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arXiv · 1907.08992

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

Motohiro Sobajima·Yuta Wakasugi
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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.

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Motohiro Sobajima, Yuta Wakasugi. 2019-07-21. Supersolutions for parabolic equations with unbounded diffusion and its applications to some classes of parabolic and hyperbolic equations. https://doi.org/10.2969/jmsj%2F83928392

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