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This paper is concerned with supersolutions to parabolic equations of theform \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 $D\in C(\mathbb{R}^N)$ ispositive. Under the behavior of the diffusion coefficient $D$ with polynomialorder at spatial infinity, a family of supersolutions with slowly decayingproperty at spatial infinity is provided. As a first application, weighted$L^2$ type decay estimates for the initial-boundary value problem of thecorresponding parabolic equation are proved. The second application is thestudy of the exterior problem of wave equations with space-dependent dampingterms. By using supersolutions provided above, energy estimates with polynomialweight and diffusion phenomena are shown.

Supersolutions for parabolic equations with unbounded diffusion and its applications to some classes of parabolic and hyperbolic equations