Weighted energy estimates for wave equation with space-dependent damping term for slowly decaying initial data

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.