  In this paper we study the behavior of the energy of solutions of the wave
equation with localized damping in exterior domain. We assume that the damper
is positive at infinity. Under the Geometric Control Condition of Bardos et al
(1992), we prove that: 1) The total energy decay like O(1/t) and L^2-norm is
bounded for the solutions with initial data in (H_{0}^{1},L^{2}). 2) The total
energy and the square of the L^2-norm, repectively, decay like O(1/t^{2}) and
O(1/t) for a kind of the weighted initial data.
