  In earlier works, we have shown the uniform decay of the local energy of the
damped wave equation in exterior domain when the damper is spatially localized
near captive rays. In order to have uniform decay of the total energy, the
damper has also to act at space infinity. In this work, we establish uniform
decay of both the local and global energies. The rates of decay turns out to be
the same as those for the heat equation, which shows that an effective damper
at space infinity strengthens the parabolic structure in the equation.
