  This paper is concerned with weighted energy estimates for solutions to wave
equation $\partial_t^2u-\Delta u + a(x)\partial_tu=0$ with space-dependent
damping term $a(x)=|x|^{-\alpha}$ $(\alpha\in [0,1))$ in an exterior domain
$\Omega$ having a smooth boundary. The main result asserts that the weighted
energy estimates with weight function like polymonials are given and these
decay rate are almost sharp, even when the initial data do not have compact
support in $\Omega$. The crucial idea is to use special solution of $\partial_t
u=|x|^{\alpha}\Delta u$ including Kummer's confluent hypergeometric functions.
