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2021 Supersolutions for parabolic equations with unbounded or degenerate diffusion coefficients and their applications to some classes of parabolic and hyperbolic equations
Motohiro SOBAJIMA, Yuta WAKASUGI
J. Math. Soc. Japan Advance Publication 1-38 (2021). DOI: 10.2969/jmsj/83928392

Abstract

This paper is concerned with supersolutions to parabolic equations with space-dependent diffusion coefficients. Given the behavior of the diffusion coefficient 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 parabolic equation are proved. The second application is the study of the exterior problem of wave equations with space-dependent damping terms. By using supersolution provided above, energy estimates with polynomial weight and diffusion phenomena are shown. There is a slight improvement compared to the previous work about the assumption of the initial data.

Funding Statement

This work was supported by JSPS KAKENHI Grant Numbers JP18K134450, JP16K17625, JP18H01132.

Citation

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Motohiro SOBAJIMA. Yuta WAKASUGI. "Supersolutions for parabolic equations with unbounded or degenerate diffusion coefficients and their applications to some classes of parabolic and hyperbolic equations." J. Math. Soc. Japan Advance Publication 1 - 38, 2021. https://doi.org/10.2969/jmsj/83928392

Information

Received: 18 December 2019; Revised: 10 April 2020; Published: 2021
First available in Project Euclid: 23 January 2021

Digital Object Identifier: 10.2969/jmsj/83928392

Subjects:
Primary: 35K20
Secondary: 35B40, 35L20

Rights: Copyright © 2021 Mathematical Society of Japan

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