Asymptotic behaviour of solutions for the wave equation with an effective dissipation around the boundary
The following initial-boundary value problem is considered: \[ \begin{aligned} & u_{tt}-\Delta u+a(t,x)u_t=0,\;(t,x)\in(0,\infty) \times\Omega,\\ & u(0,x)=u_0(x),\;u_t(0,x)=u_1(x),\;x\in\Omega\\ & u(t,x)=0,\;(t,x)\in (0,\infty) \times\partial \Omega,\end{aligned} \] where \(\Omega\) is an exterior domain in \(\mathbb{R}^N\) with a smooth boundary and outside a compact obstacle. One assumes dissipation that the \(a(t,x)u_t\) is effective around the boundary and that \(a(t, x)\) decays as \(|x|\to \infty\). The paper investigates general exterior domains under the effect of dissipation around the boundary of \(\Omega\). As ``energy nondecay problem'' one understands the construction of a solution for the initial-boundary value problem whose energy never decays. Under some additional conditions, one proves the energy nondecay problem, the existence of a scattering state and the fact that the local energy decays to 0 as \(t\to\infty\).
scientific article; zbMATH DE number 1836766