Uniqueness of kernel functions of the heat equation
type: Item
instance of: scholarly article
title: Uniqueness of kernel functions of the heat equation
zbMATH Open document ID: 0828.31006
DOI: 10.1007/BF01047842
author: Masaharu Nishio
published in: Potential Analysis
publication date: 1996-01-14T00:00:00Z
review text: For \(\alpha \in \mathbb{R}\), let \(\Omega_\alpha = \{(x,t) : t < 0\), \(|x |< |t |^\alpha\}\) in \(\mathbb{R}^2\). A kernel function at infinity is a positive solution of the heat equation on \(\Omega_\alpha\), which is not identically zero but vanishes continuously on \(\partial \Omega_\alpha\). The main result of the paper states that there exists a unique kernel function at infinity, with the value 1 at \((0, - 1)\), if and only if \(\alpha < 1\). However, a more general result, in which \(\Omega_\alpha\) is replaced by \(\Omega_\alpha (D) = \{(x,t) : t < 0\), \(x |t |^{- \alpha} \in D\}\) for certain domains \(D\) in \(\mathbb{R}^{n + 1}\) for \(n \geq 1\), has already been published by the author and \textit{N. Suzuki} [Osaka J. Math. 31, 331-339 (1994; Zbl 0824.35042)]. The paper under review is therefore useful only in that it contains certain details omitted from that joint paper.
Mathematics Subject Classification ID: 31C35
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 554338
zbMATH Keywords: positive solution of the heat equation
zbMATH Keywords: unique kernel function at infinity
reviewed by: Neil A. Watson
MaRDI profile type: Publication
cites work: Q5619343
cites work: Q5651102
cites work: Parabolic potential theory
cites work: Temperatures in Several Variables: Kernel Functions, Representations, and Parabolic Boundary Values
cites work: The uniqueness of positive solutions of parabolic equations of divergence form on an unbounded domain
Recommended article: Q4316857
Recommended article: Minimal thickness and uniqueness of kernel functions for the heat equation in several variables
Recommended article: Q4233078
Recommended article: Uniqueness in the Cauchy problem for the heat equation
Recommended article: Uniqueness for a nonlinear heat equation
Recommended article: Uniqueness of solution of a singular heat equation
Recommended article: Uniqueness of Positive Solutions of the Heat Equation
Recommended article: Uniqueness of positive solutions of the heat equation
Recommended article: Uniqueness in the Cauchy problem for the Hermite heat equation
Recommended article: Uniqueness for inverse heat equation
description: scientific article; zbMATH DE number 554338