Uniqueness of kernel functions of the heat equation
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.
scientific article; zbMATH DE number 554338