Gallay Th., Raugel, G.
Scaling variables and asymptotic expansions in damped wave equations
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ABSTRACT. We study the long time behavior of small solutions to the nonlinear damped
wave equation
$$
\epsilon u_{\tau\tau} + u_\tau \,=\, (a(\xi)u_\xi)_\xi +
\NN(u,u_\xi,u_\tau)~, \quad \xi \in \real~, \quad \tau \ge 0~,
$$
where $\epsilon$ is a positive, not necessarily small parameter. We assume
that the diffusion coefficient $a(\xi)$ converges to positive limits $a_\pm$
as $\xi \to \pm\infty$, and that the nonlinearity $\NN(u,u_\xi,u_\tau)$
vanishes sufficiently fast as $u \to 0$. Introducing scaling variables and
using various energy estimates, we compute an asymptotic expansion of the
solution $u(\xi,\tau)$ in powers of $\tau^{-1/2}$ as $\tau \to +\infty$,
and we show that this expansion is entirely determined, up to second order,
by a linear parabolic equation which depends only on the limiting values
$a_\pm$. In particular, this implies that the small solutions of the damped
wave equation behave for large $\tau$ like those of the parabolic equation
obtained by setting $\epsilon = 0$.