97-522 Gallay Th., Raugel, G.
Scaling variables and asymptotic expansions in damped wave equations (57K, (uuencoded gzipped) Plain TeX) Sep 25, 97
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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 $\epsilon = 0$.

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