### Abstract

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.

Original language | English |
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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

### Cite this

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*Nonlinear Differential Equations and Applications*, vol. 5, no. 1, pp. 53-68. https://doi.org/10.1007/s000300050033

**On the asymptotic behavior of semilinear wave equations with degenerate dissipation and source terms.** / Georgiev, Vladimir; Milani, Albert.

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.

UR - http://www.scopus.com/inward/record.url?scp=0038172457&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038172457&partnerID=8YFLogxK

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 -