Scaling variables and asymptotic expansions in damped wave equations (Q1275321)

Let \(\varepsilon \) be a positive not necessarily small parameter, \(a(x)\) be a diffusion coefficient converging to positive limits as \(x\to \pm \infty \), and let the Lipschitz continuous nonlinearity \(N(u,u_x,u_t)\) vanish sufficiently fast as \(u\to 0\). The long time behaviour of small solutions to the nonlinear damped wave equation \(\varepsilon u_{tt}+u_t=(a(x)u_x)_x+ N(u,u_x,u_t)\), \(x\in \mathbb{R}\), \(t\geq 0\), is studied by use of scaling variables, energy estimates and an asymptotic expansion of \(u\) in the powers of \(t^{-1/2}\).