Abstract
We investigate the asymptotic behavior of weak solutions to the semilinear non autonomous wave equation utt - Δu + ut|ut|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 ft 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.File in questo prodotto:
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