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

Abstract

This paper is concerned with supersolutions to parabolic equations of the form \begin{equation} \partial_t U (x,t)-D(x)ΔU(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.

34 pages

References in corpus (1)

• Remarks on an elliptic problem arising in weighted energy estimates for wave equations with space-dependent damping term in an exterior domain

Cited by in corpus (2)

• Decay property of solutions to the wave equation with space-dependent damping, absorbing nonlinearity, and polynomially decaying data
• Asymptotic expansion of solutions to the wave equation with space-dependent damping