On unique continuation in measure for fractional heat equations

TL;DR

This paper proves a unique continuation property in measure for fractional heat equations, extending classical results to nonlocal operators with potential applications in analysis of PDEs.

Contribution

It introduces a novel unique continuation theorem for fractional heat equations, bridging the gap between local and nonlocal PDEs.

Findings

Established a measure-based unique continuation theorem for fractional heat equations

Extended classical unique continuation results to nonlocal operators with $0<s<1$

Provided a framework for analyzing nonlocal PDEs with potential functions

Abstract

We prove a theorem of unique continuation in measure for nonlocal equations of the type $(\partial_t - \Delta)^s u= V(x,t) u$, for $0<s <1$. Our main result, Theorem 1.1, establishes a delicate nonlocal counterpart of the unique continuation in measure for the local case $s=1$.