| Abstract: |
| This talk is based on a joint work with Professor Motohiro Sobajima
(Tokyo University of Science).
We study 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 $\Omega \subset \mathbb{R}^N$
having a smooth boundary.
The main result asserts that
the weighted energy estimates with
weight function like polynomials
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 a special solution of the corresponding parabolic equation
\[
\partial_tu=|x|^{\alpha} \Delta u
\]
including Kummer`s confluent hypergeometric functions. |
|