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

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Author list: Georgiev V, Milani A

Publisher: Springer

Place: BASEL

Publication year: 1998

Journal: Nonlinear Differential Equations and Applications (1021-9722)

Journal acronym: NODEA-NONLINEAR DIFF

Volume number: 5

Issue number: 1

Start page: 53

End page: 68

Number of pages: 16

ISSN: 1021-9722

eISSN: 1420-9004

Languages: English-Great Britain (EN-GB)


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Abstract

We investigate the asymptotic behavior of weak solutions to the semilinear non autonomous wave equationu(tt) - Delta u + u(t)vertical bar u(t)vertical bar(p-1) = V(t)u vertical bar u vertical bar(p-1) + f(., t),where V(t) is a positive time dependent potential satisfyingV(t) = O((1 + t)(-lambda)) as t -> +infinityand f(t) decays to 0 as t -> +infinity. We show that for 0 <= lambda <= p there are initial values such that the energy norm of the corresponding solutions grows at least polynomially as t -> +infinity, while if lambda > 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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Last updated on 2023-31-07 at 00:46