Scaling Variables and Asymptotic Expansions in Damped Wave Equations

We study the long time behavior of small solutions to the nonlinear damped wave equationεuττ+uτ=(a(ξ)uξ)ξ+N(u,uξ,uτ),ξ∈R,τ⩾0, whereεis a positive, not necessarily small parameter. We assume that the diffusion coefficienta(ξ) converges to positive limitsa±asξ→±∞, and that the nonlinearity N(u,uξ,uτ)...

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Bibliographic Details
Published in:Journal of Differential Equations 1998-11, Vol.150 (1), p.42-97
Main Authors: Gallay, Th, Raugel, G
Format: Article
Language:English
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Summary:We study the long time behavior of small solutions to the nonlinear damped wave equationεuττ+uτ=(a(ξ)uξ)ξ+N(u,uξ,uτ),ξ∈R,τ⩾0, whereεis a positive, not necessarily small parameter. We assume that the diffusion coefficienta(ξ) converges to positive limitsa±asξ→±∞, and that the nonlinearity N(u,uξ,uτ) vanishes sufficiently fast asu→0. Introducing scaling variables and using various energy estimates, we compute an asymptotic expansion of the solutionu(ξ,τ) in powers ofτ−1/2asτ→+∞, and we show that this expansion is entirely determined, up to the second order, by a linear parabolic equation which depends only on the limiting valuesa±. In particular, this implies that the small solutions of the damped wave equation behave for largeτlike those of the parabolic equation obtained by settingε=0.
ISSN:0022-0396
1090-2732
DOI:10.1006/jdeq.1998.3459