The determination of caloric morphisms on Euclidean domains

Abstract

Let $D$ be a domain in ${\Bbb R}^{m+1}$ and $E$ be a domain in ${\Bbb R}^{n+1}$. A pair of a smooth mapping $f:D \to E$ and a smooth positive function $\varphi$ on $D$ is called a caloric morphism if $\varphi\cdot u \circ f$ is a solution of the heat equation in $D$ whenever $u$ is a solution of the heat equation in $E$. We give the characterization of caloric morphisms, and then give the determination of caloric morphisms. In the case of $m<n$, there are no caloric morphisms. In the case of $m=n$, caloric morphisms are generated by the dilation, the rotation, the translation and the Appell transformation. In the case of $m>n$, under some assumption on $f$, every caloric morphism is obtained by composing a projection with a direct sum of caloric morphisms of ${\Bbb R}^{n+1}$.