Quantitative uniqueness for fractional heat type operators

TL;DR

This paper establishes quantitative bounds on the vanishing order of solutions to fractional heat operators using new Carleman estimates, extending previous results to a nonlocal, parabolic setting.

Contribution

It introduces novel Carleman estimates to derive quantitative uniqueness results for fractional heat operators, generalizing existing local and time-independent cases.

Findings

Quantitative bounds on vanishing order for fractional heat solutions

Extension of Carleman estimates to nonlocal parabolic operators

Generalization of previous uniqueness results to fractional and nonlocal contexts

Abstract

In this paper we obtain quantitative bounds on the maximal order of vanishing for solutions to $(\partial_t - \Delta)^s u =Vu$ for $s\in [1/2, 1)$ via new Carleman estimates. Our main result Theorem 1.1 and Theorem 1.3 can be thought of as a parabolic generalization of the corresponding quantitative uniqueness result in the time independent case due to R\"uland and it can also be regarded as a nonlocal generalization of a similar result due to Zhu for solutions to local parabolic equations.