Stabilization of the wave equation in an exterior domain (Q699246)

The Dirichlet initial boundary value problem for the damped wave equation \[ (\partial^2_0- \Delta)u+ a\partial_0 u= 0\quad\text{in }\mathbb{R}_{>0}\times \Omega,\quad u= 0\quad\text{on }\mathbb{R}_{>0}\times\partial\Omega, \] with prescribed initial data \(u(0+)= f_1\), \(\partial_0 u(0+)= f_2\) in \(\Omega\) is considered in an exterior domain \(\Omega\) of \(\mathbb{R}^n\), \(n\) odd, with smooth boundary. The multiplicative damping coefficient \(a\) is assumed to be smooth and nonnegative. Under a geometric assumption (exterior geometric control) the authors are -- by using semigroup methods and Lax-Phillips scattering theory -- able to show exponential energy decay for compactly supported initial data.