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

## TL;DR

This paper establishes weighted energy decay estimates for wave equations with space-dependent damping in exterior domains, even for non-compact initial data, using special solutions involving hypergeometric functions.

## Contribution

It introduces nearly sharp weighted energy decay estimates for wave equations with space-dependent damping, extending results to non-compact initial data using hypergeometric functions.

## Key findings

- Weighted energy decay estimates are nearly sharp.
- Results apply to non-compact initial data.
- Uses special solutions involving hypergeometric functions.

## Abstract

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.