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

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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)) 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 languageEnglish
Pages (from-to)53-68
Number of pages16
JournalNonlinear Differential Equations and Applications
Volume5
Issue number1
Publication statusPublished - 1998
Externally publishedYes

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Semilinear Wave Equation
Wave equations
Source Terms
Dissipation
Asymptotic Behavior
Norm
Nonautonomous Equation
Energy
Absorbing
Semilinear
Weak Solution
Wave equation
Orbits
Ball
Orbit
Decay

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

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AU - Milani, Albert

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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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