  This paper is concerned with supersolutions to parabolic equations of the
form \begin{equation} \partial_t U (x,t)-D(x)\Delta U(x,t)=0, \quad (x,t)\in
\mathbb{R}^N \times (0,\infty), \end{equation} where $D\in C(\mathbb{R}^N)$ is
positive. Under the behavior of the diffusion coefficient $D$ with polynomial
order at spatial infinity, a family of supersolutions with slowly decaying
property at spatial infinity is provided. As a first application, weighted
$L^2$ type decay estimates for the initial-boundary value problem of the
corresponding parabolic equation are proved. The second application is the
study of the exterior problem of wave equations with space-dependent damping
terms. By using supersolutions provided above, energy estimates with polynomial
weight and diffusion phenomena are shown.
