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.
| Original language | English |
|---|---|
| Pages (from-to) | 53-68 |
| Number of pages | 16 |
| Journal | Nonlinear Differential Equations and Applications |
| Volume | 5 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - Jan 1 1998 |
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All Science Journal Classification (ASJC) codes
- Analysis
- Applied Mathematics
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On the asymptotic behavior of semilinear wave equations with degenerate dissipation and source terms. / Georgiev, Vladimir; Milani, Albert.
In: Nonlinear Differential Equations and Applications, Vol. 5, No. 1, 01.01.1998, p. 53-68.Research output: Contribution to journal › Article
TY - JOUR
T1 - On the asymptotic behavior of semilinear wave equations with degenerate dissipation and source terms
AU - Georgiev, Vladimir
AU - Milani, Albert
PY - 1998/1/1
Y1 - 1998/1/1
N2 - 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.
AB - 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.
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U2 - 10.1007/s000300050033
DO - 10.1007/s000300050033
M3 - Article
VL - 5
SP - 53
EP - 68
JO - Nonlinear Differential Equations and Applications
JF - Nonlinear Differential Equations and Applications
SN - 1021-9722
IS - 1
ER -