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

## TL;DR

This paper develops supersolutions for parabolic equations with unbounded diffusion coefficients and applies them to derive decay estimates and analyze wave equations with space-dependent damping.

## Contribution

It introduces a family of supersolutions for parabolic equations with polynomially growing diffusion coefficients and applies these to decay estimates and wave equation analysis.

## Key findings

- Weighted decay estimates for parabolic problems.
- Energy estimates for wave equations with damping.
- Demonstration of diffusion phenomena.

## Abstract

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.