Monotonicity of generalized frequencies and the strong unique continuation property for fractional parabolic equations

TL;DR

This paper proves a strong unique continuation property backwards in time for a fractional parabolic equation, extending classical results to nonlocal operators and developing new regularity and monotonicity tools.

Contribution

It introduces a regularity theory for the extension problem and establishes a monotonicity formula for a frequency function in the nonlocal fractional setting.

Findings

Established a monotonicity result for an adjusted frequency function.

Developed a regularity theory for the extension problem.

Performed blowup analysis of Almgren rescalings.

Abstract

We study the strong unique continuation property backwards in time for the nonlocal equation in $\mathbb{R}^{n} \times \mathbb{R}$ \begin{equation}\label{one} (\partial_t - \Delta)^{s} u = V(x,t)u \end{equation} for $s \in (0,1)$. Our main result Theorem 1.2 can be thought of as the nonlocal counterpart of the result obtained by Poon for the case when $s=1$. In order to prove Theorem 1.2 we develop the regularity theory of the extension problem for the equation above. With such theory in hands we establish: i) a basic monotonicity result for an adjusted frequency function which plays a central role in this paper, see Theorem 1.3; ii) an extensive blowup analysis of the so-called Almgren rescalings associated with the extension problem. We feel that our work will also be of interest e.g. in the study of certain basic open questions in free boundary problems, as well as in nonlocal segregation problems.