On the forward in time propagation of zeros in fractional heat type problems

TL;DR

This paper proves that solutions to fractional heat equations that vanish to infinite order at a point must be identically zero, establishing a strong unique continuation property in space-time.

Contribution

It extends previous results by showing that zeros propagate forward in time for fractional heat problems, confirming unique continuation in the full space-time domain.

Findings

Solutions vanishing to infinite order at a point are identically zero

Zeros propagate forward in time in fractional heat problems

Completes previous partial results on vanishing in space-time

Abstract

In this short note we prove that if $u$ solves $(\partial_t - \Delta)^s u = Vu$ in $\mathbb R^n_x \times \mathbb R_t$, and vanishes to infinite order at a point $(x_0, t_0)$, then $u \equiv 0$ in $\mathbb R^n_x \times \mathbb R_t$. This sharpens (and completes) our earlier result that proves $u(\cdot, t) \equiv 0$ for $t \leq t_0$ if it vanishes to infinite order at $(x_0, t_0)$.