On the stabilization of solutions of the wave equation in domains with star-shaped boundaries. (Q1432700)

The author studies the decay rates at \(t\to\infty\) of the local energy \[ \int_{\Omega\cap \{| x| \leq R\}} (| u_t(t,x)| ^2+| \nabla u(t,x)| ^2)\,dx \] of solutions to the wave equation \(u_{tt}=\Delta u\) considered in a bounded domain \(\Omega\) with a smooth boundary, and supplemented with the Dirichlet boundary conditions and with initial data. The reasoning is based on suitable estimates of solutions to the Helmholtz equation \(\Delta v+k^2v=-h\) with the Dirichlet boundary condition.