On the evolution equation with a dynamic Hardy-type potential

Jann Long Chern; Gyeongha Hwang; Jin Takahashi; Eiji Yanagida

doi:10.1007/s00028-021-00675-5

On the evolution equation with a dynamic Hardy-type potential

Jann Long Chern, Gyeongha Hwang^*, Jin Takahashi, Eiji Yanagida

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Motivated by the celebrated paper of Baras and Goldstein (Trans Am Math Soc 284:121–139, 1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, nonexistence and uniqueness of solutions and obtain a sharp lower and upper bound near the singular point. Proofs are given by using solutions of the radial heat equation, some precise estimates for an equivalent integral equation and the comparison principle.

Original language	English
Pages (from-to)	2141-2165
Number of pages	25
Journal	Journal of Evolution Equations
Volume	21
Issue number	2
DOIs	https://doi.org/10.1007/s00028-021-00675-5
Publication status	Published - 2021 Jun

Keywords

Existence
Hardy potential
Heat equation
Initial value problem
Singularity
Uniqueness

ASJC Scopus subject areas

Mathematics (miscellaneous)

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10.1007/s00028-021-00675-5

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title = "On the evolution equation with a dynamic Hardy-type potential",

abstract = "Motivated by the celebrated paper of Baras and Goldstein (Trans Am Math Soc 284:121–139, 1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, nonexistence and uniqueness of solutions and obtain a sharp lower and upper bound near the singular point. Proofs are given by using solutions of the radial heat equation, some precise estimates for an equivalent integral equation and the comparison principle.",

keywords = "Existence, Hardy potential, Heat equation, Initial value problem, Singularity, Uniqueness",

author = "Chern, {Jann Long} and Gyeongha Hwang and Jin Takahashi and Eiji Yanagida",

note = "Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature.",

year = "2021",

month = jun,

doi = "10.1007/s00028-021-00675-5",

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TY - JOUR

T1 - On the evolution equation with a dynamic Hardy-type potential

AU - Chern, Jann Long

AU - Hwang, Gyeongha

AU - Takahashi, Jin

AU - Yanagida, Eiji

PY - 2021/6

Y1 - 2021/6

N2 - Motivated by the celebrated paper of Baras and Goldstein (Trans Am Math Soc 284:121–139, 1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, nonexistence and uniqueness of solutions and obtain a sharp lower and upper bound near the singular point. Proofs are given by using solutions of the radial heat equation, some precise estimates for an equivalent integral equation and the comparison principle.

AB - Motivated by the celebrated paper of Baras and Goldstein (Trans Am Math Soc 284:121–139, 1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, nonexistence and uniqueness of solutions and obtain a sharp lower and upper bound near the singular point. Proofs are given by using solutions of the radial heat equation, some precise estimates for an equivalent integral equation and the comparison principle.

KW - Existence

KW - Hardy potential

KW - Heat equation

KW - Initial value problem

KW - Singularity

KW - Uniqueness

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JO - Journal of Evolution Equations

JF - Journal of Evolution Equations

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ER -

On the evolution equation with a dynamic Hardy-type potential

Abstract

Keywords

ASJC Scopus subject areas

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