Scaling Variables and Asymptotic Expansions in Damped Wave Equations

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