Scaling variables and asymptotic expansions in damped wave equations

We study the long time behavior of small solutions to the nonlinear damped wave equation epsilon u(tau tau) + u(tau) = (a(xi) u(xi))(xi) + N(u, u(xi), u(tau)), xi is an element of R, tau greater than or equal to 0, where epsi lon is a positive, not necessarily small parameter. We assume that the diff usion coefficient a(xi) converges to positive limits a(+/-) as xi --> +/- i nfinity, and that the nonlinearity,N(u, u(xi), u(tau)) vanishes sufficientl y fast as u --> 0. Introducing scaling variables and using various energy e stimates, we compute an asymptotic expansion of the solution u(xi, tau) in powers of tau(-1/2) as tau --> + infinity, and we show that this expansion is entirely determined, up to the second order, by a linear parabolic equat ion which depends only on the limiting values a(+/-). In particular, this i mplies that the small solutions of the damped wave equation behave for larg e tau like those of the parabolic equation obtained by setting epsilon = 0. (C) 1998 Academic Press.