On the existence of spatially tempered null solutions to linear constant coefficient PDEs

TL;DR

This paper characterizes when linear constant coefficient PDEs admit only the trivial spatially tempered null solutions, using algebraic-geometric methods based on the PDE's defining polynomial.

Contribution

It provides a novel algebraic-geometric criterion for the triviality of spatially tempered null solutions to linear PDEs.

Findings

Null solution space is trivial under certain algebraic conditions.

Characterization depends on the polynomial associated with the PDE.

Results are motivated by physical considerations of tempered solutions.

Abstract

Given a linear, constant coefficient partial differential equation in ${\mathbb{R}}^{d+1}$, where one independent variable plays the role of `time', a distributional solution is called a null solution if its past is zero. Motivated by physical considerations, we consider distributional solutions that are tempered in the spatial directions alone (and do not impose any restriction in the time direction). Considering such spatially tempered distributional solutions, we give an algebraic-geometric characterization, in terms of the polynomial describing the PDE at hand, for the null solution space to be trivial (that is, consisting only of the zero distribution).